Investigations in Philosophy of Space

Elisabeth Str6ker

translated by

Algis Mickunas

~ Ohio University Press 1 Athens, Ohio London

Translation ©Copyright 1987 by Ohio University Press

Originally published as Philosophische Untersuchungen zum Raum. (Frankfurt Am Main: Vittorio Klostermann, 1965) © Copyright 1965.

Printed in the United States of America. All rights reserved.

Library of Congress Cataloging-in-Publication Data Stroker, Elisabeth.

Philosophical investigations of space.

(Series in continental thought; v. 11) Translation of: Philosophishe Untersuchungen zum

Raum. Bibliography: p. Includes index. 1. Space and time. l. Title. 11. Series: Series

in continental thought ; 11. BD632.S6513 1987 114 86-2410 ISBN 0-8214-0826-7

Table of Contents

Translator's Preface ................................................ ix Preface ............................................................... xi

Introduction ............................................................. 1 §1. The State of the Problem ................................ 1 §2. The Aim of the Investigation ............................ 3 §3. Preliminary Methodological Considerations ........... 7

Part One: Lived Space ................................................ 13 Section One: Contributions to the Phenomenology of

Lived Space ........................................ 14 Point of Departure and Statement of the Problem ........... 14 Chapter One: The Attuned Space ............................. 19

§1. The Concept of Attuned Space ......................... 19 §2. Characteristics of Attuned Space: Fullness and

Emptiness ................................................. 22 §3. Place and Position in Attuned Space .................. 27 §4. Nearness and Remoteness .............................. 29 §5. Movement and Orientation in Attuned Space ........ 30 §6. Attuned Space as Space-Time .......................... 36 §7. Attuned Space and the Experiencing Subject ......... 43

Chapter Two: The Space of Action ........................... 48 §1. Preliminary Remarks .................................... 48 §2. Place and Regían. The Space of Action as a

Topological Manifold ................................... 52 §3. The Locus of the Subject in the Space of Action ..... 57 §4. Movement and Orientation. The Space of Action

as Oriented Space ........................................ 62 §5. The Problem of the Way ................................ 71 §6. Nearness and Remoteness in the Space of Action .... 75 §7. Summary ................................................. 81

Chapter Three: The S pace of Intuition ....................... 83 §1. Terminological Clarifications ........................... 83 §2. The Space of Intuition as a Phenomenal Multitude

of Points ........... ·~· ................................. .' ... 85

V

vi Table of Contents

§3. The Lived Body as the Center of the Space of Intuition .................................................. 89

§4. The Oriented Space of Intuition ....................... 90 §5. Spatial Depth and Perspectivity ........................ 93 §6. The Finitude of the Space of Intuition ............... 109 §7. The Other in My Space of Intuition. Questions of

Homogenization ........................................ 113 §8. Open Questions ........................................ 118

Chapter Four: Modally Distinct Sensory Spaces ........... 120 §1. Visual Space ............................................ 120 §2. The Visual Field ........................................ 124 §3. The Problem of Tactile Space ......................... 126

Section Two: Questions of Space Constitution ................ 138 Chapter One: Corporeity and Spatiality ..................... 138

§1. Methodological Survey ............................ ' .... 138 §2. The Lived Body and the Physical Body in their

Relationship to Space .................................. 140 §3. The Lived Body and Consciousness .................. 148

Chapter Two: The Space of Movement and Objective Space .......................................... 152

§1. Spatial Structure and Corporeal Facticity ........... 152 §2. The Problem of Empty Space ......................... 160 §3. Concluding Observations on Lived Space ........... 169

Part Two: Mathematical Space ..................................... 173 Introductory Remarks ............................................ 174 Section One: Preliminary Phenomenological Observations . 176

Chapter One: Space as a Thematic Object of Consciousness ................................. 176

§1. The Space of Intuition as a Limit Case of Lived Spatiality ................................................ 176

§2. The Topological Structure of the Space of Objects . 179 Chapter Two: Basic Trends of Mathematization ........... 184

§ 1. Morphological and Mathematical Determinations of the World of Things ................................. 184

§2. The Problem of Mathematical Ideation .............. 190 §3. Symbolic Intuition (Pictorial Symbolism) ........... 194 §4. Signitive Symbolization of Geometry ................ 200 §5. The Constructive Character of Geometric

Objectivity. Geometry as a Demonstrative Science . 211 §6. Summary ................................................ 221

Table of Contents vii

Section Two: Euclidean Space .................................. 225 Chapter One: Phenomenological Access to Metrics ........ 225

§1. Formation and Relationship. The Primacy of Relationships ........................................... 225

§2. The Line Segment as a Fundamental Metric Formation ............................................... 228

§3. The Line Segmentas an Invariant of "Movements" ........................................... 231

§4. The Concept of Movement as a Leading Concept of the Theory of Invariants ............................ 236

Chapter Two: Euclidean Normal Space ..................... 239 § 1. The Concept of Mathematical S pace (Preliminary

Conceptual Clarification) .............................. 239 §2. Normal Space (Euclidean Space of the

Topological Type of the Open Plane) ................ 246 §3. The Question of Intuitability in Euclidean

Geometry ................................................ 252 Chapter Three: Euclidean Spaces with Topological

Anomalies ................................................... 258 §1. Extension of the Mathematical Concept of Space ... 258 §2. Clifford-Klein Spaces .................................. 259 §3. Clifford-Klein Spaces as Euclidean Normal Space.

Founding Relationships ............................... 260

Section Three: Non-Euclidean Spaces .......................... 265 Chapter One: Fundamental Questions of Non-Euclidean

Geometry .................................................... 265 §1. The Parallel Postulate. Historical Origin and

Development ............................................ 265 §2. Constitutive Problems of the Parallel Postulate ..... 269

Chapter Two: Foundational Problems of Hyperbolic Geometry .................................................... 2 7 5 §1. On the Metrics of Hyperbolic Geometry ............. 275 §2. The Kleinian Model. Phenomenological Analysis

of the Model Conception .............................. 278 §3. Hyperbolic Geometry and the Space of Intuition ... 281

Chapter Three: Riemann's Geometry ........................ 285 §1. Riemann's Point of Departure. The Metric

Fundamental Form ..................................... 285 §2. Riemannian Spaces. Brief Mathematical

Characterization ........................................ 287 §3. Curvature and "Curved Spaces" ...................... 291

viii Table of Contents

§4. The Question of the Existence of the Mathematical Point ..................................................... 296

Concluding Observations ........................................... 304 Works Cited and Consulted ........................................ 309 Register . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319

Translator' s Preface

Professor Dr. Elisabeth Stroker is a noted philosopher of the history of science and a leading thinker in the current development of philosophy of technology, inclusive of ethical issues attendant upon science and technology. I am grateful to Professor Stroker for the trust and patience she maintained in me during the protracted process of translation. This is more so in face of the importance of her work in the philosophy of space. It contains the best possible phenomenological and critical analyses of issues concerning the "experience" and the "being" of space, and the most outstanding and precise theoretical investigations of the problems in conceptu­alizing space. Professor Stroker's theoretical investigations include the development of the entire Western philosophical tradition concerning the conceptions of space, ranging from Euclid all the way to the geometric-mathematical problematics of Riemann. To say the least, Professor Stroker's work is both philo~ophically and scientif­ically fundamental. It sets the standard for any future investigations in the philosophy of space.

The success of the translation must be attributed to Dr. Elizabeth Behnke who, under the auspices of a copy editor, accomplished a monumental task. Her indispensible cooperation, her inexhaustible patience with every term and phrase, are completely intertwined in the translation. Indeed, Dr. Behnke was part and parcel of making the text intelligible toan American reader. Many thanks.

Preface

This investigation is m y habilitation work, revised for print, which was presented to the philosophy faculty of the University of Ham­burg in the summer semester of 1963. It was begun thanks to a generous stipend from the German Research Society. Its preparation was enhanced by professor Dr. Wolfgang Wieland of Hamburg University. 1 am grateful for his interest and valuable suggestions. 1 also owe my gratitude to Professor Dr. Carl Friedrich Feiherr van Weizsacker and, not the least, to Professor Dr. Günther Patzig for their enhancement of the work through their encouragement and critical remarks.

The first conceptual inception of this writing reaches back toward the years of my study at the University of Bonn. The lectures and works of Oskar Becker impressed me by their masterful subtleties of philosophical analyses of problems and pointed me, for the first time, toward phenomenological questioning. Less obvious is the imput in this work that was acquired from Theodor Litt. He has impacted and swayed my first philosophical studies; he played a vital role in the proposed investigation. While the path followed in this work re­mained distant to him, this distance allowed him to survey the work more intensely, allowing him unrestrained and open criticism. Hence the will of a teacher to demand, in which the power of his thought was most vital, became a most conspicuous part of this work. His frequently expressed wish to see the work attain its proposed aim, was not granted to him. As his last student, 1 dedícate this work to his memory.

Hamburg, September 1964 Elisabeth Stréiker

Introduction

§ 1. The State of the Problem

The philosophical treatment of space faces a problem whose tradition is as old as the entire history of philosophy. As an object of metaphysical and naturalistic philosophical speculation since the pre-Socratics, space seems to offer an almost inexhaustible variety of aspects throughout the historical changes of cosmogonies, systems and conceptual positions. What is given in them to date as a "theory" of space turns out to be a sublimated sediment of histori­cally acquired concepts from whose entire intellectual content space assu'mes its decisive outlines. This is true for the finite central­peripheral spatial cosmos of antiquity and its modifications in high scholasticism, as well as for the infinite, homogeneous space of Renaissance philosophy; the same is true for the absolute space of Newton no less than for the much acclaimed "renaissance" of the Aristotelian conception of space in the field theories of modern physics.

With the development of modern natural sciences, thinking con- ~d cerning space acquires new impulses from experimental research. Ñ0\- eLt- _ ~~~ Non-Euclidean geometries and their extension of the concept of ~(tY\VVit"'!J space have contributed to the discussion of the problem from the mathematical side since the end of the eighteenth century. History shows that from then on philosophy and the exact sciences not only diverge, but also continuously confront one another in the contro-versies concerning epistemic claims arising from the treatment of the problem of space. The fact that since the beginning of the nineteenth century this controversy has been oriented almost exclusively by Kant's conception of space indicates, on the one hand, the extraor­dinary fertility of the Kantian a priori and, on the other hand, has its basis in the fact that the latter bears unmistakably within itself the properties of the universal structure of Newtonian space whose validity has become increasingly questionable. The most recent and in tense encounter between philosophy and science was precipitated 1 [Ar¿oi:J 1}­by the theory of relativity at the' dawn of this century. It is { tAp\IY'~ h1 revolutionary: it transformed the entire physical world image and )

1

~ tl'~ ~VL\J~\ ~M'-P( ~ rcW[\'Of\g'vv.-1['6. ~ <S fU U. ~'>t , .. , "'f"fA .t( .

2 Introduction

renewed the demand for a reply to the philosophical question of space.

Subsequently, it seems as though the problem of space has become merely an occasion for a reciproca! glance across the boundaries between philosophical and scientific research. Meanwhile, the phi­losoiJhy of our day begins the discussion from another point of view. With inc;!~~§i!!giJ11~@s_t__ij:_!_lg:ns Jpward the world ~~nc~ prior to. ~-n.<! ·ªgart f!Q~ __ ::;,c;i§!lG.~_é!!!.li dis~gy.e.uUhe_space_of_eye.J.:y.day things. ~nd .tbeir rel~üon~JJ.i.llS,_th.e..spacJLOLhu:u:unllLªª·fu.JLplac~ their everyday--existence. Wherever existential philosophy deals with space within the framework of m~ons, it only recognizes as philosophically significant the space of "Dasein"

---? experienced in its everydayness apart from scientific conceptions. ~'ttif' Heidegger's differentiation of "being in" from being "in" something

attempts to distinguish the traditional categorical conception of space from space as existential; Merleau-Ponty's alternative concept

:...rk6~Joo-fi2A.~ of a "troisieme spatialité", in contrast to physical and mathematical spatiality, strongly suggests a domain of spatial research inaccessible

ru.~_ "$

)/;.(L.::

to the mathematical sciences. Perhaps the solution to the riddle of )! space should be sought here, ~ solution airead y clearly seen by Kant, V..?l for in arder to elucüiate the gossibility of things, he presented space a~Ce§.§m. co1J.!!i1!2!1 for su eh .~si6ilgy. - ·

Influences of existential-ontological and existential-phenomeno­logical efforts are particularly obvious in the area of contemporary psychology. Although the naturalistically oriented scientific psy­chology had begun to investigate space perception experimentally and genetically over a hundred years ago, more recent research follows the various analyses of "space experience" under the aspect of holistic psychology, phenomenological description, and method­ological hermeneutics. In contemporary psychopathology there is a notable emphasis on the abnormal experience of space, leading toa multi-faceted analysis of "spatial disturbances." Such research is not too encouraging, since during the last decades it has led to a confusing multitude of various "spaces." Yet, at the same time, this points to numerous efforts under various methodological aspects to master the complex and multi-layered problem of space.

If the entire range of problems were finally surveyed, then imme­diately specific questions would have to emerge. With what justifi­cation does one speak of "spaces"? Kant was of the opinion that space must be necessarily unitary, and that all talk of spaces made sense only if they were parts of the same space. Has this claim lost its validity today? The space of intuition and the space of mathemat-

Introduction 3

ics cannot be arranged next to each other and are not conceivable as parts. If there were a more encompassing space containing them as parts, then its structure should be accessible to investigation; if not, then it would have to be seen as irrational, and the genuine, unitary space would be inaccessible to thought. Or is it that the title "space" can be genuinely claimed by only one of the regions currently called spaces? Or perhaps in accordance with its meaning, space ought to be sought and found apart from sciences in our direct presence to and orientation toward the world. In this case, what is always present in the pre-reflective, everyday consciousness, what is grasped unthematically and understood as space befare the inception of conceptual activity, would have to be elevated to reflective light by philosophical investigation. Furthermore, the investigation would have to justify to what extent any further talk of spaces is nonsen­sical; it would have to decipher the source of error, expose the speculative willfulness and inappropriate systematization where thinking, too intent on a unitary vision, grasps aspects of a most heterogeneous kind under the concept of space.

If the talk about spaces is valid, then we should ask about the differences and commonalities of their structures. Furthermore, it EJ.ight ~uestion of hi~r_?_!,Chy: does ~of t_h_EJU_Y.l];!i~l1s __ ~E.~S possess, per,h~a kin_cLoLpx.e.eminenca._foLJhJL.krlQwlJJ_Qg~_gf /¡ :@!!.li!Y? PhÜosophy must not only seek an answer, it must also take // a stand with regard to the sense of ~his question itself.

§ 2. The Aim of the Investigation

The ultima te aim of this investigation pertains to theory of science; it purports to be a contribution toward a philosophical grounding of geometry. The concept of the theory of science must nevertheless be taken in a broader than usual sense. Since the end of the nineteenth century, the theory of science has been understood predominantly as methodology or as applied logic. Thus, it analyzes the modes of operation of a science, observes its specific concept formation, and investigates its conceptual assumptions, its modes of deduction and the type of its laws. Ideally, each science regards itself as method­ologically unified. That is to say it views itself as an embodiment of statements whose interrelationship is based either on a finite, even if numerically and typologically variable, quantity of irreducible prin­cipies or on a limited number of distinct, although factually and logically unifiable types of operations. The first is concerned with axiomatic ·disciplines, the second with experimental sciences.

4 Introduction

Within each individual science, it is possible to investigate and compare the structures of relatively closed partial systems of prop­ositions, "theories," according to the type of their basic presuppo­sitions and procedures. If the comparison of methods is extended to different sciences, then we would devise a common methodology as a theory for a specific group of sciences. It could function as a methodological framework for such groups as natural or human sciences. As a general and inclusive methodology of the sciences, it is subsumed under the idea of the unity of all possible forms of theory and thus under the essential explication of science in general.

Yet a theory of science is not exhausted by pure methodology. As a reflective effort toward the structural clarification of the sciences in the broadest sense, it must remember that sciences are ultimately sciences of beings, oriented toward what "is," what "exists." Indi­vidual scientific questions asking what is, and how it is, are oriented ontologically toward qualities, quantities, relations, and causal and functional laws. In contrast, a theory of science requires another position. This position is concerned with a factual area of research, not taken in the simple objective sense of its immediate "being," but in a mediated sense: something existent that is "on the way" to a specific methodological exposition. Only in this modified sense does this area become an object for the theory of science. The efforts of a theory of science go beyond a mere methodology in yet another respect. Method is not merely about something existent that is to be known; it is equally a method of the interrogator who knows and wishes to know more about what there is. Conceived in this manner, a theory of science must take its point of departure from an entirely different ground than from mere methodology. In contrast to meth­odology, a theory of science assumes at the outset a more encom­passing task in a twofold manner.

Taking science in its objective relationship to the subject matter as well as in its relation back to specific forms of the activity of subjectivity, a theory of science faces a problem that can be called ontological in a specific sense of contemporary philosophical thought. What we mean here is a problem emerging froma continu­ing interrogation of the subject matter whose aim is a universal explication of the "sense" of being. This process must pay constant attention to the being of the interrogating subject (Husserl's transcen­dental ego, Heidegger's Dasein). With regard to our present investi­gation and its precise limitation to its thematic object, this means that a theory of science is necessarily confronted by the questions as

Introduction 5

to what space might be and in what sense unqualified assertions of its existence can be given.

Moreover, in contrast to methodology, a theory of science cannot begin with the established sciences in arder to discover their methodological structure. Rather, in its attempt to make the method comprehensible, it must return to tl;w already given relationships and states of affairs from which the scientific questions originate.

With specific regard to space, this means that its treatment in a theory of science cannot begin with the abstract mathematical spaces-assuming for the time being their plurality-but with the spatiality given and experienced prior to all sciences. Thus this \lve-J investigation begins with a detailed treatment of the "lived" space.

li)f(). ce._ lt is based on the view that a solution to the question of the meaning a_QQEl§.!t:lnce of the "abstract" spaces of the specjal sciences and dis~jplines can be sought only on the grounds of a prior investigation Qtthªt"~patiality__w]lich is_J;:Q:::given ~!'.~_lJ:R.I!QP-!;!d_w.b!'l_UeY.~L~Uace b~QJ1les an object of scie~search. This corresponds to the " concepÚ~n~-Ín late Husserl: all objectivity of science is necessarily related to corresponding elements found in the "lived world." This relationship founds the validity of the sciences. lt thus remains to be explicated "how all the self-evidence of objective-logical accomplishements, through which objective theory (thus mathematical and natural-scientific theory) is grounded in respect to form and content, has its hidden, source of grounding in the ultimately accomplishing life, the life in which the self evident givenness of the life world forever has, has attained, and attains anew its prescientific, ontic meaning. " 1

The following work is justifiable only as an attempt to clase a gap that has not yet been successfully filled, despite a series of valiant attempts. lt propases to develop the problems of space under the main notions suggested above. lt will provide a critica! illumination and mutual evaluation of the many scattered concepts found in diverse branches of research, without subsuming their heteroge­neous results and diverse theoretical positions under a forced unification.

The work does not make any claims to completeness. lt deliber­ately limits its analyses to the problems of lived space and the spaces of geometry, i.e., the mathematical spaces as free geometric mani­folds. Thus it excludes an entire domain of questions concerning the practica! use of geometric structures in physics. "Physical space,"

1. E. Husserl, Krisis, § 34d; see also § 38.

6 Introduction

which actually should be added, remains here only as a requirement to be fulfilled by a subsequent and more exhaustive investigation. Our work takes into consideration tasks whose complexities by far

~ surpass our efforts. At the very outset we are not certainvvhether and t()_~P.~2~_t!lnLQucchos_eQm~th~;:[i~-ii~iíY &<:i_eq1J,a,t~;J. The -free ~anci ideal formations of geometry, with regard to their pure geometrical existence, to be investigated in the second part of this work, are extremely difficult to explicate, especially in their relationship to the things of "real" space. This relationship is particularly difficult to explain with regard to physical bodies and their spatial interaction in physics. The term "application" not only does not explain this relationship, but actually hides various problems. If geometry could be equated with a tool of physics that is constituted and, so to speak, prepared by the mathematician, then the problem could be resolved quite readily. Yet the relationship between geometry and physics is not that simple. The elucidation of such a relationship must remain the task of a more specialized work.

Our investigation may be called systematic only to the extent that it is not conceived as a history of problems and is therefore not to be subsumed under the criteria pertaining to the latter sort of study. While such a distinction is justifiable for science, we admit that it may be dangerous and misleading in philosophy. To propase a sharp separation between the systematic and the historical aspects of philosophy would result in an obfuscation of the fact that each "system" contains at least as much "history" as the conceptuality it uses rests in historical origin and retains the heritage of a long tradition. At the very outset, the systematic development of a problem is not primarily determined by its historically evolved status as a scientific problem; rather, it is already conditioned by a situationally determined horizon of the pregiven comprehensibility and unproblematic self evidence of everyday life. Though we may not recognize or even admit such obvious and everyday intelligibil­ity, it is no less historical and historically constructive.

This investigation intends to deal systematically with the problem @ of space as it is articulated in contemporary views and made

accessible by current methods. Yet it is to be remembered that the investigation does not merely rest "on the ground" of a tradition that

'ft!l~i \:)(]IV\ is to be surveyed in its temporal unfolding, useful perhaps only for dexographic interests; rather, the work is "taken up" with a tradi-tion, to the extent that the tradition eiífers- into ___ the ¡)i()lJTéms fi1Ve-stigated, as is the case for the second part of this work, and to the extent that the mode and manner of questioning are shaped by the

Introduction 7

tradition. After all, we raise questions about space not because we know nothing of it, but because we always ha ve a prior acquaintance with it. It is already grasped and laid out in sorne sense befare we begin to ask questions about it. Its presence must be such that it is open to our questioning. Although our effort fails to do justice to history, it is aware of its indebtedness to history. 2

At this juncture we must briefly indicate a specific difficulty. The plan of our investigation must be limited to space without due consideration to time. At first glance, leaving out time seems to avoid an unnecessary complication; yet a closer look reveals a genuine lacuna. S_llilce and time constitute a unified whole and thus are Lk~ 1

related essentially. Such-a relatíl:msiifp-does--not appear siiíiPTY because both are "forms of intuition." Their relationship does not become obvious by a simple declaration that space is "the arder of one next to the other" and time is "the arder of one after the other." This usual conception mer~~!!!!Y of SJ2!!Ce and ti~. The unity does not lie in the simple fact that each case of one next to the other includes a specific "temporal point." The language and the aims of modern physics, expressed in Minkowski's space-time continuum, has done proper justice to this unity. Yet the intertwin- ,() ing of space and time is more intimate and fundamental in the domain of daily life. Without going into greater depth, the present work can only indicate the crucial domains of such an interrelation-ship.

§ 3. Preliminary Methodological Considerations

An investigation whose intent is to unfold the problem of space in its entirety requires, first of all, methodological guarantees.

Though our endeavour will subsequently be delimited more precisely, it can initially be characterized as phenomenological. Today phenomenology flourishes in a multitude of trends, and no

"- - ~- .-" -....._" -~-... -•r--" ---

2. For the historical treatment of the problem of space, we can mention the works of W. Gent, H. Conrad-Martius, K. Deichmann, H. Heimsoeth, A. Koyré, and M. J~~I¡('In particular the latter presents an excellent treatment of the physicalistic history of the problem, ranging from the natural philosophy of the Greeks to the general theory of relativity. The work of Koyré is limited to the historical period between Cusanus and Leibniz, although it is most trustworthy for this historical period. The intent of E. y,)) ~L.-o Fink's worMÍs not primarily "historic-doxographic"; rather, it takes up the --­entire ontd'(ogical tradition und_E)!'_ t_hg le_ading aspe_<_;t1; Q.L~P-!!f~· time, and movemenc------~-~- --- -

hf'k

8 Introduction

longer admits of a unitary procedure. Nowhere has it taken over the He¡_..,1 conception of its celebrated past master Hegel. For Hegel, phenom- J

enology was a way of "apparent knowledge," comprising a prelim-inary step with which philosophy should begin in order to attain true knowledge as the end phase of a dialectical movement. But Husserl did not follow in Hegel's steps. Since Husserl, phenomeno- Hv\\

1

logical philosophy considers its task to be "pure description" of a "given." The given ranges from apure act phenomenon,. within the framework of the Husserlian reductions, to the "being-in-itself" within the unreflective attitude of naive realism. Even existential ontology is concerned with seeing entities as they are "in them­selves." The efforts of existential ontology are centered on "allowing that which is to come to manifestation." In its "hermeneutic" }-).¿., l, procedure it adheres consciously to the Heideggerian formulation of "the tautology of a descriptive phenomenology."3

To elucidate the value of phenomenological procedure for its own sake would be like carrying coals to Newcastle. Certainly it has established a method that is responsible for subtle analyses and far reaching insights in all areas of contemporary philosophy. Yet this

; raises a serious question: can philosophy continue to use this procedure without running the risk of eliminating the possibility of self-critique as the fundamental element of its life? Philosophy requires self-critique more than does science, for in contrast to many of the individual sciences, philosophy does not have a constant corrective for its statements in experience. Philosophy can realize such a critique only if it elevates reflective thinking to its ultimate possibilities. Thus it must continuously submit the capacity and limits of its own method to the light of critical self-reflection. Even if at the first glance the given seems indubitable and self-evident, philosophy must admit at the very outset that its "given" is completely conditioned by its formulation and thus exposed to specific limitations, which are justifiable only by the choice of a point of departure; the given can only be challenged at the point of departure. Thus the Husserlian "principie of all principies" for phenomenology, namely that "every originary presentive intuition is a legitimizing source of cognitio~,'' that "everything originarily ... offered to us in 'intuition' is to be accepted simply as what it is presented as being, but also only within the limits in which it is presented,"4 must not only be maintained most precisely while

3. M. Heidegger, § 7c. 4. E. Husserl, Ideen I, § 24.

Introduction 9

working with specific descriptive analyses, but must be respected more precisely than was done by Husserl himself in the process of his own phenomenological investigations. Moreover, we_m~_¡;t~sk a critt!e_é!Lqges_tiop.:ar_el19!_!_h~ lil!l:it1l .. ()f th~_glY!:lll a~ the same time the limits of phenomenology itself? Since by its owitenets it is.orierite(f to~ard ''the thingstliemselves'r, and devotes itself entirely to indi­vidual researches in accordance with the prescript of a science of science, it is caught in the idea of a lineal progression toward the given. Yet it easily becomes blind to the fact that by far not c2 everything belonging to its phenomenal region can be explicated ].5

completely in pure description. This is true precisely of those analyses of phenomenology dealing with noetic-noematic clarifica­Han of scientific modes of knowledge.

The difficulties emerging in this problematic area must be clari­fied: the investigation will ha ve to begin with lived space, show that the geometric spaces ar~ founded in it, and show how they emerg~ out of it in progressiv~lyJlight;Jl'_kY.eLª,ch!evements. ATfn()'ligii ai first ghinceNtliifi'"áiiproich may- seem simpf;-'~mf7traightforward, it appears entangled when it is viewed by a thought that is required to reflect u pon its own position. What must be kept in mind constantly is that the subject, who faces the theme of space analysis in its entire range of problems, can obviously only be a subject who is somehow ~ already in possession of all spaces. That mode and manner of proceeding "from" the lived space "to" the geometric manifolds is re-flexive in the literal sense, as determined by a retrogressive view, does not present any special difficulties; in fact, it is the commonly used method in phenomenological procedure. Yet this movement is disrupted when we recall that the "way" of the subject through the various spaces that we investigate is our own way, that the subject, who is being viewed retrogressively in space, is in truth the observer himself. While this revealing state of affairs may initially seem self evident, its meaning must nevertheless be fundamentally rethought , in arder to surpass a merely lineal or progressive mode of phenom- V' enological observation.

Thus it would be naive self-deception to "describe" the space of sensory intuition while disregarding all determination of measure. lt is indeed true, as will be explicated in a more precise sense, that mathematical space has its foundation in the space of intuition. This will be justified by our analysis insofar as it will deal with the space of intuition prior to mathematical space. Yet as phenomenological, } and mindful of a complete consideration of the "given," the analysis must also mention that the space of intuition, in the only way it is

10 Introduction

) accessible, already contains aspects of mathematical space. After all, the space of intuition is not merely a space of a sensibly intuiting being; its investigation cannot be kept free from concepts whose sense belongs to a "subsequent" context in which they can be considered thematically and explicated more precisely. This is valid mutatis mutandi for all "spaces" to be thematized. Thus the subjectl does not traverse them as if they were a suite of rooms~ ~lCinust always be kept in mind that the subject, of whom we speak in the third person for the sake of clarity and exposition, is in truth none other than ourselves.

The investigation is placed in a difficult situation: on the one harid, it must be adeq~ate to the factual necessity of proceeding progressively from one spatial form to another; on the other hand, from whatever vantage poirit has been reached, it is confronted by factors that by their very nature can be made transparent only in subsequent analyses. Yet such factors cannot be banned by any decree from a particular moment of investigation. Our investigation knows no other way to master these difficulties than by disrupting the progressively advancing analytical work through self-reflection. This would allow the incorporation of the results brought to the fore into the next step of the process. Such a movement resembles a spiral more than a straight line. The latter certainly remains in the foreground of the work, yet it is not the sale constituent of our method.

If by the term "phenomenological" one means all that is required for the display and explication of phenomenal structures-i.e., the "given," along with additional structures belonging to a specific mode of access, and inclusive of the conceptuality required by such a task-then our attempt can only conditionally be called phenom­enological. The investigation intends to justify this characterization in those parts dedicated to the descriptive and analytic work. Such a point of departure, to be determined shortly with more precision, will allow direct observation to speak for itself. lt will completely avoid all constructions and derivations and will exclude all avail­able scientific opinions "about" this area. This means no more and no less than that the latter may .not function here with their statements as necessary presuppositions; rather, they must attain their meaning only within and "in the course" of the investigation. Either they may be discussed for the sake of their own clarification, or they may require phenomenological analyses to the extent that as phenomena they belong within our framework. This would be the case, for example, with geometry. This means at the same time that

Introduction 11

our investigation is exempt from any refutation by a theory that deals with the problem of space constructively. Nonetheless, it remains exposed to the possibility of missing the phenomena as well as to the dangers of one-sided exposition of other opinions, and it obviously remains open to critique concerning its own point of departure.

In what follows, the traces of Husserlian inflúence will be manifest only mediately. Yet the investigation is indebted to him even in places where it is far removed from Husserl's phenomenological position. Husserl himself had not thematically dealt with space within the framework of his phenomenology. It was the merit of O. Becker's work5 to have applied phenomenology in an analysis of the constitution of geometric objectivity. Our proposed investigation is similar to Becker's in both theme and motive. However, our inves­tigation differs from his in that it does not accept the Husserlian brand of transcendental idealism as an incontrovertible assumption, .1( as does Becker's.

To the extent that Husserl's own investigations-i.e., his analyses of the spatial thing-have any significance here, they assume the framework of his transcendental-phenomenological reductions and serve to exhibit the constitution of a thing in pure transcendental consciousness. Such analyses are based on the criterion of a process of reduction whose sense and methodological correctness are based on the display of objectivities in terms of the "how of their modes of givenness" for pure consciousness.

For all that, the Husserlian transcendental reductions will not be performed here. This does not imply a metaphysical presupposition, one favoring o~tological realism. Rather, it primarily implies a simple conviction that a dubious step can be avoided in Husserlian transcendental phenomenology, a step that otherwise would be a hindrance at the very outset to the mastery of the problem of space. While the radical accomplishment of his reductions also leads to the reduction of the factual-empirical ego to pure consciousness, we shall counter this by pointing out that the subject, strictly under­stood, resists in principie any reduction in the Husserlian sense. I.he . subject must be reestablished in his full and concrete su]Jject_iv:ity ... i.e., in Qisf(1SÜJ2!Íy .. iiflél_, cqiÉ!J~~1JZ~~~~~~Fa~:,~eClE~~ITi.~~.qu_tred ~ for the phenomenology of s~ace. It is justifiable and indeed neces-

•. ·'''' -·-:..>·-0'"''""- -· ,,,_,o.,...~-,.c-,-.=~

5. O. Becker (1). Numbers in parentheses after an author's name refer to the works in the bibliography listed by such a number under each author's name. Only the frequently cited works of Husserl will be given with the standard abridged title for the sake of convenience.

12 Introduction

sary to exclude all that is "merely" factual and to disregard all that is conditioned situationally by the individuallived body; neverthe­less, it must be maintained that to neglect facticity and corporeity entirely would mean a premature move toward metaphysics and away from the methodological correctness of descriptive efforts. Thus the subject after the reduction would not be grasped phenom­enologically as a subject, since in Husserl it functions in the sphere of pure transcendental consciousness as constituted· in accordance with the requirements of a spatial thing. Toward the end of his works, Husserl noted these and related difficulties and recognized the subject as something that cannot be included in the process of reduction in the form in which the process was initially inaugurated.

Our arguments against Husserl are not in tended to be a destructive critique of his phenomenology; due to an entirely different point of departure, such a critique would be fallacious. A transcendent position can serve only to confront two theoretical stances with the aim of clarifying their own thoughts. Nevertheless, a critique of specific Husserlian points is justifiable when we can take exception to mistakes within his phenomenology, and specifically when such mistakes could have been avoided in terms of his own point of departure. Husserl himself would not have denied us this right.

PART ONE

LIVED SPACE

SECTION ONE

Contributions to the Phenomenology

of Lived S pace

Point of Departure and Statement of the Problem

The point of departure of this investigation is prior to any decision J concerning the independence

1 of the external world, reality, or

being-in-itself from the subject. It begins with the observation of a form of consciousness that as "natural" or "everyday" conscious­ness it is nevertheless far removed from being realistic. Its position toward the world is nothing other than one of being straightfor­wardly given over to the world, of direct respect for the world in its demands and assurances. Its statements contain neither realistic nor any other kind of "istic" meaning; only reflective thought is exposed to the danger of interpreting them in such ways. To grasp entities as beings-in-themselves oras merely subject-related is to presuppose a reflective attitude. This attitude requires that the subject extricate himself from the stream of events and experiences and differentiate himself from the world. However, the ontological sense of prereflec­tive expressions is metaphysically indifferent. It consists in nothing other than an immediate being "by" the things and being "in" a world.

A spatial relation is therewith already expressed, although the illumination of its nature is left for later pages. Here it cannot yet be assumed in advance insofar as the subject, still to be considered, does not yet grasp himself with respect to the meaning of this spatiality. For him the living being is justas spatial as are things, and like them, he too exists "in" space. If asked about the nature of space, he takes it to be something empty that appears filled with worldly things, events, and states of affairs, and thus to be simply a "world space." Yet su eh a conception arises from a theoretical, even if primitive, attitude; strictly speaking, it is an answer to an explicit

14

Contributions to the Phenomenology of Lived Space 15

question, but not a primordial relationship to space. To test this answer in terms of its meaning and justification is reserved for subsequent discussion. Of crucial importance here is only that such a "position," based on a response to an explicit question, is clearly removed from all immediate otientation to the world. In this latter c9W case space is not thematized. It is pre-reflectively there in the process of corporeal and intellectual activities without becoming an object for consciousness.l

Our investigation ought to begin with the analysis of this pre­reflective world-posture. The subject must not be interrogated pri­marily in terms of his judgment "about" space, but in terms of his comportment "in" it. Of course, the subject does relate to space by making judgments "about" it. This constitutes his very being as a subject: to know oneself in "opposition" to the world, to have space objectively and to be able to express judgments about it.

This ambivalent awareness of space obviously plunges the inves­tigation into inextricable difficulties at the very outset. How can it adequately grasp the relationship of a being "in" space when this being is essentially constituted by being "over against," outside of, space?

The formulation of this question reveals at the same time another problem. Even the notion of "over against" suggests a spatial relationship, which, as one is apt to say, is nevertheless meant non-spatially. A brief reflection on language readily shows that thinking, regardless of its' efforts to escape the power of spatial metaphors, can express itself only by succumbing to their misleading force. This is not only the case when thought employs them deliberately for the sake of vividness, but even when, with full insight into the inadequacy of such images, it attempts to exclude every spatial meaning transmitted by a word. It would require far-reaching philosophical explications of language to develop the problem of space from this viewpoint. Here we must forego such considerations for obvious reasons. We are excluding this type of semantic reflection and shall take extreme care to provide the greatest possible clarity and terminological univocity.

1. This is why we speak here of lived and not experienced space, specifically when the concept of experience is used in phenomenology in an · almost unlimited terminological sense without being appropriate for our subject matter. The concept of lived space was already used by K. v. Dürckheim in 1932; however, our investigation follows different method­ological principies and thus results in different formulations.

b lA-\ O.t prv \vlr-\.... ~~~\elét

~~ JY

16 Lived Space

The return to the question raised abo ve concerning the proper point of departure requires a closer look at the constitutive relationship between the subject and space. It is based on the subject'smode of beingas a corporeal subject. 'I'he lived body must be understood in Le­a strictly phenomenal sense. This means that it can be taken neither asan organism nor as a physical body infused with a soul. As special K& sciences, as methodologically established arrangements for research- .l

ing corporeity, physiology and psychology imperil the view of what \..et is given directly and originally: primordially, corporeity is neither a system of organic processes nor a body "inhabited" by a soul, nor even a "unity"-regardless of how conceived-of body and soul. It is also not primarily one's own lived body with specific sensations or inner content ("states"); immediately and originally, the lived body is presentas another living being and, more specifically, as the com­portment in which this being's relationship to the surrounding world is announced in both a sensorial anda senseful way. The lived body always appears as lived body only in such comportment in a situation and is understandable immediately from -~ithin the situatwn; this mode of understanding is prior to all scientific explanations. No analogical inferences, empathy, or other auxiliary constructions of a sensualistically oriented theory used to grasp the other are of value herei Rather, corporeity is _ _given in a way that is indifferent to any regional distinctions between the Q§YChic and the 12hysical, the inner andthe outer. A.lllioug~ity ¡~- also quite accessible to the 1nvestlg~tionfrom either aspect- even if the conceptual structure of this "also" need not concern us here- we must maintain a strict distinction between the corporeal phenomenon given in immediate presence and corporeity as determined by specific methodologies. 2

2. The anthropology of the twenties has insistently pointed to this "psycho-physical indifference" of corporeity. See M. Scheler (2); see also F.J.J. Buytendijk, H. Plessner (1), (2), andE. Rothacker. This anthropological research has served well to rediscover an aspect of corporeity overlooked by the special sciences ever since the Cartesian dualism of body and soul. If anthropology insists that its position is prior to all specific researches, nevertheless the ensuing conception of the neutrality of its point of departure cannot be construed as indifference of philosophy toward the individual sciences. In its own progress, philosophy must interpret scien­tific results with its own possibilities of reflective thinking; it must la y open their sense by introducing all the methodological precautions of the indi­vidual sciences, which in their turn-as specific "aspects"-require structural elucidation.

In the latter aim, anthropology is distinct from existential philosophy.

;~·)·tr-~ • ,....,.., .,_...,.., ..

V~ a f\ eeJ .... \ur w ~

Contributions to the Phenomenology of Lived Space 17

Corporeity is understandable only frorn its cornportment toward the world, and it lends itself to phenomenological observation only when approached in terms of differing styles of its relationship to the surrounding world. We can indicate three such styles: asan attuned corporeity, it is a carri_er of expressive content; as a practica! corporeity, it is a point of departure for goal-oriented activity; andas a unity of senses, it is the center of perception. L

01·b ~ e,kJ

It will become apparent that each mode of being of the corporea\ subject corresponds to a specific spatial structure, or, to be more exact, that "the" one space is obtained, structured, and filled differently depending on the cornportment of corporeity. It is quite clear that we are not dealing with three separable forms or three temporally and genetically distinguishable spatial steps or stages, as if the homogeneous space of an objective space-consciousness were capable of being deduced or developed from them. Rather, this singular space presents a contingent presence in objective con­sciousness that is completely unavoidable, given our point of departure. The point of departure must be accounted for, although its elucidation must remain a matter for subsequent considerations. The following investigation of the three spaces is concerned only with this: to elucidate and present the variously structured ways the subject, as corporeal subject,/oo~ space in accordance with the ? various corporeal modalities~

It is important to note that in this context the differentiations are ( ~p~~ch'?log~al,but ontological. The corporeal modes of comport-rnent are not psychological. Although there is a psychology con-cerned with the genesis, structure, and performance of expression, of pragmatic processes, and of perceptual activity, this does not mean that ontological questions concerning the nature of expression, action, and perception and the understanding of their whence and whither are superfluous. In fact, it is assumed that such questions are already answered insofar as they provide clues for the proper interpretation of the results of individual sciences. Modes of com- \j¿,,_,\J,~I.-{ portment are here not conceived as objects of investigation of an

While the philosophical _pr~ble~-~f--~~rp~_J~~~tyj~}~!!l~ ex_~l~ded_ by H~~). in French existentialism the lived body assumes a preeminent place as a theme of exhaustive structural analyses (see the works '52: of J.P. Sartre and M. Merleau-Ponty; see also the survey by A. Podlech). Their emphatic rejection of all efforts by the special sciences provides an occasion for critical reflections to be offered later in an appropriate place.

)(, 1..-vC\<-"',;o;" vJ ~(\vJ v{- ~\(_ W<\t

18 Lived Space

extremely broadened theory of behavior, but rather as a kind of senseful relationship of corporeity to the world.

The differentiation suggested above is not episternological, just as the relativity of lived space with respect to the corporeal subject is not epistemological relativity, but a relativity of being. The three emphases suggested have to do with the modes of being of givenness, not with modes of givenness of entities. The latter might assurne that within a particular position toward the world, entities could be given otherwise than their type of being would require. This is nevertheless excluded by the strictly correlative relationship be­tween world and cornportment toward the world . .Thus in the attitude of expressive understanding there are no things given "for the sake of .... " Where expressive understanding suddenly "breaks forth," then, taken frorn the side of the things, the thing is "given" ontically otherwise than befare; yet the transformation of the rnode of givenness is ontologically grounded in an altered orientation of the subject. While such an orientation is comprehensible in terms of an entity of a particular kind, the entity is only understandable frorn the subject's comportment.

The relationship between space and subject cannot be sirnply accepted as a reciproca! relationship without further reflection. The old, enrooted-cü-n.c;;ption of receptaculum rerum, assuming space to be identical and indifferent with respect to the changes of its contents, is contrary to such a relationship. A more detailed expo­sition of this will be offered subsequently. With the proposed division into three styles of corporeity, the investigation is forced into differentiations that are justifiable only rnethodologically. While our investigation separates the attuned space frorn the space of action and the latter frorn the space of intuition within the lived space, according to the three modes of corporeal cornportment, in arder to show their characteristics and properties separately, we remain cognizant that we have assumed separations and divisions that are valid only if the analysis, with its increasing precision of distinctions, is capable of adding to the clarity, visibility, and cornprehensibility of the unity and intimate relatedness of what is being analyzed.

Chapter One

The Attuned Space Jl-v ~ ~f\VI wt-k ~ u.,M_

§ 1. The Concept of Attuned Space

~vdt The attuned space is hardly accessible to conceptual thought.

Certain misunderstandings that could make such thought more difficult must be avoided at the outset.

W e must first defend our thesis against the conception that space is gr ven üñly iiTtTs~áCcessibie tü-llie-éls-ure~ei;Csi>;-é:effiüsrceifaíñly isa-cünamóil-JüiffieaSU'rellien:-t-.--y~t~it-Ts --~ot a mere medium of 1 (Q) measurement. In its most primordial ontological form, space is on -:>-­the~ side of numerical and quantitative determination; its , primary characteristic lies in its being a quality and an expressive fullness. Grasped in its unique immediacy, it is all enveloping and Cüinpromises the "atmospheric" dimension of an attuned being. It is a space of labor, of leisure, of festivities, of devotion-a space that is loved, hated, feared, and avoided. As a medium where human life is realized, it has its own proper visage. It means calamity or seclusion, a foreign place or a home, a place of transient residence or an enduring stay. It is different in accordance with the differences in the being who inh~its it. W '-" A~ /) Ff~~;¡f'!M- ~

The un~rr'J't~nmfigof this space is not peri:eptwn, a.gd awareness 'tx-\-f<?f{.ettüt-1. of space is not cognition; it is rather a way o~.r{g moved and affected. S pace indeed exercises an "effectivity," yet its relation to 1W, rK."'I\~ 11

experience is not casual; rather it "addresses," "imparts." Space is not primarily an object for a subject who performs acts of spatial understanding. Rather, as attu~ed space, i!J~_as an ap_prQ_IJ_ri_~~!E:~c!.e of coexistence withJb.~l~y-~d-~o. Such coexTsfence escapes all the 1 conceptua.Tdeterminations of a thought founded on the opposition of v'

object and subject as a "relationship" or "connection." All these in their turn are founded on the primordial and intransgressible bond (} between the_c_9rp~e. Thus lived experience here Sr \e.~W­does not mean an oriented engagement in the sense of act phenom-

1¡

<or¡o•Tu{ .~~~ ?

20 Lived Space

l 1\~t kw.,~}t;-¡ &- kb~ 't d' t' . h . lf f eno ogy; as an ai uned llved' expenence, 1 1s mgms es ltse rom oriented engagement by a lack of intentionality. Neither does it mean a corporeal state within the subject; thus, strictly speaking, the concept of sensation must also be avoided. Here lived experience means a unique communication of the living- experiencing ego with another, with an expressively animated space. 3

A further objection must be countered; namely that the attuned space emerges only during a particular temporal phase, during the "exuberant moment." When our subsequent discussion occasionally touches upon such a moment, it will be for the sake of exemplifica­tion, as an appropriate and powerful manifestation of salient fea­tures. Yet such features are not restricted to these examples. Expres­sive understanding is a unique mode of orientation toward the world; expressive understanding has its own sense-context, and it must be taken in this uniqueness and interrogated with respect to its space.

This does not contradict the notion that "spaces" may appear to be different. The transition from one to the other, from the space within a church to the "animated" street, does not imply a dissolution of the atmospheric space as such, but rather merely a change of the expressive content. The difference between such "spaces" is itself a

~~----- - __ • ..___ _____ .--•------=.,. __________ --~-----------------

positive determination of tb.ª attl1:Qe(LSI@_G!Lfl.S such. · ~------ ~-- ------~~---------- ---------- ~-- ----

Expressive Understanding is also not contradicted by the fact that it is disregarded when the attitude of the subject is determined by practica! or theoretical aims. Rather, having a purpose makes it necessary for one to perform a specific shift, or-to speak metaphor­ically-to step out of the attuned space into a space with a com­pletely different structure, consisting of goal-oriented activity, of sensory intuition, or of pure thought.

Attuned space is encountered in a pre-reflective orientation to­ward the world. Such orientation to space is an immediate affinity with the world. For an attuned being to be in another "space" and to live in another "world" are expressions identical in meaning. They testify, on the one hand, to the fullness of sense and meaning characteristic of the space discussed here and, on the other hand, to the specific difficulties involved in the conceptual determination of attuned space. While at first the attuned space is bound to limited spatial surroundings, it appears to be capable of multifarious exten-

3. Concerning the concept of attunement and its relationship to feeling, see O.F. Bollnow, S. Strasser, and P. Schroder. The concept of sensation will be discussed subsequently in greater detail.

The Attuned Space 21

sions. We speak of the space of our future and our past, of our wishes and our hopes. Home and away are not only significant wholes, with appropriate feeling and mood accents; they are at the same time understood and differentiated spatially. Within them the human also

]

"moves," and things and events have their "place"; they are "near" or "remate," getting in our way or lending us a free "way."

Seen from the vantage point of objective space, all these are obviously mere metaphors, spatial images for relationships that "in reality" are not spatial in kind. Yet this objection does not touch the modes of ex eriencing such spaces; after all, our lives within them V are not merely imaginistic. The objection cannot explain why space plays such a preeminent role in lived experience and why specific unities of meaning offer themselves spatially to lived experience. A more exact investigation in this direction would lead us too far afield from our present purposes. We shalllimit ourselves to attuned space in a narrower sense.4

4. It will be shown that this space includes temporal determinations. This relationship between space and time is stressed by E. Straus (2)-at least through an application of an ambiguous and misleading concept of sensa­tion. Yet the relationship between space and time should not lead us to encompass everything within attuned space, within whatever is designated by "space" in the metaphorical use of the word. Phenomenologically, a clear distinction must be made between the attuned space in a genuine sense, i.e., between a space given as "real" under specific and always freely actualiz­able changes of attitude of the subject who can find himself in it as real corporeality, andan attuned "space" in a transferred sense. The "transfer," in its turn, is a clearly demonstrable datum of lived experience; even when we "live" in such spaces and have "resettled" ourselves in them, then, given a corresponding attitude, the attuned space can become in principie a specific datum of lived experience. This in turn is possible on the basis of an already understood spatiality in the genuine sense as defined above.

The complex of phenomena constitutes a unique field of interesting and specific analyses wherein it is possible to distinguish various types of transfer. Here is not the place to investigate them further. Only different metaphorical meanings of space should be mentioned: they are found in ?-­~üioms, in----püeiry,ínthe illusory spaces of drama and their numerous realistic-spatial surroundings on the stage space of the theatre, in the imagined and imaginable spaces of fictionalliterature, etc. It is remark­able that the contemporary theory of literature pays specific attention to the theme of space (see Bachelard, Blanchot). The motivation for a typological reevaluation of the literary aspect of space stemmed from W. Kaiser. This dissolved the traditional principies of articulation in the theory of fiction, principies that previously pertained purely to the subject matter. See Kaiser

22 Lived Space

§ 2. Characteristics of Attuned Space: Fullness and Emptiness

The primary access to attuned space is offered by the character of Jhings_l!0.J;. This does not constitute a prejudgment concerning the relationship between space and spatial things. No matter how this relationship will be determined, it is in any case clear that space as}. such is accessible only from its fullness. It is not readily obvious at{ the outset whether it is possible to investigate an empty space. What ~~~~!Q~~!!'_~9!VIªl-12h~~~ft~!!~Jt<;:§,_Qf_.~p~g~_j_s_t-ªJ&en from the spatiality of things. The characterization of space is ac~essil:i1e"to-~refTecTivé ~-añáiysis only through a delimitation of various kinds of thingly attributes. But the notion of "thing" in attuned space must be taken with a grain ofsaTf lestwe~touch-u pon it, leaving our thought with its customary although insufficient categorical distinction between things and properties. Yet strictly speaking, it is precisely in the lived experience of expressive things that such a distinction does not occur . .Jlw_dllii-.Y.ªg~ Q!it.I!Yf:)!J_IlS_l1b1~-~ and object, _ which _ thj_s_ disti:qgtion presupposes, is a __ recent ont~log{c'afpr'ódu-ct:·The thÍng as -;-bearei.~f ~~pr~ssion do~;-;ot líavé prÓperties ~to be perceived; ratl;~.Jl-~ik¡:¡J;'~- .with - its "character."5 It is no accident that our attempts to describe the lived -~----r-;----· ·-·-experience of expression lead us to specifications suggesting, and drawn from, the psychic domain. This implies neither a misuse of language nor an inappropriate anthropomorphization, even when we do not take things in · their objective color- or form­characterizations, but rather in their own unique and momentary "toning." Exuberant or sad, hard or tender, soft or severe-these are both sensory formations and sense-formations, symbols in Goethe's sense, which lend us "specific dispositional moods" anél which at the same time possess "sensible, moral, and aesthetic purposes"; one can "even employ them as a language when one wishes to express primordial relationships."6 Goethe's hesitation, his suspi­cion of being exposed to the ecstatic, seems to be baseless. The lived

(1), pp. 360-65, and (2), pp. 24ff. See E. Stroker for an entirely different typology of imaginative space in painting.

5. The distinction between property and character was first made by L. Klages. Similarly, J. Konig, following the fourth of E. Husserl's Logical Investigations, differentiates between determining and modifying predi­cates (pp. 1-16). For a characterization of the latter and its specific kind of relationship to the subject, see especially pp. 32-41.

6. J.W. Goethe, Sixth Section.

The Attuned Space 23

experience of attuned space expresses itself readily in such primor- S dial relationships. Prior to objective recognition, its forms are 7 already understood "physiognomically."

In attuned space, therefore;tñeaísfinction between primary and secondary qualities does not exist. The form of things communicates as expressively as does their color. Both are equivalent in their relevance for mood; in their physiognomic content they are capable of mutual support, enhancement, or dissolution, and of eliciting dissonant experiences. Even the size of things is here far from being a mere quantity. The experience of the powerful or the sublime, as ~ well as the graceful and elegant, belongs essentially to size. It reaches complete fulfillment where its harmony is sustained and maintained with the rest of the characters. Regardless of how it may be determined, regardless of what aesthetic criteria may be sought for it, it is decisive that even size remains incorporated as a moment of the expressive totality, and thus it cannot be extricated and given an independent character without the loss of its valence of mood. The

imp_:~~ ~~~~~~_;g~~-~~~~er of J.Q!~!L2.I.l<L .. ~!~ _is . espec1ally convmcmg m a trend of p1ctonal art that mcorporates m C>f(I':SS.:í.~ 0 '1t

its formation the effectivity of precisely these two moménts. "Expressionism" owes its expressive power, not to mention its sovereignity, to the presence of form and size factors not bound to any objective attributes of a thing.

Form and size are constitutive factors by which we apprehend )-, . +, generally n_ot o_nly the perspe?tivity of space: ~ut als_o all the rest of tr•~ l~t ~Jt\: the determmatwns related to It, such as centnclty, onentedness, and M-~~ finitude. What can be said about them with respect to attuned space? ~ •n• By raising this question, the subject betrays at the same time the fact that he already belongs to another space; no other state of affairs underlies the claim that the spatial perspective is conceivable in terms of the size and form of things thaii that the size of things is observed comparatively as greater or smaller in relation to others and that their forms are consciously perceived in certain truncations and intersections with others. But this means that they must be regarded as purely quantitative determinations. As such they make their first appearance in the space of intuition. P_erspecti_y~ is ~a_f§l_~_Ol_IJ.ercep- ') tion. If we infer from this that "there is" no perspective in attuned space, then we require additional explications in arder to preclude closely related misunderstandings.

Indeed, in attuned space things are encountered in a specific order of being behind one another. Yet while size and form vary in accordance with perspectivallaws, the constitution of attuned space

24 Lived Space

does not depend on them. That the perspectiva! arder can become physiognomically significant and participate to a great degree in determining the spatial atmosphere is not contested here; this is in accord with the functions of form and size as expressive characters. Thus it is sought in pictorial presentations of this space for the sake of its mood-bearing moment. Yet it is essential to incorporate this phenomenally incontestable state of affairs appropriately. Attuned space "is" perspectiva!, but not because perspective is a structural property of this space; the claim is rather that the lived experience of expression evoked in attuned space by the perspectiva! arder of things can only be given because the experiencing subject in this space does not live without the objective space. While living in attuned space in a most primordial world-posture, prior to all purely objective orientation, the subject already has the objective space "behind his back."

Attuned space therefore bears determinations that allow it to appear as profiled against the background of the pure space of intuition; thus it is never free from the determinations of the latter. Even the perspectivity of attuned space is a characteristic present by virtue of the space of intuition. Yet perspectivity does not belong to attuned space as such. The resulting consequences for the orienta­tion1and centering of attuned space will be pursued in subsequent paragraphs. Meanwhile it is essential to touch upon a phenomenon that, along with the color, form, and size of things, possesses great power with respect to atmospheric space.

Sound plays a uniquely significant role in attuned space. In space-determining power it surpasses even color and form. At first this appears dubious. After all, the arder of colors and form is an arder of one next to the other, i.e., the genuinely space-constituting arder, while sounds are present one after the other in time. More­over, color and form can be more precisely located in space than can sound, which only indicates a direction.

To counter the first doubt, it is necessary to become free from the presupposition that every space must be defined as an arder of one next to the other. It is not yet certain that the space investigated here can be circumscribed by the classical Leibnizian definitión. If the claim about the importance of sound could be confirmed, then there would be an indication of a structure entirely different in kind.

Although it is a matter of gradation, localization is attained with a differing precision for the colored object and for the source of sound. In contrast, between color and sound there is a complete qualitative difference that touches upon the spatial characters of both.

The Attuned Space 25

The color is attached phenomenaliy to an object, not only as a lc\or /~u property, with all its nuances and differentiations of intensity, but also asan expressive presence. Color can never appear except on a colored object. It is otherwise with sound. Sound detaches itself from its soun;e. It is not a property but an event; it is not attached to something but draws nearer and recedes into the distance. It can indeed be said that it is a sound of something, that points to a source. But this character of originating with something is, strictly speaking, Souttdjflb,·'i not appropriate to sound but to noise. Noise is always noise of something and is always perceived as such. Sound, in contrast, has an existence that is detached from its source; it becomes a sound precisely because it is capable of such detachment. This state of affairs rea·ches its completion in music. Here it is the sound itself that fades away and strikes up, not the cello or the first violin. Of course sound can be heard as the sound of an instrument; but in this case we are dealing with an entirely different mode of lived experience. Not only does the expressive content attain a purer and more complete presentation and effectivity in the lived experience of the pure tone formation, the sound experienced in its free and appropriate exis-tence has a different spatial character. It will be experiencel more intimately than a sound of an instrument; it pervades space and determines its atmosphere more completely than it would if it were perceived in relation to its source.

The same process of detachment can be traced with extreme clarity in spoken language. It too is primarily a sound formation brought about by specific organs of the body. Yet the experienced sound is not merely the sound of a speaker. By virtue of the sense content of the word, the detachment occurs here so completely that one requires a specific shift of attention in arder to grasp it as a word of the immediate speaker. This shift will be spontaneous when someone speaks a language of which · we "do not understand a word." Here the word will be come noticed as a word spoken by someone, who emerges into the foreground with all his personal accentuations. With the emergence of the shift of signification from the spoken to the speaker, the experience, in its conceptual sense, approaches noise. As a sensory formation, the word is subsumed under laws, making possible a theory of language, just as sound obeys the laws of harmony. The clatter of a motorcycle, however, does not follow any musical syntax. Noises are not objectifiable; ~ their "sense" exhausts itself in being noise "of something." This is not the p ace to ela orate furhher relationships and differences between tone and linguistic son~ d. We have memly suggested how

¿ Cli ~í J

+/5

26 Lived Space

the sense-fulfillment of a sound formation goes hand in hand with an emancipation from its source.

What sort of determining forces noise and sound have for space becomes clear when the same space is experienced once as sound­less and then as filled with sound. Colors and forms maintain the same order and distance, remaining remate forms that change only when dusk and night blend and dissolve their contours. The sound

i coming toward us not only fills spa_~_bl11 also contracts it. A distant church moves closer wiTha sudde~ resounding of its bells. lts inner space becomes noticeably smaller as soon as it is pervaded by the sounds of the organ; in a siient film, the screen attains a peculiar distance.

A relationship of this kind between space and sound presents a new problem. The sound formation taken in itself is a temporal formation. What remarkable relationship exists here between space and time? We shall return to this question in another context (pp. 36 ff.).

The qualitative content of attuned space is not exhausted by being an expressive content of color, form, and sound. lt is possible to suggest a domain of lived experience in which the experience of expression has remained in its purity. This purity must have persisted not only across the objectified and reified world, but has remained inaccessible in principie to any other apprehension except the physiognomic. Literally speaking, animate creatures also belong among the "things" of the attuned space; they constitute its atmo­sphere more clearly than do inanimate things and their spatial valence is easier to detect. From the standpoint of objective space­consciousness it must be complete! y inconceivable that ill. J)}.I'ÜJ' ¡ . . presence "space would become immense" for us, or "too stt'ílin~·l 01/r '¡(i that they could allow us "leeway" or could squeeze us out of our "place," compelling us to leave our location and to move away from them spatially. This sphere of the expressive experience of fellow creatures shows most clearly the remoteness of attuned space from all measurable determinations. lts distances are other than measur-able. ~serves the concept of. di~~J~:L the region of the measurable, then g_cag_be said that there are no cfist~n~~s--in attuned space. --- ----------------~-----~----

----Tlirough-Itsfiíl1ness, attuned space has become visible, at least allusively. A question could be voiced concerning with what justi­fication we infer space from fullness-what might space otherwise be? The sense of such a question implies that space is something other than its fullness. lt assumes an empty schema capable of being

The Attuned Space 27

seized in itself. Perhaps this question js influenced by conceptual analogies of the space of intuition. Yet despite the manner in which the relationship between space and thing may be structured in intuition, the essence of attuned space excludes the differentiation between empty and full space. To wish to extricate an empty form of attuned space from its formations would be senseless. Indeed, it too contains a lived experience of emptiness, the encounter of "noth­ing," the dissolution of its formations, which "have nothing more to say"; yet this emptiness, experienced as a mood, is not to be confused with the empty form of space sought by abstract thought. Attuned space is not merely given and lived in its fullness, but rather it is this fullness itself. Its loss is the loss of attuned space. This is the reason why we speak of fullness as we would of attuned space itself.

§ 3. Place and Po sitian in Attuned S pace

Attuned space has its general characteristic in being a form of expression. As such, it is nota manifold of positions, nora system of dimensions. Place and position in it are not determined by an assertion of quantitative relationships of a there to a yonder: Even m y own place in it is never merely a point determined relative to the place of things. I am in my workroom-this does not mean being 1 somewhere "next to" the chimney. 1 allow the inside of a Romanesque church to affect me - yet my "place" in it is not at a comer of the quadrature. Certainly, these positions belong to the space in which 1 reside; its formation, its architectonic articulation, are part of its fullness and shape its atmosphere. Yet they do not determine my place in it; for my lived attunement, no tangible relationship holds between the lived experience of space and the position in which 1 find myself as a corporeal being. M y phenomenal place in attuned space is not ascertainable. As an attlineabeing,I have no determinable locatioñln tñTsspace.-----~-------- ------~ ---~ Thus in attuned space my lived body cannot be a point of

reference for a relative determination of the position and the place of things. Attuned space has no center of reference from which it would be possible to arder and separate the experienced things and determine them as there in relationship to a fixed here. lndeed, in m y vision, hearing, and touch, things are constantly present and thus are related to me in my corporeal organization; yet this relatedness not only does not play a part in the actual experience of things but even in reflection on experience it does not appear as a facet of attunement. It is a relatedness of things as pure perceptual objects to

28 Lived Space

me as a corporeal-physical body, and this relatedness as such belongs to another space. Relationships from the oriented space of intuition reemerge here quite readily. Yet here this space is not yet thematic; in attuned space it remains veiled in the sense described above. Its perspectivity is present in attuned space without being part of attuned space. That attuned space does not possess a center must be taken in the same sense. The centricity of space requires its relationship to a corporeity that is necessarily of a type that is localizable, i.e., a corporeal-physical body. Indeed, as an attuned corporeity I am not without location, yet it is not through location that 1 am an attuned corporeity. Corporeity as a bearer of expression has no spatial position.

Lack of orientedness is closely related to the absence of a center in attuned space. More precise results will appear in the analysis of expressive movement. In the attuned space "there are" no preemi­nent directions to be taken in arder to attain something, as would be the case in the space of action. When one follows directions of roads on a map, one does so in arder to orient oneself-yet one does not orient oneself in the mood space of the landscape.

Attuned space yields itself fully and completely first of all in a purposeless lingeri;g w~e-tliT~gs r~~ealtlieir-genuTiievísage:~Tlieir there ancf-yonderarenotpu~;-¡;-~8itiülls Tii- spacé ,-ai:bifiarlly and externally interchangeable. The thing, as a bearer of expression, has "its" place. The place belongs to its expressive power and accentu­ates or diffuses the physiognomy not only of the thing but of the space as a whole. An exchange of place of two things not only abolishes their genuine expressive fullness but also noticeably

)

disturbs the atmosphere of the e.ntire spac.e. P. ieces of_f~-~-ni.tu.re do (]!) not "belong" in the marketplace, nor church windows in an office. 7

It is precisely wi~~~sattuneínent:·witíí--Süch clashes of style, that the place of the thing appears most strikingly; on the one han~ they reveal that ¡~la~~_Qelongs __ t~--~~- e~press_ive

-·; ~~~:_;:~ti?~t~~!~~~ ~::~t~e~~i!~ii:g~~~sY~~f!Jll~~~-h~ appropriate relationship is capable of transforming attuned space as a whole and of disrupting its atmospheric unity. This shows clearly how little it can be characterized as a mere proximity of places and locations. Its structure deviates markedly from a mere manifold of positions. Rather, it is given as a closed unity of expression that can

7. K.v. Dürkheim, p. 406, already indicates these lived experiences of "belonging there."

The Attuned Space 29

be affirmed or negated, accepted or overlooked as an inseparable whole.

§ 4. Nearness and Remoteness

Sin ce in a ttlJlle~d.c~rpace_plac.e..an.d._p.ositioiLdo...noLallow_deterrni­JI~_ti_on cl_eüYJ;l.dJro:m....mat:riJ::s.~.am.J) iJLYiilid.iOL.its ... d i sta nces. In attuned space there are no measurable distances. This is understand­able from the qualification of things by place and from the holistic structure of attuned space. In.metric space, distances between things can be increased or decreased without incurring the same changes in the things themselves. In contrast, in attuned space such changes do involve the things themselves: their "in-between" is not a mere \1'!-\.t~we!-•,., relationship of arder to be considered apart from, and in comparison with, other things; rather, it is a qualitative, expression-bearing characteristic of the thing itself.

As such, it is related to the "place" of the experiencing subject, i.e., to his being attuned here and now. The nearness and remoteness of things open up to the subject. The difference between them is not a mere matter of degree; nearness and remoteness differ qualita- <J

tively. A distance is composed of smaller segments. Yet nearness does not consist of other nearnesses, and remoteness is not com­posed from various remotenesses. Distance can be separated into partial distances, but remoteness cannot be divided into remote­nesses and nearness, into nearnesses. A remoteness does not contain another remoteness, just as nearness does not contain another nearness and an addition of nearnesses does not result in remote­ness.8 Distances subsume things as perceptual objects, but not nearness and remoteness. The latter are expressive phenomena, although expressive in another sense than color, form, and sound. The nearness of a thing in attuned space does not have its meaning only because its content appears clearly, but because it is near tome. Nearness and remoteness are not attributive determinations of the S thing grasped, but of my mode of grasping. J

Nearness is pure being-present, lingering here and now or being threatened or beset by things, which, by their irritation, intrusion, and oppression, leave no "space" for one's own relations and movements. This space can be established in two ways: in yielding, fleeing, and removing oneself; or in surpassing, overpowering, and

8. In another context the irreducible characteristics peculiar to nearness and remoteness are mentioned by E. Straus (2), pp. 288ff.

30 Lived Space

overcoming. The constitution of the remoteness of attuned space takes place in this dual movement; thus remoteness is that which is "no longer," it is "there" where I no longer am, where I have ceased to be present- a limit of my attuned space. It lies "behind" me, it has departed from my view and vanished from my actually present lived experience. Yet it is also something else: what is "not yet," what is still veiled from my view, and sois also a limit of space that lies "befare" me. It becomes an aim of my search, an orientation of my departure. Nearing is not the dissolution of a remoteness. Distances can be traversed, yet remoteness can never be reached. With each nearness appears a new remoteness. It is only nearness as an attained here-now in a coming from a no-longer and lingering befare a not-yet, just as remoteness is only the surpassing of nearness. Both are reciprocally related.

It is striking how spatial and temporal relationships pervade one another here. Nearness and remoteness are spatio-temporal phe-

Q nomena and cannot be conceived without a temporal moment. While yielding a horizon for attuned space, they also contain another aspect of conceiving this space as "space-time." It is only where nearness and remoteness are abolished-i.e., in metric space-that space and time are sundered. Distance is a purely spatial quantity. ------

§ 5. Movement and Orientation in Attuned Space

Nearness and distance are relative toa motile living being capable of approaching or distancing itself. It may seem that the problem of motion is secondary iir significance. Since all motions . occur in space, then obviously space must be assumed as the "play-space" of motion. Hence the analysis of motion could be avoided for the understanding of space. Space appears as a precondition of any possibility of motion.

At the outset, the following investigation will be based on the phenomena pure and simple, in a specific sense-i.e., we shall observe the phenomenon of the self-movement of the subject. Self-movement always served as a sign of the living: if something can move "itself," can follow its own impulses and not mechanistic causes, then it belongs to the living region. To the extent that life articulates and unfolds itself in its richness of gradations, it also grows in the richness of its forms of movement. The richness oflife's growth and the increasing development of movement variations are

The Attuned Space 31

mutually implicatory movements of the same unitary development.9

The bodily dynamics distinguish themselves from mechanical motion by more than the vitality of inherent drives. Were this the sale distinguishing characteristic, then the perception of movement would be reduced to an organism traversing a basically meaningless

i series of linear points. But in fact each apprehension of a bodily movement is constantly transcended toward a specific intention: the ~ movement is a searching, defensive, furious, tired, happy movement and hence is already grasped as a dynamic mode of relationship to the world. It is understood from within and in relation to its situation; both limit one another. Thus the question as to whether a pre-given situation motivates the movement or whether the move­ment constitutes the situation cannot seriously be asked.

This means that expressive movement unfolds itself and is com­prehensible in its total fullness only in and from the space wherein the attuned being lives. That one moves differently in a cathe~l than in a factory:, i~.2J!lJI.SégmJ!.l.ffer~E!:ly t~~Il.~verqL can !!!l~ be understood solely by understanding the coordination of reflexes ano muscularccmtfficfions. Tlíis is not to deny the fact that such coordii1at1oñon1leinovéiñent of parts may constitute a necessary condition. Nonetheless, expressive movement must be grasped from the experience of the space "in" which it takes place. But what is the sense of this being-in?

In its singularity, attuned space is always a specific space of motility. It allows and requires specific forms of movement and excludes others. The movement of my lived body is a comportment toward the expressive content of these spaces, and it either fulfills or () rejects their demands. At the same time, it is a space of movement in another sense. If my attunement is created by the atmosphere of space, then my movement appears to be formed by it and attuned toward it; yet this is only one side of a reciproca! relationship. There is also the other side. Space is not just a space fo~ mi?~~I11-~~!!~i!.~ becomes a space, in its specific attunement through my movement. Through m y movement 1 cañnegafe-If, atfeSlTo1T;'O'fCl'Üse mys8Ifoff

9. The contemporary theory of science of biology and physiology and its "introduction to the subject" (J.V. Uexküll, A. Portmann, F.J.J. Buytendijk, V.v. Weizsiicker) has freed itself from the path taken by a purely physical­istic approach to physiology; it explicitly sees itself as an effort of "under­standing," without devaluing the results of earlier researches. The work of Buytendijk, in particular, is outstanding in its strictly delineated conscious­ness of method.

32 Lived Space

\

from its content. Thus space no longer remains what it was, but is immediately transformed. In a reverse manner, I am not only a receptacle for its contents but a co-carrier, and first of all a shaper of its atmosphere through my movements.

Attuned space is not just an expressive form "within" which my movement occurs. lt arises as the space of my own movement. My expressive movement is exactly what its spatial content is as mine; in turn, the spatial content itself has a moment of my activity. If we keep in mind this strictly reciproca! relationship between space and expressive movement, then we have a principie for attaining a more detailed exposition of the structure of this space. lt was already noted that this space is not a pure manifold of positions, but a formative whole. The phenomenon of movement allows further determinations of space that are hardly graspable by any other means. Here we are confronted specifically with the problem, touched upon above, of orientation in attuned space.

First, a question emerges with regard to the lived body and its expressive movements. What are its characteristic movements in attuned space and what can be inferred from them for the body's orientation?

Most decidedly this is not a question of how the lived body as one's own is experienced in attuned space. In being attuned I am one with things and with space. My lived body is included in such an attunement without any intention on my part toward it, or without its condition being somehow present thetically to my awareness. Such an awareness can emerge suddenly with qualitatively unique sensations-data characterizable in accordance with "postura! situ­ation," "depth," and intensity as typical modes of the inner giveness of one's own body.t0 Yet in attunement as such there is no positional consciousness of one's own body. lt is reflection that first places my body, with its modes and manner of intertwining with the world, back into the sphere of judgment. Yet the lived body of attuned space is offered to reflection in a unique manner: here it is phenomenally unarticulated. Torso and limbs constitute a complete unity in the attuned body. They separate only in an oriented space where the limbs assume specific functions. They can specialize themselves to the extent that the limbs assume an almost independent role from the trunk, which, determined by the limits, is as it were only set into

10. A bodily "inside" with "postural situation" and "depth" presents the body itself as spatial. For the relationship between lived body and physical body, see p. 52 of the present work.

The Attuned Space 33

motion by them. In expressive movement, however, limbs and trunk are mutually interactive. In this mutual attunement and harmony of the individual dynamic forms, we experience charm and grace. These appear in walking, striding, and dancing, as well as in resting and standing still. E ven pointing and grasping in attuned space is not simply the protrusion of the arm from the trunk, as is the case with the conscious intent of getting something in the space of action; rather, the limb remains completely caught up in the dynamics of the whole body.

If we consider such an undifferentiated body in its modes of motility, we notice salient differentiations from its purposive move­ments in the space of action. Buytendijk rightly criticizes Spencer, pointing out that by equating charm and simplicity of effort, he completely misses the essence of expressive movement.u Its unique­ness and richness is precisely in the unintended power, with its fullness of possibilities undetermined by any rational point of view. When Schiller characterizes those movements that "correspond toa feeling" as charming, he expresses precisely the effortless, aim-free movements consisting of "the beauty of form under the influence of freedom. ''12

What does the absence of any economy in expressive movement mean for attuned space?

We take a couple of steps back in arder to see something better, turn around, move to the side in arder to allow something to have an all-sided impact. In the space of action "there are" indeed such movements and, seen from the side of an organism, there is no difference between them. Expressive and purposive movements do not differ in terms of their course but are distinct in sense. In the space of action, a step back is an "unnatural," compelled movement; a step aside to avoid an obstacle or to make room, or any turning around, are avoided if at all possible. In attuned space all these movements are noncompulsory, unintended, and yet obvious. This is not because in our attunement we are not oriented toward our movements- even in the space of action we are not explicitly turned toward the activity of the body, but to our aims and the means available to attain them-rather, it is the space that constitutes itself without compulsion, while the space of action is experienced as a demand.

Expressive movement offers an answer concerning the orientation

11. F.J.J. Buytendijk (1), Section F. 12. F. Schiller, p. 254.

34 Lived Space

of this space. Attuned space has no specific orientation. This does not mean that it is completely without orientation; such a space, as space of a lived body, is impossible. Yet it lacks a qualitative differentiation of values among orientations. The movements all essentially occur with the same spontaneity, equally effortless and obvious. Attuned space is atropic.

This state of affairs is supported by yet another phenomenon. If one assumes that the endless multiplicity of movement orientations of an oriented space remains related to the triad of the elemental pairs of opposites-up-down, right-left, front-rear-then it appears that orientation can be differentiated only to the extent that the pairs of opposites are differentiated. Yet just these differentiations slip away in attuned space. The differentiations mentioned are, as differentiations in orientation, primarily functional; they are condi­tioned by the functional articulation of the lived body, which lends space an asymmetrical appearance, We shall speak of this while discussing the space of action. In attuned space the body is "prior" to such articulations. This can be seen in the case of powerful expressive movement and posture·revealing a high degree of sym­metry.13 At any rate, what has too often been repeated is also valid here: the experiencing subject observed by us can hardly discover the atrophy of pure attuned space since other kinds of ontologically grasped possibilities of comportment toward the world are already accessible to him. Y et the attuned space as such do es not contain differentiations of orientation.

In attuned space these differences contain another differentiation of value, i.e., valence of mood. The elemental pairs of opposites are qualitatively distinct as bearers of expression and significance. Hence the opposition up-down is strongly pervaded by· an atmo­spheric and specifically religious difference, constituting a mythical residuum carefully preserved in all cultural development. In the early Christian form of life there were also magical meanings of left and right. Thus the left side meant the honored, the holy, yet also the demonic and despised. It had to be protected by adornments.l4

13. In addition there are the conspicuous ethnological discoveries that the known religious rituals of prayer, devotion, and contemplation are symmet­rical in posture. Yet let us also recall the bodily posture in sudden fear, in abrupt astonishment, the gesture of supplication, the gesticulation in doubt or in sadness, the exuberant joy, the posture of staring off into space, etc.

14. See J.J. Bachofen. In expressive experience there is generally a noticeable residuum of myth, to the extent that the mythical structure of

The Attuned Space 35

The relationships prevailing between the mode of movement and the determination of orientation are particularly clear in dance. As a paradigm of expressive movement, dance reveals in a greater relief all the moments that are given only weakly in attuned space, since they are intertwined with aspects of movement belonging to oriented space and thus cannot be grasped in their purity.15

consciousness can be considered as an early form of historical conscious­ness. As E. Cassirer has attempted to show, the space of the primitives possesses its own arder and articulation solely within physiognomic char­acteristics. Certainly for him mythical space is merely a primitive precursor of scientific space; the modern consciousness of space is specified by its total liberation from such mythical elements. For him the expressive experience of modern consciousness remains limited solely to the sphere of I-Thou encounters, and thus is inappropriately narrowed.

It would require an entire investigation to deal with the problem of the space of primitive peoples living today. These issues must be avoided precisely because it is questionable whether the phenomenological­descriptive method is adequate for them. The old demand of Schelling to understand the mythical world not "allegorically" but "tautegorically" is being reassessed by ethnology. The world of the primitives cannot be simply delimited as primitive in comparison to ours; rather, it must be interpreted as a self-contained whole with its own structural regularities. The magic and myth must be understood from "the presuppositions characteristic of peoples of these times" in arder to acquire a categorical system befitting the differing consciousness-structure of these peoples. This is the requirement proposed by A. Gehlen (2), p. 10, who contrasts it sharply to the method of "understanding" which merely "starts with the present and moves toward the past." Undoubtedly, this demand is applicable not only to ethnology but to all historical sciences which take history seriously as history and as such manifestly present phenomenological description in the broader sense with a diversified field of work.

Nevertheless, we must not overlook the limitations of phenomenological description in the specialized ethnological areas. Our contemporary con­sciousness of the primitive mentality is notably an alien consciousness, which always arises in the medium of its own categories. Even where we employ methodological procedures that take seriously "the presuppositions of such a mentality ," these eo ipso en ter the "understanding" of a conscious­ness that requires, according to Gehlen, a structural change of its own. It is only with such an explicitly self-critical limitation of its claims that an ethnographically oriented "phenomenology of space" would be justified. At any rate, phenomenology today would find a rich accumulation of ethno­graphical materials, especially in the areas of the pictorial arts (see the works of E.v. Sydow, H. Tischner, H. Kühn, H. Read, H. Werner).

15. P. Valery interprets dance poetically and F.J.J. Buytendijk (3), pp.

36 Lived Space

Dance is a motile and complete oneness of torso and limbs, a playful exuberance of dynamics and a beautiful · aimlessness of specific movements for which there is neither a point of departure nor an aim, neither a beginning nor an end. It is an entirety of movement. Just as it is indivisible into parts and pieces, so also its space is not graspable in terms of a series of points on a path and a multitude of locations. That dancing lacks any fixed and determined orientation is quite obvious: while turning we are moving forward, while moving backward we are stepping onward, while moving onward we are returning-all this appears "impossible" in an oriented space. The movement continuously assumes its space and, so to speak, tenses it anew with each phase.

The previously underscored state of affairs áppears here from a novel side. Each movement occurs not only in a spatial, but also in a temporal-rhythmic, succession of movement phases. If we remind ourselves of the role that movements have for attuned space, if we keep in mind that attuned space is accomplished as a movement­space, then we are once again offered a reason to speak of it as a space-time.

§ 6. Attuned Space as Space-Time

There are three states of affairs pointing to an interconnection between space and time: attuned space as a form of executing

139-49, analyzes it psychologically. Its relationship to space was first emphasized by E. Straus (1). Straus takes the movement of dance to be expressive movement as such and attributes to it a particular _"presenta­tional space." Since he sees an essential relationship between dance and music, he concludes that it is through music that the structure of space is first "created." For him, the "presentational space" is one of sound and thus has a structure determined by sound. Straus's treatise is distinguished by its refined observations and nuances and appropriately shows the general interrelationship between space and movement. Yet it lacks sufficient methodological strictness and results, on the one hand, in an unclear and unrelated multitude of "spaces," and on the other, in a premature absolutization. Thus an independent audial space is not guaranteed just because the space of tone and noise ca-determines the structure of such a space. That Straus could attribute to them a space-constituting function lies in his inappropriate narrowing of the space of expressive movement to "presentational space." Only subsequently will it become clear wherein lie the fundamental conditions of space-constitution (see Part One, Section II, of the present work). ·

The Attuned Space 37

movements, theii co-determination by something temporal (sound), and their horizonal limitation through nearness and remotness as spatio-temporal phenoínena.

In arder to examine this interconnection more closely it is neces­sary to touch briefly upon the problem of time. Yet it must be explicitly emphasized that within the framework of our problem we offer only a rough survey, a few hints and nota complete presenta­Han.

The interconnection between space and time is usually limited to the notion that space, taken as a location of points next to one another, is related toa now, a temporal "point." Conversely, space belongs to time insofar as the "flow" of time is represented as a one-dimensional formation, as a straight line, i.e., as a spatial continuum. Yet so conceived, the relationship between space and time is only a loase proximity, a mere "also." Space appears "in" the temporal point only as a postulate of conceptual completeness. It reminds us that "next" to space there is also time, to be thought of in the mode of now. In any case, the now allows everything spatial to remain as it is. Space "is," while time "flows," and in each temporal point space remains the same. Things change in it with time, but space itself remains timeless.

M. Palágyi can be considered to be the first thinker who has gane into the problem of the interconnection between space and time in a persistent and original way.16 With his conception of the "flowing" space where each temporal point has a corresponding world space, and each spatial point a corresponding temporalline, there appears a new point of departure for subsequent space-time researches. It is remarkable that it was not philosophy but physics that took posses­sion of it! 1Palágyi sa:w such an appropriation of his conception as a crude misunderstanding and turned against it with indignation. Yet in fact the alleged misinterpretation of his "flowing space" into Minkowski's space-time continuum was accomplished so easily only because in truth even in Palágyi the space-time interconnection is seen as nothing other than a mere coordination. His space is "more flowing" only because it is in the flow of the "adjuncted" time. Nevertheless, there is a difference whether space is more flowing because of time or whether it is flowing of its own accord and thus conditions the flow of time itself. Palágyi fails to notice that his chosen point of departure for this question is conceptually unfavor­able. Like the physicists, he begins from the assumption that space

16. M. Palágyi, pp. 1-20.

38 Lived Space

and time are mathematical manifolds of points, that they confront the thinking subject as objects and can be coordinated one with the other after they have been grasped separately.

But what if we are dealing with a space that is not merely a manifold of points, and that is not merely posited as the object of a judgment, but is rather lived? What if time is originally not a homogeneous, mathematically differentiated series of points, but a lived time, a present that binds future and past? Would space and time be merely coordinated, or is there an entirely different kind of connection?

If our investigation of time were to correspond with our planned investigation of space, then, in accordance with our method, we would first observe how time is possessed by thé subject----rnot as an objective thesis, an object for consciousness, but "ekstatically" in his lived comportment toward the world. From there we would move to the analyses of objective time. It will become clear in subsequent contexts why we are following the reverse procedure. At first we shall discuss briefly both the question concerning the mode of givenness of time in consciousness and that of the constitution of this time consciousness.

Time is experienced as flowing: all events, all changes, and all duration, occur "in" it. "In" it the subject knows himself and the beginning and end of his corporeal existence. As such, time is primarily given not as a change of the contents of consciousness but as a happening of the world. Everything happens, occurs, runs its course in it; in it there is enduring, abiding, beginning, and end; Time is originally given as being-conscious of an event in the world. A momentary occurrence is able to show · the flow of time more clearly than an enduring one. Experienced as now, in the mode of the now, in the privileged givenness of originary vividness "in person," the now is already past and has become something that has been sinking continuously and irrevocably into the "depth." Finally, it is extinguished in a completely empty background, inaccessible to consciousness. Consciousness is aware of this vanishing on the basis of its capacity to follow the now in its modifications, to trace it retentionally, although not as far as one would wishP What is it here that persists, what is it that in a specific retentional phase does

17. See E. Husserl (4), # 77, for the distinction between retention and reproduction. Subsequently we shall deal with Husserl's hyletic data. Protention, for which there are essential analogues, must be left aside in this brief sketch of the time problem.

The Attuned Space 39

not allow the now to change to another now but holds it as just having been and thus as the having be en of a now? This persistence in the stream of time would be incomprehensible if the simplest world event did not have a specific sense-content, if the conscious­ness of the now were not a being-conscious of a sense-bearing now, understandable through all the phases of retentional changes. If the now of the sound of a bell were to become the whistle of a locomotive, and thus to be experienced differently in each point of the retentional continuum, then it would not be something that has been-it would not be the retentional modification of a now.

The identity of the sense guarantees the relationship toa now of something that is passing and makes comprehensible the "now" and the "having been." Furthermore, the identity of the sense-content implies that the given time has a holistic structure even if it is distended across the ekstatically lived time. Although distinguish­able in terms of past, present, and future, this triplicity of temporal modes is not separable; the just-passed and the just-coming are co-determined by the now, and it by them. Moreover, the phenom­enal now is not a discrete point; taken chronometrically, it can persist. Further, there is another remarkable characteristic that distinguishes the originally given time of consciousness from chron­ometric time.

The previously observed singular event as something given is an abstraction. Were we to follow a number of successive events in retention, the su'ccession itself would also be obtained in retention, and thus the originary time of consciousness would assume an unequivocally directional determination. The latter event cannot surpass the earlier (which can be the case, for example, in a genuinely false reproduction). Earlier and later are retentionally irreversible; time in its flow never reverses its direction. Neverthe­less, outside the narrow sense of retentional modifications, its temporal distance undergoes a specific change. With progressive obscuration, the distance collapses toward an "infinitely remate" nebulous point where all distinguishable contents vanish. The originally given time is oriented, finite, and perspectiva!, as is oriented space.

But does not the retentional continuum continuously lose its orientation precisely because its point of orientation, the living now, is itself in "flux"? Does it not require in turn an orientation toward something that "holds" this continuum? This something as such must have extremely contradictory characteristics: on the one hand, it must necessarily be changeless, fixed, and identical; and on the

40 Lived Space

other, it must not remain outside of the temporal flow. Here appears the riddle of what we call consciousness in the form of self­consciousness. This consciousness does not vanish temporally with the changing objects but maintains itself as the selfsame. 1t outlasts all changes and retains its own identity in the flow of time. Yet it is this very consciousness that can detect its own identity only in the flux. This does not mean that self-reflection arises along with the being-conscious of time, such that consciousness thereby gains access to itself, but only that the conditions for the possibility of a time-consciousness as such are constituted in it. Time can thus be given objectively only to a being who knows itself.

Reproduction must be distinguished from retention. Reproduc­tion, as a recollection, is the way that natural consciousness moves as time-experiencing. While retention is a flowing after along what was once present, reproduction is a representation of what has been. lts self-orientation to the past is a spontaneous accomplishment of consciousness. Consciousness is capable of "transposing" itself to a locus or a duration of the past that is illuminated anew as if it were in the modality of the now. Since we live "in" remembering, this new now is neither the originally given now, present "in person" in the lived experience, nor is it present as existent, since what is "in person" is the now of a past. Meanwhile, this now has long since been surrendered to the stream of the retainable; it is repossessed in reproduction and through it "made" into a living present. Yet here the retentional perspective undergoes a characteristic transforma­tion.

Since recollection is phenomenally a fulfillment of the retentional continuum, whose temporal stretches are seen perspectivally, it seems that reproduction cannot give us any absolute size constancy. Yet with representation as a spontaneous act of self-transposition there appears a new problem. Taking the fact that in the act of representation the newly attained now doe~ not have the mode of givenness of originality, "perspectivally" then means: perspectiv­ally to any re-presented now, i.e., in principie to any arbitrary point of retentional continuum. Yet each duration in proximity to the now, the "freshly" reproduced, points to a slight perspectiva! shift. If it were possible to fulfill each position of the retentional continuum with a "new" (re-presented) now, then the perspective would thereby be removed, the "actual" duration of the individual tempo­ral stretches would be regained in (re)experiencing, and time would be homogeneous. lt would be necessary that any point whatsoever of the retentional continuum be reproducible, whic;h in principie is the

The Attuned Space 41

case; yet, in addition, each of these positions would have to be reproducible "now," which in essence is not attainable. Instead, it is possible to repeat the recollection of a reproducible event an · arbitrary number of times; the reproduction can turn back at will to the same event of the pastas often as it wishes. In such a repeatabil­ity there is a corrective factor for the perspective of time. Since two recollections are never the same, in that they are enacted from a different now and are continuously motivated by diverse factors, such repetition of reproduction offers the possibility of freeing an event of the past from its perspectivity. Moreover, when an event is reproduced, each of its recollected now-points can be rendered into a new recollection and this in turn can be repeated. This means that the structure of reproducible time is similar to that of homogeneous time. Regarded phenomenologically, the free motility of the repro­ductive glance prepares the ground for the construction of chrono­metric time.

Despite the many simplifications of the states of affairs, the discussion presented above should not deceive us concerning the real complexity of the temporal problem. Essentially it had to do with the modes of givenness of time in natural consciousness. It remains in a sphere of temporal givenness that is not the most original.

Time offers itself to reflective analyses in a mode in which it is not "given," or consciously presentas flowing, but in a way in which it is appropriated and lived, in the primordial sense of the word, without thetic awareness of it. Heidegger developed this "ekstatic" mode of time appropriation from the care structure of Dasein.ts Time in this conception of an enraptured ekstasis is even farther removed from the time of consciousness as a continuous, homogeneous series of the one after the other. Unlike its appearance in objective consciousness, where time is constantly related to the present in its three modalities, time here is a unity of the three phases of "ekstases." Here the future is not later and the past not earlier than the present; rather "Dasein is temporality as past presencing future"; it is constantly "contemporaneous." As will become obvious, this contemporaneous structure of lived time can be grasped only as the "time of Dasein," but not as an inner-worldly time. It has its ontological ground in the temporality of Dasein as the sense of its being.

Undoubtedly Heidegger was able to articulate structural charac-

18. M. Heidegger, §§ 65-71.

42 Lived Space

teristics of lived time that had to escape Husserl because of the latter's orientation toward the constitution of objectively experi­enced time in pure consciousness. But how is the temporality of Dasein as contemporaneiety to be understood in relationship to the inner-worldly time? What the latter means for the being of the subject, and what it is in its own right, was never interrogated by Heidegger. A more exhaustive investigation of this question would, however, lead us too far afield.

After this digression into the question of the problem of time we renew the discussion of the interconnection between space and time.

If Palágyi's proposal turned out to be inappropriate for showing structural unity between space and time, the analysis of time­consciousness could only show that the "time of intuition" is complete! y free from spatial moments. Its relationship to the space of intuition was only one of analogy, implying a deeper layer of interconnection that was not yet itself revealed. In arder to discover a plausible interconnection between space and time, showing not only that they have a relationship of coordination but that temporal components are traceable in the spatial structure and spatial com­ponents are traceable in the temporal, the interconnection must be sought in the forms of lived spatiality and temporality. And these cannot be grasped objectively but must be presented unthetically, not known but lived or accomplished.

That such a search for space-time unity did not arise only from a speculative need was evidenced by the phenomena in attuned space. What leads further into the question raised above is precisely its characteristic form of accomplishment in expressive movement. If one seeks the mode and manner in which not only spa:ce but also time is obtained "ekstatically," one finds it precisely in living movement. While this is true of both expressive and practica! movement, there is, nevertheless, a noticeable difference between them. While the latter realizes the "ekstatic" unity of the three phases of time in its accomplishment, in objectifying reflection it allows this unity to be incorporated into objective time. The former is completely incapable of such an incorporation. Even reflection upon the expressive movement does not succeed in grasping it as "occurring in" a specific time. Expressive movement is nota process that begins and ceases, commences "now" and breaks off "then." It is upsurging and resounding without fixed limits; it has no disrup­tions in objective time. Even its stillness is an arrest in the whole of movement, which not only contains or includes past and future in

The Attuned Space 43

the present, but is pure presencing, pure contemporanaiety. Subjec­tively and objectively, expressive movement is the paradigm of an "ekstatic" temporal wholeness to be thought prior to any differenti­ation into temporal modes.

Any state of affairs must be grasped from its time characteristics in such a way that attuned space, as a specific space of movement, is constituted through something temporal. Temporality does not mean here a process in an already present objectifiable time, but a grounding of time, time as "ekstatic," projecting. Corporeal move­ment can be formulated in total indifference to space-time. At the same time, it is corporeal movement that first of all incorporates time into space and the latter into the former. This is a highly inadequate way of spmiking, stemming from a thought that is dominated by two separate "forms of intuition." Yet in corporeal movement there seems to be an ontological ground for an originary unity of space­time (pp. 145ff.).

It is quite remarkable that the traces of time in space show up much more clearly, phenomenally speaking, when the subject lives in it more "timelessly." In attuned comportment toward the world there is no time for the living subject; a being who is essentially only an attuned corporeal being knows nothing of temporal flow. Time is dissolved in the experience of attuned space--and in the reflective analyses of this space, time will be grasped as a moment of space. It is distance in particular that allows the cognition of the temporal moment of space. As spatio-temporal phenomenon, distance determines the motility of the horizons; as spatio-temporal, it is the limitation of the "metaphorical" spaces. Distance delimits the attuned space both as temporal space and as attuned time-space. lt is on this account that in the nearness and remoteness of at­tuned space, spatial and temporal determinations are mutually pervasive.

§ 7. Attuned S pace and the Experiencing Subject

The structure of attuned space appeared in our investigation of its fullness and emptiness, of place and situation, of its nearness and remoteness. Its essential characteristics presented themselves in the reflective analyses of the experience of space.

At this point we can offer two valid objections. Philosophical consciousness since Husserl has taken care to trace the distinctions between something and the experience, perception, and cognition of that something. We must not overlook that in our special problem we

44' Lived Space

are not concerned with noetic-noematic unity, insofar as attuned experience is not an intentional orientation toward something. Still the question could be raised concerning the relationship between the experience of space and space itself.

Is space itself inaccessible to the analysis of the experience of space? The answer to this question was already prepared in the considerations of space and expressive movement; it requires com­pletion and deepening.

lt appears that experience is here to be understood only with a grain of salt. Attuned space does not confront the experiencing subject as something independent, a being in itself that must first opera te in arder that one may "react to it." S pace does not ha ve an existence of its own, separated from the subject, to which the subject should establish a relationship; as a space of my movement, it is much more space through me than my experience is through it. The strictly reciproca! implication prevailing here between space and the experience of space can more easily be shown through its characterization as an event than it can be subsumed under fixed concepts. What can be grasped in immediate perception as an encounter between subject and space appears all too easily to be a paradox.

Attuned space offers itself in its fullness.lt appears that the fullness is nota mere methodological expedient, a specific approach to space that could be grasped on another occasion by other means; rather, space is this fullness itself. Its vanishing is thus not the disappearance of something in it but a loss of something as a whole. The phenom­enon of disappearance best reveals the reciproca! relationship be­tween space and the experience of space. While attuned space, with its full physiognomic content, corresponds on the experiential side to the uninterrupted fullness of psychological impulses announcing themselves in the richness of expressive movements, the obliteration of attuned space is only one si de of the reciproca! relationship. On the other side, the loss of attuned experience extends all the way to the "emptiness of heart" of which Scheler spoke and for whom it is the "originary datum of all concepts of emptiness as such."19 The indif-

19. M. Scheler (3), p. 298. In addition, see H. Tellenbach's report concern­ing the spatiality of melancholy. According to him there is frequently found with these patients a disturbance of the relationship to the space of action, which Tellenbach, following Heidegger's Dasein-analysis, suggests is a loss of "nearness" in the sense of making room for equipment (p. 292). However, the "emptiness" of melancholy does not correspond in a phenomenologi-

The Attuned Space 45

ferent or callous person fails to notice anything that addresses him; to the extent that he feels ernpty himself, he stares "into ernptiness." With the phenomenon of absolute emptiness as well as with that of concrete fullness, the reciproca! relationship between space and the experience of space is visible, a relationship that can hardly be thought in sufficient intimacy.

Unlike things, attuned space is not "outside" me. It "surrounds" me, it is about me-this is its mode of givenness. But I am not in it in the same way as things are in it. Through my experience, I am spatial on the basis of my possibility of being an experiencing being that is an expressive, motile, living being. Attuned space is with me as an accomplishment of my attuned being, relating to it in mutual conditioning and fulfillment-this is its mode of being. In this sense its being exhausts itself in being a being for an experiencing subject and above this it is nothing "in itself."

An immediate objection arises: attuned space must be "merely subjective." The concept of subjectivity, as well as its correlate, is engulfed in a multitude of meanings. Even if such concepts, are gnosiological and not ontological, the purpose of our investigation requires their brief discussion.

For one, subjective means that which is appropriate to the ego; subjective is everything that is in me. This relationship is determined and limited through the (unreflectively experienced) relationship of my ego to my body. What is decisive here is not the body itself but this relationship. Bodily possessioJ).s-limbs, organs-are nothing subjective. Subjectivity requires a relationship to an ego that is distinguished from its lived body even by a non-reflective consciousness. That this relationship of the body signifies essentially a subjectivity means only that the ego grasps itself as an ego of a lived body. It is not that it is a lived body as such, but that it is a lived body of an ego, of a self, that makes it capable of distinguishing what, in an ordinary sense, is one's own and what is alien. Subsequently it will be shown that the sense of the relationships "in me" and "outside of me" is based on this relationality of lived body to ego (See pp. 140 ff.). In this sense my perceptions, surmisings, and stirrings of feelings are subjective; they belong to me, they are my "own." The correlate to this subjective

cally precise way with the ernptiness we are refering to. Tellenbach's observation that the inner ernptiness "corresponds" to the ernptiness of the world and the ernptied space "intrudes into the inner ernptiness" (p. 16), is nevertheless worthy of notice.

46 Lived Space

side is what is alien to ego, alien to me. Things and their relationships do not belong to me; even "alien" persons are in this sense not subjective.

Only that which is m y own can be "merely subjective." By this we mean deceptions, errors that appear as such in the disruption of a coherent understanding, in the cancellation of an experiential con­text, of a continuity of sense. They can be discovered by me or by others. Because of the possibility in principie of such a discovery, there appears a further meaning of the subjective. What is subjective is that which an ego calls its own, what is with me and m y ego. Thus in a primary sense the other, the alien, ego is subjective; it is a "subject." l':go-ownness is then not only my own but an ownness of each ego, all egos. lts opposite concept is all that is aliento each ego, the totality of the "objects."

This subjectivity is of a singular kind. Each ego is one by virtue of its lived body; the Hved body is a mode of givenness of "my" ego as well as of "another" ego. Another ego and my own ego mean corporeal ego and nothing else. But as such, ego has surpassed its own, and I have surpassed my own, corporeity. This transcendence of the body grounds the subjectivity of the subject as universal. lts correlative concept is objectivity as intersubjectivity. Language, art, history, and science are objective in this sense.

What sense can then be attributed to the claim that attuned space is subjective? lt cannot be subjective in the first sense; it is not "in" me but "about" me. The serenity of a landscape is not that of my sensations but is something in the landscape. I can experience it in

· contras't to my own (ego-own) subjective sensations. In this sense . attuned space is also objective.

Obviously this claim does not mean that attuned space jg objective in the sense of intersubjectivity; it is not a homogeneous space that is the same for all subjects. lt must be admitted that in its expressive fullness the attuned space, with all of its specificity and uniqueness, appears "always" as my own. Yet I know myself in it at the same time with others-or without others-alone. Solitude is comprehen­sible only as an absence of others. To become aware of solitude is in its own way to co-experience the other. Thus attuned space is comprehensible only as a possible space for the others. lt retains an intersubjective moment and therefore an objectivity in the latter sense indicated. lts relationship to an experiencing subject, which is certainly its general characteristic, cannot be confused with subjec­tivity in the sense described above.

What can appear as doubtful conc¡3rning the existence of at-

The Attuned Space 47

tuned space is, in any case, the fact that in experience it cannot be grasped objectively. This is not due to attuned space, but to the subject, which, in the mode of being of attuned experience, relates itself sensibly to a world without standing over against it as a "sub j ect."

J ¿¡ 1¡ {l \Q"'--\, !( 1

'v ¿, ,~¡.\Q(Jv\of (:J 1

\ '('

Chapter Two

The Space Of Action

§ 1. Preliminary Remarks

Lived space is not exhausted in being solely an attuned space. This does not yet encompass the totality óf spatiality, which, as experi­enced, has its specific characteristic of being related to a corporeal subject. Attuned space turned out to be a space of expressive movement. As such, it has the uniql.ieness of being free from differentiations of orientation. This is its profound difference from

< both of the other forms of experienced spatiality. The concept of orientation is characterized by two factors. Orien­

tation presupposes differentiatable zones, determinable loci, posi­tions, a here and a there; orientation is always an orientation from .... toward. Furthermore, it includes the possibility of move­ment appearing as "directed" and oriented. Expressive movement

--l. has demonstrated that not all movement is oriented. It has also turned out that this is closely associated with there being no point of refer­ence in its space. Thus lived space can be oriented only to the extent

1/11\f,-that there is a formation of a center in it. Orientation and centering of space are only two different terms for the same state of affairs.

If there is to be a form of lived spatiality manifesting a univocally determined orientation, then the corporeal subject must exist in it as a lived body that can be grasped univocally as located here in distinction to each there. In this way the corporeal subject appears in two respects: as acting body it is the point of departure of goal­oriented activity; as unity of the senses it is the'point of reference of sensory intuition. In accordance with these two modes of comport­ment there appears a distinction between the space of action and the space of intuition.

The primary formal determination of the space of action is the "wherein" of possible activities. The concept of activity will be understood as the realization of a project through the lived body and

48

The Space Of Action 1 49 .-/

its members. Thus we avoid the as§u~ptl-on that the lived body is an / implement, a means in order to .. : . This frequently employed instrumental image is factually incorrect; the lived body is funda­mentally distinct from all i:hstrtiments. It stands in an irreversible ., relationship t~ them by having and manipulating. ThiSISiíOtThe ~o into the problems indicated. Yet at the outset the acting body must be seen as capable of manipulating implements. Subse­quent observations will consider the form of bodily activity employ­ing certain tools as means. But is the way in which bodily activity is understood of signifigance for its space of action? This question introduces into our investigation a specific prejudice concerning the relationship between the space of action and the acting subject. It is the task of the subsequent analysis to take a stance toward this question and to show what it means to say that all activities occur "in space." It cannot be decided at the outset whether the en tire structure of this space is indifferent to the fact that its center contains a being who employs implements and instruments. .

· But the subject does not merely employ the implements, which leads us into a problem. The subject of this space understands the use of equipment in whose manufacture the mathematical construc­tion and the laws of exact science play a role. These instruments, strictly speaking, "apparatus" presuppose the geometry of measure and the theory of physics. It may be objected that this unexpected intrusion of the natural sciences into the corporeal sphere of the subject disrupts the basic methodological principie of this investi­gation. This principie requires that the mode of comportment of the lived body be the sale point of departure of our investigation. Nevertheless, it cannot be denied that the observed subject's behav­ior does not differ phenomenally in the least when he handles a fS constructed apparatus instead of a simple implement: in the actual process of activity, both are equally means toward the attainment of something. Otherwise he would not be an acting being oriented toward the utilizability of things.

At first glance this may suggest that technical construction need not be considered in the space of action. The silent assumption would be that nothing is changed through its structure. This assump­tion has been always made whenever the space of action entered the ( field of philosophical interests. Credit is due to existential philoso­phy for having pointed out this assumption within the framework of its problems. Yet its conception of the space of action does not recognize an apparatus as a problem. Of course no methodological inconsistency lies therein; the existential concept of being, i.e.,

50 Lived Space

"Dasein" (as Heidegger's "care" and Merleau-Ponty's "Etre engage") and the limitation to the sphere of everydayness is sufficient for the existential point of departure.

lt is different with us. Indeed, we retain our point of departure; we observe the subject in the space of action, in which he comports himself and understands himself in an unreflective attitude. Phe­nomenological completeness requires that we consider his manipu­lation of apparatus; this is no less a phenomenon than pre- and extra-scientific praxis. That the active subject engrossed in the world of work may have no insight into the specific mode of mediation between the lived body and constructed apparatus does not justify our discrediting it in philosophical investigation. Yet we face a specific problem: the manipulation of apparatus suggests a state of affairs whose signifigance can be understood only after the investi­gation of geometry, i.e., its so-called application to lived space. The appearance of fact at the head of our investigative progress intro­duces factors of technology the comprehension of which assumes that the aim of our progress has been attained. Moreover, the clarification of such factors assumes that the investigation of the lived spatiality as a whole has been completed.

This phenomenologically most disquieting situation places us befare two alternativas: either we must curtail discussion of those phenomena that can be sufficiently articulated in the attained level of investigation or, while including all the phenomenal contents, we must make anticipations of areas to be analyzed subsequently and declare such anticipations explicitly through reflection. Previously discussed reasons suggest the appropriateness of the latter course.

The access to attuned space turned out to be its fullness. The expressive characters of its things constituted its atmosphere. The expressive world dissipates in the space of action. Here the expres­sive characteristics of things vanish into the qualities required for their utility. Thus they lose their effective and communicative physiognomy; now they reveal their suitability or resistance "in view of " a goal. This is not to say that the glance now penetrates a kind of expressive level of an entity and reaches its genuine determinations, its "in-itself." Expressive characteristics and prac-

® tical properties of an entity do not relate to one another as a surface to a kernel. An entity is, in its mode of being, phenomenally whole and complete in its determinations; it has them as a whole. Further­more, they do not comprise its different "sides." This way of

j speaking obscures the fact that we are confrontad by determinations of two distinct regions of sense, constituting themselves only in the

The Space Of Action 51

subject's orientation and perspective. The applicability and the usefulness of entities are first given as such in a particular grasp; they are opened in projects of activity, and apart from these projects they remain incomprehensible. Furthermore, this is not to be under­stood as if an entity were apprehended thematically as the being­there of Dasein or the coming-before-us of things. Active engagement with things has a specific mode of seeing, appropriately character- . . .. \ ized by Heidegger as "circumspection." Its discovery rests in the e 'f(' '···~\ ~ ' '

implement, whose mode of being is ready-to-hand.20

The structural analysis of implements in· the space of action, nevertheless, runs up against sorne difficulties. This space is ontologically relative to a temporary project, to a specific situation of the acting subject. Thus it contains a temporal moment; the space of action has a dynamic texture. Is it possible to discern anything about o it? In its constant transformations, must it not continually escape conceptual boundaries? It would be so if each project were an absolute beginning, if each actual space were completely distinct, and if between them there were no constant transitions and perva­sive regularities. Asan acting being, however, the subject appears in his historicity. He finds himself in an already formed world of work that is not his own creation. Immersed in it, he also participates in its constitution. While transmitting and at the same time shaping what he has acquired, he realizes a relatively enduring project-such as in his vocational decision-from which the actual attains sense and meaning. The spaces of action of greater historical dimensions, formed in common cultural labor, comprise at the same time the framework for momentary individual projects and their "spaces" which are neither rigid nor motionless but can be supported, extended, or negated in actual activity.

It is not necessary to follow this problem any further; rather, we must ask: how does a space of a historical being present itself, a

20. M. Heidegger,§§ 15-16. We appropriate these differentiations without following Heidegger's ontologicaL conceptions. Specifically, we do not accept the validity of th![ bntol,o,gi_cal j)riürl!Y>of circumspection, in contrast to Heidegger's notion of the"free comprehension" of pure sight The claim that circumspection is more primordial and that sight is founded on it has for us no sufficient phenomenological grounds. Heidegger overlooks the fact that circumspection already implies sight, and that the latter is co-constitutive of circumspection. Moreover, the comprehension of some­thing as present at hand must place in brackets its qualities of being ready-to-hand.

----

52 Lived Space ?

space that has in its content a being who is no longe~ immersed in ~-~-...

attunement but is oriented toward the world and strives toward aims?

§ 2. Place and Region. The Space of Action as a Topological Manifold

Like attuned space, the space of action is not justa mere multitude of points in three dimensions. Yet structurally the latter does not conform to the former. It will be shown that the texture of the space of action has become looser, that the role played by the part in the whole has become different, less far reaching.

The space of action is articulated according to places and regions. Place is thelocus of what is usable, discovered by the acting body. In essence: H1ese delimitiú:ions agree with Heidegger: place is the lo.cus where implements belong. (Included are the privative modes of not belonging there, lacking, being in the way.) In belonging thf)re, tl}e thing possesses a relationship to "its" place. The same c-h~r~cteristic appeared in attuned-'-';pace~~the1~s8j1ere it must be understood differently. In attuned space the place. belongs constitutively to an entity not only in the specific mode of being as a carrier of

-::¡ expression, but as an essential co-determinant of, space as a whole. In the space of action this is true only in a very restricted sense.

That something ready-to-hand has "its" place is determined prima­rily by a moment of its duration: it is found there "customarily," most of the time," and it has its "usual" place there in accordance

l~f"-) ó :t~t~u..t "t... S

with the requirements of the acting subject. Yet its belonging to a place is not identical with its appertaining to a place and is not a constituent of its utility as such. Its place is variable within a broad limit, without a loss of its character as a specific instrument. Within limits to be specified more closely, t~_E)_pl¡¡ce of the ready-to-haJJ.d things can in princ;ipl¡}_~_e ~hgs~11 with()l!t t}¡e lqss of!he mocie()f being ofthe-ready-to-hª'u.d. -This variability is rather a constituent of the place of an instrument and opens the possibility for "dealing" with things.

This comprises a fundamental difference from things as bearers of expression. One may compare paintings in a gallery, arranged for "proper viewing," with the same paintings in a workshop, merely lying "to hand" for framing. Things that are bearers of expressive

a value receive fewer and less carefully selected places when they become mere objects to be used for a purpose! Of course, the place of the ready-to-hand is not sorne arbitrary location. In its mode as being

The Space Of Action 53

IJ)(iér v' {:lt\j·r,

ready-to-hand it is relative to the subject's mode of being asan acting subject and thus to a great extent independent from place; yet as ready-to-ha~d in an ac~ thuSit must be handy.

Handiness can have two meanings. On the one hand, it means an adaptability to the organization of the body. That which is handy in this sense is what is "tailored to the body," such as hand tools and instruments of daily practice. It is a handiness of means for the realization of projects. On the other, handiness means "having in hand" something light and comfortable, reaching for the useful in a shorter way, with lesser hindrance, etc. It contains a specific principie of economy that will be considered subsequently. Both meanings must be separated: a thing that is handy in the first sense

~{¡.\11~·1\( ~~~

\cv¿\ ~;l\"' í(

can be unhandy in the second sense, as being "there" at this ~ _ _ moment. The handiness in the second sense is a function of place. ( ov..N) IC\

This handine~llow__s_the_allo_c;ation of places; it is decisive for the \I'PI' ~ space-of action as a manifold of places:~---~-- - ---- -- - - - --- - ,

For a thing to llave "its·,; place ils-an instrument means that the place is constitutive not simply for its mode of being ready-to-hand per se, but for its handiness within a project. In order to be handy, to ~ refer to a profect, it must be empower~d_ by the choice of a subject. ·¡

Yet its place is notan arbitrary T(;cil:s:l;-~~ntrast, the ready-to-hand . as such can in principie be anywhere, even if its whereabouts is not an arbitrary place in a system of locations. Heidegger mentions that the place of an implement is discovered when it is missing. In such an "absence" the implement simply "comes before us" as it vanishes from being ready-at-hand. Nevertheless, it is the place that becomes obtrusive as having something missing, even if what is conspicuous by its absence is something ready-to-hand. "Something missing" can only mean a missing implement, the search for which is precisely a search for it in its ready-to-handedness - a search motivated by its having this mode of being as such.21 The phenomenon of the search reveals the "being somewhere else" of what should be ready-to­hand. Even while missing it remains ready-to-hand with the possi­bility of place variations; it will be missed precisely because its handiness has assigned a specific place to it as its "there." It is

21. Heidegger does not distinguish between ready-to-handedness as a mode of being of an implement and handiness as a characteristic of an '1 ~l ~ \., implement in an actual project;-thus the phenomenologically unjusti-(.---- e \ fied transition from ready-to-handedness to present-at-handedness in the \ 1 _ \ "search." )''V<lO} ~ tC!f

. \.A u"~

54 Lived Space

prepared, placed appropriately, accommodated, misplaced-all these characteristics which are not given for things in the attuned space include a certain Qrovisionality of place.

An important note must be added. The place of an implement . J,; 1 prescribed and determined by handiness in an actual project is

rc'""'ú!!'c '1 variable within specific limits, i.e., it cannot be precisely fixed. This ¡¡. v;v-\+~~ \·-1 -pi-eflgureSt~nomenon of the region.

ot ~ltuc Until now the places of the ready-to-hand were seen in isolation as individual there and yonder. The abstraction inherent in this view must be revoked. In a project of action the individual implement is always transgressed toward something further; it obtains its appli­cability from a 1~~ and in turn points to a possible totality of involvement. The place of an implement in its "there" is determined by the range of other places. Its "there" points toa specific surrounding of other "theres." Its "where" is at the same time "whence" and "whither."

Finally;-th~~wlíence and whither is the whole field of action; it is ~----? t~-~Pl1,C:_~ __ Qj'_J!.~J!~n it§~Ji. It is possible to extricate relatively

independent areas from it, regions as entire places of relatively closed connections. Each space of action is divisible into such regwns. To demonstrate this more strictly would call for a more exact analysis of the notion of a project than can be offered here. We limit ourselves to thesuggestiorithafan activity presents itself in terms of its stages of fulfillment, determined in accordance with partial aims. This has todo with the manipulation and use of things, building relatively independent and limited functional contexts within a totality of instrumentalities of the entire project. Within the entire space of action, these factors constitute relatively closed

1

space-manifolds, comprising what we have called regions. Their relationship to the place of an individual thing leads to new and important determinations that make the structure of the space of action transparent.

The place of the ready-to-hand is determined through its regions, although it is not precisely determined. It is not a punctiformal

1\) "where," but a somewhere within the limits of its region. The region is the "leeway" for the free variability of place within which it remains changeable in a restricted sense: it remains the same only when it remains within the region. The ready-to-hand is allowed certain displacements without its place ceasing to be the designated one. Two things find themselves "in the same place" only when they are found in the same region. Region is definable in this manner: a place-manifold whose specification adequately satisfies the ques-

Cllv.A ,(1 V-l~'l- ~-,~

1\W\N\ The Space Of Action 55 ,'e NL ·

tion concerning the whereabouts of an implement. "At the work place," "in the desk," are such specifications, which are topograph­ically exact as to the region of an implement in the space of action.

In principie, this includes the possibility for further specification of any region. The extent of each region is relative to a project and to the possibilities of activity to~_!.§.f!lized~:LIIJ.t. Regions as such a~e natfírst eStaolisrum-·ana opened but rather arise with what is encountered in them. What is encountered determines the extent and the limitation of regions~i)leludin~ their further articulation and the event~al possibility of ~e~ing" j)f d~mai~s and thereby the structuratwn of space. For a wanderer a sea 1s a smgle homogeneous region confronting him as impassable, while the fisherman, the swimmer, and the seafarer, in their differently motivated actions, know how to discern its various regions; for them it is structured differently. In addition, each individual project allows in principie the nesting of regions through progressive articulation. Of course, this articulation cannot be extended arbitrarily. The articulation determines the degree of structuration. In this regard, different cases must be distinguished.

In a fully structured space any region can be neste_d at will. We can characterize such space by the fact that the nesting df regions can continue in it without restriction. A thoroughlyiliuctured space would be a space where each arbitrary sequence of nesting of regions could converge toward a determínate place, and, conversely, where every place could be reached through at least one such sequence of nestings. In the thoroughly structured space, the process of nesting determines place univocally as a point of this space.22 Due to its mode of establishment, it, too, is a region, although the smallest region of this space. This do es not coincide with the definition of the region as the "leeway" for the free variability of place. Yet it must be 1!9i~cl that _a titQ!_D!lghly strl.ictlifecl._siJa,ce~Q_l!_Stitute_~ an unre~ limit of the space of action. As a whole, it can never be completely structured, but -~-u-¡y-:riic,r; or less structured. We are talking about a structured space that is neither completely structured nor totally unstructured. This implies that it can have completely structured parts and also totally unstructured partial spaces. The latter pre­cludes distinguishable places and hence does no~llow sequence

22. As an extreme example one could think of a surface covered by tile. Each tile in it has its place from which no shifting is possible. Each such place is a "point" of the tile-surface, i.e., in its space there is no "there" to be established with greater precision.

'V\()1 ~(/'!'

(/1

56 Lived Space

of nestings of regions. What is essentially unattainable is that the space of action as a whole be completely structured or completely unstructured. The latter is excluded due to its referential relation-

1 ship to the acting subject. As a bodily being in this space, he · understands his locus from other loci, his here from distinguishable

theres and yonders. This differentiability determines the orientation of space. This means that the sp¿ce of ªº-t!on, asoriented space, must ~~c_e~~arily __ ~~..SE'2E!l1Ifl.c!:~-· ~ ~IMt~ck-tjre~-.- . -~------ - - -·

It must be mentioned why the space of action is not completely structured-more precisely, why at least one partial space must be given in which the nestings do not converge toward fixed places. Why must it have places that are not "firmly" established but, on the contrary, can be determined only up to a space with leeway? The basis for this was already suggested: the ready-to-hand attains its place as required by our dealings with it. These dealings not only allow but demand the inexactness of the locus given iritlie space oí' leeway~as the regtonc)fposs1blé'shlfts.~fhis'space of leeway is nota topographical inexactÜude to ~be-·e-íimlnated by progressive activity; rather, it is a positive determination accruing to the place of an

, , instrument. If leeway were lacking the things would be too dense; ~ ·'0¡J .. "'"í: (1 the acting subject would not be in full possession of his possibilities

8-f'r \ of action. Rather, he would object to the "overfullness"-without ~\:11-. • \ the leeway of its place, the ready-to-hand would be completely [L

0 unhandy! The variability of place within a region is constitutive of the locus of the ready-to-hand. Place is thus topographically graspable "only" up to a region-yet in the smallest region, the space of leeway, place, can always be established with sufficient exactness.

In any case, the leeway as such is not given thetically to conscious­ness in activity. lt is co-posited circumspectively in the management of so:r;nething without there being any intention expressly oriented toward it. The acting subject is not primarily attending to a region,

,¡' (.vl b~~!~!.~fl_ll_S~-()flbjllg~~}Vithlll_t~e}~<lmework of his project that fi~t' ~ ~~ 8!~uctt1r~~ _the -~~?i~~· Only the encroachment of the too-dense

f~"-:'f(C fl!\ revea1s the lack of leeway. Conceptually, this involves a view which l\-.'-\\\ 'lt,1~~~ i~ entirely different fr~m mer~ c~rcumspection. It is.only the intuitive o¡\cL\'1 {v; \ v1ew that preserves d1fferentlatwns of place allowmg us to speak of

_ ¡ . ~ ·~ame" place in activity and circumspection. In pure contem-V (l V -1d" plation space appears with another structure. However, pure percep­

tion disrupts activity; the usable thing of our dynamic dealings is rigidified into a static, isolated object. Robbed of its utility charac­teristics, it is torn from its functional context. The project is annulled

The Space Of Action ?

57 o

and the space of action collapses. The pure space of intuition comes into view.

The space of action turned out to be a manifold of regions where possible nestings constantly converge toward one region. The small-est region, as "leeway," determines the place of the implement; place is univocaly determinable only up toa region. The constitutive inexactness of place requires the region to be the smallest topograph- J 1 ically exact element of the space of action. What significance does V this state of affairs have for the understanding of space?

This question is related to the final intent of our investigation which is concerned with the phenomenological origin of geometry. Meanwhile, taking the concept o~ in a sense yet to be discussed more precisely, we canno~f the space of intuition as the source of geometry. The previous analyses demonstrated that v the source must be sought much earlier, namely in the space of action. Indeed, the space of measurement has not yet emerged. Y et ,¡ geometry presupposes a mathematical discipline-one which, al­though exact in a scientific sense, still remains on the side of all determinations of measure, which functions without point­geometry, and whose basic element is the region: topology. Staying with the metaphor selected, it could be said that the source of topology is to be sought in the space of action ..

§ 3. The Locus of the Subject in the S pace of Action

As with other things, the subject also assumes a place in the space of action. Yet he has space differently than they. Things of utility have a place alloted for them, while the subject assigns his own place to himself. This distinction must be brought phenomenologically into sharper relief.

Ready-to-hand things are discovered "there" and "yonder." A there is determined univocally only in relation to other "theres." The totality of distinguishable theres is nevertheless related to a here, from which things are first ordered and space first articulated. The here has the notable determination of being singular and non-relativizable, although in a specific sense it is freely selectable. While the places of the space of action are all distinct from one another, they are equivalent among themselves: altogether they are the places of something useful. The he~e of the lived body: is the sale ~lace th_a!jsnotª p}ac~-~ an instriiñlent;--raih~~.Ttis11fst from itth~t each implement appears as located "there." Here and there are essentially distinct; there and there are interchangeable, here and

58 Lived Space

there are not. In the space of action, the here is an incomparable locus; it is the center-fromwhichitis-what itis:-Hielocus ofthe act!ng subject wh.cij'r.QJi}lisj)Iace, .úiíiüici~--the--sp(l¿e of action-.--

The non-equivalence of here ancr there is --the basis for the non-homogeneity of the space of action. This is its constitutive property which cannot be eliminated without eliminating the space of action and the acting subject himself. In this activity the subject is certainly not aware of the role of his locus. His primary orientation is not to himself and his lived body but to the things; the primary

\vQ \:¡~~ reference to his body der!_y~s fro_rg!_hings. In the attitudeofacHvity; the there is prior to the here. If the latter comes thetically to awareness, it does so inevitably from there and from those indices that dissolve typical modes of comportment, disrupt the original

07 activity, ~_l:!!_!he_p!Qj~__s;t. I can become positionally aware of my locus when the front of my car has come too clase to the edge of a precipice; it can become an occasion for my anger when my train departs from another platform while I am still waiting for it "here."

Although the here is the point of reference in the space of action, it is exclusively self-related, an absolute here to any there, incapable of becoming a there in all there-relationships. It is the locus never abandoned by the subject, who always incorporates it. While moving toward there he never makes his own locus into a "there," but

~-~-,;ü.l ~J J always takes up a new here. As a physical body, the subject is "in" space just as things are; his locus is thus also a there-yet he is not

1 grasped as a physical body but as the corpo~a_IJ;ubkct of the space 1 f-"ff~l ¡-y1\)t:t of action. Although in the first respect it is a thing among things, in

the second aspect it is irrevocably counter to all things. Thus there arises a continuous ambivalence of the subject's situation-the ambivalence of being a physical body and yet being beyond the physical body. The ambivalence is reflected in the simultaneous non-relativizability and free selectability of the here.

The acting subject is his cor12oreity; he cannot choose to "have" it or ~e is not justa corporeity; he is not ananimal completely caught in, and never capable of, transcending his corporeity.23 The

23. French existential philosophy here employs etre as a transitive verb (Marcel, Sartre). Thus the opposition to the older conception of the relation­ship between physical body and lived body is placed into sharp relief. We remain with the usual use of the auxiliary verb, the only one customary in German, chiefly because its transitive meaning is fully understandable only within contemporary French philosophy, which we are not following here. Concerning the concept of transcendence meant here, see H. Plessner (2).

The Space Of Action 59

human body is essentially only a surpassed body; the subject is a body only to the extent that he has it-and has it at his disposal in the framework of his projects. This simultaneous being and havil}g a body aQpears in its most originary form in bodily movement. The subject, caught in the here, nevertheless surpasses his here. While moving himseff in his here~-he nmches the things there, in the world.

The phenomenon of movement will require special treatment. At the present we are concerned with grasping the center of the space of action in its specificity-as the place the subject chooses "for" his body. The entire problem of the localization of the subject in.space \ is found here. While the body in its c_orporeal limitation may_find itself "here," where am 1 who is constantly "beyond" my b2Qy? The turn of phrase can only be metaphorical, an image for my being able to orient myself toward the world, and this orientation is not s~ but intentional. The being beyond the body ofthe ego does not mean

r a spati~g outside of the lived body. Thus we cannot ask "where" one is to seek the subject who transcends the body; the question must rather be phrased: how does the lived body of a subject capable of transcendiJ.l.g_!ll.~_bo2:Y3l!P~ª!JP.Jts spa~?

Ontologically speaking, the being-spatial of the corporeal subject has a dual aspect: on the one hand he is oriented toward a there, and ~G~cK}l\ on the other, he finds himself at a there, "being exposed." That -t\.i u.Jve~1

which is ready-to-hand is not only a useful means for an end, but is c-JtJ

also highly resistant, detrimental, threatening. The lived body is not 1

merely a condition for the possibility of seizing the world-in its vulnerability it is constantly exposed to the danger of being seized by the world. Although a thing aman,?. th~11gs, the corporeal thing is nevertheless located on the hith~~ 'tJlde of the thing-world. The subject is not only a body, but has it and must have it. This dual determination of the lived body is the reason why its locus in the space of action is graspable only as a region.

While the subject determines his location by intending the secu­rity and maintenance of his own lived-body through implements ready-to-hand, what is more important is the manner and mode in which this is accomplished. Reduced to vital activity, he may appear with reactions of flight, defense and protection, such as are also known in the kingdom of animals. But what characterizes the human subject is his knowledge of how, in principie, to transform these

Plessner sees it in two ways: "beyond" the body and "into" the body. "Positionality" is offered as the decisive category of an organic body, while "eccentric positionality" belongs to a thinking being.

60 Lived Space

reactions through instruments. An implement serves not only to subjugate the resisting world, but also to surpass the subject's corporeity, its limits and fragility. In the project of activity both are "taken into account" and deliberately "calculated."

The use of an implement is informative for the meaning of the here and the there. At first, the implement is suitable for the progressive structuration of the regions, the more exact determination of the there. Furthermore, the possibility of instrumental refinement of spatial articulation offers a new possibility for the extension of the here-region. 1t is an extension of a specific kind. lt does not expand the borders of the here, but incorporates the there into the here. Something ready-to-hand over "there," out of reach, becomes "here" as soon as an implement, functioning as an extension of bodily members, touches it. A dangerous thing situated "there," and prudently held there at a distance as dangerous, becomes incorpo­rated into the here-sphere as soon as the armed body can deal with it. The use of the implement is constitutive for the near-remote articulation of the space of action; it varíes the here-there opposition and controls the leeway of the acting body.

The interconnection of the phenomenon of leeway with the in­strumentally refined articulation of the space of action is most sig­nificant. The reciprocity predominating here can be traced further in two directions. If the leeway is too limited, if the subject selects his location "too near" to the things, they will become unhandy or "over­looked," and their there will remain hidden from his view. Factually this reveals a structural deficiency of the space of action which can be eliminated only through the change of place and the here-region. This destructuration of the space of action becomes preeminent w hen things, in their resistance and with their threats, are too ·near. The place of the subject can thus become a limitation, dissolving each there and allowing no "room" for action. The limit case of such destructuration of space, its elimination, corresponds to the subject's corporeal dissolution. In the space of action there is not only the too-dense for the there, but also the too-near for the here. The "stable" space requires for both, a leeway, a region of free motility; the region in turn must be regarded as its genuine spatial element.

What is the other tendency in the process of progressive structuration of space? There is a possibility of an attendant expan­sion of the here-sphere, an extension of the boundaries of its leeway

') through the incorporation of the there. lt appeared that the function ~ of implements can.Ee conceived as a süccessive differentiation of the

there and the yofer; with the ~ of technology,

\\' ~ ? q \ a\\{ o. \~ "'~ -

<V

The Space Of Action 61

the leeway approaches the ideal limit case of mathematics. On the side of the subject, this would correspond to the ideal case of an extension of its locus to encompass totality-but this means its dissolution. A mathematically, completely structured space would not have a center-as a matter of fact, it lacks such a center and is a fully homogeneous space. Since in its fine structure it is an ideal, never attainable, limit case, it cannot be a space of action. The "point" reached in the space of action is always only a region, which, while arbitrarily limitable, can constantly be thought as still smaller. Our suggestions anticipate subsequent discussion: if we can ? meaningfully say that the being of mathematical space is relative to • the subject, then the latter cannot be the subject in his corporeity, but

'- rather must be "outside" this space. ~ ,f It is decisive for the space of action that it be constituted through ;l Í a corporeal being. Were it a manifold of points, with its center r ~ "e~eryw~ere," it would no longer be r~l~ted toa corporeal being. As , ~~·.actmg bemgs, we know no other spatlahty. Nevertheless, we recog- J ~~ nize that in his activity the subject employs something other besides 9 }~ his own corporeal functions in his modes of comportment toward f _ ~ the world. Thus his space. Q(g_ctiQJl ca_n_o_nl.y_]:¡fl._ conceived f!'Q!!!Jhe ), possibility of another ~pace~ the mathematical i¡;q,_ce._This.is_valid ~ indepei1deí!tiy~ofwh~tber t]J.e acting ]Jody -iS~-~~grlizant oLitor_not.

Thus for the conception of his space it is essential to see it from the possibility of mathematics and the exact sciences.

Speaking purely phenomenally, such a mediated mode of being appears with the things of the space of action. Each relativization of the h«:tre and the there through instruments may reach a degree that can simply be equated with a dissolution of the limits of the lived space. Things beyond the horizon can be influenced and used from "here" through instrumental means. The significance of the horizon for a corporeal being will be used subsequently as an approach to interpreting this state of affairs. This is supported by yet another phenomenon: with the dissolution of limits there appears the homogenization of the space of action, the dissolution of the natural center into the "central," the leveling of oppositional orientations by the reduction of individual actions to their own corporeal center in '' automation.''

This is not the place to trace out these states of affairs. We are merely suggesting that th.{}_ spªg_~ gf actior1 revllaJ; th§.. :RJ:{}_s_e.rr.cJLQL another space that is~y~!_to _})_() igye_~t!g_ated._:I'he subject of t)J.e space oCacHon -afready-has it ·at his _ back, and .hi~~-:sP.ª~E1-.of ~~ti(}n_ is ca-:.defermi:lled by it froill tf18-cíU:tseC · ·· ·

62 Lived Space

Most importantly, we must maintain that the space of action is a topological manifold whose texture is determined by regions. Spa­tially speaking, even its center is a regían determined on the one hand by the limitations of corporeity and on the other by the capacity of corporeality for transcendence. Regarded ontologically, it is a space constituted in the project of activity, a space whose being

_ is relative to the situation of the acting subject. The situation turned ' out to be in a continuous crisis, holding open divergent possibilities:

from bodily dissolution, and the complete amorphousness of space, to continuously refined structuration and stabilization, stemming from the accomplishment of thinking that has transgressed the body while remaining immersed in the body's own instrumentality.

§ 4. Movement and Orientation. The Space of Action as Oriented Space

The non-homogeneity of the space of action is not identical with its orientation. Usually these two characteristics are not distin­guished, either because it is assumed that both determinations are given together-which obviously does not imply their identity-or because their fundamental difference is not even noticed.

What makes the space of action into an oriented space is the unequal value of the orientations, i.e., its anisotropy. Because a centered space is necessarily non-homogeneous, the identification of both aspects is obviously possible. Closer scrutiny shows that their relationship is determined by anisotropy. Since the latter is essentially a quality only of non-homogeneous space, it determines the differing values among the spatial loci. The reversal of this relationship is not essential and mandatory; it is valid in the space of action only defacto and is in no wise perceivable as necessary. Thus although the space of action of a spherical being would be non­homogeneous, dueto its center, it would nevertheless be an isotropic and therefore non-oriented space. The orientedness of space is determined not so much by its center as such but rather by the mode and manner in which the subject at its center is capable of articu­lating space in terms of qualitatively differing orientations in accor­dance with criteria based on his corporeal organization.

In attuned space, the lived body was given as unarticulated. Thus we characterized the balancing and attunement of torso and limbs and the preeminent role of the whole body which dominates expressive movement and subjugates the projective and locomotive motor functions.

e pa- "1( w\rice~.ll/\~6 '"P (t~{;y((fl (_éJr'.J\1<', 'l'fi'Q~1 (¡( 1 (D'fVO'\'J \'VII(,.d )_

The Space Of Action 63

In the space of action, the lived body finds itself in a changed situation. Indeed, seen ontically it is the same, yet it represents a different sense-content. Though it is still a body with the same modes of functioning, with the same movements, they are under­stood from a different world context. In attuned space they were ~ taken solely in their expressive content; here the sense of the moving body lies in the aim of its activity, in the where-to of the movement.

Upright movement is typically human. Thus it is mandatory to relate it to the verticality of space. It is usually taken to be the direction of the body's axis of symmetry.

Undoubtedly, verticality is a dimension of the human space of {') action. The animal does not live with things that are located

_ f>' "above," and standing is typically a human bodily posture. Only in f the human space of action is the vertical a dimension: it is a

continuum of possible opposition of orientation between "above" and "below."24

One distinction must be observed. For the sake of exactness, we must distinguish the designation of locations such as "above" and "below," "in front" and "behind," "left" and "right," from the dimensional character of these opposites. Indeed, each designation of location 5ncludes a relationship of orientation. For the acting subject, the separation is not to be understood as a distinction in the modality of givenness, as if at first there were the presence of an opposition of locations and subsequently there emerged a moment of orientation. Phenomenologically speaking, this could not be main- ® tained. What the analysis intends with the distinction suggested is rather to understand, in the first place, the extent to which above and below are possible as opposites and, secondly, how oppositional orientation emerges on the basis of the opposites.

Concerning the opposition of location between above and below, the first question is already answered: upright posture must be seen as the final, irreducible condition for the possibility of this opposi­tion. But how are above and below conceivable as orientational opposites? How is it that the acting subject distinguishes between above and below?

The fact of upright posture is obviously insufficient asan answer. Moreover, the conception of it as "upright" already assumes the

24. By dimension we understand a continuum of possible transitions of orientationaJ oppositions; thus not every opposition is dimensional. Fur­ther, it ought to be pointed out that this concept of dimension has nothing to do with the mathematical concept ("degree of manifoldness").

64 Lived Space

opposition between above and below. Indeed, the subject relates both aspects to the lived body-otherwise above and below "in themselves" would be senseless. This does not clash with the fact that among all the oppositional pairs, above and below is the most stable; it is not "taken along with" the body. (This stability results from the univoca! orientation of falling things, pulled by their weight.) This oppositional orientation remains the same even in different bodily positions, such as the horizontal posture. Indeed, in this case the head remains "above," yet the physical location is different from the indicated direction of orientation. In addition, this fact shows that directions are not present only with corporeity and its organization. If the unequivocallocalization of bodily zones were possible through sensations, through a "felt sense" of the body, or with the aid of the body schema, no orientation in two distinct directions would be given. This suggests that the bodily sensations themselves must be determined by orientations.

Orientations are neither corporeal nor in or of things. They are relationships of the lived body toward the thing; these relationships are neither causal nor telic, but primarily functional relationships first constituting themselves in the interplay between, the acting body and the world to be acted upon. And they are given to the subject in no other way than in the subject's dealings with the world. While dealing with things I first experience their weight, experience the "above" as a direction in which I exert my force against their weight and the "below" as a direction in which 1 follow their weight or the direction "wherein" I must bend in arder to lift something. Obviously such oriented dynamisms are variations of standing, specific movements from the upright posture. This is most signifi­cant for the above-below dimensionalizing of the space of activity; this interrelationship must not be taken as though something like orientation is already given by the mere fact of upright posture.

The functional founding of the dimensions appears most strik­ingly on a pair of opposites whose oppositional character is less apparent than that between above and below.

The externa! aspect does not offer any criterion for the qualitative differentiation between left and right. It is not derivable from the arrangement of the body. This seems to be contradicted by the longitudinal symmetry of the body, which appears to be divided into two halves extending equally into two sides. Yet this division is not something given in the active body; even externally it does not have the character of a qualitative opposition. This symmetry means rather a completely externa! equality of both bodily halves, but says

The Space Of Action

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f•V' V'~ , ~{~ 'o y ./rP )d/ .. \re

\f)" \f { . L' ~ 1 (' rrr.t')• r·lp 1 (' 1 ' o:r1f' e·(;''

t (!{{' '¡) e if ,; j-.'> 6 5 '&·} ur nothing of right-left dimensionalizing. From mere image of bodily appearance it is quite thinkable that in the activity of grasping, contacting, or touching, both hands may be used at the same time and act in a mirroring fashion. Such a two-handed being could not construct a left-right dimensioned space. The left-right differentia-tion does not inhere in the visible symmetrical physical features, but in a functional asymmetry of the lived body:t'To "look at" my hands asn1emnersofmyEody is to find two completely, equally formed structures; for the eye alone, the one is distinguished from the other in a freely reversible relationship. It is quite another matter when j they are no longer experienced as parts of a physical body, but as functioning organs of the lived body! Seen, it is merely either the "one" or the "other" hand that grasps; in activity, the right proves the more active, the stronger, while the left is more unskilled and weaker. The functional non-equivalence of both hands is the origin of the qualitative opJ>Dsition between left and right. 25 (It is quite remarkable that the lived body is asymmetrical only in its hands, the specific organs of its activity. All the sense organs are not only symmetrical in structure and arrangement, but also in function. This will be taken up in greater detail in terms of the space of intuition.)

The individual differentiation in the preference of the one or the other hand seems to demonstrate that the differentiation can offer no support for the right-left dimensionalizing of space, although the latter remains true for all human corporeal beings. Yet this objection does not say anything against the functional non-equivalence in general; it reveals precisely what is at issue here. It is not because the right hand prevails as stronger "for the most part" that it is significant for our investigation; this can only be attested to by experienc®' What is decisive is that the right and the left hands

25. There is a striking pathological discovery: in the impaired ability to achieve left-right differentiation symptomatic of apraxia two-handed activ­ity is not disrupted. (See P. Schilder (1), pp. 44ff.) In the anthropomorphic region of activity the latter represents a reduced activity conditioned by the loss of a function. That in all cases there appears a functional disturbance of the body with a felt disturbance of practical spatial orientation in the left-right dimension is comprehensible from the interrelationship we have delimited between bodily function and the space of action.

26. That the functionally higher value of the right hand is not all­pervasive constitutes a problem for individual research. We may forego the explication of the developmental researches (Pye- Smith, E. Stier, A. Weber, etc.) as well as the attempted explanations of physiology, which reduces the functional asymmetry of the body to the asymmetry of the inner organism.

66 Lived Space

generally act as independent functional members of the entire body. In human activity the left "need not know" what the "right is doing"; the lived body is not only both-handed but two-handed.

In any case, the states of affairs here are more complicated than in the above-below dimension, which is determined by univocal and non-interchangeable movements proper to it. The opposition of left and right is subjective in quite another sense than that of above and below. The latter remains the same in all bodily positions. Positional alterations of the body are characterized precisely as deviations from this directional opposition. It is not so with right and left; there is no bodily position that would deviate from the right-left dimension. This opposition remains constant. Right and left is meant "from" me or from someone else. Befare we discuss the orientational moment inherent in the "from," we must first become clear about the specific lived bodily situation of the one who enunciates this orientation. It is only with regard to him that we can obtain orientational univocity of right and left.

In the space of action the lived body knows how to seize things in their utility by complying with a principie that dictates its move­ments. The physiology of movement regards it as a principie of minimal tension or the minimal expenditure of energy. But regarded purely phenomenally, this turns out to be nothing else than an obvious functional preference expressing itself in the maintenance and preservation of a favorable bodily position. This preference illuminates the passive role played by the torso in the space of action. In the movements of practica! activity, the torso appears to be only a part of the body, which follows the seemingly independent actions of the organs extending from it.

It is in light of the principie of economy that right arid left first become univocal specifications. That something is "on the right," I confirm by certain movements characterized by that principie. Thus "right" is the appropriate direct~n tor pointing or grasping. Al­though my left hand is not hindJreH'; only my right hand can grasp successfully and without effort. Yet if specific motives call for the activity of the left hand, thus disrupting the principie of economy, the body will pivot immediately and "involuntarily," moving what is found on the right into a new front-back dimension.

This is nothing out of the ordinary. Yet more careful observation

The reversed understanding of the inner-body organization with respect to the unequal functional activities of the right and left hands is offered by F.S. Rothschild.

The Space Of Action 67

shows it to be a most unique achievement. It is not so much that each right and left are brought into a "third dimension," but rather that the latter remains omitted. Besides being related to things in front, the practically engaged human body is also related to things left and right; it also reckons with those lying "sideward," and knows how to get them in its grasp without turning the torso.

While left-right dimensionalizing is grounded in corporeity, none­theless it is nothing corporeal. The directional moment laid out from me to ... characterizes the spatially polar relationship between the lived body and the intended things. By means of left and right 1 relate myself to them and them to me. They obtain their being on the right and the left through my designations, yet my capacity for such designations depends on my being oriented to them. Orientation is always orientation from ... toward ... If either pole is lacking, orientation collapses.

The functional non-equivalence of orientation in the opposition between front and back is more pronounced than between the dimensions of right and left. The "third" dimension is not only qualitatively distinct from the other two, but is also the most heterogeneous.

What establishes this opposition is primarily: the organismic ? constitution of the body. Although muscles, tendons, ligame~nd ·.q -~-------~---joints are so organized as to possess a strong tendency toward frontal movement, the most decisive factor is the frontal position of the eyes. While the space of action is distinguished from the pure space 1

of intuition, for which another mode of seeing is constitutive, nevertheless things in their utility remain equally seen and require visibility in arder to be operative in a project. Only the frontal sphere of the space of action is surveyable in accordance with place, situation, and region, and is open to planning; the back-space, in contrast, is not surveyable and thus is uncertain, dangerous, and tricky.

With respect to locomotion, the organismic organization of the body, with the orientational opposition of front and back, contains yet another strongly formed functional non-equivalence. The for­ward movement is unequally favored; the space of action is essen­tially frontal space. That it is not limited completely to the frontal plane is apparent from the fact that, to a limited extent, actions are· also possible in the back-sphere. This is present for the length of one's reach and one's stride and is "utilized" in yielding and retreat. In any case, this form of movement is realized in stretches that are infinitesimal in contrast to the anticipated forward stretch. The

68 Lived Space

back-space is not actually given consciously; rather, it is there as a possibility, andas such it exists at all only in this mode of "having." If a situati9n demands its actualization, for example, in escape, an immediate turning around occurs and the space of action is trans­formed into frontal space.

Though the principie of economy present in the left-right organi­zation may be disrupted by the turning of the body, the principie is nevertheless maintained in the dimension that comes into view with the turning of the body; a longer backward movement would be counter to this principie. This comparison shows most clearly the preeminent heterogeneity in the directional opposition of front and

)

back. lt results from a superimposition of two mutally supportive aspects: on the one hand, the organismic structure of the physical body, and on the other, the principie of economy of the acting body. The acting body does not live with things behind it as it continu­ously does with those on its sides. To be sure, the back-sphere is still co-given and the subject is not set at the periphery of space as in the space of intuition. Yet he is no longer located "among" the things as he was in attuned space; the latter literally pervades even its unperceived back-field with full effectivity.

The prevalent organization toward the front has a specific mean­ing. lt constitutes the genuine orientation of forward movement, continuously discovering a new there and at the same time opening a new front. The acting subject is a being striving toward a goal. lts activity consists of planning, taking into account, completing, leav­ing behind, and stepping forward anew-without hesitation and

. l n without a backward glance. Th~xme&sive ex_R&!ien~s ¿~:.. • discarded in the space of action. lt has been subsumed by an

1 _L _L eSSeiiffiííiYaínf-Drreríted-b1riñgana thus it too beco mes an oriented S ~ l)'' ~ space. In this sense, the specific dimension in the space of action is ~ . ,~ ';1 its third dimension.

--~:f (~~ lt reveals yet another characteristic. Above and below, left and r \.~ -f.L- ·.,, right are purely spatial orientations.lt is different with the front-back ::... d dimension. lt derives its significance from the "already" and the

~"not yet," and thus indelibly bears a temporal moment. The space­time nexus differs here in kind from that of attuned space. Indeed, like the attuned corporeal being, the acting subject "has no time," yet with the latter the meaning is completely altered. This "having no time" is affirmed here in a judgment that reveals that one does have time at one's disposal. His lack of time does not mean an absence of

~ thetic time-consciousness, as in attuned space, but is grounded in an explicit positing of determined time. The objective "time-table" time

The Space Of Action 69

dominates space and regulates the movements of action. In distinc­tion from expressive movements, these proceed in a chronometric time, and in their execution are submitted toa measurable temporal duration. In practica! movements, the active subjeCt appears to be concerned primarily with time. The road to the work-place is not five hundred-meters long but eight minutes "away." Thus in the space of action, spatial and temporal determinants pervade one another in a unique way: the subject here brings time into space, although he can establish this connection only on the basis of a previously achieved dissociation.

The elementary directional oppositions turned out to be condi­tioned functionally; they are oppositional movements of a corporeal being who, within his established yet motile framework, was capable of exhausting all the possibilities of his primitive orientation.

The space of action grasped until now as a manifold of regions becomes accessible to yet a closer scrutiny. From here each there will be obtainable topographically by means of these directional oppositions. Up and to the right, down and to the left, above, below-each are indications of places for the regions; they are differentiated less through "mere" topological determinations than through the moment of activity common to them. Even while meant as pure indications of location and as an answer to a search for a locus, they are comprehensible only to a being that can move itself in their mutually posited directions. Indeed, the actual process of movement does not inhere in their conception, yet they contain an anticipation of the co-posited path based on the fundamental capac­ity of a corporeal being to move itself. The problem of the path or way will be discussed in subsequent paragraphs. The primitive topological ordering of regions in the space of action was founded, as we saw, on the originary functional givenness of the bodily orienta­tions. In this ordering space appears not simply as a pure manifold of places, but as an oriented and directionally determined space. This is tied in with specific problem. Since these elementary pairs of oppositions are related to and accompanied by the body, are they not merely subjective principies of orientation, mere means to find one's way in a space existing prior to corporeity? Does not orienting oneself mean getting one's bearings in a world already spatially ordered in its own right?

"The" space is represented in natural consciousness as the "wherein" of all things; its three dimensions are conceived as being perpendicular to each other. It seems as if these three spatial extensions "recapitulate" the movement orientations of the body, as

70 Lived Space

if they corresponded to the anatomical arder of the body. The three semicircular canals of the inner ear, which are responsible for the balance of the body, also turn out to be orthogonal one to the other. Thus we could speak of a "natural coordinate system" borne by everyone.

In arder to incorporate properly this uncontestable state of affairs into our investigations, it is necessary to indicate the methodological position of the particular science from which such statements stem: anatomy. For it, knowledge of the lived body is impossible. When­ever it becomes an object of research, the body ceases to be lived body; anatomy obtains its results from the "dead." While results obtained by these means can indeed reveal partial conditions of corporeal functioning, anatomy is in no position to comprehend the functions of corporeity in terms of a living being's body. Rather, it approaches the object of research with the knowledge of such functions airead y assumed. Thus the semicircular canals of the inner ear are related perpendicularly to one another, but that they condi­tion the sensible orientation of the moving body in space not only means that the concept of perpendicularity is already presupposed and understood from elsewhere befare anatomy can employ it; it also assumes an insight into the interconnections between the mathemat­ical structure of the organs of balance and the modality of movement of the lived body. This is possible only on the basis of having moved and having oriented one's own lived body. Phenomenological anal­ysis must not adhere to scientific propositions about corporeity, but strictly and exclusively to the lived body itself, as it is present in the immediate comportment toward the world and presupposed in each special science dealing with corporeity.

A phenomenological viewpoint offers no evidence for the assump­tion of space prior to corporeity, a space in terms of which the lived body would have to orient itself. Corporeity would have to perceive space as a pure dimensional system to which it would subsequently adapt its oriented movements of activity. If this dimensional system were pre-given, then an account would have to be offered for the possibility in principie of the intrinsic orientation being continually mistaken. That such misorientation does not take place factually would have to be explained by an accident or by a kind of preestablished harmony-if indeed there is anything at all to explain here. Instead, we must seize upon what is purely and simply pre-given, a spatial arder whose constitutive determinations are inherent in the lived body itself-which means, however, inherent in what it is in and for itself, lived body of a world, a lived body

The Space Of Action 71

whose possibilities of orientation can only be understood from its being with another, and whose very spatial principies of arder are already principies functioning in corporeityP

Further reflections suggest themselves. The above arguments appsar to imply that the three dimensions of "the" space ought to be reduced to the three elementary pairs of opposites. But then will not the arder of space be left in the hands of the contingence of our corporeal constitution? More precisely, would not the "essential insight" into the formal structure of space, which pretends to be a priori and necessary, be placed on the flimsy foundation of bodily facticity? The mere raising of the question may give rise to the suspicion of a reflexive circle: one speaks of three elementary pairs of opposites and attributes to them a preeminence whose justifica­Han obviously seems to lie nowhere else than in our unsubstantiated "a-priori" intuition of space. These and similar questions are fun­damental. In the investigation of the space of intuition they will reappear in a modified form, although here they cannot be dealt with fully. They require critical treatment in a subsequent context.

§ 5. The Problem of the Way

The three elementary pairs of opposites constitute a general framework, and each specific direction of proceeding can be deter­mined in relationship to them. Each where and there is discoverable and locatable through ways "in the direction of" right, left, above, below, in front, behind ... , although the frontal dimension turns out to be the genuine extension of the space of action. lt attains its privileged significance through its forward movement of the body. Seen in terms of a body at rest, it is only one of the directions mentioned above, even though it is a preeminent one. lt becomes the sale direction of forward movement by incorporating directions lying laterally and in back, for these can be transformed into frontal directions. The space opened in forward movement is a purely frontal space. The remaining directional determinations in no wise

27. In the pathological realm, the problem with disrupted orientations is not in the failure of cognitive capacity with respect to the three dimensions of "the" space, but in the functional failure to project a field of action. The deviation as such will not be gauged primarily in terms of objective space, but rather as a disruption of harmony in the sensible processes of activity. A.A. Grünbaum's attempt to grasp apraxia as an agnosia of spatial relation­ships has not been clinically confirmed. See the works of P. Schilder.

72 Lived Space

lose their significance for the topography of its regions; yet the inner oppositions of the individual dimensions, as well as their non­equivalence, balance out in frontal movement. This relativizes the directional oppositions and isotropizes space.

A brief discussion of the problem of the "way" or "path" illumi­nates the results presented concerning the space of action from another vantage point and clarifies new determinations.zs The problem of the way assumes its unique position in the space of action, insofar as the latter is a manifold of regions. The complexity of the latter is determined by the differing structuration of this space in accordance with project and situation. Thus sorne limitations must be assumed- otherwise it would hardly be possible to say anything universal concerning the phenomenon of the way in the space of action.

Since the space of action is structured in accordance with regions, there are many ways from here to there. They create an ambiguity in specifying orientations toward something. The acting subject is ~ capable of freeing himself from this ambiguity in the space of action insofar as he distinguishes one of the ways; this is what is meant when he speaks of "the" way. It is very seldom that it is the shortest

j connection of a visual path. In the space of action, the "direct way" seldom has the meaning of a path along a straight line. What is meant is tlm fastest wªy_to_[.f@_Q_l!AI} ?Jl!Jo,_The greatest speed does not mean the maximum physical velocity of one's own forward movement, but, rather, in accordance with the principie of economy that orders one's activities, the minimum of time required. The way in the space of action is measur~tempo~aTunits; this~separates it in principie from a connecting line segment in visual space. It is subordinated to a specific viewpoint selected, which in turn is co-determined by a

kv-fo given project of activity. mJw This also structures the "byway." This does not mean a visually \Á C! oA 1¡_' ·,yr---_

28. K. Levin has discussed the problem of the way in "hodological" space with the intention of basing a "vector psychology" on a mathematically clear concept of orientation (pp. 251ff.). His interesting individual results are valuable for our investigation, even if we deviate sharply from Levin on fundamentals. Levin begins by asking whether "hodological" space is Euclidean or Riemannian. Yet the subject considered by us in the space of action is still "on the way" toward attaining the possible sense of such a question in the first place. For his vectors and differentials are not phenom­enally present and require first of all the phenomenological elucidation of the variously founded concepts of mathematics.

The Space Of Action 73

longer way. Where, for example, an orientation to an instrument located "over there" is a deviation for the sake of acquiring a tool, in the space of action it is a direct way. (In this space the path does not have any points from which it branches off. These relations of connection between paths, unlike those in the space of intuition, are essentially determined by temporal components. A curved path having a double point in a timel~s--a-il_'(fgeometrical regard loses this characteristic in the space of activity. For it, two points of a path are never "the same.")

We assume a further topological simplification by observing only those regions that have paths in the above delimited sense, which need not be the case. Obviously the structuration of space plays a role here. Unstructured regions are characterized by a lack of any possible path; the previously offered definition of the nondifferenti­atability of places is identical with their impassability. After all, each (l distinguishable there in the space of action is in principie attainable through paths and passages. That these are different for each situation, that the best way is always different, is compatible with their relativity to a given project.

The relationship of a path to a situation brings about further deviations from the space of intuition. In the latter, place B is situated "between" places A and C if B can be reached without changing direction on the path AC. This relationship is valid in the space of action only if each partial segment of the "best" way, AC, is also an excellent way. The relationships ordering the space of action do not correspond to those of the space of intuition. Additional deviations result from the determination of the counter direction. If one defines the direction of the way back from B to A, it can deviate anew from what would, strictly speaking, be the reverse of the original direction, since the way back, BA, is not necessarily the best way. (For example, if areas are passable in only one direction, then it may depend on the condition of the areas themselves, or it may depend on the situation of the subject, for whom the condition of the path properly assumes its meaning. Hence there is no relationship of equality between the ways AB and BA; the counterdirection, i.e. the way back to be chosen, need not correspond with the reversal of the direction of departure. The way back, in the space of action, is not the reversal of the way, but rather the way "back" to the place of departure; both ways encompass a regían in accordance with struc­tural relationships, while in the space of intuition the way there and the way back merely constitute a line segment between two points.)

lt must be observed that the best way also contains the moment of

~r.rco.J í~'(VÍ~

t .;-v,rw-L

N'\<>kiNJ> J¡¿~-fV..e.t....

(

74 Lived Space

inexactness and ambiguity constitutive for this space. The region is the smallest topographically graspable element of this space. In the space of action, all the places within a region are indistinguishable, yielding for this space a specific concept of the sameness of a place. The same can be said of the paths and directions to be taken. Both are fixed only up to the regions of departure and goal; they are determinable univocally only if all paths are seen as equally running from sorne place in a region to the region of the goal. If one designates the multitude of such topologically equivalent paths, a domain of paths, then it is true that the space of action, as a manifold of paths, constitutes a topological manifold.

The definition of equivalence of the space of action, which is based on topological equivalence, is significant for the determina- ;<

tion of distance in relation to the "best" way. Yet at the same time the space of action ceases to be a mere topological manifold, since the distances are determined only imprecisely. The size of a· pace and the extent of a reach are derived from cor oreal orders of magnitude; they vary with the lived body and are determme situationaily. We are nevertheless confronted with primordial deter­minations of measure. Distance is a nontopological concept.29

The fact that the remoteness of A from B can be quite different from that of B from A is most significant for the problem of measure. There is nothing unusual in this as long as the fundamental difference between distance and remoteness is not observed and one does not exclude, meanwhile, the possibility of the "later" sciences of measurement. Their achievement is to be sought in their ability to change the remotenesses that can only be traversed into measurable distances. This results in a quantity whose validity no longer remains related toa subject who is here and now. Rather, they make it into a pure relative position between indifferent and equivalent

29. This does not mean that it must already be seen as belonging to geometry. It is distinguished from an exact mathematicalline segment by its constitutive inexactness and its changeability in terms of the situation of activity. It is only in accordance with the idea of measure that it is present as a mathematical quantity. Thus such expressions as the amount of land covered, or "in a day's work," in a "bushel-sowing" of seeds, by a "morning's" plowing, among others, are used even today as estimations, although they can be given in a geometrically determined measure. The old measures are more meaningful and expressive for a people of a nation. Besides, they show most admirably the moment of action and the temporal components of measurement. On this issue, the work of E. Fettweis is most instructive.

The Space Of Action 75

places. The geometrically measurabl~_ dis!_ance is a transformable quantityThat can only be conceived in accorda~ewitl1places-má space- Wlüi_"üüt corporeHy~ tli-"cülifiisC-remoteness~-Is--;;;sentially refatecrtoa-cÓrpor~al h.~;~· that has the distinction of being the place

1 of the subject. It is utterly untransformable: it is not a universal

)~~ relationship of measure between two indifferent "theres," but a l strictly singular situation between me in my being "here and now" -~.-.. and the things to which I orient myself . .JJ' In the elementary space of action, there are only such remate­-~ nesses, which are eliminated and projected anew by one's own ~ corporeal movements. There is only a directing oneself from a here -i .. to a there, never from there back to here: it is always from a new here ..f"to a new there. Thus if the space of action were to remain open only ~· to the measure of paths and passages, then it would contain only

~)' remotenesses but not distances. ~ To what extent such a spatial form is lived by primitive peoples -{¡ cannot be investigated here more closely. For the space of our 0----~ culturallevel, it is only a fiction. Even for the great floods of the Nile in

) ancient Egypt, the riddle is manifest of how this space, while being ~ ~ space .of rernotenes'~. ~n also be come ~ space of distance~-how it

~ 1s poss1ble to "ap~ geometry to th1s space.30 Ever smce, the sJ subject reflec~pónhis activity, sees himself in a position that { \\' suggests a paradox. He is a subject of a space that is projected from 1!

unrepeatable situation, and simultaneously he is shaped by tors that transcend the situation. Here the question reemerges

ncerning the appropriate interpretation of this ''being-in.'' We can ~ e a position toward this problem only when the problem of space

" 1s fully unfolded.

§ 6. Nearness and Remoteness in the Space of Action

In the space of attunement, nearness and remoteness are charac­terized by their qualitative difference. This accounts for the fact that in it there are no distances, no pure measure and size determina­tions.

The space of action is distinguished from it. In the space of action,

30. The Egyptians and Babylonians did not "apply" geometry but dealt with measures obtained and limited to the empirical. Such a way of putting it is comprehensible from the later scientific consciousness; it is only after the founding of geometry as a free science by the Greeks that the question of its "applicability" as such acquired a precise sense.

\

í ,,

76 Lived Space

the near-remote articulation has altered and carries a different meaning by virtue of the changed situation of the lived body and its differently constituted possibilities of transcending. The foregoing discussion has indicated that the space of action is characterized by a._lJJDm!!LI!eam~Jn contrast to_Jhe spac~]![Iii!illtiüil;TfisTrulya E<l~~()f_ nf!arne~Jhi~Eequires a !!!Ore P!~_s:_i__s~_e_x_QJI9illQ.il:'~~---

Indeed, in the space of action nearness and remoteness are not determinations attributable to things in their ready-to-handedness, but rather relationships to me in which they are near and remate. From these relationships it is clear that precise propositions con­cerning nearness in the space of action can be established to the extent that the location of the acting subject can be determined in its here. The sense and limits of such determinations have already been explicated.

To speak with Heidegger, the originary sense of nearness lies in the proximal ready-to-hand, in the being-to-hand of that which is at our disposal. 31 This shows a specific kind of determination of size. Not only is it inaccessible to exact measure, but its standard of comparison is different than in the space of intuition. Something

~~-ª~-i~~~e[~rf}_ me -~-~t~«:-~pac;~_g_t~~ti_Q.I!~ii-!_@~~~,Ql.§D some~hing by me, and this _ again is nearer than something _ behil},d me~This-re-sulfs from fhespécific strUCtúraffüií'üHhe-spaceófaction

·as"frontal space. The measuraEle quantity does not-determine fue distance; rather, ~~t~~~ll.l_e~ !~e_ remoteness. In contrast to distance, corporeity is given in it in a doUble sense: on the one hand, it is a corporeal body with specific extensions proper to it, and on the other, it is a lived body in an altered situation. Thus nearness and remoteness of things themselves become fleeting.

In activity, the remotenesses among things that are near are not thematized; they are present circumspectively, without any deliber­ate glance being oriented toward them. lt is otherwise when direct reaching fails to grasp something present "there." First of all, such a situation points directly to one's own corporeity. In an undisrupted course of activity, corporeity knows nothing of itself; only the "critical" situation throws it back upon itself. lt is only when something I need is "too high" or "too far away" that I experience the height of m y body or the shortness of my arm. The consciousness of one's own corporeal extensions is not given primarily through measurement, but rather it is experienced through the dimensions of things in immediate dealing with them.

31. M. Heidegger, no. 12.

The Space Of Action 77

Moreover, it is significant that the crisis situation can be answered by two characteristic modes of behavior. One consists of the creation of an implement capable of appropriating the distant object. Here it is less important to include the estimation of the size-relationships prevailing and the choice of the implement than it is to bring the remate to nearness or to create new remotenesses through the subject's removing hi'mself. What in perception is given as a turning from the intended there is the condition for the possibility of a relativization of nearness and remoteness which is characteristic of the human space of action.

An animal has forward movement as the sale possibility for drawing nearer. Its relationship to the things of its space is always immediate. Seen anthropomorphically, this immediacy impover- .@ ishes it. The remoteness of the animal space is always only a vital distance, dissolving and emerging anew in the fluctuation of its own dynamism and being resolved by the momentary needs of its body. The human subject does not only bring the remate to the here by reaching far beyond the compass of his corporeal structure; it is rather decisive that he knows how to bring the remate near in a negative act of turning away. The charactei="~This-actioi18-;-nú)dTated thiou-gh the specffic-u-se oflmplements, is reflected spatially in the phenomenon of corporeal turning away and renewed turning to- ..-­ward. This obviously creates the distance from things that enables him to be a subject.

As a subject, he also specially manifests a second behavior in the situation sketched above. Its brief discussion leads to a new aspect.

The situation will be characterized as "critical" because it does not motivate univocal behavior. Besides continuing his activity through mediating instrumental apparatus, the subject in such ª~ retains the option of breaking off, _ giving up. Which one will be chosen depends on--a-g-iven. sftua.Ífon.CWnaf Ís essential is that the latter possibility can call forth a new attitude: instead of indicating one's own corporeal limitations, there can be a specific orientatioh toward the remate thing, not grasped circumspectively, but discov­ered as a thing in pure sight.

This change of comportment is clearly marked in the reduced corporeal dynamics. The hand, as a grasping organ, is set out of function and orients itself only by pointing to the unreachable thére: "the there" becomes desired, or the arm can become absorbed into the trunk while the gaze "tenses" inquisitively to discern how the thing escaping one's grasp "really" looks.

Suddenly the thing emerges as apure object. This "mere gazing" ------ -

utfJ.-)1\) \

78 Lived Space

is constitutive of a new objectivity. Indeed, in the process of manipulative dealing witntliliigs;lliey are seen, recognized, and recalled, but their specific perceptual qualities remain behind the qualities of their applicability. The activity skims over the pure whatness of things toward their purposiveness; it surpasses them toward others and toward their context of application. This is the calm glance attaching itself to a thing that first discovers the individual in its proper species. Thus simple, fixing observation is equal to the cessation of activity, and, conversely, the situation of activity can motivate this new mode of seeing through its specific structure.

The corporeal manifestation of this structure is found in the posture of pointing. Indeed, the lived body is more actively disposed than in the mere viewing, since it maintains the proper organ for the appropriation of the world. Yet the transformed function of the hand clearly represents the change of one's position in face of the there. It rests, and while resting, it fixes the object without reaching it-it remains empty and reveals a specific possibility of the subject's being; in its emptiness, it lends "free space" for other modes of apprehension.

1

The animal does not have hands, it does not point. lts grasping

1 organs are always filled with the proximate and the needed, which it must appropriate. What is remate for it is only a vital distance. The hand can be empty, and it would not be wrong to say that ontologi­cally it is empty, i.e, it must be understood from those activities and orientations of the subject that allows it to remain in its emptiness. Moreover, pointing is comprehensible only for a corporeal being who is with others~-Altnougn'Itactlially- occurs-oiily-hi Tne presence ofotfíers-"íííthefl~sh," the sense of pointing must be understood from the possiqiJ_it.}'.oLa.commonalitY~wi.tlt o_ther ªubtec::ts. A solip­sistic sub}ect does not point. P<Enting corpor_~al_ly presen.!~__lb..e

~ / intf¡rSl}~~cti yi!Y__gf _ the-PQigt~I}g _s,gºj~t. - In pointing, corporeity surpasses what is palpably near by; it is at a there of its space of action yet extricates itself from it in turning toward it anew. Insofar as corporeity does not reach for the there actively but lets it remain without any purpose, it posits the remate as the remate. Pure sight constitutes a different space.32 Yet the

32. In clinical observations K. Goldstein encountered a very "remarkable difference" in cerebellum patients who could grasp correctly but were mostly wrong in pointing. Corresponding observations are in P. Schilder (1); see also the work of J. Zutt. It is to be noted that similar failures of

The Space Of Action 79

near-remote articulation of the space of action is not different to sight. Each being beyond the graspable nearness does not only mean a distantiatioh as a "posture" on the part of the subject, but also and above all, asan extension beyond into remoteness no longer reachable in activity. Indeed, the possibility of the relativization of nearness and remoteness through implements does not eliminate in principie the horizon structure of the space of action; all activity leads somewhere to the limit region beyond which space is only space of intuition. Phenomenally speaking, it constitutes at the same time the outer border of the space of action lying beyond the project of activity.

It should not be overlooked that the relationship between the two

sp.aces.is. det.ermi.·ned b.y th. e subje.ct_ .. In t~. n .. ea.-.~.r-r~.giP.nJh.e~.su.bje·c_t.is .. j active, circ~umspectively a_warf1,_ :whí]B-in th~ rmnqte-region h~ _i~-iL. theoretically positioned being, precisely because his corporeal orga­nlzfttlon does not aÚow -him to be imything ~l~~. But as the ánalysls cift:Iúi crft1Ciilsitliatíoii.Kaisííown~ ñeárnes~ can also become a pure object in pure vision and is thus extricated from the space of action. It should have become clear that the choice between these two possibilities is at the same time a choice between two modes of comportment, or attitudes in the phenomenological sense; each constitutes a differently modalized objectivity.

Their distinctness appears yet with another phenomenon, to be ~ understood as the most controversia! in spatial theory: perspective. r-~ y-e M:_

Usually it is explicated only in terms of the space of intuition. While perspective is taken up here, we do not intend to offer its complete description. It will become clear that the understanding of perspec-tive is possible only to the extent that we can achieve an abstractive separation of both spatialities and can also subsequently make their unity.

In the space of action "there is" no perspective. It first becomes phenomenally attainable with the thing of intuition, not a useful thing. The circumspective vision overlooks light and shadow, per- \..)M_I{;~­spectival foreshortenings and intersections, and the absence of all-sided visibility; such vision transcends them "in view of" sorne-

accomplishment are paralleled by a lack of cognitive achievements in the space of intuition. Goldstein stresses that pointing and grasping present activities that are different not in degree but in principie, corresponding to completely distinct modes of comportment. He suggests that this distur­bance is a "lack of an objective space confronting the subject," which is "not required for grasping" (p. 456). This confirms our exposition.

80 Lived Space

thing else. Furthermore, even the size and form of things in the space of action present determinations that are not thematized in the sense in which they are for the latter. In activity they are not conceived as primary qualities, but are incorl!orated in the characte~ utilit~·-Th~ re'!dy:í_Q__:-,~@_g is big,:'!)!J.gllg}lt~!92~Emall_''for the -ªJ!.k.!'l of ... ": its form is "appropriate for ... "; it is surpassed things. Thus tliefhTngs!ñtfie~space of achonare~~~parable, offering themselves as equal or similar to one another. These relations are not strictly morphological; they are bound to situation and application and include relations other than formal ones.

The phenomenon of size constancy belongs in this context; seen strictly perceptually, it would be a disruption of perspective and thus would become problematic. lts legitimacy is not to be estab­lished in the space of activity merely because in near-space the sizes of things are submitted only to small variations. This argument overlooks the fact that the near-space, as a zone of action, possesses

---¿ a differently constituted objectivity than that of a purely visual region, and it is projected from another center. While acting, the subject does not see himself in opposition to the world of objects, to be determined from a specific position in terms of form and size: he finds himself with a world of usable and resisting things in a situation in which the individual is continuously surpassed toward the goal of the actions. Size constancy does not dominate the space of action because the distances of the near-region are only imper­ceptibly differentiated, but, rather because in it size as a visual datum is not at all thematized.

FÚrthermore, there is a general question: why is it that size constancy dominates the visual near-region? lt is well known that perspectiva! changes of the sizes of visual things are not constant; in a specific near-region (ca. 500 m.) there appears the remarkable phenomenon of size constancy which is incomprehensible in terms of the structure of the visual organs and the nervous system or the laws of light optics. To explain this, the psychology of perception has recourse to supplementary contributions of memory and sup­ports itself only with objects one is acquainted with. To bring our investigations into conformity with the indicated experimental discoveries of psychology would require specific analyses. Although in this context they are not required, we shall offer a few suggestions. Size constancy appears with the perceptual objects of the near­region because even if they are thematized as pure objects in pure vision, they are ~y--ª.!so conceiveQ_in !_h§_se_ciimented mode of the ready-to-hand. Here pure Vfsi~ill contains a viriua.l -momeil.t

The Space Of Action 81

of circumspection of dealings with things; thus size constancy, as a pure datum of vision, can only be understood from an originary, active relationship of the subject to things.

§ 7. Summary

The previous observations concerning nearness and remoteness in the space of action were limited to the corporeal subject in the framework of his immediate comportment toward the world. They took their departure from things located "proximally" in the sur­roundings and thus from a subject as he appears in his immediate and active· corporeity. Yet a remarkable mode of comportment toward the world also carne into view: the circumspect immersion in the world of ready-to-hand things was abandoned and the things lent themselves to pure sight.

Ontically speaking, this is a late attitude. The question is, how­ever, what is its ontological status? Speaking ontologically, the relationship of the two spaces discussed here already excludes the notion of priority of one over the other because in the center of the space of action there is a being who already has the pure space of intuition as a possibility. Seen more precisely, this modality of comportment toward the world is an orientation that is not only given in specific possibilities of being, but is also an already actualized possibility and as such ca-determines the space of action.

In our opinion, a limit of the existential-ontological interpretations is found here. The limitation of Heidegger, and of the French school that followed him, to Dasein's "circumspectively" being absorbed in the world-coupled with the claim that the human is thereby grasped in his genuineness-allows only the space of action to be considered as existential space. This led to that strange curtailment of the problem whereby a differently structured space was no longer a tapie for philosophical investigation. W e do not con test in the least that-in Heidegger's terminology-orientation and remoteness are existentials; yet we take both concepts in a more encompassing meaning which must be taken in a strictly phenomenological sense. Regardless of how we may conceive our descriptive analysis, it must deal with an unavoidable issue. Not only does the acting subject orient himself in terms of his elementary directional opposites; rather, in his own "existential" space he also has the capacity to orient himself even if the possibilities of the elementary directions are lacking. And he establishes remoteness not only through the coming-into-nearness of distance but also mediately by the use of

82 Lived Space

technology that presupposes the mathematically oriented sciences. These do not comprise additional gear, but are based on an entirely distinct mode of vision no longer subsumable under simple "care."

The latter form of lived space leads immediately toward the crest of mathematization, without, however, surpassing it. More precise analysis is needed, on the one hand, to show that it is still lived space, and on the other, to recognize in it the basis upon which the edifice of metrics can be erected.

Chapter Three

The Space of Intuition

§ 1. Terminological Clarifications

The space of action as a manifold of regions for the ready-to-hand does not comprise the only kind of oriented spatiality. The subject does not exhaust his corporeal achievements in active dealings with things; his functional space is not completed in its being only a space of action. Besides being a functional unity of aim-oriented activities, ~ corporeity is at the same time a unity of sense-accomplishments; it is not only an active but also a sensibly intuiting corporeity.

Nonetheless, this does not prove that in the latter case a new kind of spatiality corresponds to corporeity. It is conceivable that what is meant here by the space of intuition is nothing other than the consciously perceived space of action, which, in its structure, is not essentially distinguishable from the latter, but is, rather, brought to clearer illumination through a specific paying attention. The func- ~

tional unity of action and perception seems to favor this view. Two things are to be noted. With regard to the ready-to-hand, it is

touched, seen, heard, and at the same time it is an object of sensory perception. In a strict sense this is valid only in a vague sense of the concept of object. Indeed, the ready-to-hand is never outside of sen­sory perception, yet for this very reason it never be comes its thematic · object. The perceptual qualities are merely "also perceived" on the ready-to-hand. This "also-perception" does not intend a consciously posited presentation of the perceptual object.33 The emergence of

33. We are using this clurnsy terrn in contradistinction to Husserl's "co-perception" (Ideen 1, §§ 27, 113). Indeed, the "also-perception" is a "perceptiva co-appearance lacking particular position in factual existence," although it is not identifiable with Husserl's conception. Husserl explicates co-perception on the basis of the relationship between foreground and background, where in fact an attentional turning of regard plays a role.

83

84 Lived Space

pure perception from it is not sufficiently accounted for by attention. Pure sight is not merely an attentional transformation of circumspec-

0 tion; t~_re.s.ent-at-hand is not simiJlY a p~ceived ready-to-hand t~~~-but rather has its correlate in a ~ifferel!~~ttitu@_Qf..fue subj~~t. This does not say anything about the structural differences between the two spaces. It is furthermore thinkable that despite the distinct modes of being, the ready-to-hand and the present-at-hand "fill" the same space. This means that the traditionally followed method of representing space through the things within it fails with respect to

Q the space of intuition. The treatment of this space will require meth­odological reflections. The sense of the being-in of things is here apparently derived from the fact of the material containment of one thing in another thing, such that obviously the container conception immediately emerges. If this were valid, then this type of space would be completely indifferent as to whether it was filled by the ready­to-hand or by the present-at-hand.

It can be shown that such is not the case. The present-at-hand also has its space with its own determinations and structural laws, through which it deviates in decisive ways from the space of action. In accordance with our fundamental phenomenological principie, let us again interrogate the subject comporting himself in space and attempt to shed light on the structure of his space from the side of a

~ corporeal being as a sensibly intuiting being. This requires precursory terminological clarification. The reason

we are speaking of the space of intuition (Anschauungsraum) and not perceptual space (Wahrnehmungsraum) is not because the latter term could suggest the conception of space itself being perceived in the same way as are the things in it. Especially since Kant, this question no longer requires any discussion. It is necessary to show

(Correspondingly there are further psychologically distinguishable rnodes of intuition, such as becorning aware with an "absentrninded glance," the fleeting view in a hasty orientation, etc., analyzed more exactly by C.F. Graurnann.) The also-perception rneant here has nothing to do with atten­tion, but is rather a title for a potential perception corresponding to changes of the total attitude. In contrast, our subsequent discussion of co-perception ought to be taken in accordance with a second rneaning of Husserl: as potentially itself present, in the sense of categorial intuition, although fulfillable only through additional actualizing positions (Ideen, §44). 1t relates to a horizon of "non-presentive" co-givenness and "vague indeter­rnination," in relation to the full object within the various series of adurnbrations. Husserl was not fully aware of this equivocation in the concept of co-perception.

The Space of Intuition 85

the distinction between sensory intuition and mere perception purely phenomenologically. A thing of intuition means a "com­plete" thing in its totality of real qualitative fullness, and particu­larly in all of its sensibly perceivable qualities, inclusive of the actually not perceived, "covered," co-given factors. It is the unity of what is perceived and what is grasped along with it that first grounds the conception of the "thing" as an identity in the fluctuation of perceptual manifolds. In this sense the thing is "intuited" as a thin&.. while perception, in contrast, offers only changing views of its sides. Thus each sensory intuition contains categorical moments; never­theless, it remains sense intuition, i.e., straight forward presence, "in person," "in the flesh."

Subsequently, the space of intuition will consistently mean the space of the sensibly present "in person," although categorically co-determined things and thing relationships. More precisely, it will1 mean the perspectivally and horizonally limited space related to the intuiting corporeal subject as its center.

In addition, the concept of perceptual space has its own meaning. While the conception of space as perceivable in itself, or the naive theory of a spatial sense besides the other senses, no longer requires discussion, nevertheless a question remains as to whether among the sensory functions there are sorne whose objectivity reveals relation-ships of a spatial kind. If this is the case, then the discussion dealing with space perception acquires an affirmative, although modified, meaning. Such a space, correlated to this kind of space-presenting sensibility, will be designated as perceptual or as sensory-space. "'·~"~CJJ1 Two such sensory spaces, the visual and the tactile, will be come (, l significant for our line of inquhy. Tlie 1irsfiri~Pa.rticular1s not to be t&<c. . ( confused with the space of intuition. Although they share certain formal elements, it is nevertheless a space of visual things that can be more closely specified, but not of intuited things.

§ 2. The Space of Intuition as a Phenomenal Multitude of Points

In the space of action, the ready-to-hand has its place. The place to which it belongs is indicated for it within the framework of a project. The place of the ready-to-hand can be fixed up to a region, which is the sole topographically graspable element of the space of action.

In the space of intuition these relationships are different. The present-at-hand things in it have ceased to have a place of their own. Removed from their contexts of utility, they stand extricated from any functional relationships with other things. An isolated entity

86 Lived Space

there and yonder has its location merely as a "position" in space completely external to it; an accidental, arbitrarily exchangeable somewhere no longer having any relationship to the thing itself.

In complete contradistinction to a visual thing-which, in all of its qualities, is a function of a location, changing with its changes-the thing of intuition remains the same in all changes of location. This phenomenal disattachment of the thing from its position reveals two properties of space that first appear clearly in objective space: its homogeneity and its emptiness. This will be discussed in the next section of our investigations.

This indifference to location of the object of intuition dissolves the totality of the spatial texture. It is striking that, in the attuned space, a thing may lose "its" place and be able to change the encompassing atmosphere; in the space of action this is so much slackened that it is only a matter of a change of region, while the location of the ready-to-hand is freely chosen within it; the space of intuition remains completely untouched by the movement of things existing in it and remains indifferent to any change.

Indeed, within the space of intuition, regions can be delimited consisting of parts of space and united under sorne point of "view." Yet their relative closedness no longer rests on the factual connec­tion of things, as was the case for the regions of the ready-to-hand things. The basis for this lies in the changed context of things. In place of the functional context in the space of action, there appears, in the space of intuition, the differently structured causal nexus. Thus its division in accordance with regions is more or less arbitrary. The regions here are no longer members of a spatial whole, but merely parts of a space capable of being thought summarily as contiguous.

However, the decisive distinction appears in the meaning of the phenomenon of "play-space" or "leeway," and thus in a different degree of structuration of the space of intuition.

Play-space was defined as the smallest region within which the place of an implement is freely variable without ceasing to be the same. Its determinateness contains possibilities of displacement whose extent depends on the degree of structuration. It was already pointed out that as play-space of circumspection, this leeway remains hidden and it is first revealed by pure sight. But this means that in truth it is a play-"space" of the space of intuition; it is a part of it and, as a part, accessible to further intuitive differentiations. The space of intuition reveals a refined structuration unattainable in the space of action. The process of nesting of regions can be

The Space of Intuition 87

advanced much further than in the space of action, and the sequence of regions of the space of intuition converges toward a fixed place as a limit value.

Yet the nesting here is not reiterable arbitrarily. The intuited object may become smaller and smaller through sorne influence or other, and its place in space may shrink increasingly, yet this process is not protractable "into infinity" in the space of intuition; rather, it reaches a finite limit no longer surpassable by intuition, a limit beyond which the reiteration can be arbitrarily protracted in thought but cannot correspond to any sensory intuition. The supercession of this limit leads to an abrupt disappearance of the object and its place. If one designates the given convergence value of a nesting of regions in the space of intuition as a phenomenal point, then the space of intuition presents itself as a manifold of phenomenal points. As positive limits of the nesting of regions, they are the smallest parts of this space, its ultimate topological elements.

Due to its limited approximatability, the phenomenal point is always a very "rough" formation. The refined structure of the space of intuition is dependent on a series of factors conditioned by the organismic order of the corporeal subject-such as the threshold of stimulation in physiology, the possible impairment of the sense organs, etc. The degree of exactness is always an empirically determinable quantity and is the object of experimental researches.

The space of intuition is completely filled by phenomenal points. It constitutes a (simply) connected set of points whose elements lie "thickly everywhere."

The concept of thickness in the space of intuition is .different from that of the space of action. The possible "too-denseness" of the places appeared in a lack of leeway when the ready-to-hand ceased to be handy. The space of intuition has no analogue for this. The intuited is never too dense; it is where it is, as intuited. Its "play-space," if it has one at all, is one of movement, although it is not as though the intuited object required a "play-space" of movement in order to be intuited, in analogy to the way in which the ready-to-hand requires its play-space in arder to be handy. The play-space is not a constituent for the present-at-hand, as is the case for the ready-to­hand. The ultimate topologically graspable factor given in intuition is not the region, but rather the place, the phenomenal point; "behind" it there is nothing that could offer a conception ofitas a region of another extensive manifold to be occupied in a way that would be relative to a corporeal subject. Indeed, its further structuration can be thought up to a mathematical point. But though

88 Lived Space

its intuited elements lie thickly everywhere, the denseness of the phenomenal manifold is not that of the mathematical continuum.

This process of further nesting not only surpasses the limits of sensory intuition, but leads mathematical thinking into difficulty, even into an aporetical situation. Underlying the mathematical concept of denseness, of the multitude of points, there is the problem of the continuum. Its mathematical discussion cannot be offered here ahead of time. The sense and nonsense of a mathematical "continuum of points" can be discussed subsequently in a pertinent place. Yet it cannot remain unimportant for mathematics that, and in what manner, the concept of the continuum has an intuitive signif­icance prior to any scientific treatment.

In addition, it must be strictly observed what really lies in the unqualified intuitive conception of the continuum-and more pre­cisely what does not lie in it. Above all, it must be stressed that for sensory intuition the continuum is not a problem but a fact. As such it means that all parts of space are interconnected and without gaps-that, in accordance with a radical formulation of Husserl, space can never have a hole. 34 The expression loses its appearance of naivete when it is understood from the givenness of natural intuition by which it is motivated. This givenness does not possess points of space but things in space in their own limitations and freedom of movement. A directly perceivable movement is for the natural consciousness-which has not yet reached the level of reflection leading to the Zenonian paradoxes-the originary paradigm of a continuous event. As long as a thing moves, it is a moving thing in ea eh phase of its movement. Yet grasped as a moving thing in space, such movement, apprehended as a change of place, implies that space itself must be continuously connected. The change of location of a thing brings about at the same time a displacement of its limits, i.e., phenomenally speaking, of its surfaces, edges, and comers. However, these limit formations constantly and necessarily coincide with the corresponding spatial formations. In particular, each comer point of a moving object continuously coincides with a phenomenal point of space, and, conversely, each phenomenal point of space can in principie be a comer point of an object.

It is the latter possibility that is primarily responsible for the naive conception of space as a (phenomenal) "continuum of points." It means only the intuitive possibility permitting any arbitrary place of space to correspond to a comer point of a thing, thus signifying in

34. E. Husserl, Ideen II, § 13.

The Space of Intuition 89

this sense the division of the spatial continuum into phenomenal points-more precisely, its successive divisibility into mere individ­ual theres and yonders. This conception does not imply that the spatial continuum "is" an (actually given) multitude of points and "consists" of mere points that when combined constitute space. In natural space-consciousness what is present is not what the contin­uum is, but only what can take place within it. Strictly speaking, there is no "atomistic" continuum as a given in natural space of intuition. Rather, there is a theory of an atomistic continuum derived from a conceptually reiterated extension of the intuitively given; the paradoxes of this theory are grounded on the latter (see p. 296 ff.).

§ 3. The Lived Body as the Center of the Space of Intuition

As with the space of action, the space of intuition has the corporeal subject for its center. Thus purely formally this space possesses all the properties resulting from its being centered. The singularity and uniqueness of the here, and its radical incompara­bility with any there, constitute the nonhomogeneity of the space of intuition. They articulate it into nearness and remoteness and condition its finitude and horizonality.

The situation of the lived body in the space of intuition is different from that in the space of action in two respects.

In distinction to the space of action, the space of intuition is present as a space of remoteness. This determination also has a dual meaning. First, it means a spatial remoteness between the things and myself. In this sense, its remoteness in fact surpasses that of the space of action, although the latter is also characterized by an articulation into nearness and remoteness. The second meaning consists of a characteristic remoteness of the body in relationship to itself. Indeed, in the space of action the lived body is not present to itself, but is rather at the things, is oriented toward them; yet it has them only in immediate contact with the body. In the space of intuition the body is beyond itself in an entirely different sense, and this by virtue of the distinctness of the visual function which plays a decisive role in sensory intuition. To pick up something ready-to­hand and to "fix one's eye upon" something present-at-hand, grasping it visually-here not only two fundamentally distinct attitudes of the subject, but also two entirely distinct relationships of the relevant thing to corporeity are manifested. The latter expres­sions are meant metaphorically, since the thing has no way of being in the eye as it is in the han d. Vision do es not occur through the body

90 Lived Space

as corporeal in the same way as with handling. Indeed, the eye is a member of the corporeal body, insofar as it is sensitive to touch and pressure, but not insofar as it sees. As seeing, not only is it not given as it is when seen, but it cannot be given-as seeing-through any kind of objectifying observation. In seeing it determines itself not as a member of the physical body, but as a corporeal organ in function. And since this function is an objective one, it means that the eye does not have its object on or in itself, but in space. While intuiting, corporeity appears in a mode of functioning that has its fundamental conditions in its own body, but does not bring corporeity itself into view. The higher value and the higher power of achievement of the visual function lie in that the intuiting corporeity looks away from itself, since it can effectively assume its function only in a specific spatial remoteness from corporeity.

This transformed role of the lived body in intuition is presented in bodily corporeity itself. Its own dynamics are reduced qualitatively and quantitatively; they sink to a minimum to the extent that those movements in which corporeity appears as active, in a narrower sense, become redundant for perception. In this space, limbs and torso are deactivated in their specific functions and remain motion­less as an undifferentiated whole; phenomenally, they are nothing more than mere bearers of the senses. What remains essential in corporeal movement in this space is, strictly speaking, only the movements of the sense organs and the equivalent forms of dynam­ics such as the movement of the head and locomotion. What is noticeable in intuiting corporeity is above all its organic and functional symmetry. All organs are superposably related to the bodily axis, and the functional differences of the sides are dissolved.

How does this altered situation of the intuiting corporeity appear in its space? What does this situation mean for space?

§ 4. The Oriented Space of Intuition

The intuited space is also an oriented space with its center in the oriented corporeal being who emanates rays of intuitive intention­ality to things and encounters them above or below others, on the left or to the right of them. The topography of this space acquires its sense only under the assumption of a standpoint; it is a description of a location in the framework of a corporeal system of relationships whose point of departure is indeed freely choosable, yet within which any given location is anchored. Thus the arder of things among themselves is here not a pure relation of position, not a mere

The Space of Intuition 91

constellation, but rather still a situation-an ordering of the there relative to the here and now of a corporeal subject who apprehends what is sensibly intuited. Each relationship of arder obtains its meaning from its associative relationship. It is not merely a factual arder of things in their pure places but an emplacement of accom­plishments of the subject at the present in a here and now.

This system of relationships consists of the three elementary orientational oppositions. To discuss them in detail once again would be a repetition of the essential traits delimited in the previous section. That they were already discussed earlier was not due to an arbitrarily systematic procedure but, rather, was necessarily based on the state of affairs: the orientational oppositions are primarily functional oppositions in a narrower sense; they are distinctions whose understanding is to be sought with corporeity as an active subject. The consideration of the oriented space of intuition as such would be unnecessary if its directional oppositions consisted merely of those of the space of action. Factually this is the case; the sense of the directional oppositions in the space of intuition is the same as the space of action. Yet with the former, certain changes appear. They will become comprehensible from the interrelationship of orientation and movement. As has become obvious in the previous chapter, the directional oppositions are comprehensible only from the dynamics of the corporeal subject. Each where that is determined in orientation is bound to the whither of the movement and the possibility of movement. The contrary directional oppositions result from the qualitative difference and opposition of the singular dy­namic forms. In the space of intuition these are leveled out to a great extent due to the fact that the movements of intuition are reduced to the minimum. The characteristic feature of the space of intuition in contrast to the space of action lies in that the anisotrophy of the latter is here equalized.

This is most conspicuous in the weakly formed opposition be­tween left and right. As a functional difference, it is completely meaningless in the space of intuition. The differentiated activity of the left and right hands in grasping disappears completely in the space of intuition. When the hand is used to point-in its only function within the space of intuition, a function intermediate between grasping and seeing-left and right are equivalent. The remaining dynamics in this dimension, the movements of the gaze and the corresponding movements of the head, are externally equivalent; they are also felt "from within," kinaesthetically, as qualitatively different.

92 Lived Space

This is correspondingly the case with the pair of opposites above-below. In the space of action the emphasis is placed on qualitatively distinct movements engaging the entire corporeal body, while in the space of intuition we notice only the dynamics of raising or lowering the glance. The movement is slighter the farther away the things are and the more clearly the space of intuition presents itself as a space of distance.

The most pronounced and significant change is undergone by the third pair of opposites: front and back. While in the space of action it has a most pronounced opposition, in the space of intuition it is eliminated-not by balanced, qualitatively equivalent movements making it isotropic, but rather through the fact that the dimensional tension is radically annihilated.

Indeed, the space of intuition, like the space of action, does not completely clase itself off with the frontal plane. The plane can extend with the "wandering of the glance," and the horizon appears closed at the limit from which the line turns back upon itself. Thus the actually intuited part contains a conceptual moment of being a mere segment. Nevertheless, the back-field disappears-and indeed on the basis of the exclusively frontal organization of the visual function. Strictly speaking, seeing "toward the back" is functionally impossible. Even if we were to turn our head around, considered functionally, we would still be oriented forward. In this sense, the space of intuition is always frontal space.

While attuned space is experienced as surrounding, effective in its all-encompassing fullness, and the space of action allows limited possibilities for action in the back-sphere, in the space of intuition the back-sphere is finally lost. The lived body moves toward the periphery and is no longer "among" the things, but has them exclusively over against itself. It is no longer located in space, although as corporeity it is itself spatial. Thus in this placement it represents a dual polarity consisting, on the one hand, of the inner-spatial there of things and the here of its position, and on the other, of the non-spatiality of this polarity, since it is tensed between the spaceless subject and space itself as an object.

This is a most remarkable fact. The subject has attained a mode of being that truly legitimates his title as subject: he not only intuits things in space as pure objects, but also confronts the world as an object and regards space itself objectively. "The" space means primarily a space that is regarded as one for all corporeal subjects; the subject knows that his space of intuition is a "mere segment" of it. This specific mode of conception is not first introduced into the

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perspectiva! space of intuition through reflection, but is rather present from the very outset in the everyday awareness of space. No priority or posteriority can be demonstrated here. Only a subject who has the space objectively is at the same time corporeally located at the periphery of a perspectiva! space of intuition as his own and, conversely, finds that his perspectiva! space of intuition is not given to him otherwise than as a "part" of "the" space. It is only by respecting this state of affairs that we can enter into the variously stated question of how the corporeal subject can derive a conception of the one space from the oriented space of intuition given only to him from time to time. Certainly he does not move from the first to the second or from the second to the first. His monadologically spatial world is nothing else than the mode and manner in which he as a corporea1 subject already has the presence of "the" space.

But this space is no longer relative to a corporeal subject in his monadological isolation. If it has any relativity of being, then it is relative to all corporeal subjects. "All" here does not mean a numerical multiplicity but, rather, allness as a unitary form of intersubjectivity with a specific consciousness-structure. That the latter is historical, that the objective consciousness of space is historically changing, is not being overlooked, although it can be explicated in greater detail only subsequently (pp. 156 ff.). In any case, strict reservations must be maintained concerning assertions about the structure of this space. Such reservations do not imply a postponement of a problem but correspond precisely to the state of the phenomenon of the natural space-consciousness. At this level of space intuition, space is not yet given thematically but is present in the singular space of intuition in the form of the "continuation" of space "beyond" the horizon.

The following investigation is valid primarily for the monadologi­cal space. From the foregoing it is obvious that we cannot answer the question of how objective space develops, unfolds, etc., from it. Rather, following our method, it is necessary to show how the one intersubjectively conceived space appears in the perspectiva! space of intuition.

§ 5. Spatial Depth and Perspectivity

The corporeal being located at the periphery of space has become a subject; he finds himself in a position of absolute opposition to the world as the totality of sensibly intuitable objects. But they remain only what is present of them "in the flesh": they are related to the

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position of the lived body, whose outlook toward the world is monadological.

This intuiting subject has the world befare and for himself, while the problem of spatial depth appears in the having of the world for oneself.

How is spatial depth present to me in my here and now? It is not an object of my intution in the sense of immediate self-givenness, like the things befare me. Only they are there "in person," are found there and yonder. (We must not overlook that, strictly speaking, they are present "in person" only from specific "sides," while the other sides are only co-meant. At this point we still exclude perspective and accept the things only in their being behind one another, i.e., without regard to perspectiva! changes.)

Between the things of nearness and remoteness there is a charac­teristic relationship of priority inconceivable as mere differences of distance between things among themselves.

The things nearby are open to my glance and are given in full actuality. Their dimensions can be completely traversed from here with a wandering gaze. This is not valid for remate things. The near and the remo te stand in a relationship of covering and being covered, of the hiding and the hidden. This relationship is reciproca! insofar as the one calls forth the other; it is irreversible insofar as it is unequivocally determined by the single privileged point of my space, my standpoint. This space is exclusively frontal space. However, 1 do not have things only befare me, but also befare other things. The remate object is haunted by an unresolved residuum for intuition. The open, the obvious appears on the dim background of intuition of the alien, the questionable, the undiscovered but still to be discovered. The hiddeness of a distant object is not something irrevocable, but a momentary fact. It is conditioned by my being here and now and can be abolished through locomotor movement.

Locomotion in the space of intuition is motivated solely by what is hidden, undiscovered; its sense derives from the aim of discover­ing. In its being still undiscovered, the present-at-hand is the proper object of the self-moving corporeal subject in the space of intuition.

In covering and being covered, space appears in its depth dimen­sion. It is essentially distinct from lateral extension. 35 Extension has to do with objects; it is a pure characteristic of things, graspable in

35. The distinction between the surface- and depth-extension of space is clearly worked out in detail in the little-noted but excellent work of H. Lassen (1), specially pp. 124ff.

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pure relationships of size. It is not so with the depth dimension. Depth cannot be measured as a spatial interval. Measurable dis­tances are reversible and transposable. Depth, however, is not a measurable interval but a remoteness, a polarly tensile relationship between the things and me; it is irreversible and strictly singular. Corporeally 1 am constantly here and now, a lived body in a situation. Thus depth can never be abolished; not only can it never be surpassed, but it is always created anew in locomotor movement. Therein lies the phenomenal state of affairs: 1 can never wander in the oriented space of intuition, I always take it with me. In all my traversals of space it is always befare me. Although from moment to moment it is different in a continuously changing plenum, it nevertheless remains structured in accordance with surface and depth, belonging to me as long as 1 am sensibly and intuitively oriented to the world.

My stance in opposition to the world and my capacity to turn toward the world are intelligible in terms of spatial depth. As the genuine dimension of my being, and in complete ambivalence concerning m y being as lived body oras consciousness, depth is first comprehensible in its own equivoca! sense: first, it is the sole dimension into which it is possible to move forth; second, it is precisely the dimension that cannot be surpassed in progressive movement forward. On the one hand, it is a space that requires me as a motile being to occupy the center, and, on the other, it is at the same time a space in which 1 cannot wander. This apparent paradox can be resolved only through an ontological explication of the spatial depth of the subject- of the subject in his corporeal restriction to the here and now and in his intentionality of consciousness, which transcends corporeity. It is important to take the latter in the narrow sense of an ability to orient oneself in pure opposition to an object-world, free from the effects of space from concernful absorp­tion in a merely "proximial" world of ready-to-hand things at our disposal.

Grave arguments have been raised against the phenomenon of depth. Berkeley's theory of space became the starting point for a series of discussions concerning the question as to whether or not there is an original depth perception, and Berkeley's arguments elicited a multitude of treatments of this problem in philosophy, psychology, and physiology.

The central conception of Berkeley is that real space is solely two-dimensional. It is associations of a determínate kind that first create the "idol" of the three-dimensional space of depth. In con-

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nection with Locke's theory of association, the emphasis lies, on the one hand, on the linking of tactile and visual data; and, on the other hand, on the unification of acts of perception and thoughts. Thus through the experiential knowledge of the all-sidedness of the spatial thing in the individual intuition, extension in depth is associated with the surfaces. Furthermore, the spatial depth was explicated physiologically through binocular perception, which ought to unify stereoscopically the non-corresponding two-dimen­sional retinal images (a later term). With each of these arguments the peculiarity of the depth dimension became clear. lt is characteristic that, during the following periods, one followed one of these three trends of thought, and that the theories of space evolved during the nineteenth century were systematically devloped in three direc­tions.

The last of the arguments presented became the point of departure for physiological and psychological observations of the senses. Their results, especially since the discovery of the stereoscope by Wheatstone (1838), led toa hardly surveyable profusion of individ­ual analyses and quite extensive experiments, which cannot be explicated here in any greater detail. What is relevant in this context is the total methodological milieu of the individual researches concerned with the phenomenon of depth. 36 In brief, the main difference between their mode of observation and ours is this: theirs is concerned with explanation, while ours with understanding. They inquire into the physiological conditions that must be fulfilled in arder to result in depth perception: they find these conditions in the form and arrangement of the visual organs and in neurological processes. In contrast, we inquire into the sense of seeing for the subject; we do not take seeing as a process functioning in accordance with physiologically and physically determined laws of the organ­ism but as a "function" of the subject, as a mode of his having a world. The immediate question is whether a causal-explanatory science, with its constantly repeatable experiments and its verifica­tion of statements by an individual case, does not deserve preemi­nence over a point of view having nothing better to offer than sorne dubious "evidence." Such a question should be a warning that

36. A good overview of the historical development of this research is given by F. Sander. The work by W. Arnold, the title of which is misleading, deals with the experimental method of the psychology of depth perception. For individual problems see N. Guenther, E. R. Jaensch, F. F. Linke, G. Lintowski, F. Mayer- Hillebrand, and B. Petermann.

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the fundamental method prevailing here has been misunderstood and misinterpreted. It considers results that have been disassociated from the context that grounds them and sets states of affairs against one another where only methods are to be compared.

The individual scientist must be cognizant that what he seeks to explain can be interrogated causally only because that something is in sorne manner already given and open to understanding. This fact is no less than self-evident. With regard to our problem, it means that the eye must already see befare it can be investigated by a special science as a condition of seeing. Seeing as a function of corporeity makes the very question of the structure and mode of functioning of the visual organ understandable. The lived body is presupposed by the organism; it was already a body capable of functioning and acting befare it could be come a "re" -acting body, an object of research for the special sciences. What decisively determines their view, how­ever, is a rigorously controlled methodological procedure that con­tains measures for the reduction of the corporeal subject to a "corporeal" object; this is precisely why the lived body as living, i.e., as a functioning corporeity, escapes them. The special science, must necessarily divest corporeity of its situation and place it in a factual situation that is repeatable at will in arder to obtain its results. Yet it gains its propositions at the price of the living body. Strictly understood, corporeity radically escapes any objectifying scheme of investigation. Irreducible in principie to constant and arbitrarily reproducible experimental conditions, it is rather already itself a participant in establishing such conditions. As the lived body of a subject who visually investigates the visual process in light of physiology and optics, this lived body stands outside of the postu­lated methodological scheme and leads unavoidably to a circular structure of the objectivating observation of vision: observation must presuppose the capacity that it seeks to investigate.

Obviously the claim of the special investigation is justified if it does not pretend to be more than an explanatory science, a system­atic analysis of the organic conditions of vision. To the extent that the regularities investigated are those of vision itself, the investiga­tions are legitimate; after all, lived corporeity is not without a physical body and constitutes an ontic unity with it. Yet such investigations can never be ontological. They cannot conceive of the living corporeity as a mode of being of the subject. The regularities of the latter are different from those of the organism.

Phenomenology and ontology cannot criticize the results of the individual sciences so long as the latter do not infuse the copula of

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their expressions with metaphysical meanings. They must conceive of themselves as sciences of entities limited by a specific method­ological view. Philosophy remains primarily on this side of individ­ual investigations. Yet insofar as it documents and also establishes such this-sidedness, it transcends the individual sciences toward a philosophy of scientific entities themselves, thus becoming a theory of science. While it attempts to become an explication of being, it is mistaken if it pretends in this manner to become a "fundamental" ontological endeavor. lt cannot simply remain "on this side" of the individual sciences, unaffected by their accomplish­ments. If it is to become "fundamental," it will have to take a position not befare all sciences, but behind them in arder to illuminate their ontological foundations.

Regarding our problem, we must acknowledge that diagonal disparity "must be" recognized as a causative factor in depth perception. That depth perception is possible in this manner does not, however, affect another way of looking at it: "seen" primally, depth is a clue for the intentionality of the corporeal subject, its being as consciousness.

The thesis of the association between tactile and visual data has frequently been discussed. Among the newer philosophical theories, Husserl's early phenomenology revives the old Berkeleyan concep­tion, though in a greatly modified guise, in the theory of constitution. According to Husserl, the constitution of the thing occurs in various syntheses, and, above all, in the sensuous syntheses of visual and tactile primordial objects; such constitution takes place at many levels.37 Following the pre-spatial fields of the first level-where the visual and tactile fields consist of completely separated, quasi­extensive manifolds in which only the hyletic data are individu­ated-the second level extends the field through the motorics of bodily members, in which the visual and tactile data are united in a determinate manner. Oriented space is constituted only at a third level. lt is built from the oculomotoric field through "reinterpreta­tion" of the visual depth into a third dimension. The basis for the reinterpretation is found in kinaesthesia, which presupposes carpo­real tactility.

The notion of reinterpretation is based on the view that space is not originally "given" but, rather, is "constituted," such that unbe­knownst to naive observance, it contains unnoticed traces of preced­ing constituting "achievements" of pure consciousness. Behind this

37. E. Husserl, Ideen JI, § 9.

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question there is a fundamental problem that completely separates the spirits of the new phenomenology. Thus existential philosophy, specifically the French style, sees its irreconcilable opposition to Husserl precisely in that for Husserl, phenomenology is irrevocably bound to the transcendental question. From the very outset, phe­nomenology for Husserlis constitutive phenomenology, and specif­ically, it is a "systematic uncovering of the constituting intentional­ity." The discovery of its typical character of accomplishment reveals the hidden origins of the objective phenomena in transcen­dental subjectivity. The object itself is then ultimately only a "transcendental clue" for the analysis of its constitution in transcen­dental consciousness.38 In sharp contrast, the fundamental principie of phenomenology according to Merleau-Ponty is expressed as follows: "mon acte n'est pasoriginaire ou constituant, il est sollicite ou motive." ["my act is not primary or constituting, but called forth or motivated"]. 39 In place of the constitution of objectivities in pure transcendental consciousness, there emerges a motivation of the acts by the "in-itself" (en soi), which is neither a constitutive nora casual process but rather a "medium" between both. This is not the place to pursue this topic. Here the question is how these two contrasting conceptions relate to the spatial depth phenomenon.

The phenomenon as such is not the least denied in transcendental phenomenology. Husserl's methodological reductions reveal that he intends to interrogate the purely and simply given in order to illu­minate retrogressively the activities in which the given became es­tablished as a result. This requirement need not be rejected a limine; rather, we must ask how such a layered constitution is to be under­stood, and what kinds of ultimate givens are revealed. For Husserl these are the hyletic data, which are formed intentionally by the animating, sense-conferring conception that makes them objective.4o Yet it is repeatedly objected that these cannot be "phenomenologi­cally brought to light." However, this assertion is notan adequate counter to Husserl without further comment; he does not challenge the immediately given, unreflective attitude. Here, what it means to bring something to light phenomenologically has been shifted. In accordance with his conception of transcendental phenomenology,

38. E. Husserl, CM, § 41; FTL, § 97. 39. M. Merleau-Ponty (1), p. 305. 40. E. Husserl, Ideen I, § 41 and §§ 85-88. (In this context it is irrelevant

that for Husserl the hyletic data are not the ultimate givens; rather, they in turn are constituted in originary time-consciousness.)

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Husserl is not concerned with the immediately given but with the demonstration of the constitutive conditions for the possibility of the given. This demonstration moves at an entirely different level of phenomenal elements than those his critics presuppose.

If we do not concur with Husserl's conception, it is not beca use we agree with the arguments of his critics. Rather, the states of affairs require Husserl to bring to light the content of the individuallevels of constitution; he must offer us meaningful access to how the constitution of objects must be thought. It must be possible for the constitution of objects in pure consciousness to be insightfully traced by the consciousness which reflects on this constitution. Yet this demand is plainly not met by Husserl. The hyletic data and the sense-conferring acts of noeses belong to the really intrinsic [reelen] composition of the experiential stream, while the noematic object constituted by them as intentional is foreign to the acts. This constitutes a sphere of objects whose phenomenal transcendence is no longer maintained in its originality, but, rather, whose origin is transposed into the really intrinsic conditions of experience. Never­theless, such an object cannot be comprehended in this transcen­dence in terms of its assumed origins. And the aforesaid transcen­dence itself is incomprehensible when it does not mean a really [reell] existing object, but only a transcendence immanent in con­sciousness (incorporated into the really intrinsic immanence of experience). 41

It appears that no matter how insistently Husserl's distinction between hyle and morphe is maintained, Husserl himself could not adhere to it with the strictness and consistency demanded by his own theoretical strictures. Whenever Husserl purports to operate with "hyletic" data, it becomes obvious that, strictly speaking, these are already "animated" contents. Since according to him, they constitute themselves in originary time-consciousness and are next to one another and, one after the other, are unities, units, and multitudes, they are already categorical formations even in a Hus­serlian sense.

41. Husserl's use of the terms "transcendence" and "immanence" is confusing insofar as the intentional object is characterized by him with both terms in relation to the unreduced, to the reduced object, or to the really intrinsic composition of an act. We shall not trace this problem any further here, since it hardly concerns our theme. Concerning the critique of the constitution of objects of intuition in Husserl, see H. U. Assemissen; on this problem see especially pp. 66ff.

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As our suggestions indicate, we cannot follow Husserl on these points. If our reflections are correct, then the pre-spatial field is not maintainable phenomenologically. The reinterpretation of the visual field into an oriented visual space, the apperceptional shift from one-next-to-another and one-upon-another on the surface into one-after-the-other in space, which must necessarily be introduced in arder to attain the intuitive spatial depth from the merely quasi-extensive manifold, seems to be an intellectual construction whose individual stages of constitution are not clear and which is inappropriate for the facts of the case. (Indeed, there is an ap­perceptional change in a reverse sense, namely from space to field; yet this transformation is only possible on the basis of the originarily given space. It is a specific, abstractive achievement of the intuiting subject. See pp. 124 ff.). In contrast to Husserl, we accept spatial depth as primordial, as a given that can no longer be interrogated in terms of sorne hidden associative activities. 42

Merleau-Ponty also conceives space in this sense. It is understand­able that Merleau-Ponty develops his conception of space against Berkeley, and it is instructive that this occurs in his very first arguments. For Berkeley the impression of spatial depth results from the association of the frontal and lateral surfaces of a thing. Depth for Berkeley is nothing without this association; the depth surface of a physical body is in truth once again a frontal surface that is seen from the side. The impression of a thing as three-dimensional thus only appears when this lateral aspect is co-present in thought. This argument contains two presuppositions: on the one hand, the possibility of presenting a three-dimensional thing in a surface and, on the other, the inclusion of the corporeal capacity of locomotor movement. Both presuppositions are discussed by Merleau-Ponty. Through examples of perspectivally presented objects, he demon­strates his theory of the freedom and facticity of perception, from which it follows that "la profondeur nait sous mon regard, parcequ'il cherche a voir quelque chose." ["depth is born beneath my gaze because the latter tries to see something"].4 3 His point of departure is deliberately taken from represented figures. If the observer already sees a thing as a spatial thing in what is really a

42. An indication of the originality of spatial depth is present, according to C. Stumpf, in the fact that within an intuitable surface of any kind, planarity and curvature, already include the third dimension. (See also pp. 155 ff. of the present work.

43. M. Merleau-Ponty (1), pp. 394 and 304.

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two-dimensional, flat figure-an observation that is completely different from a genuine depth perception-then this appears to be possible only on the basis of a primordial intuition of depth. Meanwhile, one here overlooks a newly emerging problem. The distinctiveness of depth does not lie only in that it is the "most existential" of all dimensions (la plus existentielle), "locating" me befare the things (par Jaquel je suis situe devant les choses); it no less retains its exceptional position precisely because it can be abolished on the pictorial surface while retaining its depth-a surface is capable of presenting "space" that is given "in person" and is nonetheless illusory (see pp. 124 ff.).

Berkeley's second presupposition maintains that depth perception is nothing other than an associated lateral perception. Merleau­Ponty counters this notion with the objection that an incorporation of the view from the other side, and thus from the other corporeal subject, would transfer the space univocally polarized by corporeity into a two-sided, indifferent relationship. Space would lose its center and depth would lose its sense. This objection follows as a consequence of his fundamental position as ,an existentialist. He gives special consideration to spatial depth becausl;) the "mineness" of the world comes so obviously to presence in it: it is the depth of a space whose center is in me, and only in me, and which can never be a space for another. This is a philosophy which stresses the exclusiveness of the "mineness" of Dasein and of the world. Even if it does not discredit the entire domain of intersubjectivity, such a philosophy truncates it considerably in its existential relevance. In its conception of space, existentialism evidently has "no space" for the aspect of the other. Thus it must necessarily Overlook what is tenable in Berkeleyan thought, which, although veiled under an untenable psychology of association, is nevertheless the underlying basis.

If one lets this veil fall, then the core of the argument remains: The position of the other, and thus the other himself, is incorporated. To think that this incorporation would lead to a rejection, to a negation of depth given "in person," is the mistake of Berkeley. He overlooks the purely phenomenal state of affairs. In wandering about the thing, in the discovery of the initially given depth as width, there is no experience of deception, as though the three-dimensionally meant thing were suddenly to appear as a surface. Rather, it is seen in a new depth, and instead of abolishing depth the wandering continuously reaffirms it. The ordering in accordance with surface and depth is a pervasive structural moment of the space of intuition. That in

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wandering a thing will be grasped as three-dimensional, and that the apprehension of the thing does not fall apart into a sum of exten­sional moments, these make sense only because with the view of each side the thing is "intuited" in its entire volume. The single three-dimensional space is present in my depth dimensioned spate of intuition, as it is in that of any other corporeal subject. It is in no wise devalued simply because it is oriented toward me in my being here and now; the space of intuition is deployed for the intuiting consciousness in a multiplicity of possible aspects. It appears as a "perspectiva!" space because it presents the hic et nunc view of the one space for all corporeal subjects. What Leibniz has claimed for the relationship between the monads and the world is valid for space: they all reflect the universe in their own way; yet it lives in them all as a whole.

Given the state of affairs-where the subject intuits space perspectivally and conceives of himself as its center-the subject ceases to be the subject only of his space. The subject possesses a world that faces him, relative to his lived body, and a space exclusively his own., Yet he knows the world, he knows space as possible for others and theirs as possible for himself. This prefigures a particular kind of substitutability still completely immersed in the corporeal and stilllimited to the purely spatial meaning of "taking the place" of another while remaining with oneself. Here the subject in his corporeity already represents his intersubjectivity. It is impor­tant to note that this is not something that can be attributed to the individual subject nor can it be arbitrarily accepted or disregarded. The depth of space is not the "most existential" of the dimensions because the momentary location of my lived body provides a center for an otherwise all-encompassing and positionless space, but be­cause it affirms my participation in space as a corporeal subject­with an ever different view and fullness.

The phenomenon of depth is closely related to that of perspectiv­ity, and the observation of the latter may be newly illuminated and complemented by what has just been said. The previous limitation of our discussion to the things as being merely in front of and behind one another may now be dropped. It may be noted that beside this arder, they are also given in specific intersections, foreshortenings, and "disruptions." Thus we must touch upon the central Husserlian concept of spatial adumbration in the phenomenology of perception to the extent that it implies the phenomena of form and place as relative to the position of the observer.

The phenomenon of adumbration as a psychological, mathemati-

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cal, or artistic problem must be left out of consideration. We shall accept only the phenomenon of sensory intuition, from which all subsequent problems originate. We shall consider only the subject who is found in the center of perspectiva! space. He knows nothing of linear or central perspective, nothing of geometry and wave optics; he has the things for himself, and nothing further.

Being befare me, space offers itself in its depth; it is revealed in the phenomenon of covering and being covered. The spatial thing is not hidden solely because it is hidden by others but also because, in an individual intuition, it is never completely discovered. The thing as a whole is never given "in person." M y view is always an aspect and encounters the thing from specific "sides." Each individual intuition is essentially one-sided and is therefore inadequate in terms of what it can accomplish in grasping an object.44 The thing is accessible all-sidedly only through motion, whether the motion is a succession of one's own progressive movements ora spatial displacement of the thing. Thus its all-sidedness first unfolds in successively going through a series of positions.

The following problem then emerges: how can the successive aspects give us "the" thing, which shows itself in them in different modes of appearance and at the same time is present through them all as something identical? How can an all-sidedness result from many one-sidednesses? Can partial intuitions ·integrate themselves into a total intuition of the thing? These questions become important with regard to perspective: here the true form of the thing results not merely from the sum of individual views but also from disrupted side views; each successive one certainly complements the preced­ing one, but also falsifies it and changes it anew.

Seen more exactly, such a formulation of the problem assumes a specific standpoint and accordingly has a readily available point of departure for its solution. In reality it does not permit the emergence, through perspectiva! aspects, of an identical form of the thing, but rather already presupposes it: the true form is one that is given independently of position, unperspectivally, given in an intuition that is no longer sensory intuition, no longer directly related to the body. All changes experienced in the course of a series of aspects in the intuition of a thing must be seen as disruptions of the true form of the thing. That naive intuition does not progress from error to error but can speak justifiably of grasping "the" thing means that the

44. Concerning the concept of inadequacy, see E. Husserl, Ideen I, §§ 44 and 143.

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subject somehow has the true form of the thing. While traversing a manifold of places, the subject is not continually required to verify the thing in experiential "material" in each individual case, much less to generate it.

This notion contains two main theses and accordingly requires two kinds of explanations. (1) Each individual sensory intuition is supported by a unifying moment; the contents of the individual aspects are bound by the unity of sense of the "full thing of intuition." (2) The thing as a unity of sense, retaining its identity through the changing aspects, can be nothing other than the geomet­rical physical thing. It can be counted as the true thing, thus the sciences can determine the thing independently of position and can conceive of itas it is "in itself."

The grounding of the first thesis usually has recourse toa specific function of consciousness. Since Kant this has been introduced in various ways under the rubric of synthesis in arder to account for the constitution of unity. Using a house as an example, Kant has presented the conception of an identical thing as an achievement of "the power of imagination" in three syntheses.45 Apprehension must first collect the impressions into an "image," although the latter is in no position to bring the object to presentation. This requires the addition of reproduction, which connects the individual apprehen­sions one with the other. Finally it requires recognition, which orders the continua! successsion of reproduced images in a superordinate synthesis, connecting them sensibly one with the other, "in accordance with a rule," which allows the individual "images" to become "sides" of an identical object. While this is based on the strict Kantian distinction between matter and form on the one hand and between forms of intuition and categories on the other, the thing-conception is developed as something that is added to the individual images. This clearly states that the sensibly intuited thing is not given solely through sensibility but is partially achieved by understanding. Thus apprehension and recognition are called a synthesis in intuition, while reproduction is called a synthesis in concept.

This conception requires a variation in Husserl's theory of contin­uous syntheses.46 In distinction to membered syntheses in which discrete individual acts unify themselves into an articulated unity­such that this unity is to be grasped as a new act of a higher arder

45. I. Kant, Vol. III (Transcendental Logic, second analogy of experience). 46. E. Husserl, LU, Investigation 6, p. 47; Ideen 1, § 118.

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founded in partial intentions-the continuous syntheses are distin­guishable by a blending of various partial acts into one act; never­theless, this act cannot be conceived as one of a higher level. On the contrary, the unity of the object does not first emerge in a continuous course of individual intuitions through a specific synthetic act, or through a particular form of synthesis· added to the partial concep­tions; rather, the object is constituted as a "simple" unity without recourse to new kinds of intentional acts.

In phenomenological description this state of affairs reveals that each individual intuition contains the presence "for the time being" of an aspect, and that this aspect is known as one among many possible aspects. More precisely, this means that individual phe­nomenological components contain implications of further aspects in which the thing can appear. This possibility is "open" insofar as no given aspect can claim preeminence with respect to its value in presenting the thing; thus their series can proceed arbitrarily and endlessly, though repetitions may occur. Yet the content of the series is not completely indeterminate. In each individual intuition there are anticipations of further intuitions. Their indeterminacy means positively a determinability in the framework of a prescribed style.47

Thus in individual intuition we do not merely see forms but, rather, "sides" of a thing which "belong" to the thing. In arder to observe it all-sidedly, I turn and shift it and expect something from such movements: the view of the "back"-side. A m~re succession of perceived forms ora serial arrangement of discrete, individual acts would not induce identity. The total intuition, "thing," is that which continuously directs further experiences of the thing and motivates the movements relative to it.

We must add that this unity of intuition and movement is not only a motivational unity but, above all, a functional unity. The unified apprehension of a thing is not mediated fortuitously through corporeity but has its basis in it: only a motile corporeity is able to maintain, in the one-sidedness of a view, the índices of further aspects of the thing. These aspects become redeemable only in the actual dynamics of corporeity. The sense of such índices includes the notion that they are viable solely for a corporeal subject.

The second thesis regarding the objective form of the thing as geometrical-physical assumes a different signifigance. It is striking that on the basis of this thesis, the perspectiva! adumbrations are

47. Concerning the concept of open possibilities, see EU, § 21c; concern­ing the meaning of anticipation, see Ideen 1, § 44; CM, § 22; EU, § 8.

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conceivable only in privative modes of description: disruption, truncation, deformation. While these terms are correct for formal geometrical concepts, they are inappropriate for the actual states of affairs. Thus in all the subtlety of his analyses of perspectivity, Husserl could see in the adumbrations nothing more than the subjective modes of appearance of the thing; and he quite unfortu­nately chose the example of perspectiva! adumbrations to demon­strate his distinction between the object of intuition and the concept of intuition. In contrast to the physical thing as an object, the perspectiva! adumbrations must be mere data of sensations belong­ing to the really intrinsic [ree11] components of the perceptual experience. Despite the ever greater emphasis on the distinction between the adumbration and the adumbrated, Husserl failed to recognize that even the former is not just an experience of conscious­ness, but an adumbration of the experienced thing itself. Indeed, it is an experienced mode of appearance of the thing and yet, as such, is not a really intrinsic component of experience; otherwise that component itself would have to be spatial. But such a notion is obviously countersensical. That the perspectiva! adumbration is continuously given as relative to the subject in his corporeal now and here, that in addition the subject is intuitively co-conscious of it as relative, does not change the fact that these adumbrations belong to the thing itself-certainly not to the thing given in the categories of objective science but as it is given in my corporeally-centered space.48

The phenomenon of depth makes it clear that this space is not governed merely by factual affairs. 1 find myself in a situation with things, and it is no accident that the alleged deformations resist extension into depth. The perspective appears as a "disruption" only to the extent that it is taken one-sidedly "perspectivally" on the thing, such that the thing offers itself in only one "view"-thus

48. E. Husserl, Ideen I, § 41. In the Husserlian phenomenological sense, color and form-adumbrations would have to be attributed exclusively in the perceptual noema; as really intrinsic components of experience they are not to be found. That vision corresponds to a physiological process is not to be denied, but in the phenomenoiogical attitude this process is not perceived and thus it does not belong in an act phenomenology. Concerning a critique of Husserl's one-sided noetic analysis of the phenomenon of adumbration, see A. Gurwitsch. In contrast, in another place the noematic side gains its full force with Husserl; see EU § 16. (The special position of this work must not be overlooked; see L. Landgrebe's foreword.)

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disrupting itself and its sense. The perspective is not in the subject, nor is it merely something in the intuited thing. It is a mode of the insoluble and mutual relationship between a corporeal subject and an object. It guarantees the subject his constant connection with the world and makes his orientation in space possible.

These results suggest our agreement with sorne thinkers who defend the significance of perspective against unjustified claims of the natural sciences. Yet for us this significance is not sufficiently explicated, and another question belonging to the problem of per­spective is not answered: why is it that the intuition of a die is not just a formal identity in the changes of perspectiva! aspects, but precisely a perspective of a "cube with six squares"? Obviously the mere identity of the thing is insufficient here. If the continuous synthesis assures that the individual intuitions do not fragment into a sum of individual perceptions, it does not provide the basis for the fact that the form of the thing, as independent of position, is the form of this and of no other thing.49

One cannot dismiss this problem as a mere conundrum with the suggestion that the consciousness of the object already "has" the form of a die with its six squares. Although this state of affairs cannot be contested, the question still remains whether this mode of intuition can be made comprehensible.

With the perspectiva! constellation of the die there are six distin­guishable, frontally present views. Although they are still grasped as "sides" of a die, each individual given is reducible to a singular surface. The peculiarity of these six perspectives lies in their disappearance. What does this mean for the thing of intuition and its space, and what does it mean in particular for the "placement" of the subject? Perspectiva! adumbration depends on spatial depth. The exclusion of perspective means an exclusion of the depth of the thing; as a fact, the die is given without depth in the view mentioned. Thus there is a "perspectiva!" intuition of the thing in which depth is dissolved and the thing is given as sheer extension. The die no longer stands in a specific situation relative to the lived body, but rather is located on a surface standing perpendicularly to depth, which is annihilated. It has become pure extension.

49. We chose a die as a simple model, although the following reflections are valid in principie for any form of thing. Since at this time there is no discussion of geometry, the concept of "square" is meant only morpholog­ically. (For the distinction between morphological and mathematical prop­erties see pp. 184 of this work.)

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Where depth is dissolved, however, the subject ceases to be corporeally in space-the die in the conceptual sense "with six square surfaces" is in a space in which 1 cannot wander, in which 1 am no longer a lived body in a situation. It is a homogeneous space without standpoint, a space whose subject is externa! to it. But if the space of intuition is only the mode and manner in which the one objective space is appropriated monadologically, then the subject's intent to determine the "true" form of a thing-independently of position, as identical for all subjects-becomes comprehensible. Since a corporeal subject can only have an object through the mode of individual perspectiva! intuitions, then he can validly and justi­fiably address the perspectivally ordered world as the true world. The problems of perspectivity do not arrange themselves according to truth and falsity, but sol el y around the two poles of existence of the subject as lived body and as thing-positing consciousness.

§ 6. The Finitude of the Space of lntuition

The space of intuition is a finite manifold. The array befare me of things one behind the other is not unlimited but reaches a determi­nate end of the visual range. Although its boundary marks the line where spatial things cease for my here, it is not a boundary in the exact sense of the word; rather, it is phenomenally an indistinct and unclear region of dissolving and dissolved contours, supported by the co-consciousness of the "continuation" of space beyond them. It is intuitively possessed as a horizon. The phenomenon of the horizon is dual. The far horizon is to be distinguished from the near horizon or inner-horizon [innenhorizont] constituting a limitation within my space of intuition in the sense of a boundary of the appropriation and differentiation of spatial structures.

The inner-horizon is determined by what in physiological optics is called the optimal boundary of the capacity for resolutions; phenom­enally, it is the indistinct region of limitation that still maintains the presence of the spatial structures of a thing. It is thus a horizon of the near-thing within the total visual expanse. The sizes of things are located here; they diminish with increasing distance and reach the lowest limit at the far horizon. At the far horizon of the thing the interna! and externa! horizons coincide; they are dovetailed, as it were. Yet in a certain sense both horizons are independent one from the other; they vary, for example, in movement, and no firm rule of their coordination can be established. This is most clear in the various possibilities of displacement of the horizon.

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To begin with, each approach toward the things means displace­ment of the inner-horizon. With the progressive clarification and deciphering of the spatial structures of the object that occurs with each approach, there results a continuous displacement of the dissolving boundaries. Yet there is no necessary connection between this receding of the near horizon and a displacement of the far horizon. The displacement of the inner-horizon is not perceived as though it shifted toward the "outside" but rather as a receding toward the "inside"; the movement will not be experienced as a protrusion but more precisely as an intrusion. (We often speak of "deeper" intrusion into the "inside" of things when our basic concern is merely to grasp the spatial structures of the surface more precisely, in the course of which the inner structures frequently come into view, as for example in microscopy.)

The greater stability of the far horizon, in contrast to the inner­horizon, appears when one observes the forms of approaching that are equivalent to one's own movement of nearing. With respect to the achievement of clarity, the drawing near of the lived body is equivalent to the things "drawing near"; both equally achieve a displacement of the inner-horizon. The latter can be obtained even if the lived body is stationary, while the far horizon is displaceable only when one actually moves oneself. One's own movement thereby already assumes special status; thus we must consider such movement, as a specific characteristic of corporeity, with regard to the principie of relativity.

It is important to note that there is no such relativity between corporeity and the thing. While the movement of the thing can in principie be compensated for by the movement of one's own lived body, this equivalence of both courses of movements can be grasped only with respect to the thing itself, not with respect to the movement as such. The movement of corporeity is always absolute in the sense that its specific inner experience (as kinaesthesis, corporeal sensation, etc.) reveals the moving body immediately. A relativity of movement is possible only among moving things. 1t is here that the conception of the "assumed" object at rest attains its realization; in the space of intuition, such an assumption remains constantly bound to the specific conceptual senses of the natural context of things evident in the common example of a departing train and the "stationary" station. This assumption can first attain full indepe11dence and the form of a principie in a space that is no longer ontologically relative to a lived body.

The extension of the inner-horizon within the space of intuition is

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nothing other than what was previously called the pervasive structuration of space. Within the space of action there is already an essential function of implements, which accomplishes progressive differentiation and leads to an ever finer differentiation of the there and the yonder. While in this space the articulation reaches only up to the region, in the space of intuition it can be extended further by apparatus accomplishing further decomposition and clarification up to the phenomenal point. Even the latter is not a fixed size; as a "point," its value is relative to the means used for intuition. Within these means, what is ultimately given cannot be articulated any further; rather it vanishes abruptly. Yet precision technology regards the attained level of research and instrumentation of the phenome­nal point as itself merely a region again capable of further decom­position. Its stepwise advance into further levels of precision is motivated by the claim of progressive condensation of the topolog­ical nets all the way to the "ideal," i.e., really unreachable, limit case of a mathematical point.

Nonetheless it must not be overlooked that in the use of these implements there is a metabasis eis allo genos of intuition in the truest sense of the word. The apparatus of "magnification," seen from the side of the phenomenon, is indeed named appropriately. It aids the corporeal functions of vision and assists toward magnified clarity, exactness, and differentiation. Y et its signifigance and achievement is not exhausted in being a mere analogue of sensory intuition. Subsequent reflection must clarify the meaning of a replacement of the visual ray by a light ray. These two concepts must be strictly distinguished. The latter is used in the framework of science as a physical concept having its special position in a hypothetical-deductive system whose concepts can definitively be alloted various meanings. This is not the place to discuss these issues since even the concept of the visual ray requires clarification. After all, seeing as a "raying" event in lived space does not lead to an intuitive fulfillment of its meaning and cannot be phenomeno­logically brought to light. Nowhere are visual rays encountered, but always things in space (see further pp. 248 ff.).

New determinations are to be grasped at the far horizon. We must recall the elementary orientational directions and the corporeal movements corresponding to them. While the being who is oriented toward the far horizon is at rest, it captures and encompasses the alignment of things in the distance in one glance. In the limit case, above, below, right, and left shrink into a phenomenal point. The far horizon is that place or line at which the elementary extensions of

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things vanish. It is especially the leveling of the opposition between above and below that makes the horizon a "horizontal" and intrin­sically limitless, although finite, line for the disappearance of things. Depth and perspective vanish with it; the most distant things are seen as flat and when in movement are no longer grasped as moving.

Heidegger has pointed out that the finitude of the space of intuition is related to the finitude of Dasein. 5° The limitation of this space means at the same time limitation of one's own being as living and corporeal; the finite space is a space of a being who is and ends in time.

This corresponds to our own point of departure. The signifigance of the horizon-phenomenon is grasped primarily in the intuition of motion. In the nearing and distancing of things the boundaries of my space of intuition reveal themselves not only as what they are ontically, but also what they mean ontologically for m y being as a corporeal being. In the phenomenon of the horizon, spatial and temporal determinations pervade one another in a remarkable way. Horizon is the transition of things in intuition from the not-yet to the just-already, from the still-there to the no-longer. One may look atan auto speeding away and vanishing at the horizon-at the moment of "vanishing," is it a spatial nothing or a temporal no-longer? As it "emerges" at the horizon, is it suddenly grasped there or does it already appear? Even a more precise analysis would show that both determinations flow inseparably into each other, that here temporal and spatial moments are simply not separable.

The horizonality of the space of intuition leads to the recognition of its time-space character, and its limitation points to an original unity of space and time, although quite unnoticeably and almost imperceptibly. In intuiting it is the orientation toward the location of what is emerging that remains in the foreground. This is in confor­mity with our previous observations that the space-time as time­space is experienced to the extent that the experiencing subject is not aware of time itself explicitly. In the intuitive mode of being, the subject is in principie aware of time as objectively as of space. He apprehends the horizon as horizon of space only on the basis of a separation that has already taken place. Correlatively, he knows how to determine his finitude as being "in" time only through limitation in an objectively known time. The separation into two objective "forms of intuition" is itself constitutive for the horizon as time­space phenomenon.

50. M. Heidegger, § 23.

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Yet the far horizon is a boundary of space only in an inauthentic sense. What is decisive is that the experience of a boundary contaíns a consciousness of the "continuation" of space. The emergence and disappearance of things at the horizon is not perceived as destruc­tion and re-creation, but as coming from and disappearing into the "beyond." It is precisely this limitation that reveals the space of intuition notas something closed u pon itself but as a section of "the" space: as a lived space relative to the lived body of a subject who has itas lived space, in this arder and·articulation, only insofar as he as corporeal subject has already transcended it.

The horizonality of lived space, seen by Heidegger as an exis­tential of Dasein in its corporeal finitude, is an admission of a mode of being of Dasein in which Dasein is beyond its finitude. Dasein not only has a corporeally-centered space as its own, but rather is in addition objectively aware of the one space as space for all corporeal subjects.

§ 7. The Other in M y S pace of Intuition. Questions of Homogenization

In the space of intuition things are in front of me and for me; it is my space. Through its depth and horizonality it testifies that it belongs tome. Each of its things is a counter-pole to my corporeity here and now; it is a thing with me in a given situation.

But not only things reside in it. My discussion up to now, resting on a solipsistic corporeal subject, was an abstraction attainable only retrospectively. After all, there are "things" in it that I can intuit as having, like all other things, their location in my space, and yet which are fundamentally distinct from all things. I know directly that they too are a lived body. And lived body means a corporeity like me, and thus a center of its space of intuition distinct from mine and insurmountably el o sed to me in m y own "here."

How is it that the multitude of spaces of intuition results in the conception of one space known to us in common? How do the infinitely many aspects allow themselves to be integrated in the intuition of "the" space?

There is no calculus that could achieve this, nor need there be one. Rather, the questions just posed must be unmasked as absurd and the preceding deliberations as inconsistent.

They started with a multitude of perceivable lived bodies and their spaces and inquired into the emergence of "the" intuition of space. I conceived of the other as lived body, and thereby primarily in

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purely external spatial relationship tome; if my relationship to him were based only on sensory perception of his physical body, then the questions posed could be significan t. lt is different if 1 grasp the other not merely as a physical body but as the lived body of "another" who is also an "ego". Such knowledge is more originary and precedes mere sensory perception of his physical body. lndeed, 1 never know the other without corporeity, since he is corporeity. Yet to perceive what is "there" as corporeity 1 must already know what makes it into corporeity-in arder to be another being for me, it must also be an ego.

This "also" is not that of a spatial thing external tome. The other as an ego is never to be found in my visual field, just as 1 am not to be found in this way for him. lnsofar as we both know ourselves as egos, we stand in a relationship that is not merely non-spatial, but first allows me to say something about it and to raise questions about its spatiality. As an ego, 1 am completely determined through the other, as he is determined through me. Asan ego, 1 am not befare him or subsequent to him; we are both egos, since the one is only an ego for the other.

Thus the question concerning the one space acquires another accent. lt is not to be asked how the infinite multitude of distinct spaces of intuition results in the one, identical space for all lived. bodies; rather, what is to be discussed is how the singular space of intuition is constituted in whose center there is a being who is not only a motile corporeal being, but also specifically one who tran­scends his corporeity, and whose ego turns out to be conditioned by other egos.

With the emergence of the other there also appears a difference that separates the space of intuition from visual space. We observe a thing-the other from his location and 1 from mine. He points to it: this corporeal motion means more for me than a mere outstretching of one of his bodily members. With it I immediately understand the situation of the other, his orientation toward something grasped by him in a purely objective confrontation. He further points in arder to "direct" meto something. Thus in his corporeal posture he already represents not only his being with the things, but his being with me. A solipsistic subject does not point; pointing is understandable only with a corporeity that is a corporeity for others, and thus is capable of being the corporeity of anego. What is crucial is that he can point out "the" thing to me- a senseless undertaking if he meant only the visual thing, which is different for him than for me. The aim of pointing has already surpassed the latter toward the wholeness of

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the thing, grasping it all-sidedly and knowing that I too intuit it in the same manner. Our agreement does not aim at the visual thing but at the intuited thing in its meant all-sidedness.

The identity of our being, as a consciousness of an object, is rediscovered at the thing insofar as we are oriented toward it-not in the manner of vision but of intuition. This is a categorical intuition inexplicable solely in terms of corporeal functions; thus it can be maintained that in my space of intuition, as well as in that of all others, there is an intersubjective moment graspable on the thing as an identity. But the thing is not now meant as identical in the change of its perspectiva! adumbrations, but as one for all subjects like myself capable of categorical intuition. Naturally this has nothing to do with two identities. Our previous observations touched only u pon a single subject, as if to imply that the others were not there, as if they did not belong to my components of intuition-and yet although they were actually not present, they were nevertheless co-posited. (The "real solus ipse would be completely incapable of such co-positing; he could not conceive of an alien ego-corporeity in the full sense. Even if he were conscious of his own corporeal functions, he would have no awareness that his corporeity is for others, and thus he would inappropriately assume the title "ipse.") Insofar as I recognize the other corporeity as also being an ego, 1 simultaneously allot ita place in my space, a place that is not merely a there but also a here. The here thus loses its uniqueness; my here turns out to be a there for the others. M y space of intuition is indeed mine by virtue of my corporeal being, but for me it is any space by virtue of my consciousness, and thus a space in which it is possible to have a harmonious experiential context with others.

The guarantee of such harmony is already present in the corporeal. The here of the other is not simply his here; at any moment it can become my here by virtue of my capacity for movement. Phenome­nally speaking, my movement contains the loss of a center; the movement into spatial depth, revealing in the succession of things their contiguity, announces the homogenization and thus its loss of limits. Its horizon is a horizon relative to my immobile corporeity­its retreat is my advance, and thus the "continuation" of space is nothing other than my movement.

Yet my movement creates and recreates a new center and a new depth. In this sense, the intersubjective space as a whole cannot be unfolded, since the factual movement extends only across a deter­minate part of space; due to the finitude of the corporeal being, an endless movement is impossible.

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The problem emergÍng here has already been noted by Scheler. Scheler, too, seeks to explícate space as a space of movement. For him space is not at all an object of any intuition-thus Scheler criticizes Kant for grasping space as a purely perceptual formation­but solely a vital achievement of a motile body, of a being of drives "having nothing to do with reason and intersubjectivity." Scheler poses the following question: how can the limited possibilities of movement of a finite corporeal being account for the understanding of space as infinite? The answer is that space as such remains merely a "possibility" of a capacity for self-movement, while its actuality is descriptively introduced by a consciousness capable of conceptual­izing space. Thus the "immense paradox" of the one, substantial, empty space comes to the fore; a space that the subject, on the basis of dissatisfied dynamic drives and his unlived and unlivable possi­bilities of movement, "projects outward. "51

With this point of departure Scheler has undoubtedly touched a nerve of the entire problem of space. He has discovered a defect in the traditional theory of space. The latter was accessible only to a philosophy oriented from a pure consciousness. Space is space of a being who is irrevocably its own lived body-it is a space of movement in two senses: a space for, as well as constituted through the movement of a corporeal being; this was shown in the preceding in vestigation.

Nevertheless, Scheler's solution remains unsatisfactory in various respects. First, it must be asked whether the unlived, constantly present overabundance of drives, the held-back vital powers, con­stituting, according to Scheler, the difference between humans and animals, is adequate for the comprehension of infinite space. It seems that the ego-corporeity must contain an entirely different and moved structure of drives: they must be entirely different from merely vital activities, and yet they must be understandable in terms of the ego-centric, the corporeal subject. His being as a conscious being must at least pro vide the conditio sine qua non for that specific experience of movement allowing us to seek the foundations for the consciousness of endless space not in the mere fact of my capacity for self movement, but in a consciousness of this ability. That such a consciousness should be liable to succumbing to a deception, · confusing a mere "possibility" for "actality," constitutes a thesis to which Scheler must revert without undertaking any justification for such a thesis-which would have required the precise clarification

51. M. Scheler (3), pp. 295ff.

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of the meaning assigned to the concepts of possibility and actuality in the sense meant here.52

Secondly, it remains completely incomprehensible how a projec­tion of space, conceived by Scheler as a purely vital activity, ought to bring about an identical space-consciousness for all ego­corporeities. It is also difficult to discover how a qualitatively singular experience of overabundance could become conceivable for con­sciousness as a quantifiable and mathematizable emptiness.

Finally, Scheler overlooks the problem of the topological structure of this space. The presupposition of the open-infinite manifold, accepted by Scheler without question despite the fact that it is most problematic, does not follow convincingly or univocally from his own basic conception. If the reservation mentioned above were removed, it would be possible to agree to an endless-closed structure of space curving back upon itself, particularly when Scheler repeat­edly characterizes corporeal movement as periodic and rhythmic. Yet the space constituting consciousness cannot simply, arbitrarily decide to represent space in one or the other manner. On the other hand, consciousness cannot perceive the one open-endless space as necessary: indeed, it is precisely here that it cannot be maintained as necessary, since it cannot be presented through any specific complex of phenomena. It is impossible to derive an intentional content of the historicity of consciousness from the history-less constants of vital structures.

In any case, it must be pointed out that we hardly have sufficient knowledge about the history of the unreflective experience of space. What unfolds as a history of the problem of space is the history of theories of space, but not of space consciousness itself. From the beginning, these theories were by no means articulated completely in thoroughly phenomenological description; they remained pre­dominantly speculative orientations toward cosmic-metaphysical questions. In turn, however, the theories were the conceptual sedimentation of a spatial conception, which, although not created directly from the natural intuition of space, was not in an open contradiction with it. Thus the theories mediately present the history of space intuition.53

52. In addition see pp. 157ff. of the present work. 53. Phenornenology has not yet recognized the task of investigating the

philosophical past as historically problernatic in terms of its own rnethod­ological style. We shall return to this question (pp. 157 ff.).

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§ 8. Open Questions.

The problems just alluded to lead beyond the region of what can be shown descriptively into the domain of the metaphysics of space. To pursue them would be the task of broader discussion. At this juncture only a few of the remaining open questions ought to be raised. We cannot simply blur over the specific difficulties that have appeared in the last discussion.

They discovered the subject to be in a double attitude toward the space of intuition: on the one hand, the corporeal comportment "in" it, and on the other, the theoretical attitude intuiting "the" space objectively.

While the space of intuition turned out to be co-determined by, and not detachable from, objective space, the converse was not true. Thus the question remains whether the objective conception of space, which we assumed to be contingent datum of consciousness as such, is capable of being traced further back, or whether it must be taken asan ultimate factor, asan irreducible structural characteristic of "the" space-consciousness in its own specific character. Viewed in terms of space, we must ask whether space, as a "pure form of intuition," is to be related to the space of movement, or whether its basis lies rather in the conception of space as space of movement.

To recognize space as both, to reduce it to an unconnected "both/and" of the peaceful coexistence of two distinct "aspects" would evidently run counter to the meaning of a philosophical theory of space. But to allow one of the two to be solely valid would be equal to the truncation of the perceivable phenomena discovered in the framework of our analysis. The conception of space as a mere form of intuition is powerless to explain the interrelationship of corporeal movement with space, specifically with regard to the plethora of individual phenomena present in the observation of various corporeal modes of comportment. In contrast, Scheler is exposed to the danger of a reflexive circle. His projection theory attempts to derive the objective consciousness of space from a non-conscious structure of drives, which, in arder to be appropriate for the constitution of "the" space, must have assumed non-vital, intersubjective moments of consciousness.

These doubts cannot be directed against our investigation, since it did not undertake any attempt to deduce the objective space­consciousness from previous achievements of the corporeal subject. Nevertheless, it must defend itself against another critique. Again and again our attention has been drawn to the fact that the investi-

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gation has traced the path of a subject through differing spaces, which the subject has "always already" traversed. The manner in which space is possessed at a given level of his comportment is decisively co-determined by his objective consciousness of space. Thus our investigation must take more seriously the charge that the conception of space as a space of movement of a corporeal being has been surreptitiously attained after all, that in reality space for this being remains nothing other than a form of intuition of conscious­ness.

Our investigation has not been able to bring together these two factors. The answer to this question lies beyond phenomenological demonstration. After briefly touching upon the modally distinct sensory spaces in the following chapter, the next section deliberately abandons the phenomenological domain in the narrower sense without negating it. The immediate aim is to devise a theoretical foundation capable of elucidating the phenomena dealt with until now.

Chapter Four

Modally Distinct Sensory Spaces

§ 1. Visual Space

It must be emphasized at the outset that the sensory spaces to be considered here will not be developed as fully as was the space of intuition. The latter turned out to be relative to a specific, self­sufficient mode of being of the intuiting subject. In contrast, the modally distinct sensory spaces are present only as nonself­sufficient. In arder to reveal their structure, it is necessary to assume a particular abstractive view. The subsequent limitation to visual and tactile spaces does not constitute a prejudgement concerning the existence or nonexistence of other sensory spaces-such as an audial space-but rather serves the explicit purpose of clarifying the space of intuition. The two designated spaces relate to it as two abstract, nonself-sufficient moments of a concrete whole.

Yet in no wise does this mean a construction of this whole from its parts or its breaking up into parts. Rather, we are solely concerned with abstractively seizing upon all those essential traits of both sensory modalities, which as a sum never comprise sensory intu­ition, although they can shed sorne light on the latter.

The separation of visual space from the space of intuition encoun­ters sorne difficulties. Their difference does not lie in differing objective data, but only in their differing apprehensional sense. As a thing of intuition, the thing is accepted as "full." Though it is constantly perceived one-sidedly in adumbrations, it includes ev­erything co-given with it, and is grasped as a thing. It is only in this manner that it can correlate to a possible world of the subject. In concrete intuition, the merely perceived is already and always surpassed toward the whole of the object as a specific unity of sense.

The abstraction that must be performed in arder to move from the intuited thing to the visual thing requires the bracketing of the co-perception and the reduction of the complete thing of intuition to what is visible "on" it. It is to be maintained that the visible is meant

120

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qua "visible on the thing." Strictly speaking, the co-perception of the thing is not completely annulled but only consciously separated as such from what is genuinely perceived.54 An apprehended residuum of the thing as a whole remains present and available in the view of an object at rest. The movement of the thing elicits the apprehension of the changing visual contents as "sides of the thing." Indeed, the apprehension of the "side of a thing" is pushed into the background by the intuitive attitude and its new visual content. Thus although this apprehension does not completely break down, it is present only marginally. While the apprehension of the moving thing of intuition offers merely the other side of the same object, visually speaking there is a clear consciousness of the complete otherness of the visual object. What remains of the object and its space in pure vision after thus bracketing the co-perceived?

First, there remain sorne of the characteristics analyzed in the space of intuition, e.g., its peculiarity as a phenomenal manifold of points-which have basically been revealed with the phenomenon of the visual thing (pp. 85ff.)-as well as those properties that are related to the finitud e of the space of intuition. But they clearly show that the space of intuition is not exhausted in being mere visual space. In any case, the limit of the space of intuition is the same as that of the visual space; the domain of the intuited things extends no further than that of the visual things. Yet in intuition it did not genuinely appear as a limit, but as a horizon in the more precise sense that the latter is perceived as the region of the not-yet and the no-longer beyond which space "continues." This co-consciousness of the continuation motivates the anticipation of the new experien­tial context and determines the sectional character of the space of intuition.

It is otherwise in visual space. If one remains strictly with what is given in vision alone, then nothing remains of the continuity of space. Visual space does not have a horizon but a limit, a region of the literal disappearance of the thing. The visual thing as such is either given or not given "in person"; it does not, however, have the not-yet and the no-longer. The same holds for any perspectiva! adumbrations. Strictly speaking this concept too belongs to the space of intuition; the visual thing does not hove adumbrations, rather it is adumbration. That is why with every position and after every movement the visual thing is a different one; it appears differently

54. The complete annulment of co-operation would lead to the annihila­tion of spatiality. Compare the analyses of the visual field, pp. 124 ff.

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relative to the position of the observer. Thus all discussions of the phenomena appropriate to the clarification of the monadological traits of the space of intuition were, strictly speaking, oriented toward things as visual things.

That the distinction between the two "spaces" is slight makes it understandable that they are usually identified. The lack of their differentiation constitutes the ground for the controversy among the opponents dealing with the "euclidean nature" of the space of intuition; the numbers in the opposing camps as well as their apparent power of proofs are somewhat equal. The concept of "euclidean nature" is associated with the existence of parallels in the intuited (morphological) sense of the orientation in the same direction, i.e., non-intersection of two straight lines, as well as the constancy of an object through specific movements, rotations, trans­lations, mirrorings; this means that from a space determinable as euclidean one expects the same intuitive fulfillment of those require­ments demanded of euclidean geometry as a special theory of invariants. Obviously, for the visual space these requirements are not fulfilled; it does not have parallels in all directions, and the shape of the thing and its size are functions of the location.ss In contrast, a euclidean nature can be meaningfully attributed to the space of intuition insofar as the rift between what is itself given and what is co-given vanishes and the objects are "intuited" all-sidedly in any motion. Parallelism is also guaranteed here. The controversia! question of railroad tracks stretching into depth is answered: they are visual things, two strips approaching one another, while as objects of intuition, they "run" parallel-they stretch further, be­yond the horizon, a situation which is already implied in the conception of "railroad tracks."

Preliminarily, this is the sale justification for the claim that despite its centricity, the space of intuition is euclidean-more precisely, that the same metrics can be applied to it as to the homogeneous space of objects. It is of course necessary to show that this is the euclidean metrics. Here we wish only to suggest that the solution to the problem depends on a precise formulation of the question andona clarified terminology.

55. The change of size of visual things is not constant; in a specific region of nearness, there appears the remarkable phenomenon of size constancy. Thus visual space becomes nonhomogeneous in a dual sense: it is not only centered on the point singled out as my here, but it is also articulated concentrically into various steps of distinctive distances.

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A more problematic question is whether the reduction of the full thing of intuition to its purely visual givenness accomplished here does not obliterate space altogether and destroy its depth which is implied in the things being in front of and behind one another. Such would be the case if this ordering of the things were not retained for the visual objects. If one were to orient oneself toward a specific thing-constellation and perform the reduction under discussion, and if sorne change were to occur in the region of the visual thing, one would find that despite the appearance of a formal deformation of the visual thing, one would still apprehend the change as a movement and not merely as a transformation of forms on a surface.

The perception of movement is an accomplishment that cannot be carried out solely through the visual function. The sensory basis of constitution of something seen as moving cannot be provided solely through vision; rather, it requires the complementary activity of touch.

The functional unity of vision and touch has been stressed for a long time. The perception of a thing was already conceived by Locke and Berkeley as originating from the associative connection of visual and tactile data. Berkeley in particular repeatedly stresses the strict distinction between the visual and the tactile function as well as the fundamental role of the tactile sense for the construction of the perceptual world. For him the visual is only a sign for the tactile, and touch alone "suggests" to the rest of the senses their objectivity.56

Berkeley's influence is found in Husserl's theory of aesthetic syn­thesis. Underlying the genuine thing-giving synthesis, it has, according to him, the function of uniting the tactile with the visual data into an appearing schema.57 Just like Berkeley, Husserl also maintains a strict separation between the visual and the tactile thing, to such an extent that he attributes self-sufficiency to the purely visual thing (spatial phantom) rather than to the tactile thing.

Yet it must be noted that even where the discussion focuses purely on the visual thing, no abstraction can be made from its materiality. It is a mistake to think that what is not and cannot be given in "pure" vision must completely cease as a datum on the visual thing. Insofar . as a thing is conceived in any sense as thing-like, its materiality is eo ipso co-given phenomenally in a "pure" visual datum such as color. After all, the color is not merely a color medium externally attached to the thing; rather, in its other mode of appearance-shining,

56. G. Berkeley, §§ 117-19, 125. 57. E. Husserl, Ideen II, § 10.

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glittering, shimmering, transparent, dull-it betrays something of the material structure of the thing. But the materiality of the object as such is founded sensibly in touch.sa

Furthermore, the activity of the tactile sense must already be presupposed if the vision of movement is to be possible at all. The criterion for the vision of movement líes in that the moved can in principie be "followed by the eyes." Yet such following is insuffi­ciently understood if it is apure movement of the glance; this as such must also be present to the body, i.e., it must be experienced "from within" as oculomotoric, and it must be able to be made evident as motoric. This typical inner experience of a movement is, however, necessarily accompanied by tactile sensations. The same is true of more extensive movement of things, which can be compensated for through the movement of the body, thus from within, kinaestheti­cally. The indicator for a movement is therefore ultimately the rest or movement of the lived body, or its limbs, which can only be experienced as the rest or movement of a "body" capable of tactility. A corporeity capable of vision but not of tactility would know no movement, not even its own, nor would it be cognizant of a visual space; the latter remains ontologically relative to a corporeal unity of visual and tactile functions.

§ 2. The Visual Field

It is not contradictory that through a specific change of appercep­tion the spatial manifold of visual things can be transformed into a field conceived as purely extensive. The moving visual things rigidify into a merely superficial arrangement of colors and forms, and their depth is reinterpreted as a mere next-to-one-another and one-upon-the-other of figures with specific intersections. A being­one-behind-the-other, still graspable in visual space by virtue of specific displacements, merges here into a purely extensive, merely figural change.

The bracketing of perceived movement abolishes all residuum of "thing" -apprehension and thus also the conception of emptiness as "leeway" for the movement of things. Surface confronts surface, color touches color without any intervals; the doubly reduced and thus disciplined vision ultimately no longer perceives color as a mode of appearance "on" the visual thing, but merely as extension. Gleam, shimmer, etc., "on the thing" are transformed into an extended "spot" of color.

58. W. Schapp, pp. 19 ff., for additional aspects see also D. Katz (2).

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This level of abstractive vision is required for a pictorial presentation of the painterly spatial perspective, since the latter is precisely a depiction of three-dimensional space on the pictorial surface. Seen strictly phenomenologically, this does not have to do with a presentation of space but with a transposition of the abstractively attained visual field onto the painted surface. It is no accident that the full mastery of the painting technique of the "intervals," i.e., of the empty space in between things, appears quite late in the historical evolution of painting. It not only requires the "eye of the artist," but also a vision in the sense we have sketched out here. In addition, it is necessary to have an aesthetic attitude that is first attained from natural intuition through a specific reductive undertaking. Thus from the painted figures on the pictorial surface, the three-dimensionally apperceived object imme­diately "leaps to the eyes" of an impartial observer, as long as the laws of linear and painterly perspective are maintained within the broadest limits. In any case, it is also possible here to remain in the necessary attitude toward the pictorial presentation itself or to choose between or shift among various possible attitudes, and thus to call forth the impressions in accordance with the requirements of the so-called originary experience.

A more detailed exposition of this problem of presentation of space would lead us too far astray from the true aim of our in­vestigation.59 It must be kept in mind that visual space phe­nomenologically precedes the visual field and that the former in its turn is conceivable only as an abstractive moment of the space of intuition. Since it is not a "purely" visual space, but is constituted through vision and tactility, it already appears as a relatively complex structure. A further step of abstraction leads to the visual field, which is difficult to describe phenomenologically. It is occasionally grasped when the possibilities of coincidence of visual and tactile data come into view. Y et a separation of the originary unity of coincidence of the visual and tactile thing is phenomeno­logically problematic. Even pure surfaces require the tactile function in arder to appear visually as surfaces. Where the surface is not really touched, but in a transferred sense is "touched" with the eye, the tactile sense is still required, just as the kinaesthesis of a lived body capable of tactility must be present in the perception of movement. Thus a further question appears: whether and to what extent touch in its turn is to be conceived as a function that presents

59. On the pictorial presentation of space, see E. Stroker.

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objects and space, and particularly whether it is constitutive of an independent tactile space.

§ 3. The Problem of Tactile Space

It is an old insight that the tactile sense plays a constitutive role in the cognition of objects. Kant points out that it is the sale directly perceiving sense, and thus the most important and instructive. Subsequently, Palagyi emphasizes its fundamental role for all per­ceptual knowledge. In the same sense Katz, who is the first to offer a precise psychological analysis of touch, ascribes to it a decisive character of reality and primacy for all sensory knowledge.6o

This remarkable activity of the tactile sense still does not guaran­tee the existence of an independent tactile space. This question can be decided only through an exact investigation of the objectivity given in tactility. Precise analyses must be undertaken concerning the manner and mode in which tactility and its presumed objectivity are given with its respect to function as a sense and its unique position among the rest of the senses.

Katz had already recognized that tactile phenomena are bipolar insofar as they result from a subjective ( corporeally-related) and an objective (oriented toward the objective characteristics of things) component. As a result, the terms tactile perception as well as tactile sensation have a justifiable meaning. In the scale of sensory func­tions, the tactile sense assumes an intermediary position between sensory impressions of a state or condition and sensory-perceptions yielding objectivity.

In accordance with this bipolarity, our investigation will have two directions. On the one hand, it will survey the mani:qfold of locations of a tactile there with regard to specific spatial structures, and on the other, it will reveal the specific manner and mode in which corporeity is given tactility. The latter in its turn is twofold: in tactile sensation and in kinaesthesis. This complex problem consti­tutes the specific difficulties of the following investigation. The problem inheres in the essence of the state of affairs. Although our observations are concerned primarily with the question of tactility and the possible discovery of tactile space, these distinct aspects

60. I. Kant, Vol. VII (Anthropology, Part One, § 17); M. Palágyi (2), 3rd Lecture; D. Katz (1), pp. 255 ff.; additions to D. Katz in R. Pauli's work.

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cannot be strictly separated in arder to show their subsequent intersections. B1

The existence of a tactile space long remained indubitable. Yet it is characteristic of its particular problems that its first thematic investigation also led to its denial.

The first answer to the question posed here did not come from the area of philosophy, but from that of pathology. In the investigation of brain damage, the neurologists Gelb and Goldstein were led to the view that "spatial characteristics do not appear through the qualities mediated by the tactile sense .... Spatiality enters the tactile domain only through visual representations, i.e., there is only one genuine visual space."62 Their investígation, which gained a great deal of attention in neuropathology, became a catalyst for the revision of basic principies acquired in other ways. For example, using their research Schilder placed his conception of a "body schema" on a new basis. Their investigations are worthy of mention on two grounds. First, the complex of problems relating to tactile space was confronted for the first time; second, they share the fate of many initial works: with the development of a new problem, they also reveal their own felt lack of a solution. Thus in the work mentioned, it is not clear what is to be really understood by "tactile space," specifically when it is taken as something that is not on hand. What Goldstein and Gelb bring out is the inability of a patient to localize tactile impressions of his own body. Yet nothing is thereby decided concerning a tactile space. Above all, the method used by the two researchers to attain their results must appear dubious. If their work did not propase anything more than an attempt to describe and explain mere behavior, remaining an internal concern of neurology, then it would be justified, and philosophy would have no right to assume a critical stance to it. But their case is expressly different: from individual analyses of extremely complex pathological find­ings, they make a claim that purports to be valid for the tactile phenomenon as such, and thus pretends to be an essential proposi­tion in a phenomenological sense. How little both investigators are clear on their own methodological stance already follows from the view, for example, that the execution of arbitrary movements by the patient is similar to that of a normal child. Besides, the exposition moves in various circular conclusions. The most significant one is

61. The next section will investigate more closely the aspect of touch as sensation and the constitution of the givenness of one's own body in touch.

62. K. Goldstein and A. Gelb (2), p. 73.

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found in the localization hypothesis: the inability of the patient to localize tactile impressions on his own body is reduced to the lack of optical perceptual capacity, and from there it is concluded that spatiality does not pertain to the manifold of tactile data themselves, but rather that spatiality is constituted solely by the visual sense; yet it is assumed in turn that the latter is mediated by the kinaesthetic residua awakened through the tactile impressions.

Concerning the necessity and universality of the optical capacity of representation for tactile localization, it has been conclusively shown by experiments with persons who were born blind that the assertions of Goldstein and Gelb are contradicted by facts. 6a In any case, it is assumed here that the patient lacks all optical representa­tions and that for localization he is reduced purely to tactile data. Other experimental researchers working in psychology also reject the question of tactile space.B4

We mention these individual scientific results because they are significant for the present problem. If the question concerning an independent tactile space is raised, then it can only be approached through an abstractive separation of vision from touch. The behavior of the blind provides an appropriate phenomenal basis for the investigation. The anomaly, particularly of those born blind, is such that they have already bracketed vision in a quasi-"natural" way; in normal behavior such bracketing is a specific methodologi­cal arrangement.

Nevertheless, the rejection of the tactile space from the pathological side does not univocally mitigate against the existence of a tactile space. In the evaluation of pathological discoveries, one is not al ways consciously clear that the observer confronting the patient finds him­self facing a radically alien consciousness. He can only describe the

63. See S. Monat-Grundland, who repeated the experiments of Gelb and Goldstein on fourteen persons born blind. The work is published only as a fragment and leaves the basic question open.

64. According to J. Wittmann, the blind person who does not move feels "completely space-free" and "in no manner related to space"; he experi­ences himself "as it were only as a thought function" (p. 433). Even his own movements are not given for him as a change of place. Wittmann's student W. Ahlmann reaches the result, on the basis of self-analysis, that the blind person constructs his surrounding world purely temporally on the basis of touch. A. van Senden, who gathered more than sixty reports from persons with cataract operations, thought he could claim that the temporal schemata of the blind have no spatiality. The spatial concept of the patients is to be attained exclusively intellectually and without any sensory foundation.

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patient's behavior with the aid of a privation, with the assumption of a "breakdown" of normal functions; as a point of departure for de­scriptive pathological work, this private approach is debatable in contemporary research.65 Again, it must be kept in mind that what is presented here in "pure" description is always and ineradicably an already interpreted comportment, especially when such a description unavoidably in vol ves the observer's specific conception of space. The complex of unavowed presuppositions leads to unclarity concerning what one here simply rejects as "tactile space." lt is remarkable that the researchers cited never explicitly raise the question of what is here to be genuinely understood by tactile space. Obviously, abnormal behavior permits two entirely distinct aspects of observation not to be confused with the other. On the one hand, the patient is seen by the doctor as located in the space of intuition pre-given to the doctor himself. The doctor asks how the patient finds his way in it despite a breakdown of the visual function, and how the space of intuition is presented in the latter's tactile space. If one asks about "tactile space" in terms of this aspect, then it is given as nothing other than the space of intuition filled with tactile material. The nonexistence of such a "tactile space" could then only mean a lack, or rather, a failure of a spatial structural order determined by visual space, which is alien in kind to the "pure" tactile manifold.

From this we can conjure up the affirmation as well as the negation of an independent tactile space. There is a difference as to whether one raises the question concerning the structural nature of a manifold of positions merely accessible to touch in an otherwise pre-given space (this question is extremely important for the psy­chology of the blind; after all, it is of utmost importance for their orientation), or whether one approaches this problem from an entirely different aspect: the possibility of a spatiality constituted primarily in tactility itself". This would have to correspond to a manifold of places recognizable purely for its own sake as tactile space, furnished with its own structural laws and existing without the participation of the visual sense; in fact, it would not require the

65. Contemporary psychiatry-especially where it has assumed holistic psychological aspects-wants to replace the concept of the "breakdown" of the function with that of the "change" of the function, i.e., it regards the first not only as a pathalogical datum but also as a basis of motivation for the consolidation of the remaining functions into a new gestalt of activity. Obviously this new conception of the "phenomenon" is on a new basis and demands its critical revision.

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visual sense at all. Indeed, in its own right it would be able to cast sorne "light" on the specific structure of visual space.

The last point obviously leads to the genuine problem of a tactile space. Its phenomenological treatment is hindered by the pre-given interlacing of the tactile and the visual thing in all perception. At first glance and from an appropriate viewpoint, it seems that it is possible to establish the independent and essential features of touch; it is possible to conceive of a touching corporeity that does not require the activities of additional sensory functions for its opera­tions. Yet such a corporeity is conceivable only for, and in terms of a being who comprises a functional unity of tactility and vision. The attempt to extricate the uniqueness of touch reductively from the total structural activity of space-constituting sensory functions is bound to the restriction that it can only be undertaken in terms of a being who is simultaneously visual and tactile.

In touch the world is given in the primordial sense as standing in opposition. Touch touches something, and indeed something mate­rial, something thing-like, which it encounters and which simulta­neously confronts it. This leads to the experience of the thing's resistance. The tactile experience is not constituted by something that is encountered "there" and "yonder," but rather by the direct presence of something resistant. First, resistance is experienced in a specific orientation: the object resists "at a specific place," be it "there" on sorne objectivity orbe it "here" on one's own body-or on both at the same time in the characteristic double givenness of touch. In contrast, other experienced content requires a correlation with a specific directing of attention. But the primary and exclusive con­cern here is with the tactile manifold of "theres." Indeed, this manifold is never detached from the lived body, although this corporeal contact can be excluded from consideration and touch can be investigated exclusively in terms of any tactile-spatial arder.

Something touched offers itself to touch primarily as a resistance without intervals; there is no place on it that is not in principie accessible to touch. This does not preclude that the touchable something can have conditions under which touching is impossi­ble--perhaps too high a temperature or an electric charge. This conditionally determined untouchability is merely factual, given in the specific empirical case, without constituting the limit in princi­pie of the tactile activity. The suggested absence of lacunae in the touched thing is present in touch even with optically discontinuous structures (wire, lattice, etc.), since what here constitutes the "lacu­nae" even in touch is precisely the continuously touchable connect-

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edness of the edge.66 This is not to argue that the thing must necessarily be touched in continuity in arder to be present as continuous. Thus the touching of a surface with spread fingers reveals an experience not only of a multitude of discrete, singular tactile positions, but also of the surface as a continuous surface. The continuity of the tactile material-we use this term provisionally to exclude tactile formal structures without wishing to lend matter complete independence-is the first structural characteristic of something tactile.

Nonetheless, the continuity examplified above is precisely what can be contested. Obviously any co-experience of the total surface in stigmatic touching can be traced to the in te grating role of a spatial representation having an optical origin.67 Certainly in normal comportment the visual sense is of great significance. With the exclusion of vision from the full complement of functioning senses, then roughly speaking, the touching body can touch a surface only stigmatically, and yet can have the experience of its connectedness; the careful touch of the blind also seems to suggest that one can speak of the connectedness of tactile material only through the incorporation of vision. Yet the last observation speaks in truth for an independent tactile continuum-the blind person does not touch point for point because he doubts the continuity, but because he wishes to inform himself about the size, form, and constitution of an object whose continuity is already presupposed. But the other observation, namely that even persons born blind are not continuously engaged in touch for their orientation, does not speak unconditionally for a tactile continuum. Pathological observations are of little help in this question; clinically speaking, it is still controversia! whether the disturbed are or are not in possession of a kind of representational space. Nevertheless, it can be shown along other paths that a tactile manifold of locations is constantly a continuum-indeed, that the tactile continuum presents the originary form of any possible kind of spatial continuity whatso-

66. From the above discussion it is furthermore understandable that the results of radiology showing the discontinuous structure of matter lie at another level of discourse. It has nothing to do with the continuity of the phenomenal objects.

67. Thus argues, for example, H. Lassen (1), pp. 58 ff. For him all tactile impressions are "embedded" in a "continuum of representational space." Yet it seems to us that apart from Lassen's fundamental analysis, this problem is not treated sufficiently far as a basis for rejecting a tactile space.

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ever. Touch achieves this by virtue of its specific relationship to the touching corporeity.

This cannot be sufficiently understood as a correspondence of each tactile impression with a tactile sensation which, as a whole, are linked together in the unity of the lived body. What is decisive here is that the continuity of the tactile material is not given simultaneously, as with visual material, but successively and, indeed, in the succession of bodily movement. In this case the unity of sensory activity and bodily movement is different and more intimate than in the remaining sensory-functions. What is valid for the ocularmotoric in vision is correspondingly valid for the movement of bodily members in touch, yet this further has an entirely different and more important meaning for the immediate function of touch itself. In a completely originary sense, the tactile sense is a motile sense. Something tactile is present as touchable only and exclusively in actual corporeal movement. The arder of tactile positions as tactile is exclusively an arder of succession. It is given as such only in relation to the successive continuity of the movement of corporeal members experienced from "within" not only as a continuous series of tactile sensations, but also kin­aesthetically as a continuous succession of corporeal phases of movement. Here the externa! succession is simultaneously tied to a successive inner continuity. But this "inwardness," as only one particular mode of experience of one's own corporeal movement, is always a continuous series of individual phases, resulting in the property of continuity of that which is touched.

Thus it is apparently incorrect to derive the continuity of the tactile material from that of the visual material. Rather, it is obvious that the continuity of movement constituted in touch also founds the continuity of the visual material.

The thesis rejected cannot be defended with allusions borrowed from stimulus-psychology. This psychology claims that the dis­tances between the points of sensation on the skin present "in reality" a discontinuous material, and that continuity can be given only with the support of visual function. Apart from the fact that such argumentation moves at another methodologicallevel, it leads, in its own way, only to a displacement of the problem. The upholders of this thesis would have to draw the same conclusions for vision and explain how seeing a continuous surface can result from the mosaic-like structure of the retina, which in addition is

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disrupted by a blind spot.68 In all of these cases the physiological facts do not lead to the understanding of the plain and simple phenomena. The lived body, as a functional system, has its own structure and arder of activity, which deviates from that of the physical organism and is not comprehensible in terms of physiol­ogy.

The unique interconnection between movement and touch leads to further consequences and problems. lt must be added that the continuity of the tactile material guarantees the continuity of the total corporeal surrounding field. This means that the touching corporeity has no location in its surroundings that is not basically touchable or that could not be filled with tactile material. Each of its constellations of limbs and each of its phases of movement can in principie encounter something tactile; each inner kinaesthetic sen­sation can be brought to coincidence with an external tactile impression. Only one thing is essentially excluded: the surroundings cannot be completely filled with tactile material, since this would mean the destruction of the motile body.

What can be experienced of something touchable are not only specific material properties such as hardness, softness, and rough­ness, but also form and size. The distinction between primary and secondary qualities is originally accessible to touch, and only the former are relevant for the question of a tactile space. The touching of size contains a first primitive determination of measure. Due to the immediacy of corporeal contact with things, the quantity does remain attached to a qualitative property; the form a.nd size of things are here not only "related" to corporeity in its own orders of magnitude, but are completely incorporated in the multifarious qualitatively and intensively distinguishable, and distinctly felt, movements of corporeal members. Yet a new difficulty appears. Kinaesthesis is a multi-dimensional manifold, and as a sensation of movement, it does not have a determínate direction. After all, conditions and states are characterized by a lack of orientation

68. Even in the cases of hemianopsy, as is obvious from the researches of W. Fuchs, there is no discontinuous visual field, but rather, despite an increased scotoma, there are merely disruptions and displacements. Even a partial hemianopsy does not yield an empty half-space but a newly structured visual space, although viewed physiologically it should be half absent.

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altogether. How is it then possible to discover something like size, form, or dimensionality of the tactile object by the sensation of movement?

Indeed, our analysis is confronted by an insurmountable problem. The frequently defended view that the manifold of kinaesthetic sensations is reducible to three distinctive capacities connecting the tactile and visual data borrows its argument from a region that is aliento touch; this implies that there are not three measurements for the purely tactile thing.

In addition, all touching is a flowing temporal event-which, however, is in each temporal phase merely a touch of a surface. This factual circumstance only seems to have an analogue in visual space. Indeed, while there the thing is constantly seen only in specific side surfaces and is first perceived from all sides in the temporal movement either of the thing or of oneself, nonetheless each indi­vidual aspect contains an indication toward further aspects. The anticipation follows a strictly predelineated style. In contrast, the individual tactile experience lacks similarly co-given indications. Of course, successive touching takes place on an object in accordance with specific expectations of further touch, yet here the resolution of such expectation lacks the character of a mere fulfillment of some­thing already anticipated. Rather, it is like an answer tó a question where initially the response is completely uncertain. Thus it is necessary-and not only for the cautious touching in the fore-"sight" of the blind-to follow up all anticipations with the actual process of touch, surface for surface, form for form, in arder to obtain the total object of touch-and this is in complete contrast to the object of intuition.

Moreover, the purely tactile manifold of a surface structure has no nearness or remoteness. In arder to appreciate this, one must attempt to liberate oneself completely from optical conceptions. Nearness and remoteness are not tactile factors since the touched object lacks any spatial distance to corporeity. Each touched objective there is at the same time corporeally felt touch here. Here and there are thus not differentiated as places; rather, they are distinctions in the directedness of experience. At the same time there is no proper tactile perspective. There are no phenomenal truncations and disruptions such as there are in visual space. The tactile object can be arbitrarily turned without revealing tactile adumbrations. This is confirmed by our previous explications of perspectiva! appearances as objective phenomena in two senses: they appear on the object itself as well as with another corporeity.

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Hence they can only be given to a subject who in his corporeal distance "objectively" has a world of pure oppositionality. Perspective demands a radical separation of lived body and thing, of corporeal conditions and objective reality; this separation is unattainable in touch.

Moreover, perspective is related to spatial depth in the space of intuition. Yet it would be premature to infer the lack of spatial depth from the absence of perspectivity of the tactile manifold. Obviously here the question concerning tactile space attains its proper importance. Indeed, touch lacks the phenomenon of things being covered over and replaced by other things; the tactile thing is where it is and is constantly accessible to touch. Taken strictly phenomenally, there is no such thing as one touchable thing being in the way of another. Yet it is remarkable that the sketches of those born blind do in fact show coverings and intersections, even if there is a complete lack of perspectiva! presentation. Thus there appears an impression of depth in the plane of the sketch. It is already striking that these patients master the problem of sketching and that for them it makes sense to present the touched world on a surface at all. Such mastery assumes the differentiation of their tactile world into surface depth. Yet contrary to first impressions, these discoveries do not lead us far into the question posed. Since such sketches are capable of various interpretations, we are of the opinion that they can neither prove nor contest an originary "tactile depth."69

To decide this question, we should investigate a circumstance that did not receive due consideration in our previous deliberations: the specific manner in which one's own corporeity is experienced in touch. Closer observation of the resistance by virtue of which "something" is first present at all to touch reveals its dual nature: experience not only shows that the tactile thing resists corporeity, but that the touching corporeity resists the touched thing. Corporeity

69. It is to be noted that the researchers who have dealt extensively with the sketches of those born blind have specifically rejected tactile depth. W. Voss assumes an a priori spatial representation of those born blind as an explanation. H. Lassen (1) attributes to the blinda spatial field from a visual­like structure, despite the exclusion of the eyes, since according to him vision is nota peripheral process in a sense organ but a central event. In any case, for him the pictorial depth in the sketches of the patients remains exclusively visual; the "picturability" of the world can never be compre­hensible from the structure of the tactile world.

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thereby experiences itself with respect to its touchability: it is a "physical body," a thing arnong things-and yet at the sarne time, it rernains unbridgeably separated frorn thern in that it is a feeling body and, indeed, one that feels "frorn within. "70 This phenomenal datum is quite important. Such inner awareness of one's own body reveals at the same time an arder of depth. Depending on the kind of impression and its location on the body, tactile impressions on one's own body are experienced more or less peripherally. The intensity and quality of the effect are not only given as such, but are traced at various depth levels. This experience reveals something crucial, something that cannot be mediated by optical experience, and as a pure state or condition cannot be accessible to vision; it is founded exclusively in touch. The touching of one's own body grounds the originary distinction between surface (as touched "outer" surface) and depth (as a specific experiential datum of one's own feeling-felt corporeity), and thus a first conception of the "body" of an "object" in its full voluminosity. Insofar as the touching body is also a touched body, "objects" are structured at the outset in accordance with surface and depth. lt is not surprising that this depth of tactile space differs from that of visual space. lt is not perceived in the objective arder of things, but rather is experienced primarily in one's own corporeity. lt is a sensory function that in its objectivating activity remains halfway between outer and inner. lts objectivity remains embedded in the medium of changing corporeal conditions. The experience of depth in tactile space, deviating from that in visual space, explains, on the one hand, the lack of the phenomena given in visual depth (adumbrations of form, coverings, etc.)71 and, on the other, the positive fact that a lived body functioning and acting exclusively in terms of touch can have a space structured in terms of surface and depth-and at the same time, know how to represent it on a pictorial surface without the additional assump­tions of a purely intellectual spatial intuition, of visual residua, etc., becoming necessary.

The above analysis does not claim to have dealt exhaustively with the problem of visual and tactile space. Their characteristic

70. Concerning the constitution of the body and the experiences of outer and inner in so-called double sensations, as well as with respect to the mode of givenness of one's own body, see pp. 143 ff. of this work.

71. All perceptual phenomena are also lacking in the inner givenness of one's own body.

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properties were sketched only to the extent that they were essential for the problem of the founding of the space of intuition. Visual and tactile spaces, separable from the space of intuition only abstractively, are accessible to descriptive analysis only condition­ally. They ultimately lead to methodological difficulties befare which a phenomenological procedure in the sense previously used comes to a halt.

SECTION TWO

Questions of Space Constitution

Chapter One

Corporeity and Spatiality

§ 1. Methodological Survey

The preceding investigations remained within the framework of a phenomenological method. This procedure makes the claim that it can acquire its data from the reflective orientation toward various modes of comportment of the corporeal subject in space. What was present for description was primarily not space asan object given in thetic space-consciousness of the subject, an object that would provide the basis for the judgement that space is; rather, what was described was space as it becomes appropriated in pre-reflective corporeal comportment-in execution. This point of departure was based on the claim that this relationship of the subject to space is more originary than, and is assumed by, the oojective relationship.

This assertion is not without difficulties. While we believed it necessary to maintain a linear progression from one spatial form to another, we were also clear that although this movement was phenomenologically justifiable, it was at the same time valid only as a foreground. In truth, the correct way of thinking dictated by the state of affairs should resemble a spiral, insofar as at any given stage of his being spatial, the subject must have already traversed all spatial modes. But this subject is no one else but ourselves. In this regard, the phenomenologically descriptive work unavoidably had to be disrupted by an occasional self-reflective move.

138

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Furthermore, we were aware that what was phenomenally "given" -if at first glance it was most obvious-was not something that carne to us of its own accord and could be received by us without raising the question of its origin. Indeed, the sense of the concept of givenness was alloted its full due. We took something as present for phenomenological explication and description in contrast to its being constructively devised or deductively derived. Nevertheless, at the very outset this given was seen from a specific aspect that already determined its choice. The comportment of the corporeal subject in space remained the leading point of orientation. In attuned experience, in goal-oriented activity, and in sensory intuition, the subject appeared in three different modes of orientation toward the world, with their respective sense-orders, each of which turned out to be constitutive for the structure of its own space.

It was stressed at the outset that the anticipated division and classification had to proceed abstractively, in a very specific sense, for the sake of clarity. The analysis had to separate and to thematize what in reality was present as a single moment in a concrete totality of factual behavior. The attuned, the active, and finally the sensibly intuiting lived body, cannot be extricated purely in themselves. Rather, the lived body is to be regarded as lived body insofar as it functions in the ways we have described, and even then never purely and exclusively in any one aspect. Correspondingly, it is also true that the three "spaces" analyzed are not parts of a single space but merely structures of the one space according to the corresponding modes of corporeity. Furthermore, the work of analyzing found itself in the circumstance that we were concerned with an unseparated but nevertheless clearly articulated unity of distinguishable structures and this became the methodological justification of our procedure.

There are additional reasons why this investigation is called phenomenological with certain reservations. In a specific sense every phenomenological endeavor is precursory and points to an­other endeavor that cannot remain with the phenomena as such. Even the concept of phenomenon involves a metaphysical position. That something is "appearing" makes it depend on something whose appearing it is; it is appearance in relationship to being. It is irrelevant for us whether the latter, for example in ontological realism, "announces" itself in the phenomena, or whether it is "concealed" by the phenomena and thus justifies hermeneutical ontology. All attempted separation between a phenomenology on this side of metaphysical positions and the metaphysical positions as such could not be maintained. While dealing with the subject

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matter, each method progresses factually. Yet what is immanent in the essence of any originary method is the meaning of being in light of which the subject matter is explicated.

For the following investigation this means that the descriptive analysis of space necessarily includes an ontological treatment of the problem of space. The ultimate question is not how space is experienced, but what it is. This is the reason why our investigation had to hint at this question, and to hint at an answer ahead of time. Finally, in our attempt to provide an answer we consciously accept all reservations and restrictions resulting necessarily from the meth­odological situation suggested above.

§ 2. The Lived Body and the Physical Body in their Relationship to Space

In all the controversia! questions concerning the relationship be­tween the subject and space, it appears certain that space is somehow "related" to corporeity. In the previous analyses this type of rela­tionship turned out to be twofold: on the one hand, space appeared as the "wherein" of corporeity and thus as a condition of its possi­bility for motion; on the other hand, space appeared to be primarily conditioned by, and structurally dependent on, the manner and mode of corporeal movement. This correlative relationship of corporeity and space, seen up until now in the three forms corresponding to the modes of comportment of the corporeal subject, must subsequently be illuminated more closely with respect to its own structure. The various modes of spatial appropriation present in the different types of corporeal modes of comportment must be traced back to the con­ditions of the possibility for a constitution of space as such.

This kind of regressive procedure reveals an unavoidable point of departure reflected in two concepts whose meaning is guaranteed and established only in their mutual relationship. Corporeity and spatiality are as abstractions totally empty and are never given as data. They reciprocally require one another for their meaning. We cannot ask how one emerges from the other or how one is founded by the other, but only about specific correlations within their reciproca! relationship itself. One does not precede the other­neither ontically-genetically nor ontúlogically. Corporeity was not "before" spatiality, since to be corporeal, i.e., the foundation, is to assume spatiality; the latter, in its own turn, was not "before" corporeity, since corporeity determines spatiality as what it is­spatiality of corporeal movement.

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That corporeity is already in space is a conception of natural consciousness having sufficient justification in the simple phenom­enal state of affairs. Corporeity of a determinate nature and mode of functioning, and corporeity of a determinate concretion as "lived body" is found in a spatiality that is likewise of a concrete determi­nate structure with an already "finished constitution." Thus corporeity finds itself in a space that for natural consciousness appears as given.

But can we think of the lived body as being-in in analogy to the way one thing is contained in another, and understand such being contained from a specific mode of corporeal functioning? Can it be surmised that in the nature of its functioning the lived body is posited as the ground for the possibility of apprehending itself as being "in" another?

Let us recall the conception of the lived body as it was sketched at the outset. As it is present in nonscientific understanding, the lived body presents itself as a completely indifferent psychophysical unit. Consequently, reflective analysis cannot see in it anything more than a complex unity of activities and functions. This unity is the determining basis of its being no other except this entity with its modalities, i.e., with its comportment toward the world. This unity turned out to be tripartite: corporeity appeared in its sensible comportment in the modes of attuned experience, goal-oriented activity, and sensory intuition.

However, these modes of comportment do not reveal the lived body as primarily present to itself; rather, what is given to it in the first place and above all is a section of the world. Where the lived body itself becomes given, it does so in a way comparable to any other mode of givenness. In sensation as "corporeal feeling" it is present in a specific, qualitatively and intensively unique state. This state does not signify objectivity. It is an awareness (Innewerden) of the lived body in a nonthetic mode of consciousness which, how­ever, has two specific characteristics. First, corporeal consciousness is consciousness of the lived body as a whole. Even with strictly localizable organ sensations, the organ is not sensed alone; rather, the lived body as a whole functions as the phenomenal background of all these single sensations. Secondly, each actual consciousness of one's own lived body already belongs to the latter's entire existence (Da-sein) prior to all differentiated states. Each corporeal conscious­ness always and already assumes the prior being of the entire lived body which is not exhausted by the singular given states. If feeling (Befinden) is the way in which one becomes conscious of one's own

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lived body in the mode of having, as a mode of being of the subject it is nonetheless already prior to any consciousness of "having." The lived body is a founding and basic phenomenon for any kind of "having" as such; as will be shown more precisely, it is the first and necessary condition for the possibility of any form of possession of something or of knowledge of something.

The certitude of the lived body, mediated by its feeling, is a typical certitude "from within." But where is the "outside" of this "inside"? How is it that we make this spatial distinction between inner and outer? The boundary between inner and outer runs visibly at the surface of the lived body. Yet this expression is senseless-a functional unity cannot possess a surface. Lived body, in its strictly phenomenal sense, is on the hither side of all separations between inner and outer, as it is hither of all separation between physical and psychic. That there is such a distinction is founded in it. It is corporeity itself, then, that constitutes this separation, and indeed through its specific function of touch.

It is now necessary to investigate tactility not with regard to tactile space, but with respect to the mode of giveness of one's own lived body. In touch there occurs the originary communication of the corporeal subject with a world "outside." Something resistant is encountered in it, and the lived body is confronted by something material and thingly. But how can a thing confront the lived body? Obviously the thing does not resist it, since it is present to the touching corporeity without limitations. Resistance cannot mean a limitation of corporeal activity; moreover, in the experience of resistance the tactile function reveals a determination characteristic of the lived body, namely its being a physical body. The lived body as a material thing, as a "physical body," constitutes itself through a specific corporeal function. Here the founding role of the lived body for any kind of givenness becomes clear. Any theory that thinks that it can construct the lived body on the model of levels-material and the psychic (Husserl), or the inorganic, the organic, and the psychic (N.Hartmann), whereby the physical body ought to be the founding level of the remaining "higher" levels-fails to conform to the phenomena given. Even when they claim to be phenomenolog­ical, they uncritically follow the conceptions of the special sciences. The essence of originary corporeity, being nothing else than a system of functions, does not include being a merely "physical" body. It is possible to think of a corporeal being whose functions are so designed that a physical bodily world, and thus corporeity as a physical body, would never cometo appearance. At the same time, the corporeal

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being would not cease to be corporeal. To be sure, such a corporeal being is not our own. My lived body is physical corporeity; never­theless, 1 know of the existence of a physical thing called m y "body" on the basis of my lived body. As a physical body, the lived body is given in immediate connection with the material world. Yet the material world was not befare the lived body nor did the latter precede the former. The constitution of a resistant thing-manifold and the constitution of the lived body as a physical body are only two sides of one and the same process.

But corporeity, as a physical-bodily corporeity, is not yet suffi­ciently determined. If it is material, thingly like other things, then it must not only be touching but also touchable corporeity. lt is characteristic of the tactile sense that it has unique sensory function of the so-called "double sensations." The lived body is a subject as well as an object of touch. In its resistance it is given to me objectively, like other things, and at the same time it is experienced subjectively as receptive of sensations. The possibility of this imme­diate union of coincidence between corporeal feeling and corporeal perception is the ground for the differentiation between outer and inner. Yet this union of coincidence must become present as such to awareness. This requires a specific experience of identity wherein corporeity, in its touching knows that it is identical with itself as touched. 1t is only to the extent that this consciousness of identity is there, that the concepts of outer and inner assume a fulfillable sense. 1t is not that 1 somehow touch a thing, but that it is my corporeal thing-which while touching is given as self-touching-that makes this originary separation between inner and outer possible. With it the lived body has constituted a first spatial relationship, which is retained in all higher functional activities. But insofar as the lived body establishes this activity only in a mutual interplay with the world of things, it itself becomes exposed to them. On the basis of its physical-bodily constitution the lived body has all the attributes of things: extension, form and size, materiality and weight. All thingly qualities are at the same time corporeal, in the precise ontological sense that one can speak meaningfully of them only with respect to the strict correlative relationship of lived body and world. Thus there are no corporeal bodies or bodily things, no size or form of one's own lived body, without the possibility of its presence in the world of extended things and vice versa. At the same time, however, there is a place in space for corporeity justas there is for things. The being-in-space of the lived body solely as a spatial thing is, to say the least, a delusion of a naive consciousness; an essential determination

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of the lived body to the extent that it is a bodily corporeity, is to constitute a space-establishing function as such.

But while it is comparable to things, this does not mean that the lived body thereby becomes a thing among things. A thing does not experience its thinghood in its encounter with others; it does not have any "thing-experience" of "itself." Yet the self-experience of the corporeal "thing" is entirely different in kind from experiencing all other things. While grasping myself as a body, 1 am present to myself "in the flesh" in an incomparable way. Thus the sensory intuition of other things are given in continuous syntheses that fuse the partial intuitions of perspectiva} aspects into an intuition of a whole-this is essentially excluded for my body, since my body can never become fully accessible in the successive progression of a perceptual series. But there is no need of such syntheses or identi­fications, such fulfillment of indications and anticipations of the particular parts. Rather, my body as a whole is present to me immediately and actually; it is not given objectively but felt through an arder of depth clearly differentiated in accordance with quality and intensity. And this felt state is first constituted in the touch of one's own touching-touched lived body.

lt is important to see that touch, in its space-constituting activity, is not repeatable by any other sensory function, not even by vision. The latter is neither constitutive for the materiality of the thing nor for the differentiation between inner and outer. A non-touching corporeity would not have an external world. Indeed, it seems like tedious speculation to say anything about the world of a lived body that is essentially not a touching lived body. Nevertheless, sorne hypothetical statements are possible. We must exclude both the mode of corporeal givenness in vision and the constitutive role of touch for vision that was already demonstrated.

No corporeal feeling, no inwardness corresponds to vision alone. There are no visual sensations, as there are sensations of pressure, hardness, and warmth. Where such sensations occur in vision, for example with a dazzling and harsh light, vision becomes disturbed. One's own lived body is not given in any manner in normal vision; 1 do not see "in" the eye in the same way 1 touch "at" the skin; rather, 1 see into space. (This mode of expression already presupposes the accomplished distinction between inner and outer.) The eye is an organ of my corporeal body insofar as it is receptive to touch and pressure, but not insofar as it sees. As seeing, the eye is a member of the lived body, but it is not itself given to me in seeing. Only the oculomotoric kinaesthesia, as a sensation of an organ, gives me a

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vague and merely peripheral inner awareness of my corporeallived body. Yet there is an absence of the unity of coincidence between one's own seeing and seen corporeity, between subject and object; this is possible only in touch. Even kinaesthesis already assumes the lived body as touchable, and thus assumes the tactile function.n

Indeed, the lived body is visually distinguishable, but in a way that is different from touch. As visible, m y lived body is a lived body for others who perceive it, but nevera lived body forme. This is not only because it is only partially perceivable by me-even in this partial perception it is aliento me in a unique way. It is conceivable that this partial view can be acquired by the use of a mirror, whereas there is in contrast no analogously functioning implement for touch. The mirror which "lets me see" my lived body, in truth does not let me see it, but only its image-and this only through the mediation of an already pre-given external world. Mediated in this manner, this image can deceive, distort, or flatter, can lead to doubts whether it is really my lived body whose image I perceive. In contrast, touch is beyond such doubts. A touch, a blow, immediately convinces me of the mineness of the lived body. Yet it convinces only me. Were the other to touch my corporeal body, he could have the same tactile impression as from other things, but never "my" sensations. In a strict phenomenological sense, there is no co-feeling of corporeal sensations.

The functional uniqueness of touch appears still with yet another characteristic. What it encounters is given to the lived body only successively; the touched thing first unfolds in a succession of individual tactile impressions. But what is the ground of the experience of the unity of the touchable thing in the temporally discrete succession of tactile experiences?

The very sense of the question implies that the touched thing is already known as one. This knowledge is attained not merely through one's own movements but through self-movement or, more precisely, through the motion of a self who knows itself as moving. To grasp something as one thing in the flow of impressions is to assume that this flow is apprehended as a movement "in" which something identical is given, because it is appreheiided as a move­ment of something identical: the moving lived body. Thus this

72. In fact, it is difficult to say something about a merely seeing corporeity, not even about that which it sees. Any perception of something extended would already be impossible for it; seeing would not even be seeing of surfaces. Vision without touc,;h remains completely incomprehensible.

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question is addressed at the outset to a corporeity that is not only active in touching and is identical in touch, but that in touching also knows itself as the same and, at the same time, knows itself as self-moving.73

One must be cognizant of the unacknowledged "presupposition" that the genuine sign of life is self-movement. The unreflective consciousness refuses to grasp movement otherwise than as a relative change of place of something identical "in" time. It can hardly conceive that a movement or rather, that what it grasps as such could also be apprehended as a temporal change "in" space; although this is thinkable with difficulty, yet that it be sois rejected by unreflective consciousness even when the change indicated shows no sign of preference for either orientation. Nevertheless, when the process is grasped as a movement, as a change of place in space, it is because the change is accepted on the assumption of the identity of something in motion; it is a movement for a consciousness that grasps identity. This is the meaning of the frequently stated proposition that all motion "requires" an identical carrier of the movement-the process, which could be grasped as movement, "requires" identity; the latter presupposes the identity of consciousness with itself, a consciousness of a self maintaining itself in the changes of the perception of movement. Only such a

73. This is not to claim that there is a reflection inhering in each experience of self-movement such that this identity is given thetically to awareness. lts coming to attentive perceiving does not have the character of being-conscious-of-something, as is appropriate for every reflectio:q; "di­rect" self-consciousness is not at all an intentional consciousness. The structure of the experience of movement is not as if in it the moving corporeity were meant and in its differing phases of movement were established in its identity. Indeed, a reflection of a specific kind can intend the identity of an ego-corporeity, yet this expressed intention is first possible on the basis of an unreflective pre-knowledge of one's own identity as it is primarily experienced in motility. Perhaps it would be more appropriate to speak here of ipseity and not of identity. This term is appropriate for the unreflective selfhood of the ego as an identity given here. Identity presup­poses difference, and strictly speaking, is never a simple experience, but always a result of acts of thought that are not expressed; since they are oriented primarily toward differences, they establish sameness despite the difference, or rather they question and test the sameness of something that is being asserted as identical. The ipseity meant here, however, is not questionable or testable with respect to identity, but is rather a simple phenomenal datum.

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consciousness can experience change and transformation, as well as their differences.

The notion of the self-identity of the lived body is contained in its motile comporting itself toward the world even if no knowledge of this self is present. Animal corporeity may be present to human understanding as self-less and, thus, privative in contrast to a self. However, human corporeity not only essentially determines itself as the lived body of a self, but also knows itself as such; it determines itself as an ego-corporeity and thus it is primarily a self-moving corporeity in the genuine sense.

It was already suggested that movement as such could be indiffer­ent to space-time (p. 41). This cannot and does not mean that it is "outside of" space and time; it is grasped only in both determina­tions. Yet in both, movement indicates not only the duality of both determinations but also their originary unity. This is still present in tactile activity. With the constitution of outer and inner that founds a primordial spatial relationship, the touching lived body attains the contiguity of the tactile there only in a succession of now and now; it knows how to attain a manifold of locations,, a space in a succession of individual sensations. While bound to the identity of a feeling-felt self, they are grasped primarily as a succession, as a forward movement in time. By constituting space in the mode of its own tactile movement, the lived body also constitutes time in the manner of ek-stasis; yet because it finds itself in this kind of self-movement, this space-time unity is already doomed to disrup­tion. While the lived body apperceives its originary mode of func­tioning as self-movement, simultaneously it lends the movement the sense of a change of location of its body (which remains identical with it), i.e., of its temporal process in space. An assumption of a continuous transformation of its body in time would be tantamount to the dissolution of its self. Space-time is ontologically relative to a corporeal being as such; space and time are ontologically relative to a lived body that is also an ego-corporeity.

Without scrutinizing the specific function of time more closely, it is essential to trace the relationship between space and corporeal movement at greater length. We attempted to decipher this relation­ship in its ontological origin. We saw how the lived body, with its inner and outer aspects, founded a primordial spatial relationship and at the same time revealed its own fundamental spatial determi­nation as corporeal body. The incipient aporia of corporeal being-in can be resolved only in this way: while as a lived body it constitutes space, through a specific corporeal function it constitutes itself si-

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multaneously as a physical body in space. Thus corporeal space is that which is determined by its activity. The paradoxical structure of this state of affairs is founded on the fact that here we are confronted with the ultimate relationship beyond which thought cannot progress.

If touch turns out to be the incomparable function that grounds the strictly correlative and reciproca! relationship between body and space, nevertheless it does not solely comprise the richness of this relationship with regard to its contents. While founding every sensory performance, it remains at the same time half immersed in this performance. Due to the immediate contact of the lived body with things through touch, the lived body cannot accomplish objectification in tactility alone. By presenting an externality alíen to corporeity, it founds an initial objectivity by setting it against its own subjective inwardness; yet corporeity cannot accomplish this sepa­ration from itself since in touch it is present to itself in a unique way. The existence of space, which in its fullness of structural character­istics transcends the mere tactile manifold, requires further func­tions and forms of activity of corporeity. Indeed, the space of intuition has its sensory foundation in touch and the space of action also has a constitutive moment in it. Yet in their own arder and articulation they do not exhaust themselves in being merely a modified tactile space. The thing as ready-to-hand, as well as the thing present-at-hand, is not comprehensible solely as a set of sensible constitutive moments. Corporeity in a situation is ego­corporeity,lived body of a self, and thus its space is not only a space of corporeal functions but also a space of its performances of consciousness.

The preceding investigation is left with a specific difficulty: whether and how is it possible to unify the fact that, on the one hand, space is a space of movement "for" as well as "through" corporeal movement, and that, on the other hand, it is an object of conscious­ness. Thereby the problem already arase asto whether the solution to the problem of space could be found solely in the objective form of intuition, whether the specific being of space could be found in the categorial activity of the subject.

§ 3. The Lived Body and Consciousness

The meaning of the problem raised above could be formulated in an alternative way, i.e., whether the corporeal subject is originarily a lived body or whether it is originarily consciousness; whether, in

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an ontological sense, the lived body precedes consciousness or whether consciousness must rather be a requirement for the being not simply of corporeity but of ego-corporeity.

The obvious epistemological problem that emerges here must be bracketed. That consciousness has a gnoseological priority requires no discussion. That the lived body, in order to be known as lived body, assumes consciousness, is an analytical proposition. There can be no controversy concerning the irreducibility of what pertains to consciousness itself. In order to derive consciousness from something else, consciousness is required; any attempt to reduce it to something else involves a petitio principii. Consciousness as such is irreducible; it is something that has itself for foundation and must be accepted as such.

This does not seem to be the case with the lived body. If it is the lived body of a consciousness, then the latter is that from which it is what it is. We have done justice to this notion above, insofar as we have never regarded the lived body as something simply self­sufficient and to which consciousness was merely "also" attributed; from the outset we regarded it as the lived body of a subject, and showed that even its most elementary functions, such as locomotion, present corporeal being as a consciousenss.

While its ontological relevance was admitted for the being of the subject, there was nevertheless no prejudgement concerning the possibility of an ontological derivation of the subject's corporeity from consciousness. Even though at the outset the so-called philosophy of consciousness is designed to be essentially epistemo­logical, it apprehends corporeal functions as dependent in their being on the grace of something else, and does so in two ways. Thus in his Ideas and in the Cartesian Meditations oriented exclusively to "pure" consciousness, Husserl not only dissolves the entire region of extra-corporeal being into an intentional being in consciousness, but also submits his own lived body to the decree of the transcendental-phenomenological epoche. The lived body is, then, only something constituted in pure transcendental conscious­ness while the latter-in its manifestly unlimited and uncondi­tioned intentionality-remains the sale absolute. Thus one can arrive at such statements as these, that the subject "localizes himself" through his lived body, and that there is a possibility that my lived body does not exist. Even Heidegger, obviously bound to the transcendental-idealistic heritage, speaks of Dasein's spatializa­tion in its "bodily nature." This view especially comes to the fore

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whenever a specific problem is excluded from the domain of interrogation. 74

Pursuing the problem of spatialization, Merleau-Ponty is forced into an opposite conception. In his Phenomenology of Perception, the lived body attains an ontologically fundamental position. While inappropriate as a principie for deriving consciousness, it is elevated to the status of an unconditional being. Although this seems to offer a decisive and clear view, Merleau-Ponty's presentation moves in formulations favoring the other extreme of the ontological relation­ship between lived body and consciousness: "Il y a done autre sujet au-dessous de soi, pour qui un monde existe avant que je sois lá et qui y marquait ma place. Cet espirit captif ou nature1, c'est mon corps ... (There is, therefore, another subject beneath me for whom a world exists befare I am here and who marks out my place in it. This captivity of natural spirit is my body ... "). 75 This conditioned relationship, as it is conceived here, makes the short-circuited thought comprehensible. The latter consists of Merleau-Ponty's belief that he can completely explicate the structures of the corporeally-centered space of the "etre engagé" and make this space comprehensible to the corporeal subject without somehow introduc­ing objective space into discussion.

By attempting to avoid such truncations of the problem, we also recognize a specific methodological difficulty in our own investiga­Han. It appears that our problem lies in the particular difficulty of appropriately conceiving the being of the subject both as corporea1 subject and as corporeal subject.

Consciousness does not localize "itself" as a lived body in space-­insofar as it is consciousness it was always consciousness of a lived body. It was never without a lived body. Dasein does not spatialize itself in its corporeity as though, prior to and over above it, it had performed differently-it was always spatial in the form of being corporeal. At the same time, lived body as my lived body, and hence experienced as mine, was not "befare" or "beneath" but was always mine, was always the lived body of an ego. Without me it would never have begun, and in accordance with its meaning it was always ego-corporeity, the lived body of my self-consciousness. Even with strict adherence to the ontological significance of such temporally sounding expressions, the relationship that prevails here is such that

74. E. Husserl, Ideen Il, §§ 22 ff.; M. Heidegger, p. 108. 75. M. Merleau-Ponty (1), p. 294.

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as soon as it is expressed it becomes tantamount to the disruption of the true state of affairs. As the paradigm of a strictly reciproca! implication, it resists any effort to grasp its fundamental structure linguistically.

This not only means that consciousness is not the sale irreduc­ibility, justas the lived body is not the sale contingency, but rather that this twofold non-derivability of the corporeal subject is at the same time bound in the ontological unity that consists of a recipro­cally conditioned relationship. While this may be obvious with regard to the lived body, the reverse, the determination of conscious­ness through corporeity, will be opposed whenever gnoseological considerations cometo the foreground. As a relationship of being, the one-sided and irreversible epistemological orientation from consciousness to the lived body is irrevocably two-sided. Even where pure consciousness recognizes corporeity as corporeity, it does so only through a consciousness already containing a moment of corporeity itself. The only "pure" consciousness is one that functions independently from all merely factual, individual condi­tions of a lived body, but not from the facticity of corporeity as such. A consciousness without corporeity is not only not discoverable, but not even thinkable. Consciousness would be a completely empty concept because its total activity cannot be determined without the inclusion of corporeal functions. Even its complete composition with respect to its contents-the noetic as well as the noematic­contains corporeal implications. This is not only valid for a non­intentional consciousness whose really intrinsic contents are, as simple sensations, not merely mediated through the lived body, but are the very contents of the corporeal states and are themselves corporeal. It is also valid for objective consciousness, whose inten­tionality comprises only a specific accomplishment of subjectivity as such. That all factual consciousness is intentional-and that in particular it is doxic and thetic, i.e., in its objectifying acts it consists of various modes of conceiving, positing, interrogating "Being"­does make present, in the total complex of the performances of consciousness, a specific contingency that resists any further inter­rogation and is not derivable from the subject's corporeity. Yet even here we find no acting being whose really intrinsic as well as intentional contents did not have certain foundations in the prior givenness of corporeal-sensible factors-and these factors do not reveal indices toward sorne ultimate acts that would have to be deciphered in arder that we may comprehend the nature of such factors.

Chapter Two

The Space of Movement and Objective S pace

§ 1. Spatial Structure and Corporeal Facticity

The separation of the lived body and the physical body and the insight into their relationship has solved the problem of "being-in" in the following way: corporeity as a bodily corporeity in space and corporeity as first constituting space, should not be seen as two exclusive states of affairs. Thus all that remains is to answer the following question: how is space, as a space of movement, related to an objectively conceived space? Obviously this question is not identical with the one dealt with befare. It is not concerned with the problem of the correlation between corporeity and space (which can be explicated from an anthropomorphic viewpoint for the animal kingdom), but rather with the specific relationship of space as a form of fulfillment of corporeal movement, on the one hand, and on the other, space as space of objective consciousness.

The former is co-determined by the latter. Despite its methodolog­ical difficulties, this insight could not be avoided even by phenom­enological investigations if they intend to offer an exhaustive anal­ysis of the states of affairs. The space of movement was not comprehensible without consciousness of objective space. The latter turned out to be underivative. This led to the notion that fundamen­tally it is the only one relevant for the ontological question of space. If it is the sale space illuminating other "spaces," then ontological significance ought to be attributed only to it.

This would be the case even if the being of the lived body were dependent on the grace of consciousness and if the latter were the sale unconditioned aspect of the subject. Yet corresponding to their mutual non-derivability, it is impossible to reduce the space of movement to the space of objective consciousness and vice versa. At the same time, corresponding to the mutual implication of lived

152

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body and consciousness there is a reciproca} relationship between the space of movement and objective space. What constitutes the structure of the latter, under closer illumination of the conditions of its possibility, turns out to be a sum of determinate properties. As already emphasized, the latter are not deducible from the corporeity of the subject without going through the reflexive circle. Yet objec­tive space is conceivable only as co-determined by a corporeal being.

In modern and contemporary philosophy, there are many attempts to investigate rationally the ontological status of space, to demon­strate its structural components and to deduce it from various premises. It is striking that these arguments reach their results at a price of a complete glossing over of the historical dimension. Homogeneity, void, and boundlessness appeared in these arguments as "apriori" determination of space. Yet in historical retrospect these concepts have assumed predominance only since the beginning of the Renaissance and only became accepted as the universal con­sciousness of space in the seventeenth century. In view of this fact, our claim about "the" space acquires a specific significance. Al­though we have not defined space in any strict sense, we suggested that it has one of the fundamental conditions in the corporeity of the subject. And in view of the historical transformations of space­consciousness, it is necessary not only to differentiate the prominant and enduring problems, but also to test the historically given stock of phenomena.

If one were to observe the multitude of typical arguments concern­ing the question of three-dimensionality with respect to their com­mon denominator, then it would become obvious that despite the different points of departure, all are traceable back to factors that are based in the ineradicable conception of the subject's being as a corporeal subject.

If one wanted to derive three-dimensionality "logically" from the essential connection between surface and depth, as is done by C. Stumpf, orto grasp it with H. Lassen in the unity of two "funda­mental moments" of extensionality and intentionality, one would discover that the process of proof ultimately finds its starting point in the facticity of the corporeal modality of being.76 Lassen

76. Sturnpf (1), pp. 85, 275, explicitly rejects rnovernent asan "integrating condition of spatial representation," although sornewhat later he adrnits that the representation of depth "is least extricable frorn rnovernent." While dealing purely psychologically with the conception of space and at the sarne time, confusing it with Kant's pure forrn of intuition, Sturnpf has in view

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nevertheless contests "that the question concerning the degree of dimensionality could be answered in terms of the existence of distinctive spatial directions." "Space as a pure manifold must always already be given befare the question of its number of dimensions could acquire sense." Las sen correctly states that strictly speaking the number of dimensions cannot mean the number of distinctive directions, but rather the degree of manifoldness. Since we are at the pre-quantitative level, this characterization has an advantage in our investigations. It can avoid metric characterizations (the highest number of perpendicularly related lines, etc.) and can deal with the topological properties. lt is instructive how Lassen determines the degree of manifoldness of space by its precise derivation from the "intuitive-topological" conception of space. That spatial depth constitutes a degree is shown by the univocity of its direction, allowing only one possibility of uniting points located one behind the other.77 Since the corresponding operation on the surface offers various possibilities, the degree of surface manifold­ness must be at least two. That there can only be two results from the fact that the manifold of surface relationships of a region is so constituted that given a sketch of straight lines on the surface, there can be at most two straight lines that could connect just once in such a way that none of the remaining straight lines are intersected.7 s

Thus Lassen's three-dimensionality is proven. For him it is not "an accidental property ... but rather a formal, dosel y structured system of relationshi ps," . . . an "a priori fact."

Here, however, its necessity can only offer the possibility of an insight into its fundamental conditions from which the topological states of affairs mentioned can be understood. The latter are obvi­ously based in the corporeity of the intuiting subject. The proof of three-dimensionality can make sense only to a sensibly intuiting and

only the moving thing; yet even this presupposes the self-movement of the perceiving subject. Concerning H. Lassen see (1), p. 35. We subsequently support this presentation since it appears to us to be most careful and, within the framework of our statement of the problem most, instructive. (The proof for three-dimensionality has been established by Lassen for the perspectiva! space of intuition. Yet in terms of its numerical dimensionality it cannot deviate from the homogeneous space, since it is the manner and mode in which the homogeneous space is attained in its monadological aspect.)

77. The univocity of depth will be discussed subsequently in the context of the problematic of the visual ray. See pp. 248 of the present work.

78. lntuitable topological examples are given by W. Lietzmann.

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thus self-moving being. Who else could understand that spatial surface and depth "necessarily" demand one another? Nevertheless, the converse is also valid: only the lived body as corporeity of an intentional consciousness can perceive the sensibly intuited things in the arder of surface and depth, extension and intention. In addition, it seems that an interrelationship between the degree of dimensionality and the directional oppositions is unavoidable. Three-dimensionality, as well as its orthogonal relationship, can be founded only on the elementary orientations. Indeed, it was already pointed out that in the space of action the three-dimensional, rectilinear system of dimensions is not yet given with the elementary pairs of opposites. While it is impossible to treat the elementary directions of orientation in accordance with a system of coordinates of "the" space given in any other way, the former, in turn, are are not yet such a system. Nevertheless they contain the founding moment for any spatial coordinate system as such.

In any case, an attempt to deduce the triadicity of space from kinaesthesis misses its aim. Kinaesthesia is purely a qualitative and multi-dimensional manifold. If one were to attempt such a reduction by recourse to certain coincidences and couplings of sensations of movements, resulting in an exact arder characterized by triadicity, one would be moving in a circulus in demonstrando: in truth one posits a merely hypothetical connection that cannot be shown imy­where, while the space of movement as triadic manifold has been already articulated in sorne other manner. That it can and must be seen as a three-dimensional arder does not depend on an uncondi­tioned "pure" space-consciousness. The latter is, and continuously remains, a consciousness of a lived body in space and is conditioned by the lived body. But that it is precisely this lived body, with these movements, that the lived body is constituted in this and no other manner, possessing these factual members and functions, is irrele­vant for the triadicity of space and the orthogonality of its dimensions.

W e saw that the determination of the elementary differentiations of orientation was only possible through the inclusion of corporeal organization on the one hand and through the functional activity of its members on the other.

The univocity of this determination could be maintained only with the aid of the principie of economy with which the active corporeity complied in its orientation toward the world of the ready-to-hand-the most originary form and gestalt of its intention­ality. The same holds true for the very attribution of all six orienta­tions of movement to three originary and permanent pairs of

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contrary oppositions: right and left, above and below, front and rear. They are originary in that they are given on the basis of anticipated activity, but not primarily as "pure" spatial orientations belonging to sorne kind of theoretical intuition. Their correlation is apprehended precisely as fixed and interchangable, and in this correlation the correlated have the character of opposing one another. The equal­ization of these opposites, the homogenization of the directional differentiations, takes place in self-movement which relativizes the oppositions. Instead of the three pairs of opposites, there appeared only three functionally equivalent (constantly directed to the fore) frontal orientations in the space of movement, three "dimensions" of frontal movement in which the original oppositions are dissolved. The system of three spatial dimensions is completely conditioned by the corporeal moment and the supposedly "pure" consciousness can loca te the triadicity of its space only because this space is at the same time the space of a functionally determined corporeity organized in this and no other way.

A consciousness without corporeity would have no knowledge of the "perpendicularity" of the dimensions. The orthogonal relation­ship is indeed not topological but metric. With this we can only deal subsequently, specifically since such a relationship does not belong to the composition of the phenomenon in the sense previously delimited. Yet it will be shown that this manner of measuring angles must be understood as a formalization founded in the oppositional directions of movement.

Particularly difficult problems seem to result in our investigation with regard to the relationship between the space of movement and the space of objects, especially when one considers the boundless­ness of the latter. We are deliberately avoiding the concept of infinity in its proper sense of "open" infinity. Subsequently it will presenta specific difficulty; meanwhile, we continue to speak of the infinite in an imprecise way as the and so on of space. Natural consciousness can only grasp it in this manner. What is decisive at this point is obviously that this "and so on" of space is not comprehensible either through the corporeal structure of the subject or through any other datum. After all, the insight of the previous discussions showed that the objectively grasped space cannot be regarded as a sum composed of singular spaces of motion; rather it is always presupposed in phenomenological observation. This phenomenal state of affairs must be confronted with our conception of the reciprocally condi­tioned relationships between the space of movement and the space of objects.

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Phenomenologically speaking, we face a difficulty. The modality of givenness of "the" space for natural consciousness is debatable prior to all theory. Whether "the" space is the finite but limitless space of antiquity or the open and infinite space of the modern consciousness cannot be decided, since as a whole it is not given adequately in any intuition of a single particular. lts limitlessness, whether conceived in one or the other sense, can in no manner be brought befare objective space-consciousness. This contrasts with the finite, singular space of intuition in the perspectivally intuiting consciousness; its givenness "moves" concurrently with the clues of the "continuation" of space. This conception is supported by kinematic structures and is exhausted at the achievement of an aim.

A more refined characterization can be readily found in the Aristotelian conceptual schema of possibility and reality, of poten­tia! and actual infinity. Although originally Aristotle brought this schema to the treatment of the Zenonian aporia of continuity with regard to the "inner" limitlessness of divisibility of a finite stretch of space, the conceptions are also applicable to the problem of spatial limitlessness toward the outside. But what does the frequently expressed claim mean that space is "potentially" infinite"?

If one accepts the Aristotelian dynamei on as a concept opposite to the energeia on or entelecheia on, then the possible could be taken to mean the not yet real; but the possible can only be conceived insofar as it can be converted into reality. Thus understood, the validity of possibility cannot be excluded and its realization is its positive characteristic, enabling a conceptual differentiation of the possible from the impossible.

Yet for dynamis in the sense meant here, these conditions are not fulfilled. After all, the infinity of space is not a type of possibility that could ever become a reality. The potentiality of space is a "pure" possibility that could ever become a reality. This state of affairs might have been what Scheler meant by space as a pure capacity-in his view a capacity for self-movement. At the same time he regarded it as non-real and attributed its purported reality to the "deception" of natural consciousness. That this latter claim contains difficulties is shown solely by the fact that a deception, if it should exist, must be able to be demonstrated, and that in principie it can be abolished in the context of further or other experiential or epistemic interrelationships. Since this is excluded in Scheler's case, it must be a "deception" of a specific kind; more precisely, it must be a mode of experience that cannot be grasped with the concept of deception.

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Yet Scheler's point of departure can be evaluated positively if the concept of dynamis is taken in a second Aristotelian sense. It need not be taken as opposite to energeia, and it need not mean here the possibility of teleological realization; rather, it may simply mean the "capacity" to do something or to passively undergo something.79 This is a characterization of potentia that when applied to a specific experience of the capacity for movement, can offer useful solution to our question.

Since all factual corporeal movement constitutes only a limited spatial stretch, it takes place on the background of consciousness of "further" possibilities of movement. In a concrete phenomenological sense this possibility affirms a progression of continuous movement of corporeity that is free in principie and simultaneously the "continuation" of space. Yet through every factual fulfillment, this possibility of experience is not varied, diminished, weakened, and finally extinguished; rather, insofar as movement continues at all, this possibility remains intensively as a constant presence. This means that the possibility of progression "into infinity," precisely as this pure possibility, contains in itself the moment of infinity. As a possibility "toward" infinity it is itself phenomenally infinite and never disruptible possibility. As an attribute of space, infinity is in truth nothing other than a characterization of my possibility of movement.

This does not yet necessarily determine the open-endless exten­sion of space. What is characterized here as the infinity of intensive movement is fundamentally open to two kinds of structures of "infinity" of space-the boundless but closed, turning back upon itself ("not genuinely infinite") and the open-endless ("genuinely infinite") space.

Both structures are fundamentally compatible with the specific character of movement's infinity. With regard to the genuine infinity of space, it is necessary to have a specific mode of experience of the type of movement which, despite its periodicity and rhythmicality (as merely an endless repetition of the individual steps of move­ment), is experienced as movement forward in the full sense of the word. It is only with and in the experience of moving forward that the background of further possibilities of movement is differenti­ated -in accordance with the "already" realized and the "yet" to be

79. An exhaustive treatment of the Aristotelian dynamis cannot be offered within the framework of our investigation. Compare the initial exposition of this problem in W. Wieland, pp. 292 ff.

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realized possibilities, in accordance with what has gone by and what is to come-it is only by "flowing onward" that time, along with space, can be infinite in a genuine sense.

This does not mean that the great historical turning from the topos of antiquity to the empty, infinite space of modern times could be understood solely in terms of corporeal relationships of movement. What has determined the conception of space concretely throughout history has never been a phenomenological presentation of corporeal modalities of movement; rather, they were conceptions based on world views and metaphysics. Indeed, until Newton's intuition of space they were thoughts laden with theological content. But this does not contradict the claim of our exposition with its clearly outlined limits. Our intent is to present the fundamental conditions of possible space-constitution as such. Such conditions must neces­sarily be fulfilled if space is to be given, although their. mere fulfillment does not offer sufficient guarantee for a specifically determined topological spatial structure.

Befare discussing the question of structure, we should make sorne additional remarks concerning the character of infinity. That my possibility of movement is itself infinite suggests that the infinity of space cannot be an intended aim of my movement. This state of affairs may have been the source of the notion that infinity is "merely" potential. Thus potentiality appears to be a lower ontolog­icallevel than actuality; however, this misunderstanding can arise only if the meaning of potentiality is understood solely in analogy to actuality. Yet we are dealing with a possibility that is of a lesser mode of being not because it can never become reality, but precisely because in its uniqueness it has an ontological function that cannot be confronted by any modal counter-concept.

Furthermore, the potential infinity of space does not mean that space is "possibly" infinite. The experience of limitless possibility of movement is in its own way not merely possible; rather, as experience of potentiality, it is actual, i.e., it is an experience that is actualizable in each factual phase of movement. Space "is" indeed infinite only in the mode of potential infinity, although potentiality in turn is not something merely possible. It is not a contradiction if, on the one hand, potential infinity is attributed to a space of limitless possibility of movement and if, on the other, such an attribution appears as an actuality of awareness. The potentiality will not thereby become actuality-this would be counter to it-although space possesses in its characteristic of potential infinity a unique actuality in the objective consciousness of space. This complicated

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state of affairs may be the catalyst for the succession of constantly rejuvenated theories engaged in the controversy about the actual or potential infinity of space. At the same time, it calls for a specific type of insight that does not bring space to consciousness in the same way as objects or things, but rather allows space to remain in its phenomenal uniqueness, the mode of givenness which is inaccess­ible to closer description.

§ 2. The Problem of Empty Space

The previous investigations have shown that space is where corporeity is. They have also shown that as self-moving, corporeity constitutes space, and moreover it can first have space conceptually and objectively at all through consciousness of this capacity for movement.

In any case, the consciousness of the capacity for movement is not identical with the consciousness of space; in the everyday experience of space, this space is already conceived as pre-given and as independent of one's own movements. But the specific experience of space by natural consciousness has its own foundation in the manner and mode of possible movement. While the corporeal subject intuits itself as motile, it experiences at the same time that space is there in the movement, that each actual movement has space at its disposa1. As has been shown, however, this experience of movement is so ordered that each factual movement is directly and always known from the background of broader-more specifically, previously realized and yet to be actualized-possibilities of movement as a continuum of movement phases. Within this continuum, the movement just accomplished is only one phase-one singled out as phenomenally given in the mode of full presence.

In this context the experience of an already executed movement is significant. The essence of continuity of any movement includes not only the capacity of optimal "forward" progression, but also the experience that any chosen movement can be repeated "backward." This means, to say the least, that an absolute beginning of movement can never be given. In any case, there is a beginning of having moved, and there is a first instant of completed movement, but there is no first instant of being in movement. Each beginning of a movement is itself already the continuation of movement that in turn must have its beginning, and so forth. The experience of beginning of movement would destroy the experience of movement itself; phe-

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nomenally speaking, movement is only graspable when it is already movement.

But if all movement is "already" movement, then space must "already" be if movement is to take place at all. This allows us to understand why the corporeal subject in his natural consciousness or attitude conceives of the space of his movement as a pre-given existent. For him, space is something that takes place in, but not something that first becomes through movement,ao This conception is as little a delusion as his continuity of movement would be. That each factual movement already occurs in space corresponds experi­entially to the fact that movement has always and already consti­tuted space. Of course, we are in no position to grasp the beginning of this constituting activity. Indeed, the mere possibility of such an original determination would abolish movement itself and destroy space.

"In" space does not mean that space must necessarily be con­ceived in accordance with the view claiming that the relationship between the moving body and space is one of an encompassing emptiness containing a body. Space understood substantially as emptiness, existing independently of-although containing­things, is a relatively recent conception. lt should be emphasized that this characterization is not at all the datum of pre-reflective everyday consciousness of space. If one adheres strictly to what is given, then there is nothing that could readily be designated as a receptaculum rerum; rather, the latter is an aspect of a specifically modern theory of space (whose historical genesis will not be discussed here any further). Empty space does not exist in the thought of antiquity. For Aristotle, space was not conceivable without a body. The concept of an empty space was incompatible with his physics. lt is remarkable that Aristotle establishes his theory of topos not on the basis of speculative deliberations and intellectual constructions, but in relationship to his theory of movement. What is methodologically remarkable is that in his theory of space, Aristotle touches on the direct experience of everyday life more extensively than all subsequent thinkers. However, what is thematic in Aristotle is only the place (topos) of a body as its limit (peras). Place is a region of a specific nature, with its own power of attraction, toward which each body must tend in its natural movement. By way of natural movement, each body has an appropriate place in accordance with

80. This holds analogously for the experience of "still" in the future being able to move oneself for which there will be space.

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its essence ( eidos). Thus for Aristotle the determination of place includes the determination of essence. Indeed, the topos here is completely distinguishable from its various occupants. It is nothing like a thing, and in contrast to things, it has its own meaning-but only to the extent that the place has a dynamic influence on the body and the influence is not identical with the influenced, not to the extent that the place can exist independently and indifferently to the moved body.81

81. According to M. Jammer (p. 17), Physics Delta 4 210b 34-211 a6 should demonstrate "the reality of space." This interpretation is anachronistic or at least misleading in its mode of expression. When topos is discussed, and it is said that topos is periechon echeino on topos esti, chai methen ton pragmatos, all that is said is that topos is nota property of a body. Yet it would be a mistake to seek in Aristotle an intimation of the modern concept of space. Indeed, the place is separate from the body (choriston), but this detachability ought not to be understood in terms of the later conception of an empty, indifferent space in contrast to things; rather, as is stated in Physics Delta 1 208 b4-11, it must be conceived as a distinction between thing and a place ( . . . tonto on ton eggegnomenon chai metaballonton eteron panton einai dochei ... ). This is to be interpreted in terms of the Aristotelian theory of movement; the "self-sufficiency" of the topos in contrast to the body that finds itself in it consists of the influence of the topos on the thing: deJonsin oti esti ti o topos all oti chai echei tina dynamis.

Recently, in her interpretation of Aristotelian theory, H. Conrad-Martius has pointed out certain ambiguities and contradictions (pp. 109 ff.). Yet these, as well as the critical claims that Aristotle did not have a sufficient concept of space, seem to us to be groundless. lt would be inappropriate to accuse Aristotle of simply having a "false conception of physics," justas it is nonhistorical to miss a specific chapter in his Physics concerning "space" (as translation of chora), yet at the same time to accept that in the Aristotelian theory of topos "space is always meant there in a secondary way." In accordance with an Aristotelian way of working, the meaning would have been explicitly mentioned and analyzed if the term charos had had the specific meaning that is being attributed to it. The casual and non-terminological usage of this word *e.g., Delta 1 208 b7 and 32 suggests that it did not mean something like space "in contrast to" place in the sense of modern terminology. lt is remarkable that in this sense the Greeks did not have an expression for space. That Aristotle speaks of topos and charos but not of space, that he does not have this concept of space, characterizes his "space"-conception. Similar misconceptions are found in J. Cohn, who accuses Aristotle of "lacking a useful concept of space" (p. 41) and traces this lack of understanding to the Zenonian aporia, claiming that Aristotle "has not made space sufficiently clear" (p.42). Indeed, Cohn thinks he can

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This opposition between the space-cosmos of antiquity and the empty space of the modern physical sciences constitutes by itself an essential point for any theory that wants to assure for the latter space the dignity of an apriori necessity, of a timeless essential arder of pure consciousness of something similar. Here to raise and to decide the quaestio iuris in favor of one or the other conception would be a senseless undertaking. The only legitimate attempt is to illimunate how it is possible to think the substantivization of an empty space, which-as already mentioned-does not even become thematic for natural consciousness. Yet its thematization in science may appear to be motivated by the natural consciousness of space. These motives may once again be traced back to the originary space-constituting strata of corporeal subjectivity.

It would be beneficia! to recall the originary mode of activity of corporeity that founds the primordial spatial relationship through the constitution of inner and outer. It turns out that touch is not only responsible for this relationship, but that it is also constitutive for any possible sense of emptiness.

Touch reveals the unique mode of functioning of corporeity insofar as this is essentially related to its movement. It is not as though this capacity does not "belong" to other sensory functions; yet here it appears to be most closely connected to a function of performance, since it can create a manifold of theres only in actual self-movement. What is more important is the reversal of this state of affairs: even if not given thetically to consciousness, each singular phase of motion of the corporeal body constantly includes a touched there and there. The reciproca! relationship between the tactile manifold of places and self -movement illuminates the continuity of this manifold; while this manifold is constituted through self-movement, it is presented as nothing other than the continuity of this self-movement. This means that there is no place in the surrounding field of the touching corporeity that is not basically tactile, i.e., that could not contain tactile matter. At the same time, every phase of its movement and constellation of its limbs has in principie the possibility of encoun­tering something tactile. Each kinaesthetic sensation from within can come into coincidence with an external tactile impression. The con­tinuity of movement allows for the fact that the manifold of theres constituted through it is never discrete.

show that Aristotle has committed a "basic error" of "confusing space with body" (p. 114).

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Only one thing is essentially excluded: the manifold cannot be seamlessly filled with the tactile "object." Such filling would be equivalent to the destruction of the lived body and of the tactile space itself. That this continuous filling never takes place is not a metaphysical accident; rather, it is the touching lived body itself that in its movement first grounds the manifold of theres. The continuity of the tactile space does not mean that it is solidly filled in; it means rather the capacity in principie to be filled in the succession of constituted tactile manifolds. Thus tactile space yields the phenom­enon of emptiness, more precisely that of the empty place. Tactile emptiness does not merely presenta self-sufficient datum insofar as it is not a visual or in sorne other way determined suppositum; rather, it is also that which founds the significance of any concept of spatial emptiness. 82 Insofar as the lived body becomes a physical­bodily corporeity and thereby founds the primordial spatial relation­ship, the lived body at the same time discovers itself in space. This relationship necessarily contains emptiness as the "space" for the possibility of corporeal movement. In its ontological origin, space thus determines itself essentially as empty space.

The emptiness founded in touch is nevertheless only "in­between-space." It is not an absolute emptiness preceding objects, but is relational with respect to things. In a continuous change of place by things, each empty place can be grasped as a possible location for an object and conversely, each location may be regarded as a possible empty place. Yet it may mean that this is all a matter of "mere" conception which can arbitrarily focus on the one or the other aspect. Or, to speak with Aristotle, it may appear as if the matter is left to a conceptual choice as to whether we see space as a totality of places or whether this totality is primarily obtained from empty places "for" possible bodies. These two conceptions are not equivalent. They become equivalent only when in the first instance the Aristotelian topos is divested of its specific place character, and when the second case explicitly assumes the additional notion "for possible bodies." The conception of the equivalence of all places demands a fundamental transformation of antiquity's theory of movement. This was done gradually in later physics. It was the

82. This is most clear in the visual field which lacks emptiness. Without disruptions, color joins color, form joins form, and the boundary of one figure is already a boundary of another without there being "space" between any two of them. The latter emerges only where not the field but space is grasped, i.e., in the movement of things.

The Space of Movement and Objective Space 165

Newtonian conception of gravitation lhat ultirnately offered a strict formulation of a law. However, the strictness was "only" postulated, without experientially revealing the uniform rnovement of gravita­tion of a thing. The thing had to be conceived specifically as a "physical body" in arder for the Newtonian conception to be an adequate expression for the strict homogeneity of space.

The equivalence of all places for movement expressed here simultaneously leads to the possible conception of its indifference to the things. If location and empty space are no longer distinguished in terms of rank, but are only distinguished factually through the existing constellation of things, and exchangeable with its changes and if, in addition, the distribution of things is constantly discrete while their movement is continually cohesive, then their places must appear to be no more than momentarily "occupied" places of a continuum which, taken by itself, is regarded as completely empty.

Yet actually it is not this emptiness, conceived "in terrns of an interconnection with the world of things and their movement," that leads to genuine difficulties. What is problernatic is its absolutiza­tion. This conception assumes an empty space preceding and existing independently from all thinghood. While the role played by this "absolute" space in modern physics was tried-even Newtonian mechanics did not use it-it has obviously been influential, through Kant, in determining the scientific consciousness of the nineteenth century. Kant thought that if we were to think away everything from space, the latter could not be thought away and hence exists for itself. Indeed, Kant attempts to defeat the realistic conception of absolute space through his Copernican turn-yet what is significant is that even in his transcendental-idealistic conception of space as a pure form of intuition, space is still conceived as an independent emptiness preceding all things.a3

Methodologically speaking, it is remarkable in our context that Kant starts with the conception that "one" could never imagine that there is no space while at the same time "one" "can think that there are no objects to be encountered in it.'' Obviously Kant is arguing here on the basis of natural consciousness and develops his proof for the apriority of space by criticizing this consciousness. Yet this point of departure, attributed frequently to Kant, is by no means self-evident. And if "one" would wish to claim that Kant meant natural everyday extra-scientific consciousness-Kant himself does not use these con-

83. I. Kant, Vol. III, Critique of Pure Reason, "Transcendental Aesthetic," Section 1, p. 2.

166 Lived Space

cepts-then what is attributed to it by Kant as a content of represen­tation could at least be contested phenomenologically. In any case, it is clear that since then, this "container fiction" has been presented as an inviolable possession of the natural consciousness of space­At the same time it is also strange that phenomenology, educated in the subtle analyses of the contents and modes of consciousness, sees this "container-conception" asan indubitable phenomenological da­tum of consciousness.84 It seems that here phenomenology has uncritically posited as phenomenal "givenness" what one, so to speak with Kant, can only think. And this is possible when one, like Kant, "thinks" of the Newtonian space of physics-but not when one traces precisely what is phenomenologically manifest in extra­scientific space-consciousness. Even where space is given thetically for this consciousness, it will not be given asan empty space in the sense outlined above. There is a clear distinction in meaning as to whether things, moving freely in space, permit a conception of emp­tiness on the basis of their free movement-i.e., its determination is attained privatively from the world of things and becomes a spatial residuum so that this emptiness assumes a basically relational char­acter-or whether one attributes this emptiness as a positive deter­mination of space, existing independently of all thinghood, and lends ita substantial meaning. As already presented, the latter is a scientific conception whose basic motive is found in the natural consciousnéss of space, but which is nota conception of this consciousness. The container fiction, as a phenomenological datum of the natural space­consciousness, is itself a fiction of the phenomenology of natural space-consciousness.

This does not remove the primary difficulties. If space is con­ceived as a container of things and hence itself seen as a thing, then it must be in space, etc. Such a conception would lead toan infinite regress. Yet this conception, attained through a process of iteration and leading to absurdity, is nowhere to be discovered phenomeno­logically. Whenever it is advanced, it has to do with a polemically exaggerated "fiction" of a reflective critique attempting to replace the "mere" unreflected natural consciousness of space with a "true" conception of space-without, of course, explaining its appearance. Thus it fails in its obligation.

But how, in the final analysis, is the relationship between a body and space presented? What is truly found phenomenally in natural space-consciousness?

84. E. Husserl, Ideen 1, Ideen 11; M. Scheler (3); N. Hartmann (3).

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Primarily, this consciousness readily distinguishes the spatial determinations of the thing from space itself. For it the thing is not identical with where it is found. If this "wherein" were to be regarded as a kind of a more encompassing thing, then there would be no information given by the thing concerning what accommo­dates it, apart from the sole judgement what the thing is "in" it. In turn, however, the previous analyses as a whole were oriented toward spatial things and their distinct modes of being and, correl­atively, toward the specific modes of movement of corporeity required to read the structure of space from them.

The sole fact that space is indeed graspable with things already shows that even natural consciousness is concerned with the rela­tionship between thing and space, a relationship that is fundamen­tally distinct from one obtained between two things. The latter mode of being-in is determined by a relationship of two or more things; containment in-in accordance with its intuited fundamental mean­ing-is an inter-thing relationship. Yet the latter must already be spatial, must be in space; a containment of one thing in another already presupposes space as a foundation. That which founds a relationship cannot be one of the relata. Although graspable "on" the basis of the given interrelationships among things and determinable "from" it, space is not this interrelationship itself. We speak of extension, form, size, and movement with respect to things, and location, place, and region accrue to them. Yet space itself does not possess these characteristics; it is neither extended in accordance with position and place, nor does it move. Rather, its own structural characteristics will be attributed to it, although they will not coincide with those of the spatial.

In our discussions, the concept of representation was used in various ways to designate the relationship between thing and space. It seems that in contrast to the previously ennunciated characteriza­tions of the relationship, and with its closer delimitation, this concept is more appropriate to the phenomenon in question and simultaneously it avoids sorne main difficulties. Generally speaking, the relationship between the representing and the represented is neither between something and that which encompasses it, nor between a form and its content, both taken to be independent; rather, it is a relationship unique in kind.

It implies that space, represented by a spatial thing in it, is not given immediately, in "the flesh"; at the same time this lack of presence "in person" does not mediate space as if it required sorne kind of intermediary conceptions or indeed.modes of deduction in

168 Lived Space

arder to get to space from the side of things. Rather, space is "there" immediately with the spatial things. It is presented through things and given only as presented "through" them. It is apprehended and must be apprehended solely in them. It is not given in any other mode apart from being represented in spatial things. It has its being essentially in being represented through its contents, justas they are spatial only through it.

This avoids a substantial distinction between space and thing. The inherence of things in space is thus different from that of contain­ment of one thing in another. Things are not just in space, space is also in things. Both have a meaning of inherence that must be non-spatial in kind. Hence the mode of space as being in things enables the reified spatial containment of one thing in another. This corresponds with the assumed relationship of representation; the relationship between that which represents itself and the repre­sented is in fact not a spatial relationship.

In our opinion, only this and nothing else ought to correspond to the phenomenal composition of natural space-consciousness. But when one attempts to conceive of space as open-endless extension, this infinite space is constantly a space ofthings conceived as linear, unlimited and mobile, but not a space conceived "primarily" as empty and "then" as filled with things. Even here space can be only if it presents itself in things and thus is graspable in its structure only from them. The state of affairs is thereby established that space presents itself in things-and this "that" must be taken and themat­ized as its own existence and mode of being. But it is one thing to take space as space of things in simple infinite progression, and another to take it as a substantial, infinite emptiness and to think it independently of things. In the latter view, space appears as a result of an abstraction whose ultimate foundation, as has been shown, can indeed be found in the corporeal-sensory phenomena. This abstrac­tion, as has been obvious in the historical development of Newton­ian mechanics presupposes, nevertheless, a fundamental transfor­mation in the conception of a thing. This transformation leads from the direct perception of an "intuited thing" to a specifically be­stowed meaning of "physical body," with all the constitutive deter­minations alien to the simple thing of intuition (regularity of relationships in terms of mechanical causality, reduction of qualita­tive characteristics to quantitative, etc.).

One can quarrel about the physical usefulness of such a space and reject its existential justification for physical theory on the very grounds that this theory stems from it, thus demonstrating that it is

The Space of Movement and Objective Space 169

superfluous. But what is decisive in this context is that geometry, as a science of space, first became possible with the conception of empty space as such.

In its transition from the Egyptian art of calculation and measure­ment, where the purely mathematical and intuitive are completely interwoven with the quantities derived from things, to Greek math­ematics and its method of proof, the history of mathematics offers an impressive transformation to the strict mathematical thinking. Nev­ertheless, in Greek science, geometry could not have been concerned with space itself. The irreconcilability of the ancient conception of space-which found its culmination in the Aristotelian theory of dynamism-with the consequences of the geometry of Euclid­leading to a purely mathematical, homogeneous, and infinite space-is perhaps ultimately the reason why euclidean geometry was not also conceived as a geometry of "euclidean space." Further development was required befare geometry was not only concerned with geometrical formations in space, but constituted the concept of an independent mathematical space.a5

§ 3. Concluding Observations on Lived Space

The discussion above has reached the limits of what belongs to the problematics of the phenomenology and ontology of lived space. Concerning the latter, let us offer brief conclusions with respect to the most important results within the framework of our investiga­tions.

Space is, insofar as there is corporeity. Originarily, the latter is neither in space nor outside of it; rather, corporeity is spatial in the mode of space-constitution. The being of space is relative to a corporeal being: in its given structure, it points to correlatively given formations and concretions of corporeal movement. But space is not exhausted in this correlation. The correlation of corporeal function and spatial structure is intertwined with that of corporeity and consciousness. Beyond the corporeal correlations, the latter has a characteristic of being an intentional consciousness. In accordance with the intertwining of this twofold correlation, the space of movement and objective space present two abstractly distinguish­able, yet de facto mutually conditioning moments of space­constitution. In its full concretion, space is ontologically relative to

85. An essential although brief summary of this development is offered by J. O. Fleckenstein.

170 Lived shace

a corporeal subject whose being consists of the ambivalence of corporeal being and the transcendence of corporeity by an inten­tional consciousness. If space is not exhausted in its being related to a lived body, it nevertheless exhausts itself in being a space for a corporeal subject.

This relativity does not designate a mere epistemological relativ­ity. That space as an object of knowledge is related to a subject of knowledge is a tautological statement-that in its being it is related to the being of a corporeal ;ubject and is limited by such a relationship is a metaphysical statement of vast importance. It includes a departure from ontological realism, which attributes to space an existence absolutely independent of the subject, such that the relationship to a subject would be completely additional, exter­nal; space would have its being befare all consciousness, being what it is whether there is consciousness or not.

The very conception of space could be understood only as a depictive repetition in consciousness. With this is bypassed the demonstration of an image-consciousness which must still be searched out in a theory oriented phenomenologically to the lived content of natural consciousness; this conception would in addition have to explain how the space of intuition, seen merely as a subjective distortion of the homogeneous space, is possible as the space of senseful comportment and meaningful corporeal orienta­tion. Moreover, a further problem to be solved by the theory suggested is the following: the depicturing relationship of image and object is transposed into the relationship between knowledge and space, a relationship that is itself spatial and thus presupposes spatiality.

That these difficulties can be circumvented by strict adherence to the claims of natural consciousness and what it irrevocably pos­sesses should have become clear from the preceding investigation. It did not contest the claims of "natural realism" with respect to the conception of space-it contested it so little that it sought to legitimate such a realism by providing an insight into its motive and origin. Yet a retrogressive movement to its transcendental conditions leads in turn to consequences incompatible with ontological realism as a philosophical position. To attribute a transcendental turn to an idealistic metaphysics could hardly be justifiable exept under un­conditional maintenance of one of the traditional alternative schemas whose validity should be discussed with respect to the point of departure of this investigation. D~spite the differentiation of their metaphysical explication of

The Space of Movement and Objective Space 171

being, idealism and realism have their starting point in the problem of the externa! world that transcends consciousness. Yet if they are to be understood as final products of phenomenologically-descrip­tive efforts, represented most recently by the theories of Edmund Husserl on the one hand and Nicolai Hartmann on the other, they remain related less in the diametrical opposition of their explication of phenomena than in the place of origin of their search for phenomena. Both assume the (thetic) consciousness of being. In one case, being as being-phenomenon is a constituted being in con­sciousness while, in the other case, the being phenomenon is precisely a phenomenon of being, i.e., is taken as being in itself. For Husserl's constitutive idealism with its bold claim of "eo ipso being the true ontology," it is valid to say that being's transcendence of consciousness, as a mere transcendence of consciousness, is more precisely an intentional transcendence "in" consciousness. Con­versely, for Hartmann, who claims equal subtlety of analysis of phenomena, all possible positions apart from the self-evidence of realism are excluded at the outset-and, on the same grounds as Husserl, Hartmann claims that the transcendence of being in con­sciousness is a factual transcendence.B6

We shall not trace here the basis that ultimately leads both to one-sided interpretations of being and of being "in" consciousness, and to prejudiced standpoints. Although initially they do not admit their theoretical positions, the subsequent formulations of their meta­physical conceptions appear more like mere explications of positions held from the start. It is of little use to pursue here the question whether the alterna ti ve between idealism and realism can be solved­or whether each attempt to give a final and valid decision concerning this question must founder on the aporia of intentionality.

We turn now to our problem: whether space has a being transcen­dent to consciousness while as transcendence it is a phenomenon of consciousness, or whether while being a phenomenon of conscious­ness it has this transcendence only in consciousness-i.e., to speak with Husserl, this transcendence can be attributed to space only as a being-characteristic immanent to consciousness. This question seems to us to be completely aporetical.

Yet it could be asked whether this aporia is merely an expression of a more fundamental question, and whether the position from which the aporia stems is inappropriate for a solution because its way of posing the problem comes too late. For the subject who raises

86. See Husserl, CM, §64; concerning Hartmann (1), pp. 152 ff.

172 Lived Space

the question of the reality of space-and at the same time places the reality of space in question-has already acted in space, has under­stood himself in terms of space, not primarily in the naive certitude of his existence, on which his "natural realism" is founded, but rather in a more fundamental mode and manner wherein the certitude of and belief in reality have not yet come into play. In an originary sense one cannot be "certain" of space-and it is only when one becomes certain of space that it can be "real" or "tran­scendent." It is only when space-consciousness, with its beliefs and theses, attributes certitude to space that there can be such a thing as asserting, doubting, contesting and defending its reality, or "prov­ing" and "deducing" its transcendence.

Thus each undertaking of the latter kind-which begins with the question at the level of the problem of the consciousness of space, both in realistic and idealistic terms-is left hanging in the air. In both there is a lack of a secure foundation for an appropriate posing of the question. Modifying the concepts of consciousness and of the subject, which stem in both positions from traditionally distinct styles of metaphysics, does not lead to any advancement of the problem of space as long as the subject is not sought where he has his originary relationship to space-in his corporeity. Only a complete phenomenological illumination of this relationship and the return to its constitutive strata offers a hope of grasping the question of the reality of space as such, as a pressing question of a subject who, by virtue of his reflective capacity, is compelled to ask it. Because of this very capacity, he must self-critically trace the forgotten primor­dial questions and pose himself the problem that unavoidably confronts him as a corporeal being.

PART TWO

MATHEMATICAL SPACE

Introductory Remarks

With the discussion of mathematical space, the investigation proceeds toward a new and truly complex realm of problems. The previous analyses began with the directly given comportment of the corporeal subject in lived space. We must immediately point out that in contrast, that which is here delimited as the "given," in the sense · of geometric entities that comprise the new region of phenomena, distinguishes itself from any objectivity given directly and "in person." lt is conceivable only in a specific mode of access adhering to strict methodological requirements. In any case, even such a scientific-methodological approach toward a sphere of objectivity, though fundamentally distinct from all immediate orientation to­ward the world as given "in person," does present a genuine-­although genuinely mediated-mode of comportment of the subject. The modes of the subject's performance contain the conditions for constructing mathematical theory. Only they can clarify what is meant by the expression "geometrical existence" found in the terminology of positive science. They also permit phenomenological investigation of the specifics of the geometrical mode of inquiry, of the sense of geometric being and the unique ontological meaning of mathematical space.

The method, therefore, requires an analytic and all inclusive clarification. We must not only clearly distinguish the characteris­tics of this method with regard to its axiomatic-deductive compo­nents, but also bring into view as a whole those performances of the subject that are constitutive of geometrization as such. Thus the task of the investigation is to differentiate between the various kinds of intuition, as sensory and symbolic, pictorial and significative intu­ition-between various types of thinking as activities of abstraction with specific forms of idealization, generalization and formalization. The ontological meaning of what exists geometrically emerges only gradually; at first it remains covered over by a network of provi­sional activities belonging to the clarification of the entire meaning of geometrical being, which thus ought not to be overlooked here.

Hence it is unavoidable that the following investigation is drawn into the field of history. No special demonstration is required to show that the mathematical or mathematizing consciousness changes and is historically conditioned. That its objectivity is ideal, that it is constituted as timelessly valid, does not abolish the historicity of scientific consciousness; rather, it indicates the dis­tinctive problem with respect to its constituting activities. In this work the activities cannot be traced in their specifics. Their ques-

174

Introductory Remarks 175

tions are general questions concerned with the constitution of meaning, and hence cannot be limited to the geometrical domain; to deal with them would require a comprehensive special investiga­tion. Nevertheless, it is to be maintained that contemporary work in mathematics has its motivation and aim in historical roots reaching far into the mathematics of antiquity. A phenomenological descrip­tion of mathematics cannot avoid the historical tracing of such activities. Their historical dimension cannot be overlooked, since they belong to a region of phenomena, the scientific region, that is historically constituted in these activities. The phenomenologically oriented retrogressive questioning is not valid for the historical changes of scientific facts comprising the history of science, but rather for the origins of the meaning of scientific entities founded in the achievements of meaning by mathematical consciousness. To explícate and clarify these achievements, to reactivate the bestowal of meaning sedimented in the course of history was, it is worth mentioning, sketched out by Husserl in a late work as the task of an "intentional history" of geometry. It is worthwhile pursuing this task even within a modest framework. Here the concept of geometric existence is to be clarified phenomenologically along the rough lines of its structure of meaning.a7

87. Husserl's "intentional-historical" aspect must be understood in a way that would allow phenomenological retrogressive inquiries to reach not only to the foundations of sense of the scientific edifice of geometry, but also deeper, toward the primitive levels of sense constitution; it ought to uncover the "primordial materials" residing "in the pre- scientific lifeworld" (["Die Frage nach dem Ursprung der Geometrie als Intentionalhistorisches Prob­lem"], p. 219; ff. the foreword by E. Fink and the variant text in the Crisis volume, Appendix III, pp. 365-86).

In accordance with this Husserlian idea of history, our previous research concerning lived space may easily appear as a partial fulfillment of his program. An extensive exposition of Husserl's conception of intentional "history" would lead us too far afield. We stress once again that what has been accomplished with the three spaces has validated neither "historical" nor "archaeological" points of view (in the Husserlian sense of founding stratum and building up of sense mentioned above). Attuned space is neither a primordial foundation for the spaces of action and intuition, nor are the latter in the former, i.e., "based" in it. Rather, we had todo with the three spaces as different modes in which the one actually lived space is present for a corporeal subject correlated to them. Genuine questions of sense-founding first emerge for us in the following pages, with the transition from the natural space of objects to mathematical space.

SECTION ONE

Preliminary Phenomenological

Observations

Chapter One

Space as a Thematic Object of Consciousness

§ 1. The Space of Intuition as a Limit Case of Lived Spatiality

With the space of intuition, the constitution of the lived space is complete. Corporeity finds in its intuitive comportment a demon­stration that as a lived body, it is at the same time transcended toward its form of existence as a consciousness, in the sense of a nexus of intentional and uniquely objectifying acts. The intention­ality of the space-positing consciousness, which is always and irrevocably the consciousness of a corporeal subject, includes the ambivalence of a spatial existence as well as the breadth of the science of the comprehension of space. It conditions the double mode of appearance of space. This results in the subject's self­experience as a corporeal being encompassed by space and, at the same time, as a conscious being who has space unbridgeably "over against" himself insofar as he makes space into an object. His "position" in relationship to the intuited objectivities in space manifests the limits of his corporeal comportment in this space: he is no longer among things but has moved completely to the periph­ery of space. The lived body no longer properly constitutes a center, encompassed on all sides by space, but becomes completely

176

Space as a Thematic Object of Consciousness 177

excentric. It has a wodd of things only as re-presented, which, in their being "over against," reveal their pure object-characteristics.

As a pure object-space, the space of intuition acquires all those marks characterizing it as a limit case of lived space in the sense already suggested: it is the space of a lived body insofar as its fullness is intuited sensibly and constituted corporeally. Yet it contains at the same time determinations incomprehensible solely from the understanding of the subject's corporeity. The space of intuition did not turn out to be a self-ctmtained part of a spatial whole that could be thought as composed of the sum of such parts. Rather, it appeared to be the manner and mode in which the one objective space is already there for a corporeal subject. The co-presence of this space also appeared to be constitutive for sensible-corporeal phenomena, given "in person," "in the flesh."

Thus the space of intuition ceased to be merely the space of a corporeal comportment, the space of a lived body in a situation. From the very origin the comportment of the corporeal subject in space bears the mark of a different kind of possession of space. The establishment of space as an objectivating positing which provides the basis for the judgement that space "is" such and such, first makes space predicable, a subject of valid propositions. This establishment presents a unique, although neither obvious nor necessarily trans­parent, achievement of the objectivating consciousness. It estab­lishes the condition that enables the spatial subject to know and to designate himself; it makes the awareness and knowledge of the concrete situation of my lived body in space possible in the first place.

At the same time, in arder to be able to be claimed as mine, this establishment would have to justify itself befare each similar space­positing achievement through the other consciousness, were it not for the fact that it is the positing of existence of "the" space. It is neither by tacit arrangement nor by explicit agreement that the thesis of "the" space means an identical space for all corporeal subjects. My consciousness of space is mine on the basis of the existential unity of my consciousness with my lived body. As signifying, as objectivating, it simultaneously stands for the other consciousness and indeed for "any" other. Yet in this case it appears in a mode of universality that even transcends intersubjectivity. While for the latter the differences between unanimity, majority, and minority have not yet become completely meaningless, the concern here is with a form of universality in which any numerical moment is completely abolished. It is basically this universality that designated

178 Mathematical Space

the "pure" consciousness of the rnuch contested "consciousness as such." This is by no rneans conceivable without the inclusion of corporeal conditions and facticity and, indeed, it is only in conjunc­tion with thern as a structure of factual consciousness itself that it can be grasped. It rnay not be thinkable asan independent structure, floating, soto speak, above the individual beings, yet it can becorne cornprehensible purely functionally in terrns of its specific inten­tional activities as a corporeal, individual being. Nevertheless, even so conceived, pure consciousness presents a givenness that cannot be discredited. Deterrninations of its own can be phenornenologi­cally brought to light in contrast to factual, individual conscious­ness, as well asto all intersubjective universal consciousness. It has its history in its specific universality as the identically functioning intentionality of "all" corporeal subjects. Moreover, the totality of subjects that it subsumes rnay require delirnitation against possible and entirely different consciousness-structures. Y et all of this should not rnitigate against the application of a concept which, though rnetaphysically laden, nevertheless appears unavoidable for an ap­propriate explication and phenornenological analysis of the problern to be unfolded subsequently.

With this universal consciousness-in its ontic unity with a lived body on the one hand and, on the other, in its own specific functionality-possibilities are opened in principie for the corporeal subject. As already shown, the extrication of the one objective space already underlies each of his corporeally concrete rnodes of corn­portrnent and is co-deterrninant even for every structure of lived spatiality. The expressively attuned and the circurnspectively acting subject determines his space no less than the "theoretical," rnerely intuiting subject on the basis of a previously accornplished positing of "the" space; and the way traced by us above is more akin to a circle that incorporates its end and its beginning. What allowed us to speak of the "levels" of lived spatiality was rnerely the increasing degree of openness and clarity with which the one objective space carne to consciousness (thetically) for the cornportrnent of a partic­ular subject himself in a given dornain of his activity. And the space of intuition rnay be designated as a lirnit phenornenon to the extent that it reveals deterrninations that could not have been grasped without the assurnption of objective space in the phenornenological description of corporeal cornportrnent in the space of intuition.

In the phenornenal dornain of lived spatiality, however, objective space plays only the role of an inactual horizon of consciousness. It is "there" for an experiencing, acting, and sensibly perceiving

Space as a Thernatic Object of Consciousness 179

subject; it is "co-given" without any independent intentional glance being oriented toward it frorn arnong these world-attitudes. Where· such an intention occurs, there appears a new way of regarding space and a new view of the space-problern. The change from the mere co-presence of objective space to its full thetic-actual givenness indicates the fundamental division between a corporeally engaged subject and theorizing subject, between corporeal functions and categorial achievernent as capacities of variously constituted and predelineated forrnations in the essence of subjectivity.

Thus the one objective space becomes open for the many possi­bilities of topological and metric deterrninations. Its subsumption under a science whose laws are of a particular type, constitutes one of the major problems of applied geometry. Viewed from such a science one would have to discuss the question as to the provenance of the clairn concerning the euclidean nature of the space of intuition. From the standpoint of pure geometry, which is the sole dornain of discussion in this investigation, the euclidean determi­nation of measure will turn out to be one among many, having no privileged position within the framework of formal science. Our task in this part will be to show how the constitution of geometric manifolds is to be understood on the basis of the comprehension of space previously discussed.

§ 2. The Topological Structure of the Space of Objects

Objective space functions as the substratum of mathematical determinations without itself being a mathematical space. The space positionally assumed in natural consciousness as "real" space is primarily without metrics. It is still on the hither side of metrics and as such has a series of structural properties that are in no wise specifically mathematical.

The uniqueness of this space, in contrast to all lived spatiality, appears most clearly in its isotopy and homogeneity. In its absolute oppositionality to the subject it is without any firm directional determination, without specific valence of place. In complete counterdistinction to the spatial cosmos of antiquity, articulated in accordance with center and periphery, the space of our contempo­rary consciousness of objects does not accord any value to its locations: it is apure system of "places."

Its homogeneity immediately raises the question of its topological structure. It was already mentioned that the concepts of endlessness, boundlessness, and infinity are topologically ambiguous. In the

180 Mathematical Space

phenomenological section of the previous part an exact differentia­tion among these concepts was deliberately avoided. Whether space is open-endless, or whether its infinity is to be understood as inauthentic, in the sense of boundless yet closed connectedness, was still of no importance for the problems encountered there. Moreover, such a distinction is lacking in a consciousness of space for which the one homogeneous space is indeed "there" but only in the mode of co-given presence. Phenomenologically speaking, such co-given­ness appeared only as a going-over-further of space beyond the intuited horizon.

lt is only where space becomes thematic in objective conscious­ness that topological differences first become relevant. It is illumi­nating that when the homogeneous space of objects is conceived as a manifold of empty places, it can in principie be open-infinite, at least as a noncontradictory possibility of thought. This is also valid if for sorne reason the corporeal world should be regarded as finite. During our preceding investigation of the empty space, we men­tioned that, and precisely how, the space of this topological structure dominates modern consciousness, or at least its scientific-physical­istic thinking. For the newly emerging problem, certain trends of thought must now be mentioned briefly in arder to show that the infinity of space is not merely assumed; rather, an attempt will be made to offer an insight into this structure of infinity on the basis of phenomenologically sufficient grounds.

Husserl had attempted to conceptualize this infinity of space by analyzing determinations that are characteristic of the singular space of intuition.88 His notion of the infinity of space, which delimits phenomenologically the sense of Kantian idea, ought to be under­standable from motivational interconnections originating with the immediate experience of things. If the experience of a thing is in principie inadequately given by a limited appearance, then, accord­ing to Husserl, the complete givenness of the thing is prefigured as an idea, as a system of endless processes of continuous appearing in whose constant transition one and the same thing "is more precisely and never 'otherwise' continuously-harmoniously" determined. This continuous perception of a thing, determined "more precisely as infinite on all sides," also includes all spatial adumbrations of the thing as "movable in infinitum; thus with it we grasp the idea of space, or more precisely, the idea of its infinity. For Husserl this idea is found precisely in the factually inadequate grasp of the complete

88. See Husserl, Ideen 1, §§ 143-149.

Space as a Thematic Object of Consciousness 181

objective unity of a thing, a grasp that is only possible as limited and thus must be constantly thought as open-infinite.

A critical pursuit of this argument reveals its inadequacy for the problem under discussion, precisely because of the unnoticed am­biguity in the Husserlian concept of infinity. It is quite thinkable that each continuous transition through the various appearances of a thing is in truth not determinable as open-endless, but rather as reiteration. In contrast to genuine infinity, it may contain merely an endless repeatability of a closed and thus in principie completable series of thing- appearances. This in no wise contradicts the "idea" of the adequacy of a thing as such.89

Similar reservations were already voiced with respect to the thesis of Scheler (pp. 157 ff.). In fact, it seems that despite the substantive difference of their starting points, the argument of Husserl and Scheler formally have the same conceptual structure: both maintain a factually limited possibility which, nevertheless, is actually inex­haustible. It is of the type in which further possibilities always remain continuously "open," i.e., "open" in the sense of the genuine infinity. Yet it remains unnoticed that the openness here meant as a possibility-either in grasping things or in corporeal movement­and as a possibility in the only proper sense, namely, that of the unfulfillability in principie of a claim on the part of the finite, corporeal being, cannot lead to a univocal decision between two completely different, and in this difference "open," possibilities. That is, there is a residuum of undecidable possibilities stemming from the underlying interconnections of phenomena. It is precisely because all factual experience of a thing terminates in a finite series of appearances, and precisely because all factual corporeal move­ments inevitably constitute only a limited space, that they leave

89. A precise phenomenological analysis could, as a matter of fact, show that Husserl's infinite-continuous succession of partial intentions and respectively of their syntheses of fulfillment is only a closed-endless succession. This is related to the fact that each thing is in principie movable in infinitum. Yet for the complete display of its adumbrations, one requires rotations; translations alone are not sufficient. At this point, Husserl has not availed himself of one of his earlier insights. Already in the third chapter of his Sixth of the Logische Untersuchungen we find that in the exposition of the characteristic features of "fulfillment" within the more extensive class of identification, there is an insight into an "aimless" identification in the perception of the thing without a progressive approximation toward an epistemic goal.

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open in principie the possibility for two basically distinct topolog­ical structures at large.

Another solution is offered by Carnap. Beginning with the limited space of intuition, he regards the open infinity of the homogeneous space of objects as a guarantee for the "required" extension of the space of intuition.90 This conception is indefensible insofar as it fails to do justice to the fundamental fact that the topological character of space as a whole does not depend on the arbitrary choice of the space-apprehending consciousness. Space as it is in the natural consciousness of objects neither allows a choice of connexus nor does it leave this connexus in darkness so that something definitive could first be made of it by a specific kind of requirement. Rather, the open infinity of space is there and is unavoidably present in the already characterized manner in which it is "given."

Nevertheless, Carnap's claim can be maintained phenomenologi­cally. It is based on the justifiable insight that the infinite structure of space-even if seen topologically as one among various possibil­ities-is in a specific way a distinctive structure. If we revert to the phenomenon of horizon in the space of intuition (pp. 111 ff.), we notice that the clearly co-present "continuation" of space in the consciousness of the intuited limit, which first constitutes the horizon as horizon, attains its complete and genuine meaning only with the co-posited open infinity. This side and the hither side of the limit has here a univocal sense-fulfillment. The univocity is most clearly graspable with the phenomenal relationships of ordering. These relationships-such as the relation "between"-are univo­cally continuable only under the idea of a genuine infinity. That given three phenomenal points, one lies between the other two, constitutes a univocal arder in the space of intuition. Yet in a continuation conceived to extend beyond the horizon, the arder would lose its univocity if it were to be assumed that the line connecting them was closed. In other words, space as an open­infinite space is distinguished from any other topological connection in that "in the large" it is conceived to be topologically the same as

90. R. Carnap's concept of the space of intuition does not correspond strictly to ours insofar as he do es not accept this space at the outset solely as a space of the sensible concrete thing of intuition; rather-by appealing to Husserl's seeing of essences-he sees it in terms of geometric formations and relationships. This aspect is still lacking in our investigation, since for us this problem falls within that of geometric space. Nevertheless, what is decisive here is solely the required character of extension as such.

Space as a Thematic Object of Consciousness 183

in "the small," i.e., accessible as a segrnent of space to concrete intuition. 91 Only in this way can it be present to natural conscious­ness-indeed not as a requirement but rather as a factual and sensible continuation of the limited space of intuition. Thus the sense of that which was understood as a co-possession of the homogeneous space of objects in the individual space of intuition is now displayed from another side. If on the basis of the phenomena discussed in the first part the space of intuition had to be conceived in such a way that the one homogeneous space is there at the outset for the corporeal subject, then what has just been presented shows the reverse side of this state of affairs: the latter space can only be represented in the former because its connexus is congruent in all its parts with the space of intuition.

91. Subsequently, the all-pervasive planarity of space will emerge as its topologically predominant characteristic (see pp. 248 ff. of this work).

Chapter Two

Basic Trends of Mathematization

§ 1. Morphological and Mathematical Determinations of the W orld of Things

Philosophical thought about geometry reveals an ongoing contro­versy consisting of two main questions: one is concerned with the object of geometry and the other deals with the "origin." With respect to the first, the conception that geometry is a science of space must be countered by the fact that geometry deals with formations in space but not with space itself. The question of origin touches upon the genuine metaphysical problems of geometry; within the tradi­tion, apriorism and empiricism present two extreme attempts at a solution.

We are primarily concerned with finding an appropriate access to both problem areas from what we have already elaborated.

In arder to describe the world of objects, even in the space of intuition, the subject is led to a series of characteristics which, in accordance with their sense, belong to two completely distinct modes of observation. Characterizations such as notched, jagged, egg-shaped, umbelliform, etc., stand among others: triangular, circu­lar, spherical, cubical, etc., which are basically distinct from the first, owing to the fact that in contrast to the first, they are not purely morphological characteristics.

Such morphological characteristics-while unavoidable and even sufficient within the domains of knowledge that belong to them as, for example, the descriptive sciences-are merely "vague" determi­nations; they present themselves as "fluid" and admit of various displacements in the spheres of their application.92 As descriptive,

92. Concerning the concepts of vagueness and fluidity, see E. Husserl, Logische Untersuchungen, II, Prolegomena, § 21, and Ideen I, § 74. Let it be recalled that the Husserlian concept of vagueness is not identical in meaning with imprecision, but only with not-exactness. Its positive determination results from the above context. See also O. Becker (1), pp. 398-401.

184

Basic Trends of Mathematization 185

universal determinations they are extensional concepts and, as such, they maintain a horizon that is open in principie for possible real objects falling under them. Each singular object strengthens anew their sense of being a universal concept and the number of such objects is essential for the "degree" of universality attributable to these concepts. They are, furthermore, vague concepts because they are indeterminate in relation to other features belonging to the individuals that are apprehended under them. For example, objects with otherwise completely heterogeneous features are found under the concept of egg-shaped. The contemporaneity of these features is at a given time present only empirically and thus is merely factual, constituting a contingent aggregation but not a necessary intercon­nection. The morphological concept of the egg-shaped cannot be deciphered from these remaining features. The complex of these features, although present, remains outside of the range of the extensional concept and is not apprehended with this concept; it remains completely undetermined by it. At bottom, this state of affairs designates the language of abstraction, which purportedly leads to such concepts. On the one hand, it means an extrication of a singular feature and a "leaving," a "looking away from" all remaining features; on the other hand, it includes an "ascending" to concepts of a specific level of universality which grasp more objects in common. This fact is most likely to be visualized in the image of a pyramid of concepts, wherein the greater the extent of the concepts, the lesser is their content. Such a morphological concept, as an extensional concept, has its specification "under" itself but not "in" itself.

lt is otherwise with the determinations mentioned in the second place in the enumeration above. In accordance with their specific meaning, they are not morphological but-in a sense to be discussed in more detail-mathematical determinations. This does not contra­dict their morphological use in the space of intuition. After all, in the pre-reflective consciousness of sensory intuition the categorical difference of both types that emerges here does not at all come into view. That this is the case is nevertheless nota proof for the generic correspondence of all their form-properties or of a common subsumption under the superimposed concept of "morphological feature." The fact is that this common and consciously nondif­ferentiated usage of both form-types in the space of intuition must be understood from the side of the subject: with the use of these non-morphological features in the space of intuition, the subject turns out to be a participant in a differently construed domain of

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concepts and universality. These features are concerned with characterizations of the sensible contents by a being capable of comprehending types of forms different from those that are consti­tutive for the descriptive sciences, namely the forms of mathematics.

In sensible intuitive comportment, they are primarily and for the most part meant morphologically and are employed as extensional concepts. Insofar as they retain this meaning, the properties belong­ing specifically to morphological features are also valid for them. In this sense what is valid for the features of the umbelliform or of the jagged is correspondingly valid for the features of a triangle or a sphere. What is characteristic of the latter is that they are not exhausted in this meaning. These new determinations are not depleted by their morphological sense when such determinations are used for classification.

The fundamental difference between both kinds of determinations consists in the following: with those that are to be recognized as mathematical "universal concepts," the complex of all the addi­tional features is no longer completely indeterminate. Such concepts as, for example, triangularity, sphericality, etc., are essentially so constituted that with them are posited other determinations that have a necessary conceptual interconnection. In distinction to the morphological features, we have here a simultaneity of properties that are not present merely factually, to be apprehended through enumeration and thus conceived conjunctively only in this mode; rather, a specific 1ogica1 structure shows up: the absence of one of its remaining detreminations leads to the disruption of the entire contexture and the negation of it yields a logical contradiction. Any of such new concepts is airead y a ·conceptual structure with a completely determined logical structure; in this case it not only contains other determinations under itself in the same way as a generic concept would "subordinate" various species concepts, but contains them truly in itself. Hence it is possible to conclude the entirety of the structure from each of its members.

Their validity, therefore, cannot be strengthened by any new empirical discoveries, nor can their universality be confirmed by such discoveries. The concept of a square does not become more universal simply because more objects are discovered possessing the property of squareness. Either a present or imagined multitude of things displays all of the properties that logically correspond to the concept of a square-orthogonality and equality of length of the bisectors and of the diagonals-or the properties are not given. In the latter case, it would not mean that the concept of a square has

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become senseless or useless and has no "universal character"; rather, it would mean that those objects are not square. This reveals the fundamental difference between morphological and pure math­ematical determinations: the number of things lending fulfillment to the intuition of a mathematical "feature" is completely irrelevant for its meaning-content and does not dictate its universal validity. For mathematical concepts there is no "degree" of universality. The universality of mathematical concepts is not equivalent to that of the universality of genera.

Another related fact, not adequately considered befare, is that in the geometrical domain there is no strictly generic construction of its objects. Such a construction requires a graduated series of increasing or decreasing generality and specificity in such a way that the more specific contains the universal in a specific manner. The latter can be obtained by excluding the specific differences of the species, so that it appears to contain a lesser content in contrast to the specific and, conversely, the specification of the genus occurs in such a way that through the addition of specific differences there emerge sub-species richer in content.

In contrast, the relationships in the geometric domain are essen­tially different. The wholly elementary mode of defining sorne for­mations does indeed occasionally masquerade as a generic-logical structure. The rectangle, defined as as parallelogram with right an­gles, or the circle, grasped as a closed, curved line with a constant degree of curvature, etc., are apparently present as species under certain higher forms of genera.93 Yet it must be observed that what appears here in analogy to differentia specifica has an entirely dif­ferent form of "addition" than in the above case. Strictly speaking, it does not consist of an additive incrementation of something new, not yet co-present in the "genus," but merely in a specification of what was already present in the universal-and indeed, present variably. Let B = [B1 , B2 , .•• Bn] be posited for a geometric concept B with

93. Husserl also seems to maintain this generic scheme for geometry. As a paradigm, he mentions the triangle as an (eidetic) singularity under the highest genus polygon (Ideen 1, § 12). His polemic against Locke follows from this assumption (LU, Investigation 2, § 11). This is all the more astounding in that Husserl had already established the important distinction between generalization and formalization (Ideen 1, § 13). At best, this is explainable by the fact that in elementary geometry-to which Husserl refers exclusively and only occasionally-this differentiation is hardly formed. (In addition, see the following presentation.)

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determinations B1 , B2 , ••• Bn, with a resulting special concept B' = [B't, B2 , ••• Bn; B'1 , B' 2 , ••• B'm] stemming from B, not through the fact that additional determinations B't, B'2 ••• B'm· are simply added to the first, independently of those of B¡ (i = l ... n); rather, the geometrical specification takes place in such a manner that one each of B'¡ (j = l ... m) proceeds from an establishment of a demarcation of any of B¡. This specification can continue as long as new variations are to be encountered under the obtained new determinations B'¡, B"k etc. A most extreme particularization of this form results in the con­cept whose determinations are established completely through con­stants; for geometry this means through determinate numbers.94

What constitutes a "special case" in geometry is thus not the endowing of a universal with additional, new determinations, but rather specification of the universal itself by specific establishment of its various determining parts. At the same time, a geometric formation is capable of mathematical specifications-contains spe­cial cases of itself-insofar as it contains variables. Here the more universal concept is at the same time the richer in content. The universal triangle T = [s1 , s2 , s3 ; a, J3, -y] is no poorer in features than the right-angled triangle T' = [s1 , s2 , s3 ; a, J3, -y; a = 90°] and the right-angled equilateral triangle T" = St. s2 , s3 ; a, J3, -y, 'Y = 90°; s1 = s2 ]. Both are special cases of the universal triangle not because they are contained determinations that, in contrast to the composition of the universal triangle, possessed something new not yet previs­ioned in it, but rather because the previsioned is actualized in them

94. Specific formations can also arise from E = [a ; b] through various specifications: E = [a; b, a>b], ore = [a; b, a = b] ""'e = [a]. As a simple example for the theory of conic sections, we can use the ellipse in the

2 2 presentation of its equation of center \ + Y

2 = 1, whereby the half axes

a b are variable independently of one another. (The concept of variables is obviously used in the logical and not the analytic geometrical sense, since it only has the functional variables x and y.) By specifying a = b we get a circle with radius a and finally the specific circle, such as a = 4, no longer containing any variables.

In addition, it seems that the ongoing controversy concerning the "uni­versal triangle" of Locke, eliciting up to the presenta plethora of logical and psychological discussions, rests, to say the least, on a misunderstanding of the universal character of geometrical objects. An additional problem certainly plays a role here: the relationship between the geometric object "itself" and its "figure" at the level of signs. (See pp. 196 ff.)

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in a specific form. Thus while completely deviating from the work of the classificatory sciences, the mathematician sees at the same time a scientific criterion of value of his efforts in that they correspond to the aim of the greatest possible universality. He does not investigate singular and separate formation in arder to reach "abstractively" to higher ones; rather, he seeks the possibility of universal structures, and what he has proven for them, he has eo ipso also proven for the special cases. Conversely, where the work is initiated with individ­ual formations, the universalization does not proceed so that the specific characteristics are neglected, but rather so that they are taken as variables, or at least, no use is made of their specificity.

Such a construction of the geometrical domain of forms is obvi­ously different from that in the classificatory sciences. To be sure, within elementary synthetic geometry its uniqueness is difficult to grasp. The positing of constants of variables, constituting the appro­priate process of specification, is a logical operation. It first appears in the analytically pursued geometry and thus first attains its meaningful expression in the functional equation. There is no correspondence with the pictorial-symbolic since here it is easy for morphological elements to intrude into the conception. (Thus the structural difference between conic sections in elementary geomet­rical presentation and their projective application in functional equations!) A beautiful proof that the division of the formations in the old synthetic geometry were similar to genera and species is found in the fact that in the geometry of antiquity, the theorem of the sum of angles was proven separately for the different "species" of triangle. The process was sensible and necessary as long as the insight was lacking that none of the triangles investigated are something "specific," but rather are mathematical special cases of the universal triangle.

The autonomy of mathematical determinations, in contrast to the empirical world of things, appears when we note that the former are completely free from the role of being a mere feature of things and are grasped purely in themselves. Indeed, they must be grasped in this mode if their specifically mathematical character is to come into view at all. All further properties belonging to the characteristic "spherical" will be studied not with spherical things, but with the spheroid as such; what allows us to recognize any of the properties and relationships appropriate to the concept of a triangle is not the triangularly bounded objects, but the triangle itself. At this point let us not enter into a closer analysis of the "sphere" and the "triangle" themselves; let us first take their meanings as they are understood

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prior to any reflective individual analysis by objective conscious­ness, and let us note that here it is a question of a new and unique domain of objectivity. It gains its independence not because the objects of the space of intuition are completely sorted out in it in terms of the presence of these new determinations; rather, they manifestly bear the certainty of their existence in themselves by virtue of their factual and relational interconnections which, more­over, cannot even be exactly realized in the objects of intuition.

This means that the access to this domain of objects must be at the outset other than the access to the morphological domain. The mathematical concepts cannot be abstractively obtained from the intuited objects in the same sense as the morphological concepts. This reveals the difficult problem of their origin (pp. 191 ff.).

If we briefly characterize further, purely phenomenologically, their mode of givenness for a nonreflective consciousness, then the new objects constitute an ideal interconnection. This will mean that the essence of these objects, and of the relationships obtaining between them, lies apart from the awareness of the "flowing" time. Concepts such as development, becoming, change, and even those of constancy and continuity, become nonsensical in this domain. The new objects are what they are: trans-temporal and valid trans­temporally. Furthermore, they are apprehended in a specific mode of universality. The triangle, the circle, the sphere, are distinct from any individual interconnections. While being identical, they are singular and, at the same time, are not momentary individuals in a here and now; rather, they are ideal singularities untouched by principium individuationis. The most extreme specification to be attained in this new domain of objectivity is the ideal singularity.

However, the above delimited relationships do not yet designate the mathematical concepts as mathematical. The entire domain of ideal aspects is not exhausted by the mathematical domain. It must accordingly be asked wherein lies the uniqueness of the former.

§ 2. The Problem of Mathematical Ideation

Types of entities other than bearers of expression, ready-to-hand things, and objectively intuited entities have a different mode of corporeal comportment as their correlate on the side of the subject. The interconnection of things constitutive of the types of things in lived space, as well as co-constitutive of their relevant spatiality, provides a foundation for a special unity of sense constituted in specific apperceptive activities of a corporeal being. These activities

Basic Trends of Mathematization 191

are such that they form both the lived body and the thing in their specific mode of being thus.

This correlative conjunction of lived body and lived world became obvious with the reflectively oriented analysis of corporeal move­ment in space. The correlation appeared as a result of a retrogressive observation of the immediate comportment toward the world with­out itself being given within and in the accomplishment of the latter.

In unreflective comportment toward the world, things are available as pre-given in another mode of presence. The general and encom­passing mode of presence of things in lived space is one of direct self-givenness. For something to be given directly means that in its apprehension there are no mediate moments of conception-such as being a sign oran image for another. To mean something as itself, in a strictly phenomenological sense, means to intend it not in a merely signitive mode as happens, for example, in a merely verbal under­standing or in all symbolic thinking; rather, it means that it is intuited as "itself" present and has its fulfillment of meaning in intentions of a "seeing" kind, where various differentiations become immediately significant for the latter. In addition, each objectivity of lived space bears a character of self-givenness "in the flesh". This presence "in person," "in the flesh," means literally the mode of givenness of an object in its confrontation with functioning corporeity.

The domain of objectivity now coming into view lacks first of all this character of presence "in the flesh." The givens of mathematics have no direct relationship to corporeity. The many equivocations of the concept of intuition run the risk of enlarging the difficulties of comprehending the domain under consideration here. They contain, on the one hand, the roots of the numerous and all too extensive attempts to found geometry empirically. Such attempts think that they can obtain the geometrical formations from the sensible­intuitive factors-i.e., from what is given "in person" of the lived space-through abstraction from the morphological determinations in the sense analyzed above. On the other hand, the equations also give rise to those apriorisms that purport to guarantee the objects of geometry by presenting them in an unconditioned "pure" and purely "spiritual" vision, having nothing in common with the corporeal achievements of the subject and thus completely tran­scending all sensory functions as such.

For the phenomena to be analyzed here Husserl has coined a concept, that of ideation, and demonstrated its fruitfulness as a mediation between two extremes. As a specific case of abstraction, it participates in the usual aphairesis. lt disregards all peculiarities

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present in the individual objects in their particular hic et nunc. At the same time it is essentially different. It does not merely bring forth something common in a given multitude of objects; rather, on the basis of perception "in the flesh"-presentation and imagination of an exemplifying individual object-it intuits the universality of an essence.

Accordingly, universal essentialities are obtained intuitively "on" an individual object of sensory intuition. This seems to be similar to the notion that we are concerned with extricating something con­tained in the object and first apprehended as an empirical feature befare the ideational regard can really orient itself toward it. Mean­while, even if the ideation is enacted primarily on the basis of "direct" sensory intuition, it already presupposes the presence of this very essence as one and the same, in contrast to a multitude of objects of the same type and form. With this ideation a new act­characteristic appears in which a new type of objectivity is consti­tuted. Indeed, the acts remain founded in sensory intuition, yet they manifest their new type of intentionality in such a way that the founding acts are not included in this objectivity. It contains new determinations that are not found with the objects of the founding acts.

Ideational domains of objectivity are constituted with a specific character of universality. Their objects, as essential universalities and ideal objects, are indifferent not only to the number of empirical cases, but also to the possibility of their empirical realization at all. Nevertheles, the ideationally constituted object appears immedi­ately andas self-given; yet this self is no longer a self simpliciter and "in person," but a categorial one. The Husserlian concept of catego­rial intuition includes quite appropriately the layering of two intentionalities: the sensible and the intelligible. Yet it neither merges into unitary act-quality (as is the case with the various partial intentions in "simple" perception, i.e., within one act-level), nor is it composed of two independent and closed series of acts ("sensi­bility" and "understanding"); rather, it presents a unique complex of acts of sensible and categorial moments which, in accordance with their phenomenal composition, justify the extension of the concept of intuition to include categorical structures.95 We are well aware that ideations, with respect to their phenomenological structure, are

95. Concerning the concept of ideation see E. Husserl, LU, Investigation 6, §§ 48-52; Ideen 1, § 70 ff.; and CM, § 34. Concerning Categorial intuition see E. Husserl, LI, Investigation 6, particularly Section 2, Chapter 6.

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very special and very simple intentionalities within the total domain of categorical intuitions. We must explicitly recall that the latter can be understood and justified phenomenologically without contradic­tion only in the precise sense of the much contested Husserlian concept of "seeing essences." Furthermore, it is primarily with the help of this categorical intuition that the reference to sensory intuition as the "source" of geometry attains a more precise sense. And it is this mediation that first offers an explication of the most discussed concept of geometrical intuition. A closer exposition of this will follow (pp. 253 ff.).

But befare this, it must be pointed out that something ideationally intuited is not merely not sensible; rather, above all, the ideational abstraction as such is not yet mathematical ideation. Its designation requires additional characterizations. Thus the latter distinguishes itself from other kinds of ideational performances essentially through a unique moment of idealization-and this must be taken in the specifically mathematical sense. What is grasped ideationally is not only conceived in its pure essence as a trans-temporally valid mode of universality, but beyond this is also grasped exactly. A spherical object is never exactly spherical, and yet the discovery of this inexactness suggests that its form is already addressed in immediate intuition as spherical and thus at the outset is apper­ceived as an ideal and exact form-structure.

This state of affairs is a sign not only of mathematical ideation as distinct from any other grasping of essence, but also as the specific complexity of the act structures designated by the concept as mathematical ideation. To a great extent phenomenological analysis allows what constitutes these structures to escape. O. Becker first clearly realized the problem emerging here, with all of the difficul­ties, and characterized mathematical ideation as an ideational ab­straction with a certain transgression of boundaries with appropriate limits of structures. Through the contraction of topological nets this abstraction "sharply" lays out the mathematical formations as points of the mathematical continuum.96 What ultimately constitutes the ideality of the geometric formation in a precise sense can be grasped only from the background of the mathematical continuum. The genuine problem of mathematical ideation has its objective correlate in nothing other than the mathematical continuum. Yet its problem is no longer geometrical: it falls, with all its impenetrability under

96. O. Becker (1), Section I, Part 1. Concerning the problem of point and continuum, see pp. 296 ff. of the present work.

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the domain of mathematical analysis. Since at this point we are not yet engaged in the investigation proper to it, it is premature to talk about it. Here the primary concern is to indicate the specific difficulties appearing with the question of the origin of geometry.

These difficulties increase if it is noted that the mathematical ideation, taken in a precise sense, is still inadequate to allow the pecularities of geometric objectivity to emerge. What appears solely as given in this ideation are certain discrete forms, elementary geometric formations limited in number which not only do not comprise the whole of geometric objectivity but are not even comprehended as specifically geometric, i.e., as functions subsumed under an axiomatic-deductive science. The specifically mathemat­ical sense-bestowing-for example of an exact eidetically appre­hended triangle "as" geometrical formation with all of its constitu­tive moments of meaning-in its logical formal nature is not something that merely accrues to, but rather simply transcends ideation. It first determines the constructive character of mathemat­ical objectivity, grounds its typically mathematical interconnec­tions, and makes it demonstrable within the framework of a rational theory (pp. 211 ff.).

§ 3. Symbolic Intuition (Pictorial Symbolism)

The relationships and differences between morphological and mathematical objects must be conceived from still another view, which is usually neglected in the treatment of the theme that is here at issue. A closer look leads to noteworthy insights.

The ideational intuition of geometrical formations turned out to be founded in sensory intuition. The former thus contains clues point­ing back to the corporeal functioning of the subject engaged in geometry. This does not exhaust the relationship of geometric objectivity to corporeity. It is rather characteristic and most remark­able for this objectivity that its full complexity of meaning unfolds only gradually and can be mastered on the way through a mediation of something else that is not geometry. This something else is its "presentation" in sorne medium of the sensible intuitive world. To attain valid propositions concerning geometrical formations we require a figure or-to speak with reservation-a "picture" of it in which it "shows" itself.97 This self-showing in another is not

97. For the context under discussion here it is irrelevant whether in presentation we are concerned with a figural ("pictorially symbolic") or

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something external in the object, something merely added; rather, it will turn out to be a constitutively essential aspect of the geometrical and mathematical domains as such. All that is mathematical is such only insofar as it is productive in another.98 However, the geomet­rical object thereby becomes highly complex. No longer appre­hended in direct intuition "in person" and yet not apprehensible in any other manner except in something else given "in person," apprehensible as "itself" meant in what is given without itself being in it-on the one hand, this complicated state of affairs contains the entire problem of the geometrical sense of being and, on the other, it presents the preeminent question of geometrical intuition as well as the intuitability of geometry.

For a closer analysis of this intuitability there is an extant concept which, while enhancing the distinction between the figural triangle and the triangle itself, at the same time obliterates the genuine structure of this distinction. It is the concept of representation. The figure ought to "represent" the properly and geometrically meant: this means that the representandum has already attained a clear separation from the represented, and yet that the former is appre­hended in the latter as "itself" in its independent meaning. But this condition can be the basis for a misunderstanding. It would lead to the notion that the mathematician takes the sketch, instead of the meant ideal triangle, and studies in it what he wishes to discover in the triangle itself. But this would mean that he would only obtain information concerning the triangle that is offered in the sensibly perceived formation. Thus he would not reach geometrical knowl­edge as such, but only a morphological description of a figure­which like any morphological figure would not be exact. Y et admittedly, the imprecisely sketched triangle is sufficient for exact knowledge of its geometrical properties; even an elliptical figure may be of service in understanding the properties of a circle. Although requiring the visual figure, this understanding does not rest in it, but rather in an appropriate consciousness of the meaning

"signitive-symbolic" presentations. (Concerning the distinction see § 4 below.) Here we are concerned only with the presentational function as such.

98. Related to this is the constructive character of the geometric formation (pp. 216 ff.). "To be producible" here is a general and precursory delimita­tion of the constructive moment in geometry. In the history of mathematics its meaning has been variously transformed; here it can be traced only gradually.

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of the ideal states of affairs. In geometrical work we do not "live" in a pictorial or symbolic consciousness, but entirely in the conscious­ness of the thing itself. It is the articulating reflexive analysis that first separates both; the separation must be assumed in arder to reveal the mediating character of this new kind of self-givenness. The term of symbolic intuition, and primarily pictorial-symbolic intuition-in contrast to signitive symbolism-is thus comprehensi­ble not from the mathematical intuition, but from phenomenological reflection.99 Subsequently we shall use the concept of symbolization with the explicit restriction that in geometrical work we do not mean the co-given consciousness of the pictorial mediation of the object and of the symbol as "symbol"; rather, the sale characteristic of the intention of consciousness of the meant ideal object itself. It is this fact that first makes possible the meaning of the figure as a pictorial­symbolic figure for the geometric state of affairs; symbolic intuition, in the sense meant here, is not motivated by the observation of an "abstract'' painting such as a triangular composition.

The difficulty here lies in the difference between the ideal and the spatially real ("sketched") formation or, seen from the side of the subject, between the ideal consciousness of meaning, wherein the thing "itself" is constituted on the basis of categorial intuition, and the simultaneous and essentially necessary co-positing of its sign in sensory intuition. Self-givenness and givenness "in the flesh" are sundered here in a different way than in a mere sensory intuition simpliciter. The ideal object itself is also immediately given; the ray of ideating intuitionality encounters it directly without traversing signs as the medium of meaning "of" the object. However, the geometric object is not self-given in sensory intuition, but rather in a categorically intuiting intention. Yet to be fulfilled, confirmed, and shown to be valid, it requires the complementary activity of sensory intuition functioning as a symbolic intuition in relation to the meant objectivity.

The discrepancy between the ideal geometric formation itself and its intuited figure underlies the already suggested problem of the universal triangle in Locke. Locke starts with the fact that a geomet­ric proof is always accomplished with a specific triangle-more precisely, with a figure at the level of a sign which can obviously represent only a completely determinate triangle ( containing no variables-yet it was concluded that the proof is valid for any

99. The concepts of pictorial and signitive symbolisms used here follow E. Husserl, Ideen l, § 43.

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triangle. Locke hints at this conclusion in passing, claiming that the proof at the outset relates to the "general idea of a triangle" and hence is valid per se. Thus for him the ineradicable paradox appears between the fact that this universal triangle must have all the specific determinations, and yet at the same time could not ha ve any of them. At the same time he maintains that although this triangle is completely devoid of specific determinations, it unifies all particular triangles in itself.

Husserl criticizes this "absurdity" with the remark that the con­tents of the various "kinds" of triangles are subsumed by Locke under the concept of a triangle.1°0 A phenomenologically adequate solution of the problem is not found in Husserl. lt was not even found with Husserl's presuppositions in which the classificatory scheme of genera and species was applied to self-evident, geometric formations. As a matter of fact, it would be absurd to prove the theorem of the sum of angles in terms of a specific triangle and then to conclude that this was valid for any triangle or for "the" triangle, if the latter, as genus proximum, were a product of abstraction from "all" triangles and if, thereby, none of the particularities of the formation represented in the figure accrue to it. Yet it must be observed that here no inference takes place-apart, of course, from the conclusions immanent in the proof. No inference is present from a specific triangle to other, less specified triangles and finally to a universal triangle as such. This is already an indication that there is a special relationship between the figural triangle established as a sketch in all of its determinate parts and the triangle. This relation­ship must be considered if the concept of representation is here to provide more than a way out of a perplexity.101 A representation, after all, does not only have the characteristic of representing something trans-sensory in something sensory; rather, the problem is to show that and how all specific triangles can be represented in a

100. J. Locke, Vol. II, Book IV, Chapter 7, Section 9; see also Chapter 1. For Husserl's critique of Locke LI, lnvestigation 2, § 11.

101. C. Holder (1) advocates the view that geometric proof always con­tains an analogical or completely inductive inference (p. 12). This opinion is obviously attained regressively; the proof is valid for all triangles, "there­fore" such inferences "must" participate. Yet this cannot be maintained phenomenologically in the factual process of proving; it does not even accomplish what is required to save the universal validity of geometric proof. A formal-logical solution of the problem is offered by the work of E. W. Beth.

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singular figure! Such a representation, if it were to arder the most diverse triangles under genera and species, would be impossible­justas impossible as it would be to say that all musical instruments, be they only the string, the percussion, and wind instruments (and under the latter the wood-wind and brass instruments) can be "represented" by a single sketch of a bassoon.

In truth, even the starting point of the discussion requires a correction. It cannot be denied that the sketched figure is always specific, i.e., it has fixed sides and angles; what can be debated, however, is that it stands for a specific ideal triangle, comprising, as it were, its congruent mapping. The fact not only is that the actual size of the figure is of no interest at any moment but that in geometric work itself its factual determination of size is not actually present to consciousness; this should warn us against any attempt to import sorne meaning of depiction into the concept of representation meant here. In its project the figure is taken at the outset in its universal dimensions as a, b, e, and <t, ¡3, 'Y· While being sketched factually in quite determínate sizes in the figure, they are nevertheless meant as variables. We do not abstract from sizes for a successful process of proof or for an inference to a "genus triangle"; rather, during the proof we make no use of the possibility of their constancy in arder to assure the greatest number of variables for the result, in such a way that a variable contains possibilities of determination while encompassing all possible specific cases. Thus the process does not lead from a sensible, singular figure by way of abstraction and generalization to a universal triangle lacking all particularities. Rather, it is categorical intuition as such that first constitutes the "triangle" as an ideal triangle itself and, indeed, in a figure with variably conceived determining parts-"in sorne figure or another" -and it is the latter that plays a role, though an unavoidable role, in the process of proof. In arder to see the figure as factual and determined entirely by the length and position of chalk marks, a specific-and, indeed, seen from the point of view of the geometric process-a contrived shift in viewpoint is required. The Lockean problem arises only when the contrived viewpoint is assumed at the beginning; here one overlooks the difference between what the figure is as a formation given in direct intuition and what it means from the outset in the project as a symbolic-intuitive presentation of some­thing non-sensible. The Lockean conception of the problem fails to. pay attention to the specifically universal character of the "univer­sal" triangle and fails to see the essence and function of pictorial­symbolic representation in geometry.

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The concept of pictorial-symbolic intuition lends preliminary understanding to the meaning of the statement about intuitability in geometry. This must be clarified later. First of all, we agree that the concept of intuitability, in case it is not yet specifically delimited, must be taken in the sense of pictorial-symbolic intuitability. While reserving more precise determinations for subsequent discussion, here it is possible to establish sorne provisional outlines: to appre­hend a geometric formation means to orient oneself to the thing "itself" in a pure consciousness of its meaning, and to be able to correlate certain sensible-intuitive elements to it and to its ideation­ally apprehended properties in such a way that (1) the correlation of the ideal state of affairs and the pictorial content is reversible so that not only is there a correspondence between the state of affairs and the pictorial content representing it, but also the pictorial content, insofar as it is grasped as merely sensible in the primary intention, can nevertheless be constantly apprehended through an appropriate reorientation as symbolic for the relevant states of affairs; and moreover, that (2) the possibility must be given in principie for apprehending the pictorial symbols univocally as pictures of possi­ble morphological forms of the world of things given "in the flesh" and meant as depictions.

While the possibility indicated for (1) is as valid for a model conception-such as in non-euclidean geometry-that is distinct from the pictorial-symbolic intuition as it is for the signitive sym­bolism of analysis, the requirement mentioned in (2) is specific to the geometrical intuitability meant here. It will be shown that the latter concept is delimited in a way that precisely encompasses euclidean geometry. In accordance with the delimitation indicated in (2), the circle, for example, of euclidean geometry is in its presentation geometrically intuitable in the above specified sense as a closed and equally curved line. In contrast, the straight line of elliptical geometry is not intuitable in such a presentation of a circle. As "straight," the picture of the closed line calls forth, in direct intuitive consciousness, an experience of conflict. The circle func­tions in this case no longer as a pictorial symbol but rather as its "mode1",102 since phenomenologically it is classified and consti­tuted differently. It becomes clear that the required correspondence, co-present latently in the pictorial-symbolic intuition, is exploded. Thus if non-euclidean geometry is to be presentable in any pictorial

102. Concerning the question of the model conception, see pp. 278 ff. of this work.

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way, it requires a model-but there is no model of euclidean geometry.

§ 4. Signitive Symbolization of Geometry

Categorical and pictorial-symbolic intuition designate only two of the kinds of intuition here in question; however, they do not characterize the entire domain of acts that constitute geometric objects. What comprises their geometric nature ultimately remains foreign to ideational and idealizing abstraction as well as to the forms of intuition mentioned. What will be intuited ideationally, what will be established pictorial-symbolically in a real medium, are merely specific elementary formations, along with a few geometric relationships in them. Thus with the right triangle one can only intuit the content of the triangular asymmetry; yet that the precise relationship between sides and hypotenuse is quadratic, as ex­pressed in the Pythagorean theorem, is not at all intuitable either in ideational abstraction or in symbolic intuition. The typically math­ematical character of this domain of objectivity does not come fully into view within the modes of intuition observed to date.

The elementary geometric formations considered heretofore are not fundamental formations in the sense of being objects for foun­dational investigation. The point of departure of foundational inves­tigation is comparable to ours in only one respect, insofar as its questions can begin on the basis of an airead y pre-given and evolved individual science. Yet it diverges essentially from our investigation. In contrast to ours, it does not explicitly include the subject in its investigations and thus remains oriented purely objectively. Indeed, the foundational investigation of the geometrical domain of objec­tivity is also for its part made into an object of reflection. Thus Hilbert demands: "We must make the concept of the specifically mathematical proof itself into an object of research, just as the philosopher criticizes reason itself."103 Yet the investigations of axiomatic theory and of the theory of proofs are maintained within the context of the discipline of mathematical thinking and are accomplished with mathematical and logical means. While founda­tional investigation discovers the logical structure of a domain, projecting such structures anew through new axiomatic beginnings, formalisms, and calculus, our interrogation aims at the analysis of those constitutive activities of consciousness from which the giving

103. D. Hilbert (1), p. 415.

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of mathematical meaning is first discerned at all. Our investigation faces a problem that is prior to any axiomatic researches; it encom­passes the latter not in their results, but in the point of departure of their question. It is directed to this question: what appears, beyond the geometric region as such, as a necessity for the aim of mathe­matical researches into foundations? How must it be constituted in its ontic structure and how must it be articulated ontologically so that it would become comprehensible as a region requiring special researches?

If the geometric objects were only those of intuition, whether ideational or categorical, then no mathematical investigation into their foundations would be required. Since geometry is concerned not only with direct but also with categorical and symbolic intuition, it means that the mathematical aspect is present only conditionally in intuitive orientation. The clearly discernable discrepancy be­tween the meaning-consciousness of states of affairs themselves, on the one hand, and their sensibly perceptual presentation, on the other, led to the problem of symbolization in the region of elemen­tary geometry. For a long time it was taken as a pictorial represen­tation of individual geometric formations, which are meant when the natural consciousness ascribes "intuitability" to geometry. Yet here the symbolic process in geometry is first established only in its factual and historical origin. What constitutes the progress from the old synthetic geometry to the modern analytic geometry, as seen act-phenomenologically, has its grounds in the dissolution of pictorial-symbolic intuition by a signitive symbolization, by a rep­resentation of the meant in signs that are fundamentally distinct from the geometrical "picture."

Phenomenologically speaking, sign-symbolism in geometry presents a new and most unique problem. It is far removed from being a mere and supplementary establishment of what can be present in pictorial symbolism; on the contrary, the sign points toan entirely new kind of sense-bestowing in the geometrical states of affairs themselves. We are not thinking primarily of modern formal­isms in metamathematics, but of the earliest form of algebraization of geometry anticipated by Descartes. What is genuinely algebraic in geometry, what allows it to become "analytical" geometry, cannot be investigated here in sufficient detail, and to a great extent must escape phenomenological-descriptive efforts. First of all, such efforts would have to observe separately all those acts and their multifarious syntheses that constitute number, arithmetical laws, algebraic operations, etc.; this would require comprehensive specific

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investigations. In their stead we can suggest Husserl's Philosophy of Arithmetic. At the outset our investigation must be limited: it must show that with the synthetic and the pictorial-symbolic, on the one hand, and with the analytical and sign-symbolic, on the other, we are not merely speaking two distinct geometrical languages, whose meaning can be accurately and mutually translated; rather with the transition to analytical presentation, there appears something fun­damentally new. That, on the one hand, a curved line at the sign level represents, in pictorial symbolism, a mathematical curve of a higher arder, and that on the other hand, a combination of signs F (x, y) = O "presents" the "same" curve, except now not pictorial­symbolically but in signitive symbolization, indicates that this linguistic parallelism fundamentally hides the deepest riddle inher­ent in the latter kind of pres8ntation. The special science of geometry employs the concept of isomorphy here in arder to designate the structural equivalence between the geometrical region of objectivity and the algebraic region of operations; for its aims it is sufficient to remain with the conception of a "mapping" of one domain onto the other.104

Meanwhile, the problem we have indicated has not been solved. Even after a successful elucidation of all constitutive achievements of pure mathematics, another question would remain open: how is it possible to posit an arithmetical formation for a spatial one? To signify points of space with numbers-how is this fundamental act of analytic thinking to be conceived, a thinking that was still remate to the geometry of antiquity? Obviously the numbers of a series are present as unique individuals such that no two are alike. In contrast, the space of geometry is given as a homogeneous manifold in which none of its points have geometric characteristics distinguishable from another. It seems that through the coordination of numbers and points, the latter become fully individuated; while in elementary geometry none of them were distinguishable, now each seems to be distinguished.

If the sense of that isomorphy is to be conceived, the coordination between number and space must, as a matter of fact, remain completely incomprehensible. But what then assures the possibility of such coordination is not the circumstance that the world of numbers "consists" of nothing but individual numbers; rather, it is that we have to do with a domain in terms of its generative

104. K. Reiderneister (3) develops sorne of the rnain problerns of geornetry under this aspect of isornorphy.

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principies and its being built up purely constructiveJy from a few basic conceptual operations. In this domain the specificity of indi­vidual numbers is in no wise abolished. Rather, they are submitted toan all-pervasive order and lawfulness such that this domain can validly be called homogeneous under the opera ti ve aspect. From this vantage point the question of the application of number to geometry begins to make sense.

With respect to spatial formations, however, this application is less than self-evident. That the extended spatial formations which are distinguishable and indefinitely multifarious in position, size, and forro, should be mastered collectively by the constructive arder of the numerical domain is not an immediately obvious fact. The mathematical notion of "mapping," in its statics of coordination, hides the fact that we are not dealing here with the mutual relation­ship of two equally complete domains, but with a new methodoJog­icaJ postulate for geometry. The geometrical formation "ought" no longer to be taken as a rigid and immobile pictorial-intuitive figure; rather, it ought to be thought as emergent from pure number sequences in such a way that the arithmeticallaw of their coordina­tion determines the geometric forro. And conversely, the properties that can be brought to light pictorially in a geometrical formation, such as direction and curvature, must turn out to be characteristics of these number sequences.

Thus it seemed that initially the domain for this new geometrical mode of conception was circumscribed very narrowly. Even Descartes still had to exclude the transcendent curve from his consideration. What testifies to the inexorability of the basic Cartes­ian conception is the fact that while it was established for a limited geometric domain, and subsequently became a problem for addi­tional geometric formations, this new beginning was not discarded as inadequate. To subsume geometric formations collectively under the number domain and, in a final analysis, not even to admit anything as "geometrical" if it cannot be expressed in pure numer­ical relationships or (to use a later term) be dealt with "algorithmeti­cally," is to show that this mode still dominates geometry as a methodological postulate (pp. 219 ff.). Infinitesimal calculus and differential geometry methodologically signify merely the conse­quent fulfillment of the Cartesian demand.

With this, however, geometric objectivity not only appears here in a transformed symbolic forro; rather, what is more decisive is that its analytic "conception" introduces at the same time a complete factual modification of meaning of what is geometrical. If geometry

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has become preerninently analytic geornetry, then, for example, the Pythagorean theorern concerning right triangles is no longer valid for the measure of line segments, but solely for the algebraic state of affairs: what is given here is "pure quadratic form." Number rernains predominant in its own intrinsic rules, the equation. and its "form," which are not sensibly formative but purely algebraic.105 Purely quadratic formas the square of the "distance" shows a characteristic shift of meaning-giving whereby a geometric state of affairs appears to be nothing but an interpretation of an algebraic state of affairs. And this shift is not merely one of psychological thought or a matter of mere attentiveness, but is graspable in the very sense of the ontological meaning of 'geometry.

With this sense-modification of what is geometric there suddenly open up vast possibilities for the broadening of geometry. For exam­ple, in accordance with its purely algebraic meaning, the new concept no longer includes a limitation to two or three dimensions. Only the degree of the equation determines its geometric meaning, while the number of its members is the matter of choice. Yet at the same time this choice determines the possibility of multi-dimensional geome­tries. Furthermore, the "existence" of various geometric manifolds, the metric structure of entire mathematical spaces, depends on the pure algebraic form of a c~rtain equation, the fundamental tensor. Finally, an algebraic criterion is posited over a geometric meaning. Yet it is not abolished as geometrical. Rather, in its algebraic modi­fication, it is expanded to encompass all elementary geometric for­mations and states of affairs. What is of primary importance in this context is that the transition from the elementary synthetic geometry to analytic geometry is based on a profound transformation in the domain of symbolizing acts. These are completely novel modes of consciousness in which the "analytical" is constituted-although in this novelty they are not limited to the mathematical domain. Yet they are mostly significant in this domain, since its radically changing methods are deterrninative for a completely altered conception of the scientific character of geometry.

The older geometry remained essentially caught up in pictorial-

105. Certainly, even such a form as a combination of sign-symbols possesses its own sensory shape anda "pictoriality" proper to it. However, the intuition corresponding to it is to be strictly distinguished from the signitive-symbolic intuition to be analyzed here (pp. 209 ff.). In the present context the primary concern is with the moment of meaning; the signs function as symbols "for" something that they themselves are not.

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symbolic representations of its formations. In its apprehension of geometric existence it was still exclusively bound to pictorial means for the constructive acquisition of its objects. The analytically operating geometry requires the dissolution of the pictorial element through the sign-symbol. With the concept of constructability it modifies at the same time the existential meaning of the geometrical. In terms of act-phenomenology, we are concerned with a sequence of levels of meaning-giving with a very complex structure. It constitutes mediations of various kinds with respect to the apprehended object which, while easily achievable factually, are describable phenomenologically only incompletely and with difficulty.

The pictorially intuited figure presents the geometric formation itself, even if in a limited sense of appearing as itself through the mediation of the figure; yet in and with the figure, the ideational intention truly touches the meant, so that the figural-pictorial medium points directly to the subject matter itself, without hin­drance and without deflection of the view. In contrast, signitive symbolization contains various media of meaning with distinguish­able "indices of refraction" behind each other. Thus in accordance with the motivation of the act, either their seriality is surveyed completely or the intention remains at one stratum and there independently constitutes an interim "self." Thus, for example, any discussion of a circle, where it is meant in a pictorial-symbolic intuition, is different in its meaning from a circle when it is spoken of as a "circle x2 + y2 = a2

." Indeed, in the latter case a specific equation admits coordination to the circle as also meant with the signs, yet it is no longer represented in the sense of the term used above. What is here "represented" in the sign, in strict analogy to the above usage, is not the circle but the functional equation of the circle. The circle in its turn is mediately intended by the equation. The nominal positing of the circle in the first case has, correspond­ing to the act, not the positing of "the circle x2 + y2 = a2 ," but rather "the functional equation of the circle x2 + y2 = a2 ." Here even in naive execution of the mathematical intention-and not, as in case of the pictorial-symbolic representation, first in the reflective attitude-there is a clear co-consciousness of the positing of the functional equation for the geometric formations. In other words, in signitive symbolization, the various apprehensions of the sign artic­ulate themselves as "standing for," as "pointing beyond" in the objectively oriented course of acts; or else the acts attach themselves to the sign. Both states of affairs, again differentiated within them-

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selves and to be discussed shortly, are typical for the en tire phenom­enal complex of signitive symbolization. Included is the notion that the relationship between sign and signified-in contrast to that between picture and its object-is no longer direct. This also implies that signitive symbolism is not prescribed by the meant object. While the pictorial symbol remains bound to the symbolized, thus retain­ing the "picture" character and, if need be, allowing inexactness at least within the permissible morphological limits, the sign-symbol contains the possibility in principie of a free choice. It requires that the relationship between the sign and the signified for which it is to stand rnust be established through agreement. The signi­tive-symbolic rneaning intentions involve specific acts of choice of the positing of signs, conventions of their meaning-giving, and requirements of univocity of their usage. Here an entire sequence of distinct modes of intentionality determines a multi-leveled media­tion of the meant object. The ray of intention first encounters the object on a variously disrupted path, through distinct and phenom­enologically heterogeneous levels of meaning, before it appears "in" the sign as that which is rneant throughout.

This multi-leveledness offers in its turn specific and phenomeno­logically singular possibilities. It was already mentioned that in signitive symbolism, in contrast to the pictorial, the apprehension of the signas "sign for" is inserted into or joined onto the very course of the objective acts themselves·. The first case is appropriate where something geornetric is explicitly meant in the signs, for example in analytic geometry in a narrower sense. There the signs present functional equations, and as such they already "really" reveal geometric formations, relationships, and processes, i.e., the latter are explicitly meant in the former. Yet nothing stands in the way of either the striking out or, at the very outset, the neglect of the specifically geometrical meaning of such symbols. The intentions could rernain with the equations and functions "thernselves" and could terminate with the sense-bestowal of the functions as func­tions, as algebraic equations.

With this restriction to the purely algebraic theory of equations, the defacto relationship between something spatial and its sign is dis­solved. The establishment of signs is simply a matter of choice. In this case the mathematician can speak of a "geometrical interpretation" of algebra. This mode of observation deviates from a genuinely geo­metric observation; it is not motivated geometrically, but rather is primarily algebraic in kind. An additional although necessarily re­quired intention reinstates the relationship to the spatial formation.

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This is not to say that this "merely" interpretative observation is unimportant for geometry-indeed, with it an entirely new sense of the geometric as such is being delimited. In the first place, such interpretations yield an extension of the concept of geometry. Its various branches owe their existence only to a preceding algebraic theory, an existence that could not have been constituted without the algebraically mediated intentions, i.e., without using the alge­braic process. Thus, for example, skew field geometry is based originally on an algebraic property, namely on the non-commutative multiplication of its underlying algebraic number field.

Conversely, the mathematical concept of interpretation, which was only able to assume its full extent and develop its extreme consequences after Hilbert's meta-mathematics, yields a conception of the geometric that has been fundamentally transformed-not only in contrast to antiquity, but also to modern mathematics. What is present in it as geometry is no longer a solely geometric inter­pretation of algebra, but in the last analysis more of a geometrically interpreted calculus, a geometrically interpreted formalized theory. Furthermore, this means that we are concerned with a conception of the geometrical whose original motivation is no longer the question of how an objectivity, conceived primarily as spatial, can be mastered mathematically, but rather is dictated from an entirely different and obviously opposite question: namely, what can a system of signs offer to a purely operatively established geometry if one attributes geometrical meaning to it. This conditional statement brings to expression a completely unbound possibility. Hilbert's well known witticism that for a, a, A, etc, one must be able to say at any time "tables," "chairs," shoes," "tankard," characterizes the complete unboundedness of interpretation of modern signitive systems.106

Yet remarkably, there appeared at the same time a completely new kind of meaning within the signitive domain itself. To use Hilbert's citation, while the old symbolism was founded in an

106. A question might be raised whether we are dealing here at all with geometry or whether we should allow it to be circumscribed in the traditional sense against mere "geometric interpretations." This cannot be decided here. In modified form this question is closer to that of mathemat­ical "spaces" and we shall cometo them subsequently (pp. 239 ff.). It has not been claimed here that geometry belongs today to operative mathematics. We are only attempting to trace possibilities that can result even for geometry, in the broadest sense, with the process of signitive symbolization.

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ingenious insight that for "tables," "chairs," "tankards," one can also say a, a, A, modern operative mathematics stands the relationship between sign and signified, so to speak, on its head. The originary achievement of mathematical symbolization-which produced its signs "from" an objectivity through which something pictorial and imaginable was represented, not meant in an imageless way-appears all too obvious today. The signs of such images were meaningful and were akin to the original symbols as a presentation of an objective sense. But in calculus the sign has become removed from the signified to such a degree that not only is the intention toward the signified, the external and independent objectivity actually omitted, but its omission is explicitly demanded. If traditional symbolism stressed the representative function of signs, thus meaning them "merely" as signs for something, in modern science the sign is taken for itself. Indeed, in principie it has an open horizon of possible interpretations, capable of subsuming the geometrical at will; yet it is primarily signified as itself and for itself in independent "meaning," and only as a sign, without signifying anything that is not a sign. Having no posited semantic background, the "sense" of the sign here is no longer representative, but exhausts itself in being purely operational. The sale concern is not what these signs mean, what they signify, but that and how they can be managed.

Such a "formalism" may at first sight present an exact opposite of the older sign syrnbolism, yet considered phenornenologically, it is only the end of a path already initiated by the latter. The seeming transformation of the relationship between sign and signified in the modern theory of signs is in truth nothing else than the most extreme consequence of what was posited as signitive possibilities in the original symbolization through signs.

Thus the problem of intuition and intuitability appears once again from another side. While it could be adequately solved for the pictorial symbolism of geometry, it seems that it can no longer be meaningfully posed for a purely signitively treated geometry, and especially for a purely signitively projected geometry. If the sale concern is to interpretan "abstract" configuration of signs spatially, then any possibility of intuitability obviously becomes completely redundant. Obviously, signitive symbolization is essentially distin­guished from pictorial symbolization in that it lacks any aspect of intuition. Thus in the founding of geometry, Hilbert never evokes intuition. This is all the more remarkable since iti another place he claims that intuition is actually the most secure source of knowl-

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edge, and indeed even in the very foundation of analysis.1 0 7 Yet there is also a danger of equivocation here. To understand Hilbert's expression adequately, we must respect a twofold differentiation: on the one hand, between the pictorial-symbolic and the signitive­symbolic, and on the other, between sensible and categorical intu­ition. That the sign a does not represent a straight line in pictorial symbolization, but symbolizes it signitively, and that furthermore the meaning "straight line" does not have a sensible but only categorical-intuitive fulfillment, is quite obvious. If the concept of intuition is restricted by the character of sensibility, or at best by pictoriality, then a geometry done purely signitively obviously lacks any intuitability.

Nevertheless, a complete exposition of the concept of sign retains an intuitive moment. What remains ineradicably intuited, and indeed in the narrower sense of the direct sensory perception "in the flesh," is the sign itself. This is the case even when it is taken to be a sign "for" something spatial. However, this moment of sensibility in sign-giving attains specific accentuation if the sign stands for itself, if it becomes an object of mathematical consideration. Insofar as it no longer has a genuine representative function, but is posited in its own meaning as purely operational, the sensible intuition of these signs "in person" assumes an eminent mathematical signifi­cance. In fact, it plays a leading role in the process of proof.1°8

Indeed, the signs of metamathematics are not geometrical forma­tions, yet whatever their posited meaning may be, they are nonethe­less spatial formations. Thus no operation with them is nota process of thought that merely becomes subsequently fixed in a spatial medium; rather, from the very beginning the operation necessarily occurs through signs and not by way of the signs. Although chosen arbitrarily as to their kind, they are not externa! or superfluous to thought; they are rather constitutive for what occurs in and through

107. D. Hilbert (3), p. 158. 108. K. Reidemeister (1) speaks of a "method of self-clarification and

self-control of an exact structural intuition" in mathematics (p. 201), and clearly sees that the concept of intuition in geometry is laden with ambiguities. Yet upon consideration of these ambiguities, his view that intuition in mathematics has merely the form of a motive, a conjecture, an overview, ora plausibility (p. 198) cannot be maintained in this summary form. Even the notion that the interconnection between the modern system of "intuitable propositional signs" and the space of intuition can be found solely in that the signs can be "spatially interpreted" (p. 205) does not adhere to Reidemeister's own previously stated remarks.

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them. Thus at the same time, mathematics possesses in the config­uration of signs, a means of control of its statements. This is of a peculiar kind. Owing to the fact that signs are spatial formations­more precisely, that they are sensible-morphological figures in a real spatial medium, each logical contradiction in calculation is grasp­able in direct morphological intuition. This is not merely valid as a possibility; rather, the direct sensory perception of the signs be­comes a necessary act of mathematical operations themselves.

lt is crucial that the signs are not just spatial formations in themselves, but that they stand in a completely determined and nonarbitrary spatial ordering and are subject to strictly established rules of operation. Furthermore, it is significant that this spatial ordering is linear. This does not merely mean that it is progressively realized and at the same time a temporal arder. Every elementary geometrical, pictorial-symbolic construction is also temporal; it is established progressively in time to the extent that elementary mathematical thinking is already discursive. Yet the finished con­struction does not divulge the succession of "steps" of construc­tion-in any case, it can be regressively seen "through" in its becoming, but cannot be seen "into" progressively. lts elements stand in a temporal arder of succession only in the process of construction; they are spatial because they have constitutive func­tion for an objectivity, exclusively in terms of the requirements of the subject matter.

In contrast, the relationships in signitive symbolism are quite different. The reiteration of an identical sign ata "subsequent" place first manifests the temporal continuity of process. However, as a place for something sensibly perceivable it appears as "later" not only because it is merely "other" spatially, but also because beyond the manner of spatial ordering of signs, it is so constructed that it presents time in an irreversible direction of earlier and later. This is the fundamental meaning of the linearity of their arrangement. They are so constructed that the signs simultaneously reflect the arder of thought. Here the logical "sequence" is apprehended distinctly and immediately as a spatial sequence. The linearity of the configuration of signs manifests the discursive structure of mathematical thought in a sensible-intuitive way.

The present observations are not designed to overlook the enigma attendant upon the unique intertwining of acts of "pure" logical inference with those of direct perception "in the flesh." Obviously the ordering of signs, purely as such, can be neither contradictory nor noncontradictory. How is it possible that a mere combination of

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sensible and sense-empty signs in space can signify correct and incorrect inferences? Behind this question is a more fundamental problem concerning the conditions for the possibility of any appear­ance of something non-sensible in something sensible. This is of a speculative nature and escapes phenomenological analysis and description. The latter can only show how such an appearing factually occurs in a process of a correlative and implicative admix­ture of act-intentionalities. At least it may illuminate the sense of the "concrete" that Hilbert has attributed to metamathematical signs. By disregarding any depictive function, the sign, precisely when it has become devoid of images and has become posited in its own right, has reached a new intuitability for itself and thus a capacity for surpassing the achievements of pictorial symbolism for the construc­tion of new mathematical structures.

§ 5. The Constructive Character of Geometric Objectivity. Geometry as a Demonstrative Science

The great discovery of the Greeks was that mathematical objectiv­ity is capable of and above all requires scientific, indeed rigorous methodological access.to9 This mathematical method is more accu­rately characterized as an axiomatic-deductive method. And geom­etry becomes knowledge to be proven-thus it appears as an achievement related to a thinking of a very specific structure.

This structure is usually designated as logical and even mathema­ticians call it "purely" logical. "Logical" thinking in mathematics ought not to be taken in the narrower sense, as if it proceeded in accordance with classical rules of inference. Syllogisms can hardly be said to appear in mathematics. Various attempts to formulate mathematical proofs in syllogistic form have yielded very little and frequently were based on illusory syllogisms.110 It is otherwise with

109. Since we are not engaged in historical investigations, we shall not enter into the question whether mathamatics prior to the Greeks was given an incipiently scientific treatment. More explicit discussion is found in O. Becker (4)-(7), K.v. Fritz, O. Neugebauer, H. Scholz, and A. Speiser. The investigations of v. Fritz place a special emphasis on the extent to which Greek mathematics already has exact inceptions, so that Becker (5) could say of Euclid that he "appears more like an equalizing compiler than a creator in his own right" (p. 20).

110. Instead of designating mathematical thinking as non-syllogistic, it is more fruitful to designate it positively as "algorithmic." Concerning the concept of the algorithm, see pp. 219 ff. of this work.

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a relationship of the mathematical modes of inference to the formal logic of relations. To discuss this relationship would lead us too far into the researches of mathematical foundations. It must be stressed once again that our investigation is not designed to engage in mathematical foundations. Our reflections concerning proofs in geometry do not comprise a theory of proofs in the sense of special researches. They must rather be accepted as given; they belong to the phenomenal region just as elementary geometry do es, and must therefore be regarded in the same way as the latter. At the same time it is to be recalled that modern theories are only comprehensible in light of their historical development, insofar as their inquiries are motivated by previous conceptions of problems; thus the uniqueness of their achievement emerges clearly from their own historical background.

Such a motivational nexus appears in many forms and will be elucidated under various aspects. The problem of intuition in particular reappears with regard to the specifically scientific char­acter of geometry. Now we must investigate the role of intuition in its distinct types, as they were analyzed above, and the way it functions in the total configuration of the activities of demonstrative sciences. Furthermore, the constructive character of the geometrical must be considered. The concept of construction can be clarified only in relationship to the changes of its meaning in the course of history. Finally, the question of the algorithmic structure of mathematical thought leads to the contemporary foundational investigations, and in particular, its results must be contrasted to Husserl's charácter­ization of the algorithm.

With regard to the question of intuition, to be taken once again as pictorial-symbolic representation, it obviously plays a very re­stricted role in the process of proof, even in ancient mathematics. K. Reidemeister views the achievement of Euclid to be precisely a "transformation of the intuitable into the conceptual."111 Euclid's work is characteristic of the striving to exclude as far as possible the predominant employment of sensible-intuitive experience in an­cient mathematics and to provide an axiomatic base for it. This means that for the formation of fundamental principies he selec­tively employed only a few and obviously unavoidable facts, intu­itively required. Everything else was acquired purely mentally with the systematic recourse to these principies leading to the creation of a strictly rational method. Nonetheless, what justifies our speaking

111. K. Reidemeister (2), p. 51.

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of a transformation lays too much stress on the transformation and gives too litle credit to the role 0f intuition in the founding of Euclid's geometry. Geometry even befare Euclid was demonstrative and, as shown today by researches into the history of mathematics, the need for axiomatics has a long prehistory. Attempts at proofs appeared first and were followed by exact definitions befare Euclid's axiomatic theory emerged. It can be said that his axioms are principies in the Aristotelian sense: they are first in this science, although not first for us. Due to the lack of axioms of ordering, Euclid's system of axioms, however, was so construed that in many cases of demonstration, intuition was introduced not merely "for illustration," with the aim being a subsequent observation of plau­sibility, but rather was unavoidable in the very process of demon­stration.

In the geometry of Euclid it is inappropriate to limit the role of intuition solely to the establishment of axioms. This view is contro­versia! for la ter geometry, at least from the aspect of concrete geometrical work. The methodological characterization of geometry as a deductive discipline designates an essential trait of this science; yet this applies in general to its finished structure, so to speak, without giving adequate account of the factual process of the geometer in which this structure is first of all constructed. It assumes that in geometry one proceeds in such a way that all subsequent results are obtained from pre-given or assumed premises purely by means of logical inference.112 But even with the contemporaries and followers of Hilbert, the mathematicians do not proceed so mechan­ically in their work. Rather, in more recent geometry the procedure is as follows: with a "proposed" statement, i.e., with a statement that is assumed to be true mathematically, one seeks the presuppositions required to demonstrate the statement, and quite often the opposite assumptions turn out to yield a true statement. Thus to a certain extent the results are here given first while premises are discovered subsequently. To what extent such assumptions are occasioned by intuition in geometry cannot be stated in general and must depend in

112. See O. Holder (2), pp. 26-28. L. Nelson (p. 387) also claims that as soon as the axioms are assumed, then all theorems follow "through the mere form of inference." In any case, the non-euclidean geometries, with which Nelson is primarily concerned, present an entirely new aspect of the problem. They initiate a transformation of the concept of axiom, later culminating in the work of Hilbert. Concerning the question of axioms and their meaning, see our subsequent discussions, pp. 216. ff.

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individual cases on the content of what is being assumed. At least this could hold true when the result (a theorem) is more intuitable than the premises, which is quite often the case. As a theorem it may be valid only when it is obtained deductively as a conclusion through a chain of inferences; in such cases the function of intuition is not demonstrative but "merely" motivating, although it is relevant for the technique of demonstration. To limit the significance of intuition to the acquisition of intuited fundamental principies and to relegate everything else to "pure" thinking is to misunderstand the nature and origin of such thinking.

Even for the thought of antiquity, however, it is crucial that mathematical work must encompass the intuited aspects, to the extent that such work begins with something intuited and proceeds with symbolic intuition. Even what is immediately and intuitively evident in the figure, such as a geometric proposition that is valid for the states of affairs themselves, must in turn itself be demonstrated. Even such a simple relationship as triangular inequality is not aided by having recourse to the evidence of intuition; if it is valid as a mathematical state of affairs with respect to the triangle, it must be deduced from something else.

Even if each step in thought is "followed" in intuition, and if intuition must present and retain what is grasped in each step, the real concern is not with the intuitive, but with what "follows" logically. In accordance with its essence, the logical is so constituted that it does not require any pictorial-symbolic illustration. What is announced in the latter are at best certain factual and partial results "leading" ultimately to the conclusion (e.g., auxiliary lines of a construction). The logical consequence as such escapes pictorial symbolization. Its essential characteristic, the inferential "then," cannot be presented in any pictorial-symbolic way. Only signitive symbolism can establish it in a sign and reach not only a new kind of representation of geometric formations but also and ultimately an exact presentation of geometrical propositions concerning these formations.

It is demonstrative thought that is responsible primarily for revealing the ontological meaning of geometric propositions. That something is geometric is synonymous with the fact that it is provable; this is valid not only for the meaning of the copula in geometric propositions concerning states of affairs, but also for existential propositions. Even in the mathematics of antiquity, the existence of an elementary geometric formation-e.g., a circle or a triangle-is not assured simply because it is seen ideationally and

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represented pictorial-symbolically; rather, its existence must be genuinely demonstrated. This identity of being and being proven constitutes the uniqueness of the geometrical as well as the mathe­matical sense of being. A geometrical state of affairs does not first exist and then get demonstrated; rather, it comes into being in the process of demonstration. It has its being in nothing other than in its being demonstrated as valid in a process that in its turn is capable of rational verification employing criteria of validity and invalidity. As invalid-proven as not valid-mathematical states of affairs are nothing. They may exist for a while in an individual consciousness as a psychic construct, but it does not emerge into objective, mathematical existence. This is not counter to the circumstance that there are unproven propositions (e.g., the great theorem of Fermat). That clearly noted exceptionality in the total field of deduction confirms what is stated, not as an exception would confirm a rule-­mathematics has nothing to do with rules of this kind, and its complete interconnectedness is not such a rule--but in the way a single case always disturbs mathematical thinking, as a deviation from the ideal mathematical science. Such an ideal remains preemin­ent in its attempts at proofs. The more difficult circumstance seems to be that mathematics contains undecidable propositions, proposi­tions for which no proof is attainable concerning their validity or invalidity. At the same time, this undecidability is of a unique kind. It is incomparable to any other manner and mode of uncertainty, and on precise grounds. Here the structure of identity between being and being proven is displaced. What is demonstrable is not the mathe­matically posited state of affairs, but its undecidability in principie. The non-demonstrability of the truth or falsity of such propositions is itself provable precisely.

The identity asserted is also not contradicted by the fact that geometry, like any deductive system of propositions, also contains axioms-fundamental principies that cannot be demonstrated. Aristotle saw clearly that axioms as such belong to a demonstrative science. The intent to prove everything in a demonstrative science, including the fundamental principies, would lead to an infinite regress and thus never to a science. If a science is to be established as a demonstrative science at all, it must necessarily proceed from propositions that are not demonstrated. Moreover, the latter must also be undemonstrable; if they were demonstrable, then, in accor­dance with the concept of a strictly demonstrative science, every­thing that is demonstrable would also have to be actually demon­strated, and thus the demonstration would have to be given for the

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principies as such. Undemonstrable fundamental principies are a necessary presupposition of any demonstrative science.113

Thus a decisive question is: how is demonstration present in geometry? Even the geometry of antiquity, while being prior to any "theory" of proof, saw a problem here. What characterizes geometry is not its capacity to present its formations in sorne manner or another; rather, this presentation must follow a completely deter­mined process of establishment, using specific means of establish­ment-it must take place constructively.

This is more than a mere technical problem. Properly understood, this requirement is not concerned with precisely or correctly "imi­tating" something already pre-given and known, using sorne techni­cal means in a real medium; rather, the postulate of constructive geometry contains nothing less than the geometrical formation, first generated only in and through the construction. It only comes to existence in this generation.

This state of affairs is ontologically important. Although in a preliminary way, it contains the operationa1 sense of the being of mathematical formation; its being is a being by virtue of a specific method of generation. If the condition for its existence rests in the possibility of its constructive generation, then the ultimate explica­tion of its sense comes only from the constructive activity itself. The illumination of this activity is an unavoidable task for any theory of mathematical objectivity. Yet the illumination ought not to limit itself to mathematical thinking, i.e., to the "constructive" aspect of this thinking; rather, it must touch upon the fact that this latter concept of construction is merely a late historical transposition of a

113. The assertion that axioms are undemonstrable principies is obvi­ously more telling than that they are merely unproven presuppositions. The former had fruitful results in the philosophy of mathematics. The concept of the undemonstrability of euclidean axioms lent them a unique dignity, initiating a plethora of speculations concerning the reasons why precisely these and no other fundamental "facts" are undemonstrable. At the same time, however, it was precisely their asserted undemonstrability that brought geometry its progress, for when this assertion was placed in doubt, it led to new attempts at demonstration; from the failure of these demon­strations one did not gain an insight into the demonstrability of axioms, but rather into the arbitrariness of an axiomatic system. Thus the undemonstra­bility of the classical axioms ceased to be a metaphysical riddle; rather, it acquired a relative significance. What is undemonstrable in one system is not undemonstrable per se; rather, an axiom of one system can be a demonstratable theorem in another.

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meaning having, even in mathematics, its direct roots in handcrafts. A sufficiently extensive discussion of this suggestion would require a more encompassing investigation in its own right. Here we can touch upon sorne central points in order to suggest the specific uniqueness of mathematical constructions.

Euclid's Elements already appear to be built up completely under the aspect of constructability. For antiquity, this concept is totally bound to specific means of construction-for example implements such as compass and straightedge, which are to be employed in accordance with established prescripts. However, the novelty of Euclid's Elements does not rest, in the last analysis, on a strict and systematically applied mastery of the two constructive implements which were already employed, though unsystematically, prior to Euclid. The constructive character of euclidean geometry shows up primarily in that it employs only geometric formations whose existence, while not explicitly postulated, is nonetheless demon­strated by the possibility of construction with compass and straight­edge.114 Whether the Greeks were clearly aware of the meaning of their constructive process is obviously uncertain. In any case, this constructive aspect will have to be touched upon if we are to raise the question concerning the "essence" of the euclidean axioms. Why did Euclid "choose"-if such an achronistic mode of phrasing is appropriate here-these and no other axioms? A historically subse­quent answer, in no wise supported by Euclid's Elements, is usually offered with the notion that his axioms are "basic facts of intuition," comprising the "final intuitive evidence." Yet even with a concept of intuition that can be explicated geometrically, this solution remains insufficient, since not all euclidean axioms are adequate for such a claim of evidence; sorne theorems derived from them are undoubt­edly more intuitable than their axiomatic presuppositions. Euclid himself does not offer any "meta"-mathematical remarks. Seen from the structure of his entire system, it can be justifiably assumed that these axioms were not made into first principies of geometry because Euclid regarded them as intuitively evident; rather, the criterion for their choice was seen only in the achievement of a constructive building up of geometry, and one that is as complete as possible.

The following aspect of the mathematical fertility of axioms first attained its full consequences in non-euclidean geometry. In it the axioms are principies not only in the sense of being factually prior,

114. H. G. Zeithen has convincingly established the constructive character of geometric objectivities. In addition, see A.D. Steele.

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although subsequent for us, but rather of being at the same time proteron pros emas and proteron te physei from which the building up of the new geometry proceeds in accordance with strict axiomatic deduction. The question concerning evidence for the axioms is for it a totally nonessential, if not senseless question. The establishment of the axioms is not determined by whether they are intuitively evident; rather, only what they yield and what can be accomplished with them is of significance.

The standpoint of "producing," of "establishing" the geometric states of affairs dominates geometry in a novel way and extends the concept of construction. Without its original meaning, construction now means not merely the generation of geometrical formations by means of specific implements, but also their constitution in pure thought operations through step by step construction on axiomatic foundations. Besides the pictorial construction of a formation from intuitable parts with the aid of a few mathematical implements to be handled in a determinate and prescribed manner, there appears a construction from thought elements, which are to be connected by firmly prescribed rules. While no longer bound to the instruments of elementary geometry, the new mode of construction requires a specific kind of working tool: the sign. The more geometry is pervaded by sign symbolism, in contrast to the old means of construction, the more the purely rational concept of construction attains a fundamental significance for geometric existence.

New possibilities for the generation of geometry are thereby opened-which, however, also leads to new problems that could not emerge at all in the old geometry.

The predominantly new, in which the signitive symbolism far surpasses pictorial intuition, can capture not only the pictorial and picturable in its signs, but also the purely categorial such as logical operations. This allows not only a signitive representation of geo­metrical formations, but also a complete proposition about these formations. In contrast to analytical geometry, a new isomorphism of a higher level comes into play: the coordination of a sign system to a propositional system. It is in this possibility that a formal theory of proof can first be grounded. From its level of reflection and with its signitive means the structure of mathematical thought itself can be precisely grasped-yet at the same time difficulties brought about by this new possibility of construction come into view.

It must be recalled that for antiquity the elementary process of construction, employing the available means of construction and the simple rules of their use, must have made it completely

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self.evident that a mathematical formation could be constructively generated in a finite number of steps-indeed, self-evident to such a degree that this finitude of geometrical construction did not even have to be emphasized or affirmed in any specific proposition. In contrast, the conceptual expansion of construction toward concep­tual operations presented fundamental difficulties. Thus in arder to guard against paradoxes, foundational mathematical investigations found it necessary to posit a requirement that all formations that can be generated ought to be attained in a finite number of construction steps in accordance with specific pre-given rules of operation; if it is to be comprehended in the most strict sense, this concept of constructability contains this additional postulate of finitude. Indeed, as it turns out, the finite modes of proceeding do not exhaust all constructive means, although the finite means are completely constructive.

The meaning attained by the requirements of finitude for the constructive building up of mathematics allows the appearance of the algorithmic structure of mathematical thinking. In its original sense, an algorithm is universal, univocal by prescribed procedure of calculation through which a solution to a particular problem can be reached in a finite number of steps. If the concept of the algorithm is extended beyond the common procedure of calculating to include operating with signs in general, it can serve as a characteristic of its system of rules. Each individual rule would then present an algo­rithm, and the entire system of rules would also be conceived as an algorithm to the extent that the use of rules is established univocally and completely. It is only with the possession of these algorithms that the genuine problem of calculability, decidability, and generat­ability become present. Since they lead into specific foundational questions of pure mathematics, they cannot be pursued here any further. 115

Nonetheless, it would be beneficia! to touch upon Husserl's delimitation of the algorithm and the definite manifold. For Husserl, a definite manifold is characterized by a limited number of basic presuppositions (concepts and propositions) which determine com­pletely and univocally all mathematical states of affairs of the provinces concerned so that all the states of affairs can be deduced in unbroken sequence from the presuppositions; hence what is mathematical or nonmathematical always follows as a formal-logical consequence or as a logical contradiction from the axiomatic pre-

115. For more details see the works of P. Finsler and H. Hermes.

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suppositions. Husserl explicitly indicates a close affinity between his concept of definiteness and the axiom of completeness by Hilbert. A finite number of axioms completely and unambiguously determines the totality of all the possible formations belonging to a province in the manner characteristic of purely analytic necessity, so that of essential necessity, nothing in the province remains open, which should accordingly distinguish mathematics from all other sciences and constitute for it the decisive criterion as an axiomatic­deductive science,llB

Two years after the appearance of Husserl's Formal and Transcen­dental Logic, foundational investigations demonstrated the untena­bility of the Husserlian criterion. Since then only the noncontra­dictoriness can be maintained, and only if no limitations are allowed for the means of its proof.117 Today Géidel's proof of incompleteness (for arithmetic), showing that the arithmetic axioms yield no deci­sion concerning all arithmetic propositions, is a more serious objection to Husserl's conceptions. Moreover, the Husserlian con­ception of univocity, and the way that he has understood it, can no longer be maintained.11s

116. E. Husserl, Ideen 1, § 72; see also FTL, § 31. O. Becker (2), pp. 686-89, has presented the remarkable suggestion that Plato already had a clear insight into the object of mathematics as a "definite manifold."

117. The freedom from contradiction of the pure theory of number cannot be proven; it remains-with Husserl-"open," if we limit ourselves to the means of a given system, as was shown by K. Godel in 1931. In 1936 G. Gentzen succeeded in demonstrating the absence of contradiction for a number theory only with the aid of transfinite methods, by using Cantor's ordinal numbers.

118. In 1934, Th. Skolem was able to show that not only a finite number of axioms but also infinitely enumerable axioms are inadequate to "charac­terize" the number series. G. Martin, who has critically discussed Husserl's concept of the definite manifold (in accordance with Ideen I), uses, among others, the results of Skolem as a refutation of Husserl's requirement for finitude; at the same time, he extrapolates from Skolem's work the proof "that the arithmetic of the natural numbers cannot be grounded in a finite system of axioms" (p. 163; italics added). lt remains to distinguish between the axioma tic grounding of a mathematical domain and its characterization. That the series of natural numbers is not "characterizable" by infinitely enumerable axioms means only that its axioms are not specific for the natural numbers, i.e., they do not univocally ground merely the number series. This means that it is not possible to define numbers implicitly by their use, but rather that there are other objects equally characterizable by these axioms, such as certain functions belonging to number theory. In

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All these results are related to pure number theory and to arithmetic, but not to geometry. Yet the incompleteness of arithmetic ea ipso includes geometry since the corresponding proofs for geom­etry can be achieved as a whole with arithmetic models. Husserl's characterization of mathematics as a definite manifold of proposi­tions can no longer be maintained; and insofar as it is an objective correlate of an algorithmic structure of thinking, the latter, in light of this characterization, also requires corrections. Mathematical think­ing per se can be characterized as algorithmic only in a less restricted sense; the truth or falsity of a mathematical proposition can no longer always be decided through an algorithm; rather, the decidable problem in mathematics can always be resolved by way of algo­rithms. The only characterization that can be maintained today for mathematics is that an infinite number of theorems can be acquired as statements whose truth is based on a finite (enumerable) number of premises-a state of affairs that says less than the Husserlian conception. Yet in itself this state of affairs is sufficiently notewor­thy. Thus it deserves not to be passed over as self-evident, but must be understood as a unique contingency of mathematical thinking.

§ 6. Summary

The preliminary observations of this section saw their main task in the exposition of sorne modalities of consciousness that play an essential role in the total nexus of mathematical intentionality as well as in the phenomenological clarification of the modes of givenness of its correlative sphere of objectivity. The immense complexity of this nexus on the act side was not completely mastered, and the presen­tations concerning abstraction and mathematical ideation, pictorial­symbolic and categorial intuition, signitive symbolization and for­malization, offered little more than an indication of problems previously not yet treated phenomenologically.

Despite offering or neglecting to offer exhaustive analyses of the foregoing intentionalities, sorne consideration must be given them on two grounds. On the one hand, it was essential to extricate the

contrast, a separate question must be posed as to whether the numbers can be grounded by a finite number of axioms. Under certain conditions this question can be answered affirmatively if one envisages the axiom of induction as singular, as it is used singularly in linguistic understanding. If one wishes effectively to refute Husserl's requirement for definiteness, one must show that this axiom presents a genuine, undisputed chain of axioms.

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complex of phenomena constitutive of geometry, at least to the extent of avoiding the danger of diminishing the fundamental problems concerning the access to the geometrical domain of objec­tivity. To cover up these problems with accustomed terms such as "empirical" or "apriori," "intuitive" or "non-intuitive," "abstract" and "purely logical" is inappropriate as long as, first, the justifiable sense of these concepts is not properly clarified for geometry and, second, as long as their limitations are not outlined as precisely as possible.

On the other hand, the investigations achieved were important for us, since they shed sorne new light on the subject who, as a unity of consciousness and corporeity, pursues geometry. This unity is not only factually unavoidable for a scientifically engaged subject, but moreover obtains its structural illumination from specific activities of "pure" scientific consciousness itself. Although at first glance it may seem that in its orientation toward an incorporeal world of objectivity, this consciousness far surpasses the lived body and all corporeal functions, leading to the notion of a "pure" consciousness, the complex of acts investigated shows precisely that in the con­struction of this consciousness, corporeal functions take an active part. In addition, it is inadequate merely to secure a founding moment for mathematizing acts and their objective correlates in the sensibly given "in person," as can be easily shown for the case of ideation and categorical intuition. It can be made clear in the phenomenal complex of intuition that the role of corporeity is not given its adequate due if it constitutes merely a lower "stratum," or something similar, in the domain of geometric objectivities. If corporeity is to be taken with the full diversity of its various characteristics of activity, then it is apparent that the image of the lived body as the sale "source" of geometry is only very condition­ally appropriate. Indeed, this image does not relate to anything decisive in sensible intuition. It is not only in pictorial-symbolic intuition that geometry requires corporeal functioning for its com­pletion. Rather, paradoxical as it may seem, even in those formations of mathematics that can extricate themselves completely from this type of intuition and operate exclusively with signs, corporeity and its functions are brought toa new validity. Modern constructivisms are bound to the establishment of sign-series, not only factually but also as the condition for their possibility. Their operating with signs is not merely accidental; rather, the full actuality of this turn necessarily takes place on signs. Finally, as Hilbert has basically recognized, these signs are not "merely" and "also" sensibly perceiv-

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able images; rather, their constitution "in person" in sensible intuition is of the highest mathematical relevance. This all reveals that what consciousness can achieve in this new domain of objec­tivity is possible only with the continuous participation of corporeal activities. The lived body is a transcendental condition of these accomplishments and, indeed, not merely a founding moment that is surpassed by higher-level mathematical activities of consciousness; rather, owing to the implicative intertwining of corporeal functions and intentionalities of consciousness, it is also a condition for the possibility of the accomplishment and completion of those activities as such.

Perhaps we should speak here more cautiously: instead of "the" lived body, we should speak of "a" lived body. Is it not the case that modern mathematical theory is completely independent of specific corporeal structure and resultantly of the human? It is thinkable that at best this theory requires sorne corporeity and its functional modes of generation of, and manipulation with, signs, without presuppos­ing the specificity of these functions as those of human corporeity. In such case its relativity to the human subject would be merely factual, "also" demonstrable on it, while the constitution of such theory would be thinkable in principie in terms of differently organized corporeal being. Corresponding questions will reappear in the se­quel, specifically in the mathematically oriented investigations; we shall treat them when they emerge.

Finally, what is relevant to the specifically mathematical structure of thought has often been judged negatively. It has been maintained that mathematics is merely tautological. Insofar as mathematics merely reveals what was already contained in its presuppositions, the theorems follow analytically and formal-logically from the presuppositions. This position must answer the question asto why in mathematical deductions the same fundamental presuppositions do not always yield equivalent statements, but rather extremely varied and contentually different propositions which, in the case of geometry, are crucial for the description of real spatial processes. Indeed, the uniqueness of the geometric process consists of its strict formal inference, and yet it also proceeds factually. Should the ground for this be found in space itself? Should space itself enable the specific material content of geometry? From what has been said above, this could only mean that the first fundamental presupposi­tions must be based in it and that they in turn are no longer formal propositions, as if they were merely logical and thus subsumable under the contentless category of objectivity, anything whatever.

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Rather, they revert back to material formations containing factual propositions of a kind such that in arder to be grasped with respect to their specific material content, they require a particular reflective access.

No mathematical theory of foundations and no theory of proof can constitute such an access. Our subsequent observations will provide questions and solutions for the existing individual researches. These constitute our presuppositions. We accept them as given in the same way that we accepted the things of lived space in the previous part. Yet in the latter, we did not see our task as a mere ontic exposition of the structures of lived spatiality; rather, we attempted to grasp it ontologically in terms of the spatial understanding of the subject in it and comporting himself toward it. Even with mathematical space we are not concerned with a descriptive summary of what has been worked out in specific researches. The following presentation does not claim to be such a summary. lt can include only those states of affairs that are relevant for the ontological consideration of geometry, i.e., that aid in deciding the question concerning the "sense" of their formations, their space. Systematic reasons dictate the placing of euclidean geometry at the beginning of the following analyses. lt is not only historically first but, in addition, it is ontologically the most originary form of geometry, in a sense to be clarified below. The point of departure of the following investigation may be compared to that of the beginning of the first part of this work, as long as the comparison is seen as nothing more than an external analogy: if the investigation of actual space were to be a work that was appropriate to the phenomena. lt could not dispense with determinations belonging, according to their genuine sense, to subsequent forms of lived space-after all, what was constantly given to us befare any beginning was a subject who had already traversed all the spaces. Thus here too, if the analysis of euclidean space is to be adequate to the task, it cannot remain completely free from subsequent geomet­rical determinations. Here, too, the subject who raises the question of mathematical space as a whole, and begins with euclidean geometry, is the subject who has already "measured" the totality of geometrical spatialities.

SECTION TWO

Euclidean Space

Chapter One

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§ l. Forrnation and Relationship. The Prirnacy of Relationships

The previous section contained various hints at sorne elernentary geornetric forrnations which, while chosen arbitrarily, functioned rnerely as exernplary rneans for the dernonstration of sorne essential trends of the act-characteristics required in rnathernatics. At present our concern is to consider sorne of these forrnations in their own right. They and the relationships obtained between thern constitute for us the elernentary dornain of objectivity of geornetry to which the subject is oriented, just as the subject was oriented to the objectivity of lived space.

Obviously this analogy should not be given too rnuch weight. In lived space, the differing rnode of corporeal cornportrnent appeared to be constitutive for the things of lived space in their differing rnode of being; here the correlative relationship between the subject and the new object-world seerns to be frozen into the one intentional tension of the rnathernatizing consciousness. It has abandoned all the lived connections to the world and has retained the one ideal geornetric dornain of objectivity as its unique counterpole.

If talk about rnathernatical "spaces" is to acquire an acceptable rneaning at all, then the justification of this plural usage cannot be sought in the various orientations of the subject but exclusively in the structure of such spaces and the objectivities that constitute

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them, or in their various fulfillments of geometric meaning. This objectivity thereby assumes particular importance. At the same time, the following question must be addressed: How are we to grasp the all-pervasive and constant mathematical intentionality of the sub­ject in relationship to the multitude of mathematical spatial forms? The question is not only what separates and binds them ontically­to demonstrate this would be a matter of mathematics-but also what constitutes their multiplicity ontologically, as seen from the vantage point of the subject of mathematical space-conception.

The basic objects of geometry, in the sense of euclidean axiomatic theory, are point, line, and plane, as well as the relationships obtaining between them such as ... lies on ... (a point on a line), intersection (of lines), and sorne relationship of arder (between, etc.). The relationships can be called simple because their meaning is not primarily and exclusively mathematical. They are understood and sensefully di¡¡played in lived space; their geometric meaning is, mareo ver, a carrying over of predetermined meanings into the ideal sphere.119

lt is otherwise with the aforesaid relata. Indeed, in lived space the concepts of point, line, and plane have an intuitively fulfilled meaning. Yet their genuine geometric content is not at all perceiv­able in sensory intuition; rather, it first results from intuition through a process of formalization. Here the concern is no longer with a mere carrying over of the originally meant into another domain. Rather, the intent is first and foremost to posit new objects. What is meant with such objects intuitively, i.e., taken from the domain of sensory intuition, is retained as a founding moment, although with a decisive novelty which, in its constitution of the "geometric," requires extensive discussion.

It is not by chance that the first attempts to ground geometry aimed at determining the fundamental geometric formations through pre­cise definition while the relationships between them, in contrast,

119. In the following, we differentiate between carrying over (Über­tragung) and transposition (Transposition). The first directly accepts a meaning given and grasped extra-scientifically into the geometric domain, such that the meaning is made more precise and is given a strict delimita­tion; in transposition ("translation") of a concept into the mathematical sphere, there occurs a modification of meaning, yet in such a way that the modification does not encompass the entire complex of meaning, while the remaider maintain their unmodified meaning. (Cf. the mathematical concept of movement, pp. 231 ff.).

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appeared not to require their own definitive ascertainment. The latter appeared to be obvious in their own right; lacking any specific mathematical content of their own, they could be applied to geom­etry without any special delimitation. Perhaps this is the reason why Euclid lacks the axioms in which simple spatial relationships (such as relationships of ordering) become thematic.

For a long time serious objections have been leveled against Euclid's axiomatic theory. The insight into not only the practica! uselessness, but above all the logical impossibility of explicitly defining point, line, and plane was an important discovery of foundational investigation. This led to the implicitly understood definitions of these formations, which are possible only with the aid of a preestablished system of axioms in which these formations are not explicitly used. Hilbert demands that such an axiomatic system should only establish formal properties of the relationships that obtain between geometrical elements without explicitly stating what the latter are.

The crucial difference between Euclid and Hilbert lies in that at the very outset, Hilbert wishes to conceive of geometry as essentially a special case of the pure theory of relations, with the relation receiving primacy over its individual members. Yet in a very specific regard, Hilbert's foundations of geometry do not go beyond Euclid's. What is present in Hilbert's axioms of connection are the fundamen­tal propositions of geometry used in the form of existential state­ments ("there is ... ") of such relationships as "to lie on," "to go through," "to join," "to intersect," etc., without prior definitive delimitation of these relationships. The same can be said of the second group of axioms pertaining to the relationships of ordering whose description, as Hilbert says, "is best served by the term 'between.' " As already suggested, we are not concerned with a lack of mathematical foundation, but with one that is found in the essence of such relationships themselves. Their signification­fulfillment is present prior to any mathematical sense-bestowing and is taken over directly into the domain of mathematics.120

The subsequent analysis of signification must be directed to objects in a narrower sense, to geometric formations as such. We are

120. Hilbert (4), §§ 2 and 3. This does not contradict what Hilbert says about the axioms of ordering: they define the concept of "between." This definition is implicit in the linear axioms of ordering, in which, however, a pre-mathematical understanding of the verbal expression "between" is already presupposed.

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not hindered even if no explicit mathematical definition can be devised for them. Our sale concern is systematically to trace the various levels of meaning-giving in phenomenological description with regard to the appropriate objects of sensory intuition.

§ 2. The Line Segmentas a Fundamental Metric Formation

Seen from the vantage point of our concerns, point, line, and plane, in the sense of fundamental geometric formations, are already secondary and derivative phenomena. It is not only that mathemat­ical meaning-giving, along with all the characteristics of activity constitutive of it, reveals a higher-level intentionality in which the directly grasping intentions are already presupposed; even the latter are always oriented toward finite formations of sensory intuition, whose boundaries cannot be arbitrarily transgressed either out­wardly or inwardly. No points, lines, or planes are intuited here­rather, things with "edges," limited by straight lines and plane surfaces.

The edge is a morphological characteristic of the intuited thing. Grasped as such and distinguished from other forms of physical contours encountered on things by the touching and seeing corpore­ity, it constitutes the sensible foundation for the conception of a "line segment." The line segment must be se en as the genuine fundamental formation of geometry, lending to geometry its origin­ary sense as a science of spatial measure. The conception of the line segment is prior to those of line and point not only historically and factually, but also in its ontological meaning.

The conditions for its ideation are present in the sensory intuition of the individual thing. As already mentioned with respect to the space of intuition, its conception also includes the apperception of the free motility of the intuited spatial thing and its invariance in the transformation of its visual aspects. These are precisely the charac­teristics of a spatial object determining itas a "fixed body," and they are specifically discussed in various ways by modern physics. Here we are not about to enter the debate of the individual sciences and the easily misunderstood form in which they pose the question as to whether there are "really" fixed and invariant bodies in movement. Our concern is not with the problem of a fixed body envisaged from a specific methodological viewpoint, but directly with its indubita­ble phenomenon in the natural consciousness of objects, which is to be taken and comprehended as such befare any special scientific questions can be understood. The latter are possible on the basis of

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a previously explicated concept of spatial bodily movement and invariance. As has been shown, this is based on a possible world­attitude of a sensibly intuiting and self-moving corporeal subject who apperceives the thing as moving and, in this movement, as changeless by virtue of bis own consciousness of identity and bis own corporeal possibilities of movement. The fixed body, which "itself" does not change in being carried and which is thus suitable to be repeatedly "moved" for the purpose of measurement, is only insofar as there is a motile and self-moving corporeity that knows itself as self-moving and first experiences itself as "itself" in con­fronting the intuited world of objects.

As a founding moment the corporeal movement in no wise constitutes the entire nexus of conditions that are essentially re­quired for the concept of a line segment. The phenomenon of the edge, as boundary of a fixed body, is by itself insufficient to elicit the specific meaning of the line segment. After all, what constitutively belongs to its conceptual sense is its character of being a standard. The line segment is determinable in principie in comparison with, in its relationship to, another line segment. This means that only line segments function as relata of such relationships. This reveals, on the one hand, their ideal independence from the real world of things, and on the other, their quantitative-relational character. Such origin­ary, quasi-quantitative relations as "larger" or "smaller than ... "and "exactly as large as ... " first assume exact expression and precisely comparable relationships with the help of numbers. From these then we obtain that one line segment is "contained in" another once, twice .... Placed in such relationships, the line segment itself functions as a unit of line segments. Once chosen and subsequently taken to be the same and constant in all comparisons with other line segments, it is capable of determining the sought for relationship of size as a "relationship to itself," i.e., as a measure through repeated application.

The ideational independence of a line segment, its extrication as a pure and empty quantity from the world of material fullness, its relational character wherein it attains its primary universal func­tion in the domain of all that is measurable---.:all these cannot be · derived from corporeal functions; yet what the line segment is, and what it accomplishes, can be understood only with these functions.

The independence of the line segment does not appear only in that, bound to the conception of a fixed body, it is the means for comparing sizes within the sensibly intuited world of things. Once

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conceived and grasped as independent, it pervades at the same time the domain of free, ideal formations. What constitutes its authentic metric aspect is not its pictorial-symbolically apprehensible form, but rather that in it which presents the measurable and the measur­ably comparable. Thus, for example, "side," "altitude," "diagonal," "radius," "axis," etc., are indeed concepts having a precise geometric applicability (constituting, for example, a so-called position defini­tion of a geometrical formation). Nonetheless, in their meaning content they also bear morphological moments of form that cannot be readily exchanged. They must be understood in relationship to the whole of the formation. A square has no radius, and to speak of a diagonal in an ellipse makes no sense. These enumerated aspects can be mastered metrically only when their specific meaning within the framework of the total figure is completely bracketed and the meaning reduced merely to the conceptual sense of the purely extensive size of a part of a line of a determínate length. It is only in this respect that questions concerning the relationship between, for example, side and altitude in a triangle, between the radius of a circle and the side of an inscribed square, can become at all senseful. Such relationships express the genuinely geometric content of such formations; the relationships are given only between line segments, i.e., between what is conceptually and significationally equal. This leads to pure relationships of numbers.

The fundamental meaning of the line segment is not confined to the objects of the elementary geometry. Even higher geometric formations, such as algebraic and transcendent curves of higher arder, become accessible in their characteristic metric properties insofar as their relationships are measured against a determínate and precise line segment. lt is irrelevant whether the line segment remains invariant or is varied in accordance with sorne rule.

Even the determination of an angle is bound to the measure of a line segment. While trigonometric functions and their inverse func­tions contain transcendent numbers of relations, they do not change the fundamental meaning of the line segment for the determination of angles. After all, the problem of the incommensurable quantity first attains its full mathematical import only in light of it. In addition, the line segment retains its fundamental meaning in all geometrical calculations of length and area. Tasks of squaring, of the rectifiability of curves, have emerged, in their full extent, from a geometric method that attempts to approximate the formations in question by those of a more elementary kind. Thus the integral calculus found its crucial problem in an appropriate formation of

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limits which, in each singular case, had to proceed from the elementary-geometrical, i.e., straight-line, through partial areas of a finite size, bounded by line segments.

This founded meaning of line segment, assumed universally by geometry, sheds light on the more recently devloped viewpoints in mathematical investigations. The latter permit the expression of the essential content of a geometric domain even in a single state of affairs. What determines this content in its crucial outlines is the relationship of line segments in contrast to determined mathemeti­cal operations.121

But what does a "relationship" of line segments mean? Is it at all thinkable that one line segment can relate otherwise than as one and the same, wherever it may be measured? And what specific sense can be given here to a mathematical operation that would allow the application of this concept to ideal formations?

§ 3. The Line Segmentas an Invariant of "Movements"

The first question involves the problem of a multiplicity of geometries from which the geometry of invariant line segments, as a special euclidean case, will be extricated; in contrast to the first, the second question touches materially upon the method of geometric investigation.

The inceptions for the latter are already found in Euclid, even if its formal expression in mathematical group theory is of recent date. The meaning of the line segment is best illuminated by a theoretical aspect of geometry that is a fundamental component of geometry both historically and systematically: the theory of congruence. We are not concerned with the contents of the individual results of this theory, but with the existence of congruent formations as such. Congruence presents a more definitive relationship of equivalence; as an equivalence of "coincidence" it means not merely the agree­ment of two or more geometric formations in terms of spatial content or form, but the agreement of both determinations at the same time. However, the term not only concerns a specific geometric state of affairs, but also expresses something essential with respect to how this equivalence is to be thought and where the criterion should ultimately be sought for its fulfillment in each individual case.

121. We are still avoiding speaking here of a line segment as a distance between two points, since the concept of the mathematical point is not yet clarified. See pp. 296 ff.

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Moreover, the bringing to coincidence of the formations in this question involves an operation that at first sight seems completely unproblematic; yet it not only potentially includes a mathematical prejudice of a very specific kind, but also comprises a significant point of departure for the unfolding of the problem of mathematical space.

This prejudice consists of the assumption that during the "pro­cess" of coincidence, the formations will remain constant in their form and size, regardless of the "way" in which the coincidence may be attained. The metric formation thus exists independently of its position. In other words, congruence geometry possesses formations that are identical except for their location. This is not immediately self-evident and does not touch upon a single possibility of a geometric relationship. Hence the task is to obtain an insight into the conditions for such a relationship of congruence. What are the specific conceptual characteristics of the mathematically thinking subject that would shed light on congruence? How can the geometric sense of "congruent" formations be determined?

These questions revert back to our previous discussions of line segment, insofar as the congruence of metric formations is deter­mined by the invariability of its standards. But the line segment appeared to be sensibly founded on the phenomenon of a fixed body in the space of intuition. Freely movable and invariant as to its length, this fixed body turned out to be relative in its being to the subject of measuring and his own corporeal dynamics. It is signifi­cant that this moment of corporeal motility is constitutive not only for the apprehension of the "line segment" as a free geometric formation, but also for mathematical operations. Although within the subject matter of geometry "movement" has been explicitly considered only since the development of group theory, it has been seen as worthy of thought from its very scientific origins. This conception is already found in Euclid's theory of congruence, though implicitly, Euclid identifies superposable figures as such through "movements."122 It could be objected that the concept of movement in geometry is only a figurative mode of expression or

122. Euclid, Elements 1, propositions 4 and 8. We shall not enter in any detail into the controversy that flared concerning Euclid's treatment of congruence propositions and his method of epharmasein. The frequently discussed question as to whether with this method Euclid incorporated an empirical moment into geometry by having recourse to "movement" is irrelevant for the present context.

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only a kinematic "form" of apprehending, used by a geometrically engaged subject as an access into the geometric domain. Neverthe­less, "in itself," as an ideal mode of being, this domain lacks any kinematic and hence temporally-laden moment, and therefore must be apprehended in sorne other more appropriate way. The concep­tion of movement in geometry would be a scientific precursor, a kind of heuristic principie without, however, corresponding to the es­sence of geometric objectivity itself.

Meanwhile, our previous analyses have sufficiently shown that any concept of being in "itself" can only be meaningfully explicated through the incorporation of the immanent reflexivity of the con­sciousness that apprehends being. What the geometric entity "may" be "outside" this consciousness appears to be a futile question. However, that this consciousness conceives of ideal entities in this and no other way, as is shown through reflective analyses of consciousness of ideal objects, is not only established as an ontic state of affairs, but must also become ontologically understandable in terms of the conditions of its possibility in the subject constituting its being.

But this leads to the conception of movement in geometry and its noteworthy interrelationships. First of all, it is illuminating that the concept of movement be modified in its meaning if it is to be applicable to entities that, in their ideal mode of being, can only be thought as being outside of real interrelationships and the flow of time. At the same time, there is no doubt that despite the modified meaning, specific constitutive characteristics of movement as a real spatial process must reappear in the geometric domain if this concept is to be justifiable in geometry. First of all, the affirmative aspects of this concept must be established. At the same time, this will constitute an overview of the types of geometric movement as su ch.

Things and their movement in the space of intuition appeared to possess three characteristic determinations: (1) the identity of the intuited thing in its changes of place, i.e., its independence of place (place as a mere "location"); (2) the attainability in principie of any place by paths chosen at will ("free" motility); (3) the feasability of various types of movement. The first characteristic is equivalent in meaning with the existence of fixed bodies in correlation to the motility of the lived body as a "self." The free motility of the intuited thing manifests the continuity and the homogeneity of space; in it there are no "islands" that cannot be traversed and no places distinguishable by their inaccessibility. After all, space is simulta-

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neous for rnovernent and through rnovernent, narnely corporeal rnovernent. It is the paradigrn of a continuous process as such, which, in its capacity as a process, constitutes at the sarne time the rnultitude of co-valent spatial locations.

Finally, the third property of rnovernent, to be able to be realized in a rnultiplicity of forrns, can be understood only frorn the corporeal cornportrnent of the subject. Just as each rnovernent of a thing is relative to a self-rnoving being, and the forrner if characterized by being cornpensated for in principie by self-rnotility, each moved object can essentially acquire only those rnode types of rnovement of which corporeity itself is also capable. The latter, however, can be reduced to precisely two: rotating rnovement and linear movernent. The first, in contrast to the second, requires a distinctive locus conceived as being itself at rest, as a center of reference of rotating movernent. If only such rotating movements were possible in the space of intuition, then for each rnovernent in it there would have to be a distinctive location and all paths would have to run back to this location. Yet such a space could not be the space of intuition of an intentionally structured corporeal subject. The latter is not rnerely present sornehow as a unity of consciousness and corporeity; rather, this unity is very specific and is given in a deterrnined manner. While being precisely this lived body in its specific functional organization, it is at the sarne time the lived body of an intentional consciousness, which in its turn is co-determined from the outset, in its structure and contents, by this lived body. Such a corporeal subject would, however, find the continuous return to the same place contradictory to the structure of its intentionality. In the consciousness of a corporeity capable only of rnovement that returns to itself, there would be, figuratively speaking, no "space" for its possibilities of knowledge asan open horizon with a beyond for the imrneasurable, the infinite, and the irrevocably unknown, with its ever unknowable residuum. The difference between the known and the unknown, between one's own and the alien, between the actual and the "yet" possible, would merge. In their polarity, these oppo­sites determine the tendency of progressing onward that manifests itself in corporeity only by way of the translatory forward rnovernent.

Finally, the question must be raised asto whether sorne of these properties of rnovernent are retained in the geometric domain of objectivity. With regard to the first two characteristics of rnovernent, the unchangability and the free movement of the object, both are eo ipso ascertainable for the geometric domain. They are precisely the characteristics that are constitutive for the line segment as a funda-

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mental metric formation. With respect to the types of movement, the space of intuition reveals the translations and the rotations. But these ate precisely the types of movement that articulate geometry; the group of movements (proper) includes the translations and rotations and no others. The group property of geometric movements has its intuitive foundation in the movement of real bodies. What in group-theoretical calculus is called identity element ( or unit ele­ment), inverse, and group product means here a state of rest (identical mapping of itself), the cancellation of a movement through its reversal which, in turn, is a movement; and finally the possibility of executing several successive movements, whereby the end result is again a movement.

What lends the concept of movement in the mathematical domain its sense is the circumstance that specific characteristics of move­ment that are determinant for the corporeally-centered space of intuition are transposed into the geometrical domain. In other words, with the complicated and multi-leveled enactment leading from the sensory intuition of a real fixed body to the conception of a line segment, these moments of apprehension remain related to what can possibly happen with the fixed body and the ways that it shows itself as fixed. Yet the conception of movement in the ideal domain, as "transposed" from the real "movement," is in the geometric area no longer a givenness "in person," nor is it an immediate itself-givenness as it is in the physical domain. The line segment does not move "itself," nor "will" it be "moved" by any kind of cause as if it were a thing. But it is also not merely "thought" as moved or, in analogy to the latter, merely "represented" as movable. The latter may play a psychological role in elementary geometric work, but it is not appropriate for the matter under consideration. What transposition signifies here, and what it means geometrically, becomes quite clear in the mathematical description of such movements. It occurs in transformation equations that are subordinated to specific formal conditions (e.g., the non-vanishing determinants) whose common characteristic is their lack of temporal variables. What these transformations express is, strictly speaking, nothing other than a precise coordination of a formation with its "mapping." It is a purely static correspondence of two formations. Yet this does not simply repeat and trace over what was already there; rather, it is of such a kind that with their appearance the duality of the formations is first posited. Indeed, these equations have the character of prescriptions whose fulfillment aims only ata "mapping." There is no other way to think this except with the

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conception that one formation has been "transmitted" into another, has "moved" into it. In other words, what here motivates the comprehension of movement is not a concretely grasped process applied "on" the ideal formation itself-which would not be possi­ble, as it is without temporal components; rather, it is a specific mode of sense-bestowal of mathematically prescribed coordination. This takes place through a being who is the subject of a lived body with a specific corporeal structure of movement. That the extra­temporal entity is based on a mode of apprehension of "pure" consciousness, which nevertheless retains a moment of temporal sense even in this ideal, extra-temporal domain, can be explicated only through another specific objectivity consisting of the relation­ship between a pure extra-temporal consciousness (i.e., conscious­ness that constitutes something extra-temporal) and time-bound corporeity. The domain of being constituted in the ideal "move­ment" turns out to be nothing else than the objective correlate of a particular type of transcendence of corporeal comportment. The . kinematic mode of conception, as it has increasingly pervaded mathematical science in various forms, can be understood only in these terms. Such procedural thinking-whether it be called geo­metric "mapping" or analytically designated as "transformation," under the rubrics of covariants, contravariants, and invariants-has concepts that designate something more than would be merely psychological, sorne mere process of thought; they designate some­thing objective, signifying mathematical entities in their being.

§ 4. The Concept of Movement as a Leading Concept of the Theory of Invariants

The kinematic conception of geometric entities is most clearly understandable in the general theory of invariants. Supported by the calculus of groups, it is based on the crucial insight that the numerous geometries cannot only be grasped in terms of a unitary principie-as would be the case, for example, with the invariance of certain geometric formations in contrast to specific movements­formed from a subsequent extraction and compilation of aspects common to them. Rather, at the very outset, each geometry is completely determined through its invariants and thus through its movements. The latter constitute the metric domain in the genuine sense. In accordance with its structure, this domain is established as soon as it is decided which geometrical quantities remain constant in specific movements. Thus the invariance of the line segment

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univocally characterizes the euclidean movements, and thereby the sale "processes" in the geometric domain that justify the concept of movement in its primordial signifigance. That a line segment would change its length during a displacement contradicts the plain conceptual sense of a displacement. To apprehend this event, the conception has recourse to an additional moment that allows the line segment to be "simultaneously" stretched or shrunk. Such a process, which is the simplest geometrical example of an affine (specifically of an equiformal) transformation, is, in accordance with its meaning, no longer a movement or, rather, what is "moved" here is no longer properly a line segment. In geometric thought, even this process is constituted in no other way than in conceptual character­istics whose ultimate founding level is located in corporeal modes of comportment. In any case, it appears that these relationships con­stitute a sense-bestowing core for all the higher forms of geometric process, insofar as the meaning content of all of these forms is constructed from the previously conceived meaning of a line seg­ment and the sense of movement belonging to it, while at the same time acquiring new sense. All ideal "processes" in the geometric domain are significationally considered as variants of the primary forms of process, namely the euclidean movements. But that is why they no longer are geometric movements. In general they present "mappings"; below them the euclidean movements constitute the special group of congruent mappings.

What is significant is that each group of mappings, characterized by the invariants that belong to it, completely determines a corre­sponding geometry.

Just as the invariance of a line segment constitutes euclidean geometry, so the relationships between line segments constitutes affine geometry. Its transformation group is such that it transposes line segments over line segments while generally varying their lengths. Only the relationships among lengths remain, in other words, what is moved in this geometry is not line segments, but relationships among line segments. Yet this says nothing about the angle enclosed by two line segments. If constancy is required for it, in addition to the constant relationship of line segments, then we are dealing with a specific subspecies of affine geometry, conformal geometry. But since the angle can be changed variously by a (universal) affine transformation, then morphologically speaking, this geometry yields highly di verse formations proliferated by affine mapping. In place of congruence there appears affine equivalence, so well evidenced by the relationship between a circle andan ellipse.

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While both formations are completely different in a metric-euclidean sense ( one compares their definition through geometric determina­tion of position with the help of the concept of distance!), they are affinely indistinguishable. An "affine being" would know of no formal differentiation of such formations unless it took the relation of affine equivalence in the sense of our congruence. For us as intuitive beings, this means that affine geometry impoverishes the richness of euclidean-metric geometry by narrowing down the manifold of forms. It subordinates itself under the law of its own genesis, the group of affine mappings.

In their turn these are only special transformations of a general "movement" group comprising the projective transformations. It is characterized by the invariant of the dual relationship of four line segments, which incorporates the affine invariant of the relationship of two line segments as a special case. Here the concept of equiva­lence attains a degree of universality transcending that of the affine. The non-equivalence between the finite and the infinite that is preserved in affine geometry is surpassed in projective geometry. With the aid of the mappings in this group, it is possible to transmit projectively the "infinitely distant" formation into the finite and to make it geometrically available.

To go into more elaborate mathematical detail would be redundant at this point. Our concern focused merely on the insight that the various geometries alluded to here are justified in their multitude by the necessity of the various invariants required. Their systematic interconnection is, however, supported by the unifying methodolog­ical viewpoint of the general transformational group. At the same time they are determined ontically through certain kinematic be­stowals of meaning, which on the whole are founded in the concep­tion of mathematical "movement." The states of affairs of these geometries in their totality contain indices pointing back to the sense-constituting activities of a mathematical consciousness, which even in its higher intentional achievements reveals its connec­tion to and dependence upon corporeal functions.

Chapter Two

Euclidean Normal Space

§ 1. The Concept of Mathematical Space (Preliminary Conceptual Clarification)

The concept of mathematical space has frequently been discussed in current times. The philosophical controversy concerning its justification arase with the discovery of non-euclidean geometry. But this controversy limited itself essentially to the use of spatial concepts in these disciplines without coming to any genuine deci­sion. Lotze's proposal to speak of spaceoids in non-euclidean manifolds in contrast to the space of euclidean geometry could not take root. It presented nothing more than a terminological compro­mise; adequate clarification concerning what we are to understand by space as such in geometry was still lacking. The acceptance as well as the rejection of his proposal lacks to date any discussion focused on the subject matter itself. Whoever denies that geometry is a science of space, and wishes to see it deal only with formations in space, must no less justify the concept of space he employs than the one who unreflectively applies this concept in geometry, above all when he cannot avoid the conception of a multitude of mathematical spaces. Definite expositions and delimitations of this question can be purposeful and, indeed, are unavoidable in the work of the special sciences, yet they lead no further toward understanding what is to be explicated here. A conceptual clarification basing itself on a strict and material phenomenological analysis can and must avoid any terminological debates for or against a mathematical space. Its only appropriate question concerning space is completely neutral: its sale interest lies in the fact that in geometry one speaks of space and spaces. Accordingly its only task is to explicate what constitutes the characteristic content of the meaning of such language. The question is not whether "one" ought or ought not to concede togeometry its own concept of space. For us such an anonymous

239

240 Mathematical Space

"one," such a subject, does not exist. There is only a science of geornetry which claims for itself a concept of space. In turn, however, it is a fact that even this claim is historically quite late, and that "space" was notan object of investigation for geometry frorn its historical beginnings. The appropriate question for us is therefore: what was the rnotivating reason that led to conceiving geometric objectivity not only as an independent dornain in its own ideal rnode of being and its own structurallaws, but also as an objectivity of an ideal space? Conceived in the broadest terrns this rneans, after all, a way of speaking about geornetry as a science of space. For it, space is not sirnply the extensive manifold of real things and their relationships (regardless of how it rnay be "applicable"); rather, as a free science, geornetry is concerned only with itself. It is not directed toward possibilities of application. It constructs a concept of space that-in accordance with its sense-is free frorn any meaning of sorne real space of objects. Hence the concern here is not with any question of applicability of geometry in lived space; rather, the problem of mathematical space must now be posed only for free geometric manifolds. It must thus be taken as an ideal-ontological question.

The unquestioning and unhesitating way in which modern geo­metrical thought carried along such a conception of space into mathematical formations must not mislead us into thinking that we are dealing with something obvious, i.e., with a conception of space that is irnplicit from the outset in geometrical work. Rather, it has to do with a positing that is not yet grounded in pure geometrical objectivity as such. Thus even geometry, in its beginnings, did not conceive of such a space. Although Euclid established geometry as an exact science, a strictly geometrical concept of space was still cornpletely foreign-the geometric "euclidean space" is not a cre­ation of the founder of euclidean geometry.

Nonetheless, euclidean space is not justa complement to thought, a conception that merely supplements that of geometric objectivity and that could be completely absent; rather, it has throughout its fundamentum in re. That modern geometric thinking sees its forma­tions "embedded" in an ideal space, as we may provisionally formulate it, is a phenomenon of mathematical consciousness that not only cannot be contested-for as such it offers a critical point of departure for the rejection of any conception that would wish to discredit an ideal spatiality-but in addition signifies a factual dependence of this space on geometric objectivity and its relation­ships to it.

Euclidean Normal Space 241

From what has been said, it is easy to understand what constitutes mathematical spatiality. If the geometric objectivity is presentas an "event" in a process of executing determinate transformations, if the metric proper to this domain is conceivable only on the basis of an ideal possibility of movement of its formations, then objectivity conceived in this mode obviously requires an ideal "wherein" of such movements, an ideal "where" of their occurrence. This is posited, although not explicitly, when we speak of geometric move­ment, or even when it is merely grasped as a depiction in the process of thought. That there "is" an ideal space is equivalent to the statement that the geometric entity reveals its fundamental metric characteristics only in specific movements.

This explains a thought-provoking analogy obtained, on the one hand, between corporeal movements and lived space and, on the other, between geometric movement and ideal space. In the first case, the filledness of space in its specific mode of being-and thereby the corresponding space-is constituted in a strictly recip­roca! correlation with a dynamic mode of comportment. In the second case, the ideal domain, the correspondence to lived space is found in that geometric space is not something befare its "content," but rather is first built and opened up by the geometric formations or, to speak more exactly, by the given transformation group that determines this content. Here, as well as in the former case, the different movements are not merely processes in space, but also­when one bans all genetic conception from one's language-"gener­ators" of space.

In the case of geometric space, this concept of generation has a most palpable meaning. lt is not concerned with finding a transfor­mation group correlated to a somehow already conceived ideal space that is meant "in itself," such that the group functions to describe the relationship of the geometric formations "in" this ideal space. Rather, it is solely the group, which is primary and established in advance, that decides about the existence of an ideal space at all. Each group of mappings projects a specific mathematical space; this state of affairs comprises the meaning of all talk about euclidean, affine, and projective space.

It is no wonder that within the special sciences, the existence of a mathematical space first carne to conscious insight only with the construction of analytic geometry. The Cartesian discovery that every geometric formation can be related toa "straight line" signifies the first explicit postulation of a coordinate system. Ever since, every geometric proposition is related in a specific way to a system of

242 Mathemati9al Space

coordinates from which mathematical concepts such as "position" and "location" obtain their precise geometric meaning.

The postulation of a coordinate system shows clearly what binds the space characterized by such a system to lived space and what sharply differentiates both types of spatiality. With respect to the ontological meaning of such a system, there is an appertinent objectivation of that which allots to each thing its there, its place in relation to other places, within lived space: the corporeal here and its directions of orientation. As a coordinate system [Bezugssystem] for all topographical assertions in ideal space it is, to speak with Weyl, "the ineradicable residuum of corporeal subjectivity" in a space devoid of corporeity.123 Even mathematical space is space only for a consciousness and by virtue of a consciousness, which indeed does transcend the space-determining and primordially space­constituting functions of corporeity without setting them completely out of play. While being objective, in the sense of being exclusively an object for an intentional consciousness, even mathematical space as "space" is unable to be constituted in any other way than through determinations having their ultimate founding moments of meaning in corporeity-and in the corporeity of a subject constituted in this and in no other way.

Even the preference for right-angled coordinates and the experi­ence of their simplicity must be founded here. The right angle possesses, in contrast to any other measure of angles, a characteris­tically exceptional position. In geometry this is clear from the fact that the traditional division of angles presupposes the viewpoint of the right angle, which is posited as a primitive standard, while the rest of the angles are characterized as smaller or larger in comparison to it. Seen phenomenologically, orthogonality presents itself as a formalization of the absolute opposition of directions of movements, as they differentiated themselves in the gradual succession of corporeal dynamics, determining at the same time the structure of the corporeal space of movement in accordance with surface and depth.

Coordinate system and movement group are genuine conceptual moments founding the meaning of mathematical space. That "there is" mathematical space, that the conception of its existence is meaningful, depends completely on both moments, which deter­mine the mathematical concept of space and establish its limits. Neither moment is independent from the other; rather, they have a

123. H. Weyl (1), p. 72.

Euclidean Normal Space 243

specific interconnection with each other. lt is only with the aid of a coordinate system that the geometric formation becomes fixed with precision and its movement becomes analytically describable. The interconnection between coordinate system and geometric move­ment had already been recognized by Descartes with regard to its fundamental principie. lt could be objected that the Cartesian "straight line" introduces into the science of geometry something that is outside of and thus arbitrary, accidental to the formation. Descartes counters this objection by pointing out that with the continuous transformations of the coordinate axes (noted only by him), the formations change their position, yet they themselves remain "of the same species"-and for Descartes this appeared to be their essential geometric aspecV24

Even geometric movements are relative movements in contrast to a coordinate system that is taken to be fixed; each movement of a formation can be made retrogressive by a corresponding counter­movement of the system. In a more recent mode of expression, the invariants of a given transformation group are nothing else than invariants in contrast to specific coordinate transformations. As such they are primary for the genuine sense of the metric: in the more precise sense of the "geometric," the only property that is valid, in contrast to coordinate transformations, is the invariant.

However, what distinguishes mathematical space from lived space is also most visible in the coordinate system and the mathematical movements. That the coordinate system is freely choosable, or that the geometric formations can move freely in contrast to the system, is still a possibility presented as a transposition of the modes of corporeal movement, although the motivation of movement is fun­damentally changed. In lived space, corporeal movement takes shape in forms and orientations in accordance with the different sense-content of the lived body and its situation; the lived body attains this sense-orientation primarily from something else, which assumes a specific material determination and qualitative fullness through its own mode of being. Yet in a space without corporeity, the motive for movement is purely rational, and indeed rational in a specific mode. The simplicity of a formation, as an aim for a mathematical investigation in the choice of a coordinate system,

124. R. Descartes, Vol. VI (La Geometrie, Livre Second), pp. 388 ff. What is crucial in this discovery ought not to be overlooked simply because Descartes does not yet strictly distinguish between generalization and formalization.

244 Mathematical Space

corresponds to a specific principie of economy that is purely quantitative and in no wise geometric; it is determined by the rules of the domain of numbers. The simplicity sought for is not related to the formation as such, which is fixed in its geometrical form and its metric characteristics, but rather to its analytic "form." By accepting number as something completely distinct from the spatially struc­tured domain of quantities, geometry has introduced viewpoints completely foreign to space.

What is important here is that the analytically-algebraically con­structed geometry enables a dimensional extension of mathematical space. Previously the question of dimensions was completely over­looked, since implicitly one assumed that its three-dimensionality was similar to that of the natural space of objects. This three­dimensionality is nevertheless not accorded any special status in the framework of free geometric science. As a consequence of analytic geometry-i.e., a geometry that no longer has its "forros" in spatial imagery, but only in algebraic coordinations of signs that are investigated in terms of pure algebraic attributes ( degree of equa­tions, number of parts, etc.)-the question of dimensions becomes meaningless for its concept of space. Once a coordinate system is created, there is the possibility of symbolizing each point through numbers and of formalizing the distance between two such points (whose meaning in elementary geometry is that of a line segment) in a quadratic expression, so that only the degree of this equation has geometric relevance; there is then nothing that can hinder the interpretation of the multi-membered quadratic equations geometri­cally as "line segments." The "movements" of these "line segments" can be meaningfully extended from the pictorially intuited domain of two- and three-dimensionality to span over a correspondingly dimensioned space, within which all the propositions of euclidean geometry demonstrably hold.

The meaninglessness of the question of dimensions for geometry is most striking in the sign symbolism of vector calculus. lt functions without any fixed number of vectorial components, and its symbol­ism is so construed that these components do not even appear in the calculus. Only a subsequent limitation may freely interpret the completed calculus as "meant" for S3 , S4 , ••• Sn. For a vector­analytically constructed geometry there is always the free possibility of grasping itas the geometry of space possessing an arbitrary (finite) number of dimensions. This has to do with the possibilities of extension, which can easily be misunderstood as a generalizing universalization. Yet the geometry of n-dimensional space is no

Euclidean Normal Space 245

more universal than that of three-dimensional space. Insofar as n stands for any arbitrary natural number, it does not subsume something under itself as a genus would subsume a species, but rather contains it as a singular example of itself. A six-dimensional space is no more universal than a three-dimensional space, just as in the series of natural numbers a larger number is no more "universal" than a smaller.1zs

Similarly, the ten-dimensional euclidean space is no more abstract than the three-dimensional. What legitimates this common mode of speaking is the everyday and vague signification of the abstract in contrast to something that is intuitable pictorially or representation­ally. A closer explication will show in what sense one must speak here of intuitability or non-intuitability (p. 253). In any case, the transition from a three-dimensional geometry to one which is higher-dimensional has nothing to do with a process of abstraction. If one observes strictly what is completely and genuinely contained in the sense-bestowal of such mathematical spaces, and the mathe­matical intentions in which they are constituted, then their concep­tion as abstract spaces finds no support-just as one could say that there is no abstraction when in an act of free choice we ascribe m or n components to the generating vectors of a linear space. The mode of speaking of euclidean "spaces" says nothing more and nothing less. To reject this plurality as senseless would mean to reject even the singular-and yet at the same time it would mean to deny the existence of an analytic geometry as such.

An important supplement must be noted. Insofar as the previously sought clarification of the concept of mathematical space has incor­porated the diverse possibilities of dimensioning, it is supported by a véry specific case of mathematical spatiality. What determines the structure of mathematical space is not solely its geometry, but also its topology. Thus even if the euclidean geometry discussed till now has as such a univocal metric determination, no one specific mathematical form of space has yet been decided upon univocally, insofar as various possibilities . of connexus remain open. The

125. In a remarkable way Husserl seems to maintain the generic constitu­tion of the various dimensional (euclidean) geometries. In accordance with LU, Prolegomena, § 70, S3 should count as the "ultimate ideal unity" in the genus of "purely categorically determined manifolds," which for Husserl subsumes the geometry not only of euclidean, but also non-euclidean spaces. Here, however, it would be appropriate to distinguish between generalization and formalization.

246 Mathematical Space

simplest example is offered in the two-dimensional domain where we compare an open plane with the surface of a cylinder. Both are metrically equivalent, since the geometry of planes is valid for the latter, yet they differ in their relationships of connectedness. Hence the investigation of a mathematical space requires the discussion of metric and topological relationships.

This state of affairs does not limit the space-opening function of movements in geometry. While strictly speaking they are generators only of metrics, they nevertheless play an equally significant role in topological relationships since topology and metrics do not vary independently of one another. In the special sciences, the more detailed characterization of a mathematical space is taken almost exclusively from metrics. Thus, for example, the surface of a cylinder, despite its deviant connexus, is seen as euclidean space; the same is maintained for the Klein-Clifford forms of space of which we shall speak later. We now enter into this mathematical domain.

§ 2. Normal Space (Euclidean Space of the Topological Type of the Open Plane)

With the appearance of topological questions a new complex of problems comes to the fore for our investigations. In light of this, the investigation must face the task of understanding the appropriate connexus which would be the only one posited originally in the geometry determined through the euclidean movement group, namely, the topology of open-endless space.126 It is thus to be asked whether in historical retrospect those topological possibilities that euclidea:n metrics leave open in principie must be seen simply as fortuitous facts-i.e., whether for the formation of the mathematical concept of space, only those possibilities were originally selected that were required for the type of the so-called open plane-ar whether this "normal space" which, despite all subsequent con­structions of other mathematical spaces makes no claim for geomet­rically privileged position, can be conceived in terms of the subject's activities-which precede all individual geometric work and lend

126. Following O. Becker (1), for the sake of brevity and terminological univocity, we designate the euclidean space of topological type of the open plane as normal space; It is only in terms of this terminological delimitation that the concept of anomaly is subsequently to be understood in the sense of deviation of other ideal spaces in a metric or topological respect (metric and topological anomalies).

Euclidean Normal Space 247

intelligibility to topology, and from which the latter obtains all sense-clarification.

The solution of this task requires a deeper inquiry into the subject matter in arder to show that the elementary geometric formations are founded in sensibly intuited factors. It is necessary, therefore, to go back into this founding stratum and its specific characteristics.

Sensory intuition is depth-intuition-understandable ontologi­cally as characteristic of an orientation toward the world that is possible only for an intentionally directed corporeal being. In addition, it is a finite intuition. As a mode of comportment of a corporeal subject essentially characterized by finitude, it is limited by a horizonal domain of space. It will become obvious that the distinctive structural properties of normal space, linear and plane, are accessible from here; they lead to the understanding of its topology.

At the outset, the relationships here are different from the relationship between an edge and a line segment. Line and plane not only do not appear in the space of intuition, but there is not even a simple and obvious relationship of correspondence to sensibly apprehensible objectivities, as would be the case for an edge or a line segment which are both characterized by their limitations. Thus to understand a line we can neither have recourse to a line segment nor think of it as emergent from arbitrarily repeated line segments. Naive thinking assumes that it can take refuge in the view that it is possible to extend the line segment in both directions as often as it pleases, even while also denying the actualization of such a possibility of extension. Meanwhile, rectilinearity is already presupposed in the apprehension of a line segment; on the other hand, the very assumption of a merely potentially infinite progress from line segment to line is misleading, insofar as such a process would result in an arbitrarily long line segment, but never in a line. A line is not an arbitrarily long line segment since the difference between a line segment and a line is not a difference of measure. Regarded critically, the frequently employed terminology that a line possesses "infinite length" states nothing more than that here the concept of length has no signification-fulfillment. The line does not "transcend" but rather "subscends" all measures. It is not a metric but a topological concept. Analogously, the same holds for the concept of the plane. To think it as originating from an arbitrary addition of limited plane parts would evoke similar reflections. Like the line, the plane is also not a metric but a topological formation.

248 Mathematical Space

This reveals that we have retrogressive indices toward founda­tional moments that are not to be located in sorne individually given aspects of sensory intuition; rather, they must be sought in the total structure of the space of intuition as such.

The space of intuition is structured in terms of extension in depth and width. The depth of corporeally-centered space is formally distinguished from extension in breadth, insofar as it alone pos­sesses a univocity of orientation. While in the extensional spread two phenomenal points can always be related in manifold ways and modes, in the extension of depth this is possible only in precisely one way. This univocity is founded in the particularity of depth which is to be one behind the other. This being behind one another, in contrast to being next to and above one another, acquires its arder directly from the location of the intuiting corporeity. At the same time, corporeity has a univoca} criterion for the fulfillment of this arder in what phenomenally constitutes the "coincidence" of two points lying one behind the other. Indeed, this coincidence is not an intuitive, but a purely visual givenness. In contrast to the visual field, the intuitive apprehending includes the apperception of the being-one-behind-the-other, despite the visual coincidence of the two points. But this is the constitution of the primal meaning of rectilínearíty. The intuitive glance would encounter any series of phenomenal points in the characterized position "straightway" if it could penetrate the ones in front.

How significant is the particularity of such univoca! relationships of location, how dominant is the depth dimension in its structural preeminence for the total apprehension of vision, is revealed by the concept of the visual "ray", whose meaning emerges from the depth dimension of the intuited world. It must be kept in mind that with the ray, the domain of the phenomenally given is already abandoned. The intuiting corporeity does not have a bundle of rays befare itself, but a spatially structured world of things with a dimension of depth. It does not see rays but things in space. The ray-character of vision is not itself given for vision and, thus, phenomenologically speaking, is nota characterization of its functional structure as given in corporeal comportment; rather, it first acquires its meaning through an objec­tivating inspection of the perceptual event. However, that the visual function is the only one among the sense functions that can be objectivated in this manner, that it can be fulfilled in the objectivating apprehension as a "ray-like" event, is not at all an empty construction of experimental optical science, but rather has the conditions for its possibility in the structure of the sensibly intuited world of objects

Euclidean Normal Space 249

itself, as a world dimensioned in depth, and at the same time in the specific functional organization of the corporeal subject.

Apart from the depth dimension, the analysis of the space of intuition has shown the other essential structural moment of this space: extension, with its degree of manifoldness one step higher than that of the depth dimension. Opposed to the univocal and corporeally-related intentional depth, extension appeared as a man­ifold of pure relationship of things. Together with depth, it consti­tutes the structure of a space in which the intuiting corporeal subject exists in an ambivalent state: on the one hand, as a being "in" a situation, and on the other, as a being "over against" a constellation of states of affairs. As such, the pure two-dimensional extension is no less relevant for the entirety of the sensibly intuited space than the one-dimensional extension into depth.

Extension, to be sure, does not necessarily mean a plane exten­sion. The ambiguity of being next to one another (in complete contrast to the univocity of being one behind the other)-more precisely, the possibility of relating two closely located phenomenal points in various ways-does not motivate the apprehension of planarity, but rather constitutes the universal meaning of two­dimensionality. In contrast to it, the plane is a very special case. How are we to conceive the unique meaning of a plane surface as well as of rectilinearity in contrast to curvature? How is it that in our intuition and conception the plane functions as a kind of fundamen­tal surface and that in contrast to it, any other kind of curved surface is grasped merely as a "deviation from" the plane?

It is insufficient to point to the structural simplicity of the plane in contrast to the curved surface; this merely shifts but does not salve the problem. Still to be clarified is how the concept of simplicity, regardless of how it is understood, is related to a surface with plane determination.

This state of affairs is comprehensible only with the aid of the structural determination of the visual function. The apprehension of planarity emerges in a process of objectivation that allows vision to be conceived as a ray-like event: the "wandering" of the "gaze," as a mode of comportment of a corporeal subject, is reduced to a visual ray that "spreads over" a surface. But since this visual ray is formally generative of the surface, it also determines the degree of curvature, and thus the surface must necessarily be a plane.127 In the case of a

127. We are not overlooking the fact that we are dealing strictly phenom­enologically with determinations of a specific realm in terms of a "wander-

250 Mathematical Space

gaze ranging over a curved surface, the visual rays would have merely a tangenitial function and, thus, roughly speaking, the whole would fall outside of their range; this would not be the surface that would be generated by the wandering of the gaze.tza

In addition, the concept of similarity of the forms of things in the space of intuition thereby attains an intuitively fulfilled sense. The existence of similar things, i.e., similar in form although varied in size, is essentially bound to a plane-structured spatiality, since the constancy of the angles of the formation, despite the changes in extension, are possible only in it. It is possible to conceive of a corporeal being whose functions are of such a type that no univocal depth and plane extension would be present in direct intuition. For such a being, no forms of things would be similar in its space of intuition. In contrast, for a subject of our corporeal constitution, the plane functions as the bearer of a basic topological determination of everything extended.t29 (As an originary spatial apperception, founded in the functional determination of the lived body, it has a dominant structural movement of representation beyond what is sen­sibly intuitable, where thought turns to spatial metaphors. Thus the partners in a discussion encounter each other on the same "plane" of thought although they do not find themselves on the same "surface" of thought. And should their conversation remain merely "superfi­cial," the representation of the plane, in this phrase, also plays a role.

ing gaze." Of course, in a preliminary way this also touches upon rectilinearity and planarity.

128. The theoretically conceivable case of ruled surface, i.e., of a surface that has lines in specific directions, yet in general has a degree of curvature other than zero, is here excluded, since the wandering of the glance is not subtended by any limitation of direction and can be enacted arbitrarily.

129. The interconnection of planarity and similarity has been shown repeatedly in mathematics. Bolzano has employed it while demonstrating the plane-structuration of space in terms of the phenomenon of similarity. The mathematical equivalence of both concepts should not obscure the circumstance that ontologically speaking, we are concerned not with a mutual conditioning, but with a one-sided relationship of dependence; the meaning of planarity or of rectilinearity is prior and cannot be derived from the concept of similarity. The space of intuition "must be" plane structured not because in intuition there are similar things; rather, it can contain similar forms because it is structured by a corporeal being of the given functions. Apprehensions of similarity in the space of intuition ha ve already presupposed the apperception of the line, because without it, the signification-moment of constancy of angles, which is constitutive for similarity, could not be intuitively apprehended.

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While rectilinearity and planarity, as spatial structural moments, are completely based in the sensibly intuited world and its corporeally-related arder, the notions of line and plane themselves do not yet come into view. However, in light of our point of departure, both are easily understandable in their own conceptual sense. We recall that the mode of givenness of the space of intuition is deter­mined from the start by the co-presence of "the" one space, so that the former is originally apperceived as a "mere section" of the latter.

Yet this space is homogeneous and open-endless; it is conceivable as a univocal continuation of the thing-relationships predominating in the finite domain. The characteristic relationship among things contains the experience of being one behind the other in spatial depth. In accordance with this idea, it can be continued endlessly and for the sake of univocity it is posited as an open-endless progression. At first, this progression assures only the infinity in principie of the visual ray. Through the homogenization of this space, the center, as a point of inception of the visual ray, is reduced to any phenomenal point of space and no longer has any special significance. A point among other points, not only can it be arbi­trarily shifted, but also, in a strict conception of the homogeneity of space, it can and must completely forfeit its special character as a point of departure for something. At the same time, the "visual" ray becomes redundant; what remains is the purely formal meaning of rectilinearity. It no longer includes the co-positing of a corporeal function and, while being without any point of inception, it is conceived as open-endless in two directions, i.e., as a line. In its free motility it is generative of the all-sided and open-endless plane as a structural element of the homogenous space of objects. That it is open-endless means nothing other than that it is a space structured in terms of planes.

The fundamental meanings of line and plane condition our con­ception of a fundamental formation of euclidean geometry or euclidean metrics-the line segment-in terms of rectilinearity. This must not be taken as if the apperception "line" and "line segment" were to occur in separate acts of apprehension. Rather, the conceptual sense of the line segment originarily contains a moment of rectilinear­ity justas the marking off of line segments in the process of measuring is thought of as a continuous rectilinear process. This conception is completely independent of the factual operation of marking off which in individual cases may deviate considerably from the straight line. What is important is that in such a marking off, the rectilinearity of the measured path is constantly anticipated, and any deviation from

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it is seen privatively, to be determined entirely in contrast to rectilinearity.

This means, however, that euclidean normal space is a plane­structured space. In accordance with its mode of generation through the euclidean movement group, with its characteristic invariance of the line segment, it guarantees e o ipso the free motility of a formation structured by rectilinearity, leading from the very beginning to the conception of an open-endless space.

The question raised at the outset of these paragraphs concerning the special phenomenological position of euclidean normal space among other mathematical spaces is thereby answered. The answer consists in nothing other than in its topological congruence with the space of the natural consciousness of objects.

This congruence is explainable in terms of the simple relationship of founding between its constitutive formation, the line segment, and the intuitive form-characteristics of the real-spatial world of things. In light of subsequent distinctions, it is essential to introduce here the term of immediate founding. This immediate founding of euclidean normal space in the natural space of objects establishes for euclidean space-both metrically and topologically-a certain preeminence. Insofar as the latter must be seen as founding for any kind of space in mathematics, "euclidean space," i.e., the open-endless manifold constituted through the euclidean movement group, should be con­sidered as the most primordial of all mathematical spatialities. As was shown above, this primordiality is distinguished not only in its his­torical and factual priority, but also in its ontological determination of the very sense of existence of this space. It is grounded in the structure of the sensibly intuited spatial world, and obtains its ulti­mate foundation in the specific nature of the corporeally bound and corporeally conditioned achievements of consciousness of a subject capable of thematically meaning space as an object.l30

§ 3. The Question of Intuitability in Euclidean Geometry

The interrelationships discussed in the previous section offer the possibility for clarifying the conceptual sense of intuition in geom-

130. The particularity of the euclidean spatial structure is, to stress once again, not mathematical but ontological. In the constitution of the multitude of mathematical spaces, euclidean normal space functions as a kind of primordial objectivity from which subsequent spaces first obtain their meaning as "spaces." See pp. 297 ff.

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etry. lntuitability in euclidean geometry now more clearly means intuitability in the geometry of the type of the open plane. The intuitability meant here is hereafter appropriate for both metric as well as topological relationships.

Of course it is tacitly assumed as obvious that the concern here is with a kind of intuitability that is distinct from simple sensory intuition. We can point to our earlier expositions concerning pictorial-symbolic intuition and its relationship to the sensory (pp. 194 ff). The result of those investigations led to the crucial charac­teristic of geometric intuition: the possibility in principie of an alteration of attitude in such a way that the sensible moments of presentation of geometric state of affairs can be interpretad at any time as depictions of morphological forms of the sensibly intuited world of things.

While initially this determination of geometric intuitability was related purely to individual geometric formations as such, now it acquires its affirmation and support from the topology of normal space. Since this is identical to the topology of the natural space of objects, the "intuitability of geometry" signifies the pictorial, yet pictorially-symbolic comprehensibility of geometric states of affairs through the congruent connexus of euclidean normal space and pictorial space. Each extensive manifold can function as a pictorial space for the wherein of "depictions": the objective three-dimen­sional space (the space of stereometric formations). or one that can be regarded as pictorial surface ("sign-plane"). Topologically, both are identical with the ideal normal space.

The failure to fulfill one or the other or both conditions, which are not completely independent from each other, results in remarkable deviations from the type of intuitability characterized.

Let us consider the case in which the condition of connectedness between the ideal space and the pictorial space are fulfilled, but in which the geometric state of affairs can no longer be presented pictorially. The uniqueness of this case results from the limited possibilities that are open for pictorial space with respect to its dimensionality. Everything that is meant as pictorial-symbolic must evidently be bound to two or three dimensions if an alteration of attitude is to become possible with direct sensory perception of the image as morphological "depiction." Such limitation lies in the maximum three-dimensionality of the sensibly intuited world. Formations higher than the three-dimensional cannot become accessible in any kind of pictorial-symbolic intuition. The fre­quently discussed question concerning the intuitability of multi-

254 Mathematical Space

dimensional geometry-and this also has to do with euclidean geometry-must be emphatically answered in the negative if one does not wish to burden the concept of intuitability with confusing equivocations.

The manner in which n-dimensional formations (n> 3) are given is so complex that even the most sophisticated analysis would not venture to describe it fully and adequately. The mere suggestion of an "extension" of geometry beyond the "boundaries of intuition" (where by intuitability we understand only that within three­dimensional geometry) leads no further into the problem. It does not characterize what is specific in the nonintuitability of n-dimensional euclidean formation in contrast, for example, to the nonintuitability of noneuclidean geometries, which are completely distinct from the topologically normal Sn-geometry. This differentiation, if it is given at all to awareness, has not previously been an object of investiga­tion. We shall attempt to explicate it only to the extent necessary for the clarification of our problem.

It was tacitly assumed that normal space means the three-dimen­sional euclidean space. The concept of normalcy obtained from the metrics and topology of this space allows its extension toward Sn. It too is of the topological type of the open plane, and its metrics is determined by the euclidean movement group with the invariance of the line segment.

But what does "line segment" and "movement" mean in more than three dimensions? If for S3 these meanings would result through ideatively making them independent of any phenomena belonging to the space of sensory intuition and through merely carrying over their relations into the ideal domain, then their conceptual apprehension in Sn would still require new moments of apprehension. The latter would transcend the meanings of S3 , and while presupposing the meanings acquired in S3 , would at the same time transform them through the new meanings.

This would be the case if it were not for the fact that the mathematical data of S3 are already subtended by specific significa­tions that are not first motivated by the constitution of S4 • They do not appear in the plan as completely new moments; rather, they are already co-constituted in what is given in S3 and merely comprise the exclusive sense-bestowing moments of 84 • What is meant here are the significations that enter into geometry universally with the analytic-algebraic method. As already shown, this geometry com­pletely disregards the number of dimensions and subsumes its work purely under algebraic-formal criteria. Though it is capable of

Euclidean Normal Space 255

pictorial-symbolic illustration, in and of itself it does not require the possibility of such illustrations.

Yet insofar as it is not merely a vectorial algebra but a vectorially and algebraically pursued analytic geometry, the tendency of this endeavor is to make it capable of a spatial interpretation such that its vectorial equations must become conceivable as geometric states of affairs. Thus it happens that even higher-dimensional geometry speaks of "lines," "planes," "distances," and "movements"; it speaks in concepts belonging originally to S3 • While the meaning­content is the same in both, only in S3 is it capable of symbolic­intuitive fulfillment. The claim to fulfillment remains a predisposi­tion even with the extension of these geometric concepts toward higher-dimensioned spaces. A more thorough analysis of the sym­bolic capacity of intuition would be able to show that even in these spaces the originally pictorially perceivable moments are never completely abandoned, but rather persist as conceptual roots. Yet the latter cling only to concepts of objects taken in isolation ("line," "plane," etc.). In operating with them, in the apprehension of their given positional relationships, this residuum is dissolved. Whether aplane intersects another plane in Sn, or is touched by a specific line, is not ascertained with the mathematically required stringency offered exclusively by the calculus of determinants; every attempt to establish the plausibility of algebraic results through subsequent illustration also fails.

This specific nonintuitability of n-dimensional geometry is inher­ent in the fact that it is a purely formal "extension" of the geometry of S3 • Here the concept of extension is to be conceived purely additively. That is why a converse reduction to three of the number of vectorial components presenting the fundamental formations of this geometry will result in three-dimensional euclidean geometry. What is nonintuitable is thus present here not in sorne complex of properties that are characteristic of the basic concepts of geometry that resist illustration, but rather simply in the supercession of the number of dimensions-if one may say so, in the numerical "exag­geration" of the basic geometrical relationships, which in their own right are never genuinely determinable numerically and thus cannot be completely absorbed into algebraic relationships. Hence the ground must be sought asto why, despite the complete topological congruence between mathematical and possible pictorial space, the possibility of a pictorial-intuitive fulfillment is completely absent here.

It is quite otherwise with the case in which, according to the above

256 Mathematical Space

established division, the nonintuitability of a geometry is deter­mined by the fact that the connexus of the ideal space deviates from that of pictorial space, even if its number of dimensions corresponds with or is less in number than that of the space of intuition.

In addition, a distinction must be made between spaces in which the metric is identical with S3 and those that are both topologically and metrically abnormal.

The investigation of these spatial types must be added. Concerning the question of their nonintuitability, we are limiting ourselves at the present to the fundamentals.

That in particular this nonintuitability is constituted differently for non-euclidean geometry than for n-dimensional euclidean geom­etry is obvious from the mere fact that the latter is directly translated into intuitable s3 geometry by its reduction to n = 3, while noneuclidean geometry remains non-intuitable even within the three-dimensional domain. Even for two dimensions the determina­tion is difficult. Here non-euclidean geometry works as a theory of surfaces, as a geometry of curved surfaces-of this later.

As a theory of curved surfaces it should be no less intuitable than the euclidean theory of "surfaces," i.e., the normal geometry of planes, since the curved is intuitable in the same sense as the plane-in space. The meaning of this being-in contains a two­pronged problem. A surface that is meant as "curved" obtains its conceptual sense from the co-presence of the plane-structured space. lt will be seen as a curved surface "embedded in" this space. Thus a surface geometry with more than two dimensions cannot be intuitable as long as it relates in any way to the surrounding three-dimensional space, as is the case, for example, with the elementary spherical trigonometry.

Note that the question here has no bearing on whether the mathematical mastery of such curved spaces requires the inclusion of a surrounding space. There is a widely spread false conception that a curved mathematical space could "be" only "in" another space of a higher dimension.131 That it cannot be represented otherwise is incontrovertible. Yet if its being consisted in nothing other than its constitution through acts of representation of the

131. Thus N. Hartmann (3) has missed the problem of "ideal spaces," as well as the problem of application of geometry, through such mistaken notions. He obviously allowed himself to be led by the common and naive notion; any attempt at phenomenological analysis of mathematical meaning­giving is completely lacking in his "phenomenology."

Euclidean Normal Space 257

natural consciousness, then it would not be a specific mathematical space. The manner and mode in which it can be represented does not touch upon what emerges here with particular succinctness as the genuinely mathematical: its complete freedom from representation and independence from intuition. It owes its existence solely to algorithmic thinking, whose basis of being is in no wise located in an encompassing space of a higher dimension "surrounding" it, or in which it is "embedded"-all these are only metaphors of a consciousness requiring intuition and attached to the conception of. space as a container.

Apparently it is a new discovery that curved surfaces can not only be mastered from the standpoint of euclidean geometry, but that for their own part they are spaces with characteristic geometric regularity and "non-euclidean" relationships of measure deviating completely from the normal. The usual geometry was first constructed as a gen­uine theory of surfaces with Gauss, but only after the infinitesimal calculus provided it with an analytic method. It was only with the .conception of surfaces developed there that the relationships of mea­sure attained their extension toward the higher-dimensional space of non-euclidean geometry, in which those surfaces, as structural ele­ments, play a role analogous to the plane in the euclidean Sn.

This will be discussed in greater detail in the next section. The following chapter will briefly consider a mathematical type of space metrically akin to euclidean space, yet revealing different relation­ships of connectedness.

Cha pter Three

Euclidean Spaces with Topological Anomalies

§ 1. Extension of the Mathematical Concept of Space

On the basis of its constitutive moments of meaning, the mathe­matical concept of space allows specific extensions. Thus the sense of the mathematical concept of movement and its purely rational and algebraically formulatable determinations reveals the fact that with it-in contrast to the space of bodily movement-the ties to three­dimensionality are abolished. They are not only abolished in favor of higher-dimensional manifolds; an anologous "downward" exten­sion also becomes permissible.

At first sight, the application of the concept of space to surfaces may seem to be terminologically superfluous, if not nonsensical. What sense can one attach toa space in space? Yet once again it must be recalled that the decision concerning the being and sense of mathematical spaces does not stem from the conception and mean­ings of natural space-consciousness, but rather solely from the non­contradictory construction of extensive manifolds in mathematical consciousness. Accordingly, space is any manifold whose unique­ness could be characterized by geometrical movements of its forma­tions without regard to dimensional relationships. More precisely, each mathematical movement group opens a mathematical space independently of the dimensional number of what is moved. Even surfaces, as two-dimensional spaces, fall under its concept. Whether and to what extent they relate to a surrounding space of a higher dimension is to be discussed later. The necessity, and the manner and mode of such a relationship, can be meaningfully discussed only when the use of the mathematical concept of space for surfaces no longer appears as a merely analogous carrying over of its sense onto surfaces-when it will instead be clearly seen that surface manifolds are accessible toa mathematical treatment devoid of any mathemat-

258

Euclidean Spaces with Topological Anomalies 259

ical reference to a surrounding space. This would lead to a concep­tion of such surfaces that would completely exclude their "surface" character and grasp them as independent mathematical spaces.

The following investigation will consider the structure and metr­ics of such spaces. Here our interest has nothing in common with the special sciences, even if in what follows we cannot avoid having greater recourse to mathematical states of affairs. Nevertheless, the latter serve as clues for the noetic clarification of the subjective activities that correlate to them; thus they are considered only to the extent that they are conducive to an insight into intentional struc­tures of acts of consciousness constitutive for their existence.

Following the preceding arrangements, the investigation begins with a class of surfaces that on metric grounds still belong to the euclidean manifolds; yet due to their topological anomaly, they are akin to the non-euclidean spaces and lead across the threshold to non-euclidean geometry.

§ 2. Clifford-Klein Spaces

In 1873, Clifford discovered a peculiarly structured surface that in many respects played a role in elliptical space comparable to the role played by the plane in a normal (parabolic) space. Its mathematical qualities were first investigated more precisely by F. Klein. 1 32 It is a matter of a ruled surface of a second degree with specific properties. Euclidean metrics is valid in it and its movement group includes a genuine sub group of translations. While metrically it is of the type of a plane, what is remarkable about this surface is that it is a closed surface. Although its degree of curvature is zero, it contains a finite surface area,133 This surface shows that vanishing curvature and open infinity of space do not necessarily possess equivalent deter­minations.

These peculiarities result from abnormal situs of the surface. As a

132. F. Klein (3). For more detail concerning the geometry of this form of space, see also W. Killing (1), H. Hopf, and F. Ltibell (1)-(3).

133. More precisely, the metric congruence with the euclidean plane is not valid with respect to the whole surface; topological anomalies are also observable metrically. (This has to do with the multiple connectedness of the surface.) Thus, for example, in contrast to the euclidean plane, where each simply connected part-e.g., a line segment-can be moved in a threefold infinite manner, here it can be moved only in a twofold manner over the surface as a whole. Translations are normal, but rotation results in deviations that are related to the various lengths of the geodesic.

260 Mathematical Space

particular ring surface, it is multiply connected. lts connexus can be clarified in the following way. If one bends a rhombus into a ring surface, so that corresponding locations of opposite sides coincide with one another, then one obtains a formation from which one can obtain a continuous one-to-one mapping of the Clifford surface. (The mapping is not congruent; it is attainable only by an elastic mem­brane.) Conversely, the Clifford surface can be transformed into a simply connected surface: one starts with a point and severs the surface lengthwise into two, so that they are simply bordered and simply connected. This surface can be developed from a rhombus. As can be shown mathematically, the Clifford surface investigated by Klein presents a very limited special case of Clifford-Klein spaces. W. Killing started with the peculiarities of this surface, showing that it can be transported as a whole only by special movements. This characteristic turned out to be appropriate for a series of surfaces or spaces also having a non-vanishing (positively or negatively con­stant) degree of curvature. Moreover, the aspect of generation of such surfaces was permitted only by a specific mathematical lawfulness of movement and of characterization; likewise for its construction for more than two dimensions.134 In contrast, very special types of cases can be adduced precisely for the Clifford surfaces. Such would be the surfaces that could always be developed-as in the case of the Clifford surface-from the rhombus, from the parallelogram, or from parallel strips. The formation developed from the latter becomes the intuitable surface of a cylinder. lt is a one-sided surface and permits a congruent development on the euclidean plane. The cylinder should be regarded as a particular limit case of Clifford-Klein space. If one considers an additional surface of a specific type ( double · surface), then it is obvious, as F. Klein has shown, that all forms of space of this kind are exhausted. At the same time the topologically abnormal euclidean spatial forms are exhausted.

§ 3. Clifford-Klein Spaces as Euclidean Normal Space. Founding Relationships

The specific topology of Clifford-Klein spaces requires that their existence is not independent; rather, these spaces are dependent in a determinate way on euclidean normal space. This dependence is not to be understood geometrically; the mathematical treatment of

134. See W. Killing (1); the designation "Clifford-Klein spaces"- which can be abreviated as C-K spaces-originates with him (pp. 257/58).

Euclidean Spaces with Topological Anomalies 261

such spaces does not require reference to a surrounding space with a normal euclidean structure. Rather, it is ontological in kind and is appropriate to the sense of existence of these spaces. They presup­pose euclidean normal space, insofar as the mathematically consti­tutive determinations which comprise their mode of being obtain theü fundamental meaning in euclidean normal space, from which they then can first be meaningfully transposed into a Clifford-Klein space. In accordance with their meaning, such concepts as a finite surface area, "closed" surface, etc., imply the open-infinite space of euclidean geometry, in and for which their precise mathematical meanings were originally valid. What those spaces are, in what sense being can be predicated . of them, depends on the mathematical conception of space. This relates back to a specific geometric event which, as a movement of certain fundamental geometric formations, comprises space in the mathematical sense.

This relationship of dependence of Clifford-Klein space on euclidean normal space is one-sided and not reversible. This means that the euclidean normal space must "already" be befare the Clifford-Klein spaces can claim existence in the specific geometric sense. Their meanings are founded in normal space. This is obvious on the assumption that euclidean normal space is the more primor­dial mathematical space, having its immediate foundation in the natural space of objects.

In view of this new form of space, it is possible to discuss the question as to whether perhaps the space of the natural conscious­ness of objects is already of such a structure that a Clifford-Klein space could be regarded as a mathematical-ideative presentation derived directly from the space of objects.

W. Killing maintains that it is basically possible for a Clifford­Klein structure to be compatible with our spatial experience; there is no compelling reason for us to suppose that with the movement of a partial domain, the movement of the whole is necessitated, partic­ularly since our empirical access is limited to bodily movements of relatively short distances.

This question is discussed in detail by O. Becker.t35 Obviously motivated by the presentation that E. Klein gives specifically to the concerns of the Clifford surfaces, Becker begins with the mappability

135. See W. Killing (1), p. 258; see O. Becker (1), § 12 B, § 15. Becker limits himself specifically to the first Clifford space mentioned here, which, dueto its vanishing degree of curvature, must have motivated the question under discussion.

262 Mathematical Space

of these surfaces onto open infinite space. If they can be developed from a rhombus, then it is possible to think them as displaced to a rhombus network that completely covers the euclidean plane. It would represent all possible mathematical states of affairs of Clif­ford's surfaces as periodically and infinitely repeatable. This would be similar to the type of periodicity of the functional value of an elliptical function. Correspondingly, the three dimensional space of objects would be thought as articulated in accordance with rhom­boids, such that all spatial events would have to repeat themselves. periodically. At the same time, Becker reveals the "immense para­dox" present in this structure as a spatial structure of material things. It would not only be a space in which all physical events take place, but would also have to contain the corporeally constituted subject who apprehends space! But this would mean the abolition of the irrepeatable uniqueness of one's own being in favor of a mere exemplification of one's self, which would have to be thought simultaneously with infinite frequency in each parcel of Clifford­Klein space. This is logical suicide best relegated to the arbitrary play of fantasy, although in an ontological reflection it can be thought without becoming non-sensical.

According to Becker, this interpretation can be violated by elevat­ing the distinctive topological structure of euclidean normal space to transcendental understanding. If periodicity appears as a new rule in the case of a Clifford-Klein structure, then there emerges a limitation for spatial events, and this is incompatible with space as a principium individuationis. The sale connexus that allows freedom in space to events is of the type of open-endlessness of euclidean normal space; it does not completely exclude periodicity, but does not dictate it.

Despite the acceptability of the results concerning the distinctive character of euclidean normal space, the way by which they were attained is not without problems. The question whether the space of material things, inclusive of the lived bodies of the space conceiving subjects-i.e., the "actual" space-must topologically have the normal euclidean structure, or whether it could also be the Clifford­ian conception, cannot be decided in favor of the first, at least not with arguments in which the distinctive structure. of euclidean space is already presupposed. Yet this already tacitly functions as a premise in the process of exposition of the consequences that the Clifford-Klein structure should possess for spatial events; the latter are not related at all to the Clifford-Klein surface itself, but rather to the formations of development in the open-infinite (!) plane. For

Euclidean Spaces with Topological Anomalies 263

such development to be possible (for thought), the open-infinite plane must be conceived ahead of time. And the sole motivation to engage in such development is merely the need of a "euclidean being" to visualize the Cliffordian relationships in a rough manner­at least through means that topologically speaking are not without dangers; they intrude into the situs of the surface and tend to dissolve its ambiguities.

Finally, insofar as they cover the total open plane, the notion of the infinite repetition of such development appears to be problematic. Apart from the fact that the Cliffordian surface is not identical with such a covering, the closedness of the surface falls victim to a reinterpretation in intuition that has the greatest relevance for the question posed above. lt is precisely the closedness of the surface that suggests its infinite development and can demonstrate in this manner the "recurrence" of events on the surface. Yet there characteristically appears in the formations of development a simultaneous repetition of infinitely many (in relation to singular portions of congruently located) points. Thus the repetition on the surface is represented as a temporal process of "return" to the same point. Hence it should be noted . that the presentation of surface events as well as their development involves kinematic means, which in both cases imply the apprehension of time as a linear and open-endless continuum. Assuming this, it follows that the "periodicity" of events in a Clifford-Klein structure is not just a result of a reinterpretation. After all, the phenomena of returning to the same place within the surface at different temporal points is subtended by their simultaneous recurrence at different points of the formation of development.

If one wishes to demonstrate the priority of the connexus of the normal euclidean plane, one cannot have recourse to periodicity as a new law that does not appear in the Cliffordian surface itself. Even Clifford-Klein space, as a principium individuationis, achieves the same as does euclidean space and could not be simply rejected a limine as a space of real things.

The conception that actual space could be of Clifford-Klein structure may contain conditions that remain unacceptable to the extent that 1 find myself in this space as a corporeal being. lt was already shown that in the natural consciousness of space, the open endlessness is not a simple factual datum of consciousness that in a given case could be modified at will. Rather, its ultimate foundations are to be discovered in the sense-founding intentionality of a corporeal consciousness-the fulfillment of which can only be

264 Mathematical Space

guaranteed by the possibility in principie of an infinite progression. Indeed, a Clifford-Klein structure of space would not completely limit the possibilities of movement of a corporeal being, yet the conditions of a perpetual-although infinitely repeated-return to the same place would be contrary to the sense of the "progression" in principie of movement "into" infinity. In principie the "aimless­ness" of movement in a Clifford-Klein space, which is indeed further movement but not a continuous forward movement in the precise terminological sense, would not correspond to the sense-structure of the intentionality of consciousness. In any case, even these last clarifications presuppose open-endless time. All considerations con­cerning the possibility or impossibility of topologically abnormal structures for actual space are essentially and irrevocably tied to a being that even in the conceptuality employed already reveals that it does not find itself in the space in question, but can only attempt to think itself into it-and that it can make such an attempt rests on the basis of a conception of space (and time) that from the very outset is determined by open-endless extension.

SECTION THREE

Non-Euclidean Spaces

Chapter One

Fundamental Questions of Non-Euclidean Geornetry

§ 1. The Parallel Postulate. Historical Origin and Development

As a mathematical discipline, non-euclidean geometry is conceiv­able essentially from two sides. One was developed in the works of Cayley and Klein from the viewpoint of the theory of parallels; it remains clase to the elementary synthetic method. The other, the analytic and infinitesimal-geometric, owes its establishment and development to the achievements of Riemann. Although the latter is mathematically more significant, the first cannot be left out of consideration, on factual grounds. It permits the recognition of the leading historical trends more clearly than Riemann's theory does, and it has retained the oldest heritage of mathematical thought. In its light, the controversia! parallel postulate first becomes understand­able in its specific significance as well as in its important changes which at the same time introduced a fundamental transformation of the concept of the axiom.

The theory of parallels has its historical roots in the Elements of Euclid. The proposition that is usually designated as the euclidean parallel postulate--namely, that a given line possesses exactly one parallel through a point lying outside of it-is held to be the most outstanding fundamental presupposition of euclidean geometry.136

136. We write "Euclidean" to designate the geometry that was constructed

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Yet its unquestionability was already touched by Euclid's commen­tator Proclus, who made it into a controversia! point of mathematical discussion.

What is significant for our problem is that the parallel postulate formulated in the above manner is not to be found in Euclid-,-there is no "parallel" axiom or postulate (aitema) explicitly given by Euclid. Apart from a nominal definition of parallelism, given in his first book of his Elements as definition 23, there is under Euclid's postulates at the fifth place the following proposition: if a straight line intersects two others so that the sum of the inner angles of the same side is less than two right angles, both straight lines intersect, on the side of the angles.13 7

The existence and uniqueness of parallels is originally a very specific consequence of this statement. It results from the basic postulation of univocal relationships of ordering and, which was most consequential for the subsequent discussions, from the presup­position of the "existence" of the point of intersection which certainly might have been obvious for Euclid.

Yet the incorporation of the proposition in question into Euclid's system provokes sorne immediate reflections. Among the remainder of his postulates (aitemata) the fifth-and in another respect the fourth, concerned with the equality of all right angles-assumes a special position. While the first three postulates require the possi­bility of determínate constructive operations, and while this possi­bility simultaneously establishes with certainty the geometrical existence of specific formations, the postulate character (eitestho) of the (fourth and) fifth postulates is truly distinct.1 3B Indeed, two

in Euclid's Elements; in contrast, "euclidean" designates a geometric domain that is to be delimited and characterized by the axiom named, primarily with the emergence of the non-euclidean problematics. Thus taken very strictly, Euclid knew no "euclidean" geometry.

137. Euclid, Elements I, Book 1, postulate 5. 138. In a striking way both are distinct from the rest even in their

grammatical form; both have something in common requiring that some­thing "is." According to O. Becker (6), p. 214, the fourth postulate has the character of a theorem and is to be seen as a consequence of th!l univocity of the extension of a line segment. For the precision of the fifth postulate Euclid requires that the original unity of the fourth and fifth postulates should not be excluded. Becker holds that it is apparently possible to prove that the fifth postula te was formulated befare Euclid without angles and that only the convergence of straight lines constituted its content, while the fourth postulate was still redundant.

Fundamental Questions of Non-Euclidean Geornetry 267

converging straight lines "require" a point of intersection-precisely seen, the latter constitutes an existential staternent, yet the point of intersection is rnerely stated but is not guaranteed by a process of constructive requirernents! After all, the intersection of straight lines discussed here is not an operation in the sense of the rernaining postulates, but rather a consequence of an operation, narnely the extension of straight lines, the possibility of which is guaranteed by the second postulate. Thus the question of the "existence" of this point of intersection differs from that of the remaining geometric formations. It is not defined through construction but merely as­serted; it is not generated constructively, which means at the same time that it cannot be proven.

This remained a problem for later times. The controversy concern­ing Euclid's fifth postulate was unleashed for the first time by Proclus. For its recognition, he demands mathematical proof. With­out it he holds that the Euclidean claim for the existence of the point of intersection is a mere probability.139

As subsequent axiomatic investigations taught us, the demand for proof by Proclus placed an inappropriate demand on geometric science. Yet what is eminently positive in his critique of Euclid, which should have led to an important discovery is the detaching of the characteristic point of intersection of two straight lines from their convergence. Proclus does not challenge the convergence, but the point of intersection-it is possible that two straight lines moving toward one another do not intersect, even if they come into an unlimited proximity to each other. Proclus supports this claim by the fact that with other kinds of lines there is convergence, but at the same time it is only an asymptotic approach. He stresses this possibility also for convergent straight lines. Their point of intersec­tion is not universally contested: what is brought out is that two somewhat converging straight lines could run symptotically.

Factually, this includes the possible existence of many "parallels" for the straight line through one and the same point, and strictly in accordance with the euclidean definition of parallelism. After all, the latter is not given positively with the aid of a concept of equidistance, but rather employs the state of affairs of the non­intersection of straight lines (in the same plane).14°

138. For the discussion of the concept of aiterna in ancient rnathernatics, see also K. v. Fritz (1).

139. Proclus, pp. 191, 16-193, 7. 140. What is rneant by Euclid is obviously the (subsequently designated)

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In his commentary on Euclid, Proclus does not offer any clue that would suggest that he was fully aware of the extent of the thoughts that had led him to the threshold of a non-euclidean (hyperbolic) geometry. What later became influential was primarily only his demand for the proof of the controversia! postulate.

It is not the concern of our investigation to trace the attempts at proof in detail; a brief indication of the structure of such proofs should suffice. In all cases they started from the sum of two right angles in the triangle, a state of affairs that is equivalent in meaning to the questionable postulate. Usually they employed an indirect process of proof: they started with a premise opposite to the o'ne to be proven and attempted to obtain a contradiction. At best, the result led to the insight that the parallel postulate is replaceable by equivalent statements, as long as what is to be proven does not already function as a more or less hidden premise in these "proofs."141 At this point we should mention only the remarkable attempts of G. Sacceri (1735), who suggested two further possibilities besides the euclidean postulate. In tracing their consequences, he followed the notion that he could lead both to absurdity and in this manner indirectly prove the euclidean postulate. (Sacceri speaks of the "hypothesis of the obtuse angle" and "hypothesis of the acute angle." By this he means a hypothetical postulation of right triangles in which both angles placed in opposition to the base angle-with both being equal to one another on symmetrical grounds-taken together are either greater or smaller than two right angles. The refutation of the first hypothesis succeeded under the assumption of the open-endless extension of straight lines.) That Sacceri succeeded in refuting the one hypothesis-and this under a completely speci­fied presupposition-and that in contrast his attempts in the other failed, must have been the impulse that led J. H. Lambert to a new reworking of the problem. With him the discussion of the euclidean parallel postulate assumed its decisive turn.

Lambert no longer attempts either to prove orto disprove Euclid's postulate, but shows how parallellines follow from his own theory.

affine plane. For antiquity, the ideal elements of a projective plane were completely outside of the mode of observation.

141. Specifically, the proposition of the sum of angles in the triangle is equivalent in meaning to the Euclidean axiom of parallelism. This may have been one motive among others that led again and again to attempts to prove this postulate, especially since the opposite of that proposition is demon­strable.

Fundamental Questions of Non-Euclidean Geometry 269

He does not manipulate and reconstruct the problem as posed but modifies the question. What happens in geometry if one admits both of Sacceri's hypotheses? Lambert was the first to gain an insight into the noncontradictoriness of three "geometries," among which one preserves the euclidean parallel postulate and in subsequent termi­nology is "euclidean," while the other two are non-euclidean. Lambert deserves the credit for founding a new method in geometry and preparing the ground for the subsequent theory of proof. Since then, the provability or non-provability of a mathematical funda­mental presupposition is no longer the deciding factor concerning its axiomatic character-rather, the assumption of an axiom, or the change of an axiomatic system, determines the form of geometry.

This leads to the abandonment of the traditional conception of the essence of what is geometric. And this is incorporated by Lambert in his theory of parallels. The full insight into the logical nature of the problem should be seen as his most significant contribution. It is astounding how modern is his demand that in geometry "one must abstract from the conception of things" and must proceed "purely symbolically" (this means in our terminology signitive-symbol­ically) in geometric proofs.

The controversy over the demonstrability of the parallel axiom ends with the insight into its superfluity. While Euclid's concern was to lead back from the multitude of individual geometric states of affairs to a limited number of fundamental principies, the new method proceeded in the reverse direction: beginning with a limited number of fundamental presuppositions, it constructed new geomet­ric manifolds and built them in accordance with strict procedures. The way was thereby not only freed for work in non-euclidean geometry by Gauss, Bolyai, and Lobachevski, Helmholtz and Riemann; it also prepared the ground for subsequent foundational investigations carried on in the second half of the 19th century by Pasch and systematically developed to its full extent by Hilbert.

§ 2. Constitutive Problems of the Parallel Postulate

The inner-mathematical interconnection of a given axiomatic system and geometry is only of limited interest for this investigation. Of decisive importance for us is solely the mode and manner in which the new geometric manifolds are constituted in mathematical thinking. Thus at the outset there is a question concerning the way in which the euclidean parallel postulate is to be understood and the genuine locus of the constructive conditions of its content. Obvi-

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ously, it is not a question that touches upon the axiornatic or postulate character as such. This would be the case with the mathernatical view. Rather, it airns simply at a phenornenological tracing of its content back to those presuppositions frorn which the latter would be understood as the intentional correlate of specific mathernatical attributions of rneaning.

The postulate in the sense of Euclid obviously has the following rneaning: the point of intersection in question exists in each case of convergence of two straight lines, even when it no longer falls within the pictorial-syrnbolic presentation in the (lirnited) pictorial plane. In this "no longer" the structure of the thought process becornes visible. There is a tacit assurnption of the co-existence of two states of affairs-the convergence of the straight lines and the existence of an intersection-forrnation-simply because it is evident in pictorial-syrnbolic intuition for all those cases in which the point of intersection falls within the limited pictorial space, given sufficiently "strong" convergence of the straight lines. In these cases the point of intersection is trivially guaranteed through construction. Correspondingly, a "weaker" convergence of the straight lines can be compensated for in principie through the extension of the pictorial space. From this it can be concluded analogously that a point of intersection of two lines moving toward one another must be generally present, when in "weaker" convergence the point is removed farther and can no longer be factually constructed but only assumed. Such an assurnption must have been without question for Euclid. It was unthinkable for him that the geometrical event could in principie occur otherwise in the large rather than in the small, i.e, in a limited plane accessible to the mathematical operations of those times. As one can see, Euclid's parallel postulate-more clearly, his assumption of the existence of the point of intersection-contains in its turn nothing else than a sensible extension of a geometric event in a limited region of space toward the totality of space. It is analogous to the extension that was previously discussed for the limited space of intuition and the natural space of objects.

To be sure, for Euclid, straight lines and point of intersection are no longer things of a natural space of objects-nor are they on the other hand conceived by him as formations of a "mathematical space." To pose such a question of space would be inappropriate for the geometry of antiquity. But what is significant for the euclidean conceptual process is first the fact that here one begins with states of affairs that lead to the genuine parallel postulate (the existence and

Fundamental Questions of Non-Euclidean Geometry 271

singularity of parallels through any point that lies outside any given straight line) by a kind of consideration of limits, which unquestion­ably moves toward the possibility that a limited domain of space is capable of unlimited extension. This limit case is given when both inner angles taken together result precisely in two right angles. And secondly, what is remarkable is that this conceptual process begins with geometric states of affairs that are not only accessible in pictorial intuition but have the sale condition for the possibility of geometric existence, for the geometry of that time, in such pictorial symbolism. The existence was therefore bound to the medium of its presentation, whose sign was the "plane". With respect to its structure, the latter corresponds topologically to the natural space of objects. In accordance with its origin, the euclidean claim under discussion is bound to a geometry that can be carried out only in a space structured by planes; aild the "euclidean space" subsequently determined from it as a free geometric manifold can be such only if it manifésts plane structure and maintains the same connexus as the natural space of objects. Seen phenomenologically, this is the structural uniqueness of euclidean space. Seen mathematically, it is the fifth postulate of Euclid that expresses this uniqueness of euclidean normal space. In arder not to dismiss this euclidean presupposition of planarity as mathematically naive, as is so readily done today, one must become cognizant of its truly deeper founda­tions in arder to understand that it has long continued in the mathematical tradition. It seems that not only line segments and angles, but also lines and planes have their pre-scientific signification-fulfillment in the space of intuition. Their specific mathematical sense is first understandable only in terms of such signification. These topological formations are, as scientific concep­tions, grounded in particular in preceding achievements of the subject as a corporeal being and his co-given functional constitution. After all, the ray-like character of the visual function allows us to understand the topological structure of the space of intuition in terms of lines and planes (pp. 248 ff.). Geometry had neither reason nor possibility to assume any other form apart from plane geometry as long as its sense of existence originated from the mode and manner of a constructive generation. This was more so since the latter, in a most original sense, was still a production of its forma­tions and remained an instrumental production. They related to space only to the extent that the latter was already there as lived space, which provided the objects "on" which those formations reached conceptualization through a highly complex process of

272 Mathematical Space

achievement. Lived space was simply the condition for their spati­ality and pictoriality.

With Sacceri the state of affairs is not yet different. Led at the outset by the intent to guarantee Euclid's foundation for geometry against all attacks on the fifth postulate, he too, like Euclid, begins with the states of affairs in limited pictorial space and thinks that he can then carry it over unto space in general. Since he tacitly assumes the open-infinity of the line in his refutation of the hypothesis of the obtuse angle, his attempted proof rests on the intuitive background of the plane space.

With Lambert there appears a noteworthy turn.142 With the use of the Saccerian hypothesis of the obtuse angle, he obtains a spherical triangle, and thus relates his hypothesis to a spherical surface. Certainly, the recognition that on the sphere there are triangles whose sum of angles is greater than two right angles was not new. Menelaus of Alexandria, a contemporary of Plutarch, had already developed a geometry of the spherical triangle in analogy to Euclid's theory of triangles, and his "spheric" belonged to the classical fund of spherical geometry. It is all the more remarkable that many centuries of controversy had passed concerning Euclid's parallel postulate befare Lambert ended it with the obviously simple and illuminating suggestion of spherical triangles.

The originality of Lambert's account would only be missed if one wanted to see in it more than a renewed indication of long since known geometric results or to take his incorporation of spheric­trigonometric components into the discussion of parallelism as nothing other than a mere justification of a geometric hypothesis for a limited partial domain of geometric res.earch. What is decisively new in Lambert's thought is rather a fundamental transformation of a methodologically basic conception of geometry. He pointed to the spherical surface not because it is a reservoir for non-euclidean relationships, but solely because non-euclidean events can be demonstrated from it. Lambert stresses the hypothesis of the obtuse triangle not for the sake of the long since known spherical trigonometry; rather, he wants to offer the same justification for his hypothesis as is done for the euclidean parallel postulate. At the same time, he appeals to the sphere as merely one domain of its intuitive realization. Yet how little this has to do with the validity of the hypothesis follows unambiguously from Lambert's remark that

142. An extensive appreciation of Lambert is given by P. Stii.ckel and F. Engels.

Fundamental Questions of Non-Euclidean Geometry 273

the hypothesis of the acute angle would then be valid for an "imaginary" spherical surface (what is meant here is a "spherical" trigonometry for triangles with imaginary sides and real angles, ora geometry on a "sphere" with an imaginary radius-a formation that is no longer representable in any pictorial-symbolic sense).

All pictorial symbolism in geometry now functions merely as a heuristic means. This is valid henceforth not only for the new geometries, but also for the euclidean. The leading role is assumed solely by the algebraic symbol, the pictureless sign for the spatial, whose pictoriality has now become geometrically meaningless. This strict abstention from grasping the spatial in a pictorial symbolism is demanded by Larn.bert for the first time in all rigor for geometry. Expressed positively, the consequent possibility to signify exhaus­tively the spatial only in signs simultaneously abolishes for sign­geometry the unbridgeable opposition established by real-spatial and pictorially intuitable meaning: the opposition between plane and non-plane surfaces, between open-endless and endless-closed space. Speaking purely phenomenally, this abolition is visible in the simple fact that signitive symbolization discards any ties to a plane medium of presentation. For a geometry that is purely signitive and constructive in the specifically modern sense, it is in principie irrelevant whether one operates in a medium of open-endless extension or with a form closed upon itself. The plane of signs, in its constitution as plane, is as irrelevant for the characterization and meaning of a sign as is the color of the chalk used for the process of proof; and the latter is not changed in the least in its symbols and steps of procedure if it is somehow transferred onto a curved ( one-sided) surface.

This is apparently trivial-and yet it simultaneously shows a pro­foundly penetrating differentiation of performance-between picto­rial symbolism and sign symbolism in geometry from a new si de. The first demands that the plane of signs on which something is de­signated, and the plane indicated in the signs, should be topologically congruous with the formations-euclidean elementary geometry is not synthetically constructable on a spherical surface. In contrast, sign symbolism permits the sign-plane, on which it is merely signi­fied, to deviate topologically from the normal plane. In principie, it is thinkable that for an analytic geometry no other real-spatial for­mations would be needed for its production apart from surfaces of a higher order. Nothing could either take away its existence projected merely signitively in a conforma! geometry or somehow modify the mathematical sense of its existential propositions.

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This does not mean, however, that with the signitive construction of geometrical manifolds, "plane" and "surface" become indiscrim­inately equivalent, although they become formations of equal rank. To consider geometrically a (curved) surface as a plane means to modify its conceptual sense. This not only concerns the surface purely as such "in space," but also results in the constitution of completely new types of geometric spaces. For these, the surface then plays a fundamental role corresponding to that of the plane as a structural element in euclidean spaces. In those spaces, however, a specific geometry is consequently valid which already dominates the surfaces. This geometry is non-euclidean insofar as the underly­ing surfaces do not fulfill Euclid's parallel postulate.

But of course the geometry discovered by Lambert is not yet purely non-euclidean in the precise sense. The spherical surfaces on which the new insights are demonstrated are with him not yet strictly :rion-euclidean surfaces, but rather are still conceived as surfaces with spherical trigonometry. Even Lambert's geometry is factually still a trigonometry of surfaces in space, i.e., still in the euclidean­structured three-dimensional manifold. But it is not yet a geometry of a completely new space. At that time, the manifold did not even enter consideration as more than two-dimensional. In the conceptual sense of this geometfy at that time it could not be arbitrarily extended dimensionally. Lambert's conception of surfaces basically hides the fundamental novelty of a structure of thought that is nevertheless quite clearly present in its inceptual form.

A surface can first be called "space," however, when it turns out to be a manifold with its own rules for geometric events, and also when its mathematical conception is able to exhibit a reference back to mathematical embedment in space. This offers, then, the possi­bility to vary the number of dimensions. As Riemann's work subsequently demonstrated, such formulation was, in accordance with Lambert's demand, purely symbolic. Lambert's formulation first lays clown this requirement, although it could only fulfill it partially itself.

We are now confronted with a question: which founding intercon­nections should be traced in arder to reveal the ontological problematics of the non-euclidean manifolds and in arder to grasp them as new unities of sense of the space-positing consciousness­the same consciousness that does not simply surrender the euclidean presuppositions, but rather first grasps them in their genuine and fundamental meaning with the new manifolds.

Chapter Two

Foundational Problems of Hyperbolic · Geometry

§ 1. On the Metrics of Hyperbolic Geometry

This tapie is particular! y important for that form of non-euclidean geometry which, as the geometry of the acute angle, was related by Lambert to the imaginary spherical surfaces. Yet this relationship can no longer be simply represented on a surface like real spherical geometry. Rather, it is presented as nothing other than a purely algebraic relationship, which incorporates imaginary numerical quantities as constructed analogues to the usual spherical trigonometry.

Seen historically, it is the first geometry that was constructed precisely as non-euclidean; it is the so-called hyperbolic geometry. Proclus had already pointed out the possibility of numerous paral­lels going through one point. For this geometry Gauss discovered the functional dependence of the sum of the angles of closed figures on their surface content; in this way he obtained the limit case of the "infinitely large triangle" with the vanishing sum of angles. In contrast, the euclidean relationships are dominated by sufficiently small triangles.

The framework of this investigation does not require detailed exposition of the mathematical states of affairs of the new geometry. At present they are significant only insofar as they reveal the existence of phenomena that lead to grasping the question of the founding of hyperbolic geometry in sensibly intuited factors, and thus to the question of their intuitability. While at first glance the latter may be denied, a closer look reveals sorne interesting problems whose specific attraction consists in their remarkable interrelation­ships with the space of intuition.

To trace them we can use the work of Felix Klein as a point of departure. By following the suggested direction and by accepting the

275

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Cayleyan projective determination of measure, he was able to incorporate the hyperbolic as well as the elliptical geometries as special cases of general projective geometry.143 This incorporation became possible because this geometry is related to a specific fundamental surface of the second degree. In accordance with this foundation, the differences among geometries are formed solely in terms of the type of such fundamental surfaces. In other words, the specificity of a given geometry results solely from the specificity of such surface.

Thus to pursue geometry, it is necessary to conceive of the line segment as the basic formation of all metrics. While Euclid designates it nominally as the "distance between two points." it is in fact determined as an invariant property of euclidean motion. In a very specific sense this latter characteristic is lost in non­euclidean geometry. Through the new determination of measure, i.e., the projective, the invariance of a line segment is replaced by the invariance of the dual relationship of four line segments; in the surface the distance between two points results in a mutual relationship of location of four points (two of which can possibly be imaginary). Klein's main contribution to the determination of measure in hyperbolic geometry consisted of his definition of the logarithm of such a dual relationship as "distance" between two points in the fundamental surface.

At first glance this appears to be logical nonsense. To define "distance" or "line segment" by using four line segments for the definition obviously contradicts all rules of proper definition. Yet in a positive sense this definition contains the specific characteristic of a modification of the meaning given here to the concept of line segment. In the original euclidean sense this is a presupposition for the possibility of the structuration of dual relationships. It contains a categorical constitution of a unity of a higher arder; it is a specifically graded synthesis of the kind in which general "mathe­matical expressions" are constituted and which in this case implies the mathematically exact pre-constitution of the line-segment as such. As a condition for any determination of measure, the line segment is presupposed by all measures. Yet in no wise does it regulate the activity of measuring. Its simple "marking off" is, as a linear operation a+ a+ a+ a ... merely one among many possibilities of its application. Under the operative aspect, it presents itself only

143. F. Klein (1), (2), concerning the "Erlangen Program" see especially the formulation in (2) in Math. Ann. 43.

Foundationa1 Problems of Hyperbolic Geometry 277

presents itself only as the primordial form of any measuring, which nonetheless can be varied in principie. Thus the dual relationship, in accordance with its geoni.etric sense, is nothing other than such a variation in the manipulation of line segments for the purpose of new measuring. It does not define the concept of the line segment­which is already presupposed here-but rather it prescribes a particular manner and mode of proceeding with line segments.

While this procedure results in a new determination of measure, in the present case the projective, it seems to lead to the notion that the concept of line segments "acquired" a meaning that is different from the euclidean. Yet despite all this, the concept is "defined" with the aid of line segments, leading toan apparent contradiction in definition. The contradiction nevertheless dissolves immediately if we consider that we are not concerned with the usual definition of a concept and the establishment of an exact objective meaning, but rather with an establishment of a procedure that prescribes what must occur with the euclidean line segments in non-euclidean geometry. The new concept of the line segment, in distinction to the original, is nota concept of an object, but an operative concept. In its new conceptual sense it is constituted with, and on the basis of, an originary giving of meaning toa line segment in the acts of operating and proceeding with it in accord with the mathematical prescript of its dual relationship, i.e., of its logarithm.

This kind of prescript may astound a non mathematician. Ulti­mately what is normative for its sense is only the totality of the algorithm; it can become discerned only in terms of its algorithm. What motivates this prescript can be shown suggestively only with respect to what it can accomplish. The new determination of measure turns out to be so designed that it allows the use not only of distance in hyperbolic and elliptical geometry but also of the euclidean distance (of "parabolic" geometry) inasmuch as even the latter can be formulated as a logarithm of a dual relationship. It thus constitutes for the new metric conception a characteristically math­ematical mode of generalization. The non-euclidean distance is present in the projective determination of measure as a result of mathematical generalization of the euclidean determination of mea­sure; the latter in turn is nothing else in its projective aspect than a particular specification of the non-euclidean. The nature of this specification is more closely determined by the mathematical struc­ture of the logarithmic function. This means, in addition, that this non-euclidean geometry is euclidean in its smallest parts.

Furthermore, it is interesting that the suggested mathematical

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states of affairs have a specific type of intuitability for hyperbolic geometry. Indeed, it is fundamentally distinct from that of euclidean geometry, yet even if in a very modified sense, it shares with it a certain picturability of its geometric states · of affairs. After all, hyperbolic geometry, as a special case of universal projective geom­etry, has its uniqueness in that its charecteristic surfaces consist purely of real points. Following the far reaching implications that the fundamental surface of hyperbolic geometry is real-valued, and that it can be pictorially-symbolically presentiated, Klein created for the geometry related to it the well-known "model." In the sequel it will be sketched in its basic outlines for two-dimensional hyperbolic geometry.144

§ 2. The Kleinian Model. Phenomenological Analysis of the Model Conception

For hyperbolic geometry the fundamental surface becomes a fundamental conic section that is intersected by any line of the (euclidean!) plane at two real or two imaginary points (the latter means: pictorial-symbolically "not"). It can be conceived as a circle or an ellipse. Klein's notion was to limit the manifold of points within the conic section and to ascribe to them a logarithmic determination of measure. The logarithmic function has a character­istic such that in this determination of measure, the conic section turns out to be the locus of "infinitely distant" points; the conic section itself lies at a "logarithmically infinite" distance. The conic section chords function as "lines" of this geometry. The interior angles have real numerical measure. The multitude of parallellines through a point, has in this way a simple intuitive fulfillment: "parallel" are those straight lines that intersect at "infinity," i.e., at a point of the conic section (or outside of it). The conic section itself, "atan infinite distance," is never reachable-because of its charac­teristic interior determination of measure: a "line segment" of this geometry, approached by a non-euclidean being that can only move in the interior of this conic section, is constantly moved back in logarithmic foreshortening or, measured in euclidean time, a being

144. We limit ourselves here to a model of F. Klein for hyperbolic geometry, since we are not concerned with the enumeration of all model conceptions for non-euclidean geometry, but only with the exposition of what characterizes a model conception in geometry as such. Kelin's model can be taken here as an example.

Foundational Problems of Hyperbolic Geometry 279

inside the fundamental conic section moves ever "more slowly" toward the edge without ever reaching it. The interior of the conic section functions as its plane of movement; this being cannot know what is beyond the boundary of the conic section. (In an ideal mathematical sense it would at the most be approachable, as would be the infinitely distant line in euclidean geometry for a euclidean being.) Since all interior distances are measured by logarithmic relationships, there are only logarithmically determined triangle sides. The sum of angles in the triangle varies with the size of the surface content; for infinitely large triangles (in the euclidean for inscribed triangles), the surface content is zero. The attempted descriptive sketch of non-euclidean relationships indicated a signif­icant equivocation. On one side, it touched upon an anthropomor­phism readily used in the sciences. A "non-euclidean" being was confronted by a "euclidean" ("external") being; what for the latter is the interior of a conic section, for the former is aplane; what for one is a conic section chord, for the other is a line; what for one is a conic section, for the other is infinitely distant.

It is necessary to illuminate the type of understanding obtaining between these two beings. To do so we must divest the specific structure with which we are concerned, the structure of the "as," of its anthropomorphic cloak in arder to make it conceivable in terms of the giving of meaning of the one subject who alone is capable of grasping this description-that subject. which is not only the "euclidean" and at the same time the "non-euclidean" being, but also the subject concerned with such exposition of this mutual understanding.

The subject himself is a euclidean being in the sense that he has already opened euclidean space and has mastered it mathemati­cally-and has done so on the basis of the natural spatial apprehension founded ultimately in his modes of corporeal comportment and on the basis of the mathematically constitutive achievements of ideation, symbolization, and formalization in the sense presented above. Remaining at this level of constitution, living at the same time in the euclidean space of action, this subject experiences the conic section, chords, etc., primarily in their original geometric meaning-bestowing, i.e., as formations of the euclidean plane. Through the calculus of the new determination of measure, motivated by the mastery of non-euclidean relationships, there occurs then the change in meaning of those originally conceived euclidean formations, in their pictorially symbolized "self," into a new sense-bestowing of them "as ... " The change is

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such that a fixed and all pervasive relationship of coordination is established for all singular formations between the original and the modified content of meaning. This relationship of coordination is determined by the new regulations of measurement of line segments. This is always taken into account when a chord is taken as a line, a conic section "as" a totality of infinitely distant points, its interior "as" a new total plane, etc., and the resultant sense of the original concepts is in its turn understood as meaningful. The subject already has the calculus in readiness, in the pictorial­symbolic projection of those formations, when he approaches the pictorial plane prepared to illustrate, with its help, what has been achieved through purely algebraic analysis. Hence the fiction of a hyperbolic being becomes redundant for its interpretation of non-euclidean relationships; its relationship to euclidean being is understood as a relationship between two different bestowings of meaning to the pictorial-symbolic formations in one and the same consciousness "befare" and "after" the calculus. In a certain way this regulates a translation of meanings without allowing the translation to have a corresponding pictorial image-the figures, taken purely as such, rernain "euclidean," i.e., they are first posited by virtue of their euclidean rneaning. They are preserved latently when consciousness "lives" in re-signification of these forrnations through a new calculus of rneasure.

What is pictured and what is meant enter into a unique relationship of tension in hyperbolic geornetry. The signified here is not the original geometric rneaning of the pictorial syrnbol, but rather is constructed frorn it; the latter rernains in the background of consciousness in the acts of non-euclidean signifying, which use an algebraic calculus. The latter, however, is not itself pictorial, but syrnbolic in a purely signitive sense. This rneans that the pictorial syrnbolism of the euclidean plane is inappropriate and pictorially inadequate for what is genuinely meant in it. This, however, is no longer accessible via pictorial presentation in any other rnan­ner than the one sketched above, and thus it cannot be intuited in the previously maintained sense of this term. In place of the simple pictorial-symbolic intuition in euclidean geornetry, there enters a "model conception" as a mediating kind of syrnbolization which-in distinction to pictorial symbolisrn in the usual geornetric sense-requires an insertion of an algorithrn. This is achieved in such a way that the model must be independently constituted ahead of time in arder to fulfill its rneaning in sorne way in a sensible medium.

Foundational Problems of Hyperbolic Geometry 281

§ 3. Hyperbolic Geometry and the Space of Intuition

If the model conception of hyperbolic geometry is characterized as an algorithmically mediated type of symbolic intuition, it stands nev­ertheless in a particularly close association to the natural space of intuition. lt gains this affinity from the fact that it suggests a geometry of a limited plane ( of the interior of the conic section) in which the euclidean event is seen, soto speak, in a distorting mirror. Here there is obviously a remarkable analogy to the relationship between the natural space of objects and the singular space of intuition. This too is limited horizonally and, with. regard to the relationships of par­allelism, it specifically points to a noticeable similarity to hyperbolic geometry. Even within the "domain" of sensory intuition there are many "parallels" intersecting one and the same phenomenal point (compare p. 122), and even a being walking into the distance walks virtually ever "more slowly" toward the horizon without ever reach­ing it. In a specific domain of nearness, euclidean relationships seem to rule, while in analogy to hyperbolic geometry, the deviations are greater toward the border. The formal analogy is so noticeable that it threatens to reverse completely our previous results: it is not the euclidean, but rather the non-euclidean-hyperbolic geometry that apparently must be regarded as the more primordial, insofar as its founding strata are already immediately contained in the horizonally limited space of intuition. Thus the question emerges: should a more careful phenomenological analysis of the mathematization of space have led not primarily to the euclidean but rather to the hyperbolic geometry? In our previous presentation were we perhaps led simply by the factual development of history and not by systematic problems of constitution?

If this were the case, if the historical point of view had clearly been the leading one, and if euclidean geometry had thus "systemati­cally" assumed the first place in our investigation only due to its historical priority, then a critical revision of the preceding analyses would be in order. If hyperbolic geometry is more fundamental and is rooted more deeply in the corporeal-sensible event and its spatial world than is the euclidean, then it could appropriately claim full intuitability in the sense of the previously meant pictorial symbol­ism; in contrast, euclidean geometry should only claim intuitability of the model type.

But this obviously conflicts with the composition of the phenom­ena in the mathematizing consciousness. The difficulty in the illustration of hyperbolic geometry was demonstrably located in the

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requirement of a special calculus that first had to mediate the relationship between the pictorial symbolic intuition of geometric formations and the meaning-giving of something given in the figure "as" newly meant. Thus here the illustration could only function as a model. But this means that this merely conditional type of intuitability, which is mediated in a very involved way, rests upon the ontic priority of normal euclidean relationships. For it was euclidean geometry that provided the basis for the new determina­Han of measure and at the same time grounded the phenomenal preponderance of normal pictorial symbolism in the hyperbolic model.

It remains true that hyperbolic geometry is conceivable as the geometry of a limited mathematical space. It also has a type of morphological correspondence to a limited space of intuition. Yet this correspondence does not have a character of a founding rela­tionship. Hyperbolic geometry cannot be interpreted as a mathemat­ical idealization of the intuitable morphological world of things in the perspectiva! space of intuition, analogously to the idealization of the elementary euclidean formations (see pp. 184 ff.).

In addition, it ought to be recalled what truly determines the space of intuition as such and what role accrues to its objectivity for the intentional constitution of geometrical phenomena. Although in its being the singular space of intuition is relative to a corporeal here, nonetheless it turned out to be at the same time not merely a spatial "part" of an infinite space, but rather a manner and mode in which the latter can existas such and, as a whole, for a corporeal subject. The one homogeneous space is constantly co-present in the singular perspectiva! space of intuition and, in this mode of co-presence, it is already constitutive for the phenomena of perspectivity and of horizon itself. It is only on this basis that we comprehend such cognitive characters as size constancy, changelessness of the form of a thing in motion, and unlimited possibility of motion. Hence the mathematical sense-bestowing to be gained from these for euclidean geometry is already understandable from the space of intuition. Strictly speaking, it is not first the homogeneous space of objects, but rather its co-presence in the space of intuition, that allows the latter to comprise the basis for euclidean space. Similarly, the analysis of the foundations of the euclidean phenomena must have recourse to the sensibly intuited data of perspectiva! space.

The correspondence between intuitive data and hyperbolic geom­etry must be understood more precisely. Strictly speaking, it is not the space of intuition in which a type of hyperbolic regularity is to

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be found, but rather the viSual space. Only visual space-not the space of intuition-allows one to speak strictly of a boundary without beyond, and a multitude of "parallels" through a single point is a visual and not intuited (they are intuited precisely as parallels; (see p. 122). Visual space is determined as an abstract moment of the space of intuition. It is acquired from the latter by excluding the co-positing of "the" space and by reducing the full thing of intuition to the merely visible. The visual objectivity does not present any originary phenomenal data, but constitutes itself as such in its "thetic" modes by disregarding certain co-given factors in the primordially intuited data.

The relationship between the space of intuition and visual space has unique consequences for geometry. Just as visual space is phenomenologically inaccessible without the space of intuition, likewise the hyperbolic space cannot be conceived without the euclidean. Just as visual space is grasped first "from the side of" the space of intuition, so also the constitution of hyperbolic objectivity in mathematics first succeeds by going through euclidean normal space. Nevertheless, this formulation contains an easily misleading analogy. As was shown, euclidean normal space is founded in the space of intuition, indeed founded directly (p. 252). Since the former founds the space of hyperbolic geometry, hyperbolic geometry is also founded mediately in the space of intuition.

However, visual space does not "found" the space of intuition; rather, it is its dependent constitutive moment and comes into "view" as independent only abstractively, through a process of exclusion. That it is possible to find in it certain correspondences to hyperbolic geometry indicates that the pure lawfulness of vision, which participates in the full sensory intuition, might possibly underlie non-euclidean relationships and that hyperbolic geometry is applicable to it. But this does not mean that the latter is founded in visual space. "Application" is nota founding with a reverse sign. The founded and the founding have a necessary intentional and essential interconnection. The former builds itself in acts "upon" the latter, and requires the latter in arder to be. An "application" has an entirely different structure. Something to be applied is in principie free in contrast to that to which it is being applied. Indeed, the latter is required to allow the former "to find" application, yet the possibility of such finding application does not at all touch that which exists for itself without and beyond the application. A mathematical objectivity preserves its pure mathematical sense of existence even if it is never and nowhere applicable. Hyperbolic

284 Mathematical Space

geornetry is in its own right, even if the laws of vision do not obey it. Its sense of existence is related solely to the required rnathernat­ical rneaning-giving, and it is only frorn there that it can be introduced rneaningfully into the discussion concerning its applica­bility in visual space. That the latter rnight possibly "be" non­euclidean is a clairn whose ontological sense is not irnrnediately obvious. It rnust rather be explicated in its own sense and delirnited precisely against the ideal sense of being of geornetrical rnanifolds. This leads in general to questions of applicability whose problernat­ics rnust rernain excluded frorn this investigation.

Chapter Three

Riemann 's Geometry

§ 1. Riemann's Point of Departure. The Metric Fundamental Form

Riemann's work places the problem of non-euclidean geometry on a new foundation. The novelty of his point of departure, based on infinitesimal calculus, is not only internal to mathematics. His point of inception is also relevant to our problem. He advances a com­pletely determinate conception of space as such, subtended by its mathematical mastery. Riemann's program-designed to understand the world in the large from the infinitely small-offers in contrast to the traditional treatment of geometry (in the sense of the theory of parallels) a new methodological situation. The infinitesimal geomet­ric treatment of space is decidedly based on this methodologically fundamental requirement: to limit oneself in all geometrical propo­sitions strictly to infinitesimal domains and to refrain from any propositions concerning the structure of space in the large.145 While traditional euclidean geometry, starting with geometrical states of affairs in a limited domain of space, simply transfers them to space without seeing any special problems in such transfer, with Riemann there is an explicit requirement to abstain from such judgements. This is elevated to a fundamental methodological requirement.

What is unique in this is not only that one can successfully investigate the previously recognized mathematical spaces as a whole in light of new mathematics, but above all that Riemann's method creates t~e possibility for constructively projecting new kinds of mathematical spatial structures.

The mathematical relationships obtaining between euclidean and Riemannian geometry may at first sight blur their decisive differ-

145. The question concerning space as a whole is called by Riemann an "idle question"; for him only the infinitesimally small deserves scientific meaning. See B. Riemann, III. § 3.

285

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ence. If it is the case that what is generally admitted as characteristic of Riemannian geometry is also found in euclidean geometry­though "only" in the infinitesimally small-then obviously we must come to an appropriate conception and explication precisely of this "infinitesimally small." Since questions of the mathematical contin­uum are introduced here, subsequently we shall take a position with regard to it. Here we must primarily note that the notion of "Pythagoras in the small" does not at all touch the core of the Riemannian point of departure.

Indeed, the euclidean determinations of measure in the small hold true of Riemannian geometry; what this means in the mathematical formal language is that the Pythagorean theorem assumes the alge­braic form of a differential expression. Y et this is a form of the Pythagorean proposition that first results, for the Riemannian deter­mination of measure, from a further and more general principie that is not to be found at the beginning of Pythagorean deliberations. The infinitesimal turn of the Pythagorean proposition concerning the line segment is primarily a specification of a more general mathe­matical expression, the so-called metric fundamental form.

Euclid's geometry, and the conception of space as a manifold of planes co-posited for it, is insufficient for its attainment. The general foundation for its investigation is seen by Riemann in Gauss's theory of curved surfaces.

It is methodologically decisive that in contrast to the elementary geometric conceptions, these surfaces are no longer used for the additional demonstration of specific geometric states of affairs, still retaining the residual conception that they themselves are forma­tions "in" euclidean space; rather, such surfaces are understood at the outset as properly geometrical manifolds.

As independent spaces, they are determined essentially by a pertinent coordination. They are no longer related to the three coordinate directions of S3 • Rather, each determination of location is oriented exclusively in accordance with two proper coordinates or parameter curves chosen in the surface. In its mathematical presen­tation, the surface as a whole assumes a vectoral form, where the surface vector is a variable of two parameters. Correspondingly, the length of a line segment is no longer determined within the surface as a spatial curve with three functions but solely from the two surface parameters. The mathematical expression resulting in this manner for the length of a line segment in the surface is the metric fundamental form. Taken generally, this contains a result of a specific differential expression for the element of length. This means

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that all values in the surface are dependent on certain tensor magnitudes which, as partial derivations of the surface vector, are constituted in accordance with its parameters. They are constant functions of location and determine the geometrical event of the surfaces, determine the "metric fiel d." The geometry on a surface is determined univocally through these tensor magnitudes; the surface is already characterized by them. In turn, each mathematical expres­sion of the form just sketched can be conceived as a metric fundamental form, and can be taken as a constitutive element of a new geometric space.

At present it is not our concern that the fundamental form suggested is not yet the most general: nonetheless, the Riemannian point of departure already contains two remarkable characteristics. One is the infinitesimal turn of geometric problems, and the other is the positive definiteness of a quadratic form for the element of length. The first provides the new methodological means for the complete treatment of non-euclidean problems as such; through the second characteristic, the method is contentually bound to a number of geometric spaces which, as "Riemannian spaces" in a narrower sense, differentiate themselves from the still more general non­euclidean spaces through this characteristic of the positive definite­ness of their fundamental form.

§ 2. Riemannian Spaces. Brief Mathematical Characterization

A geometry constituted by the determination of length of the type of a positive-definite quadratic differential form is characterized first of all by its ability to subsume euclidean geometry as a special case-­i.e., a Riemannian geometry shifts to euclidean geometry when the tensor magnitude deploys its fundamental form in constant num­bers. Even with the preservation of its complete tensor character­more clearly, in its general determination as position functions to which accrue distinct values in accordance with position and surface orientation-it is not without relationship to euclidean geometry. Even in this general case the euclidean metric is valid for the "neighborhood of any point." Here the accent must be placed completely on the "neighboring" characteristic, since the infinites­imal point of departure strictly forbids the transference of mathe­matical relationships found in the small to a larger domain.

Certainly, it is peculiar that Riemann himself has not followed out this point of departure with those consequences that he had programatically demanded. Riemann begins with a determination of

288 Mathematical Space

measure. that includes the position-independence of length and allows, or at least does not exclude, the possibility of comparing distances between lengths. Strictly speaking, this is counter to his fundamental principie. According to Riemann, this euclidean resid­uum results in a determination of measure that is characterized by the property of integrability of lengths. Accordingly, a mathematical space is determined as Riemannian space if each line segment in it can be moved as an invariant along any chosen path; the numerical measure of a line segment is also here independent of any path.146

The congruent transfer of line segments is again to be strictly taken as infinitesimal. This demand contains what is perhaps one of the

146. Riemann's point of departure is capable of extension in two respects. When the metrics of a surface accrues solely to the metric fundamental form, and 'furthermore, when each expression of this kind can be interpreted as a metric fundamental form to be chosen as a constructional element of new mathematical spaces, there is nothing that prevents the surrender of the limitation by the positive definiteness of the fundamental form still maintained by Riemann and thus the construction of spaces that in sorne characteristics would deviate from Riemannian spaces. More signifi­cant than this deviation is another, a generalization of Riemannian thoughts assumed by H. Weyl. It presents a consistent and more fully developed extension of Riemann's concepts insofar as the infinitesimal mode of consideration is applied primarily not to vectors but to lengths. Thus a possibility is given where not only directions, but also lengths are not integrable and hence are path-independent. In these "general metric spaces" the characteristics of a vector of a given length at one point cannot decide about its characteristics at another place. To enable a congruent transfer, the neighbourhood of the point must be so adjudicated that the length of the vector in this neighbourhood can be maintained. The metric structure of this space also contains the specific gauging that enables exact propositions about the transfer of lengths. Mathematically, this structure is presented as a linear form for the establishment of the so-called curvature of the second kind; this is a measure for the changes of length experienced by a line segment in the course of displacement on a closed path. Like the first fundamental form, this one is also invariant in contrast to coordinate transformations. Thus, in conjunction with the former, it constitutes the characterization of the relevant metric space. The Riemannian spaces distinguish themselves from the Weylean generalization through the iden­tical vanishing of the curvature of the second kind, the "normal gauging." According to Weyl, it is necessary and sufficient for a space to be a Riemannian space. A more detailed discussion of Weylean spaces is superfluous within the framework of our problem. Concerning their math­ematical treatment, see H. Weyl (2).

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main difficulties in the process of following out Riemannian thought. Primarily, it Gan merely suggest to us what the differential­geometric point of departure brings about and what deviations from euclidean geometry it entails. Although the euclidean residue in Riemannian geometry preserves the relationships of line segments in the euclidean sense, there nevertheless appear notable differences as soon as one envisages movements not for line segments but for vectors. Apart from length, they also possess a direction, resulting in a number of specifications.

Indeed, as a consequence of its metrics, Riemannian space eo ipso possesses an affine connection. This means that any given vector is displaceable parallel to itself to any given place of the Riemannian space, although naturally this displacement is an infinitesimal displacement "from point to point." The displacement is ruled by specific transformations that correspond in priQ.ciple to the displace­ment transformations in euclidean space. On the basis of the infinitesimal metric point of departure, however, "parallel" dis­placement of a vector results in a change of its direction "from point to point," and this change of direction is dependent on the path of displacement. If one thinks of a vector (ora "localized vector group") in Riemannian space as having a closed path, returning to its point of departure through iteration of its displacement, it retains an invariant length for the length of this path. Yet-pictorial­symbolically-it appears in general as not having returned to the point of departure, and this deviation must be different in accor­dance with the path of displacement.

This non-integrability of directions in Riemannian space is, de­spite its original opposition to the conception of parallelism, any­thing but a paradox. Logically it follows as a completely univocal result of the methodologically fundamental proposition in virtue of which the surfaces are characterized by the previously mentioned tensor magnitudes, which enter into the differential expressions as fundamental magnitudes regulating the infinitesimal partial dis­placements.

In contrast, for a surface in which the tensor magnitudes are constant, these differential expressions provide a vector transfer independently of paths. Here after traversing the closed path, the localized vector group returns to its point of departure. This is precisely the case with aplane surface, as well as with all surfaces that can be developed from a plane. A space structured in accor­dance with planes turns out, from here on, to be necessarily euclidean in the sense that the components of affine connection

290 Mathematical Space

vanish in them. Through the integral iteration of infinitesimal displacement there results the finite transformations of the total space, leading back to the fundamental meaning of the concept of parallelism in the sense of elementary geometry.

Among the possible paths of displacement of a vector, in Rieman­nian space, only sorne have a characteristic that results in parallel displacement in the euclidean sense. Thus with the closedness of the path of displacement, the parallel displacement leads the vector back to its position of departure.

This is also a geometrical relationship that regulates the general transformation of displacement without transcending the framework of the generallaws of displacement of such surfaces. The apparent agreement with the euclidean meaning of parallelism lies in the geometric specificity of such paths. In other words, it is inherent in the properties of the differential laws of displacement in such surfaces, that for certain paths, they require parallelism in the euclidean sense with mathematical necessity, and such a path attains a distinction above all possible paths precisely through this parallelism.147 Such distinctive paths are given with the affine connection of the surface and the behavior of the localized vector group. They represent this connection; they are characteristic for the given surface as a whole.t4a

Their totality constitutes for each surface a doubly infinite mani­fold. This is similar to a manifold built from lines of a plane. The latter analogy is supported by the fact that those curved paths distinguished as "geodesic lines" possess in geometric measure an important property, enabling the comparison with the straight lines of plane-structured space. Geodesic lines are curves of stationary lengths; this means that for surface metrics they represent the shortest connection between two surface points. The shortest con-

147. It is also to be characterized from the side of S3 and respectively from the surrounding space of the surfaces, since under this aspect it is at the same time a spatial curve. Its specificity then consists in the incidence of the surface normal and the principal normal of the curves with respect to all of their points.

148. More precisely, it can be said that what solely conditions the kind of occurrence of such distinctive paths is not the metric, but the affine connection. They are concerned only with the differentiation of directions and not with vectors. This state of affairs becomes significant, for example, for the concept of the geodesic for surfaces with non-integrable length transference (see Riemann-Weylean spaces, note 146).

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nection for two points of the euclidean plane is offered by a connecting straight line. Analogously, one can proceed from one point of a surface to another in the shortest way only by taking the geodesic path. In a certain sense, the geodesic lines represent the straight lines in a non-euclidean space. The analogical use of the concept of a straight line is allowed even in a further respect. In the euclidean plane, each path connecting two points, apart from the shortest, is a curved path149 and the shortest is distinguished everywhere by vanishing curvature. It is possible to define a concept of curvature for non-euclidean spaces in such a way that with respect to them, the geodesic lines have the characteristic corre­sponding to the straight lines. Thus the "geodesic curvature" of a surface curve vanishes only for those curves that are geodesic, while any other connecting line in the surface, in relationship to this concept of curvature and in contrast to the geodesic, is "curved." At the same time, the concept of geodesic curvature clearly delimits the conception of "spatial curve" from that of the other, the "surface curve," for the sorne formation: whereas the geodesic-except for the straight lines of the plane space-appears in general as curved, insofar as it is conceived as a line of a "surface in space," it is a geodesic line of the surface itself. Considered in ~ non-euclidean sense, it is of such construction that its geodesic curvature is zero. But, for example, the circumferences of a "sphere in space" are circles of a determínate radius of curvature and never straight lines. Yet as lines of the spherical surface and as exclusively related to it, they are geodesic lines with zero geodesic curvature, and in this sense "straight lines" of the surface.

This results in a concept of curvature that contains completely different mathematical meanings. The indicated differentiation of meaning giving becomes possible only on the basis of two funda­mentally distinct mathematical meanings of the surface itself. It will become necessary to consider them once more when the frequently discussed and often misunderstood theme of "space-curvature" is taken up.

§ 3. Curvature and "Curved Spaces"

The brief consideration of the fundamental metric form already suggested the distinction in meaning between surfaces in space and

149. Connected series of lines {Polygonzügen} will be disregarded, prima­rily beca use as connecting paths they are not too interesting mathematically, since they are not differentiatable in all points.

292 Matbematical Space

surfaces as space. In the first conceptual sense, the surface remains tied to the intuitive representation, and indeed it is essentially representable only in this sense. It remains related to a spatial embedding medium of a higher dimension and furthermore, if it is taken as a curved space, its medium is a euclidean structure. As already mentioned, any mathematical treatment of a surface in this embedding space is to be conceived as euclidean.

It is otherwise when the mathematical relationship toa surround­ing space is abolished. This is the case in the full development of surface geometry, which must be seen as the genuine non-euclidean geometry and which, in its turn toward the infinitesimal, constitutes the essence of Riemannian geometry.

The decisive impetus for this conception of the surface as an independent mathematical space lies in that the surface is provided with its own coordination which, as two-dimensional, inheres in the surface itself and allows possibilities of dimensional extension for the mathematical treatment of surfaces-just as euclidean geometry has the plane for a plane-structured space of any chosen dimen­sions. The surface thereby becomes conceivable in principie as a structural element of non-euclidean spaces of a higher dimension.

The geometric independence of such spaces is apparently counter to the fact that they are concerned with curved manifolds. In accor­dance with their sense, the latter obviously presuppose a surrounding space of higher dimension and plane structure; it is only in contrast to the latterthat they can be seen as "curved." Obviously, the concept of curvature implies the meaning of planarity. Curvature is conceiv­able essentially only as a "deviation from" the plane, while in con­trast the opposite is not valid. The plane already revealed its funda­mental meaning in its structural distinction in the space of intuition which has its correspondence, on the subject's side, to the pregiven structure of the laws of vision (pp. 249 ff.).

Geometry justifies this irreversible relationship insofar as it ap­proaches the mathematical conception of curvature with methods that determine the degree of curvature of a curve or a surface in terms of the plane-structured space; the degree of curvature is seen as a measure of the deviation from the plane relationship. Thus the mathematical determinations of curvature by means of the radius of curvature, the first variation of the are length, etc., presuppose in principie the plane-structured surrounding space. Concepts such as a constant or a variable curvature of a surface in this first and simplest mode of mathematical apprehension have pictorial­symbolic fulfillments of meaning. In rough morphological terms,

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they designate the "regular" and the "irregular" way of being curved relative to the plane-structured natural space of objects, for which the degree of curvature is zero.

In all this, such unique apprehension of curvature would be counter to the conception of a surface of two-dimensional proper­manifold. lt can only be maintained as such if the mathematical concept of curvature can successfully be determined in such a way that the curvature would be conceivable purely in terms of the surface relationships themselves. The decisive contribution of Gauss is to have shown this possibility. His famous theorema egregium ends with the proof that the position function determining the curvature of a surface is determinable so1e1y from the tensor magni­tude of the metric fundamental form through which the internal measuring of surfaces is achieved completely and univocally with­out recourse to an embedding medium. Thus the Gaussian curvature is mathematically of equal significance with the internal relation­ships of measure of a surface. Whether it is a constant ora position function, changing from point to point at both surface parameters, is determinable purely from the geometric event in the surface. In anthropomorphic terms, a two-dimensional being existing only in the surface would be in a position to determine the curvature of space precisely and indeed solely from the relationships of measure of its surface without necessarily having recourse to a surrounding space.150

That this Gausian curvature is solely a so-called "inner" concern of the surface can be demonstrated by a particular property of

150. Thus the Gaussian curvature would be capable of yielding a principie of classification for the various classes of non-euclidean spaces. Hence the spaces of constant Gaussian curvature are characterized by their having (infinitesimal) congruent displacements, and the size of a geometric formation in them exists independently of position. Spaces of this state of affairs are distinguished as metrically homogeneous spaces. They include the ("elliptical") spaces treated by Riemann as well as the hyperbolic and eo ipso the (parabolic) euclidean space. From the totality of all possible mathematical spaces, these "spherical surfaces" are the only ones that have the distinction of being spaces of congruence. (The geometry of the sphere must be taken into this division, though it is not identical with Riemann's spherical geometry; rather, it is distinguishable from it, in the topological sense, through its one-sidedness. The spherical space is two-sided in three-dimensional space. The transition results from the topological identi- --' fication of two diametrically opposite points. Like elliptical space, spherical space is thus also homogeneous.

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invariance. If one bends a surface (without stretching or distortion), then the Gausian curvature does not change. Bending, as meant here, is not a mathematical concept; rather, it is morphological. lt is capable of sensible-intuitive fulfillment and can be found in any possible occurrence of bending. In contrast, when in the terminology of the discipline the Gaussian curvature is designated as bending invariant, this does not give any mathematical precision to the concept of bending; rather, this designation shows the sense of a non-identifiability of curvature and bending. Morphologically, the concept of bending is not at all differentiated from the concept (equally morphologically related) of curvature. Whether a real object cylindrically formed has a "curved" ora "bent" externa! surface is, within the morphological domain of signification, a useless question and merely a conflict about words. Only the mathematical domain offers an exact differentiation between these concepts. And indeed, each intuitive-morphological bending can in principie be grasped with mathematical exactness-but not every such bending is a curvature in the Gaussian sense. Thus in the example given, the bending of the cylinder may be ascertained in natural intuition; the Gaussian curvature nevertheless remains zero. Correspondingly, not every mathematical meaning-giving of curvature can be represented pictorial-symbolically as bending.

Even this most simple example warns us against our wanting to encumber a mathematical concept with ah intuitive (pictorial­symbolic) fulfillment of meaning. While its original conceptual construction is founded in the morphological domain, its subse­quent acquisition of meaning in the continuous pursuit of new geometric problems has become removed from its intuitive founda­tion. Intuition can no longer be fully sufficient to it, and any pictorial symbolism must become inadequate. lt is only by preserving this state of affairs that one escapes the erroneous opinion that curvature could be had only for a spatial manifold that is "in" another (plane-structured) space and is dependent on it, as if the concept of curvature is countersensical for higher than two-dimensional forma­tions. This concept signifies precisely that there are given specific "inner" relationships of measure that are just as little bound to the limits of dimensioning in the non-euclidean as the euclidean deter­minations of measure are bound to the euclidean plane. As always in the construction of concepts of geometry, here too we are concerned with the sense-bestowal that is built primarily upon the original foundation of meaning belonging to the lived spatiality capable of pictorial representation. In mathematical consciousness, however,

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the constitution of these sense-moments is motivated by a series of viewpoints that are not at all specifically geometric viewpoints. They not only lead to more precise and differentiated conceptual dimensions, but also to conceptual transformations, which for their part are most distinct in kind. In their totality they have their ultimate legitimacy in specific requirements of universality of the mathematical sciences. They are so conceived that they not only transcend the particular, but also simultaneously contain the latter as a special case.

It is with the aid of this conception of curvature that the discussion of "space-curvature" is first accessible to meaningful exposition. If any residuum of euclidean understanding is abolished in it, if the understanding is presented in a strict non-euclidean sense so that it mathematically masters the surface exclusively from within itself­which means that at the same time it envisages its curvature as nothing else than a specific inner structure of the measure of the surface-then in principie there is the possibility of any dimensional extension whatever of a two-dimensional non-euclidean manifold. Seen formally, the ascent into multi-dimensional non-euclidean spaces occurs in the same way as the euclidean; in both cases it fundamentally means only an adjunction of further parameters whose mastery is exclusively an analytic-algebraic concern.

In the infinitesimal basic structure of this calculus there is, at any rate, the fact that non-euclidean spaces are not first non-intuitive for higher dimensions. Rather, precisely to the extent that a two­dimensional spaée is taken to be non-euclidean, it already resists intuition. The appropriation of concepts from euclidean elementary geometry-length, vector, parallel displacement, etc.-may here pretend to have a certain intuitability, just as what is expressed in them is in fact originally capable of pictorial-symbolic illustration. Yet this picturability vanishes as soon as these qualities are consid­ered in terms of what happens with them in the non-euclidean. That a line segment can change its length and that a parallel displaced vector can change its direction, and that by returning to its point of departure it does not in general return to its position, is really not representable; this completely eradicates the intuitive fulfillment of meaning of "length" and "parallel" and evokes a peculiar experi­ence of conflict. An infinitesimal congruent transfer results ulti­mately in incongruence, and an infinitesimal parallel displace­ment-as soon as and precisely when, it is sufficiently reiterated­finally leads to considerable differences of direction in the sense of the original meaning of the concept of direction.

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This has nothing to do with meanings that are "carried over" or "transposed," or with a mere use of terms. Rather, the result seems to have conceptual meanings, which in their core maintain an apparent affinity to the euclidean. Yet this turns out to be an illusion when understanding turns to geometric events in the non-euclidean. The geometric event in these spaces, ruled purely by analytic algebra, not only no "longer" leads to intuitive fulfillment-as would be the case in higher-dimensioned euclidean geometry-but rather simply opposes it. With its basic point of departure it disrupts the lawfulness of the sense, and the construction of objects capable of such fulfillment.

The discrepancy between what occurs here geometrically and what direct intuition would anticipate occurring is grounded in the circumstance that pictorial representation always essentially in­eludes what Riemannian geometry excludes in its basic point of departure. The former encompasses a specific finite domain whose arder of magnitude far surpasses what is solely allowed by the infinitesimal-geometric point of departure, and encompasses it si­multaneously as a pre-given domain. In contrast, the differential­geometric calculus opens out this domain successively, and indeed through the infinitesimal iteration of geometric events "from point to point"; it establishes it first of all in operations.

In addition, this discrepancy may be decisively grounded upon the fact that what direct intuition takes to be a continuously extended domain, differeiüial geometry grasps in such a way that it concurrently prescribes an algorithm for conceiving this continuum itself. The usual mode of discourse claiming that something must occur "from point to point," or from "infinitely contiguous points," hides a deeper problem which ultimately is aporetic in nature.

§ 4. The Question of the Existence of the Mathematical Point

With the problem suggested, which at the same time touches u pon the problem of the continuum, we are approaching the conclusion of our investigation. Jt, reaches its limit, in a twofold sense, with the question concerning the meaning of existence of the mathematical point. First of all, this question no longer belongs in this domain. Rather, its discussion compels philosophy to engage in pure analy­sis. The problem of the continuum-as a problem of classification of all irrational numbers, well-ordering, countability by means of transfinite numbers, etc.-belongs ultimately to pure mathematics and lies outside of the geometric questions. What is relevant for us

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turns out to be merely the question of the intuitable continuum and its rational conception and mastery by way of the algorithm.

Secondly, our investigation encounters methodological limits­even in the suggested narrowing of the problem. They lie partially in the uniqueness of this problem itself. Indeed, our investigation is concerned with more than accepting a given objectivity offered by mathematics as something existent and, in accordance with our procedure, tracing it back to its constitutive origin of meaning. The question still remains as to whether we are really dealing with mathematically existing "objects." Certainly this is still controver­sia! in the discipline of mathematics.151 Yet only as such an object would the point and the continuum be accessible to phenomenol­ogy. While this attempt will be made in the concluding part, it can be justified, if need be, by exhibiting the problem of the continuum within the limits in which it is present within the framework of geometry, and thus revealing the limits of the method used to date.

We revert back to the continuum in lived space. There it presents no "problem"; it has a direct intuitive presence as a phenomenal continuum (pp. 88 ff.). For lived space, the concept of the continu­um had the meaning of denseness and non-interruption of spatial parts, sorne of which are individuated as "places" of things. Yet there is the constant concern with blurred, morphologically vague articulations without sharp boundaries. Insofar as the region in the space of action, and the phenomenal point of the space of intuition, are topologically the ultimate elements of the respective spatiality, they essentially escape precise localization. They are conceived as the finite limits of the nesting of domains prior to the exact determination of boundaries.

While for direct intuition they are everywhere dense, for the mathematically thinking consciousness the process of nesting is capable of progression that far surpasses what is sensibly conceiv­able. Mathematics structures space further "inwardly" all the way to the formation conceived as the mathematical point. This does not avoid, but simply hides the problem.

lts attainment obviously requires a further iteration of the topo­logical contraction and of "thinking" the nesting of domains "to infinity." This process of nesting immediately runs up against a blind alley. If the progressive iteration of nesting leads to increas­ingly smaller spatiallocations, which can be conceived as contigu-

151. Concerning this problem in general, see the subtle analyses of C. Becker (2), especially pp. 583-620.

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ous and touching with locally adjacent boundaries, then the iteration of this conception leads to a dilemma, especially if such nesting is regarded as progressing to the point which, according to Euclid, "has no parts." This difficulty leads directly to an aporia: how is a sum of such points, regardless of how densely they may be conceived, to constitute a continuous space? After all, a continuum is character­ized by the fact that it must be divisible into continuous divisibles and thus it cannot exist as a multitude of points, since points are indivisible. Should a continuum be thought to consist of points, the points themselves would have to be continuous; this is contradicted by the concept of a point. This conception faces an inescapable alternative: either it abandons the punctiformal conception of the continuum in arder to do justice to the point, or it abandons the continuum in favor of the point.

Yet the existence of the spatial continuum is not to be denied. Unaffected by all the difficulties of its mathematical conception, continuous extension is a phenomenal datum, indeed a "direct" intuitive datum. It is founded primarily in the fact of continuous motion. Only the attempts of its rational conception could be condemned to failure. But this does not place the continuum itself into question. What is in question is the existence of what is called a "mathematical point." Not only is it in no sense an intuitive datum, but even in mathematics its existence is burdened with all sorts of problems. We shall discuss it here only as a question concerning the extent to which its claim to existence can be given the meaning that was previously attributed by us to geometric existence in the sense of constructability.

The existential question of the mathematical point is thus prima­rily a question of its access. Whether and in what sense "there is" something like a mathematical point is to be decided solely from the mode and manner of its conception. With respect to its treatment in Euclid's Elements, the point apparently assumes a remarkable dual position. If in the Euclidean sense the existence of a geometrical formation is guaranteed through constructive activity having firmly prescribed means, then the geometrical point cannot exist for it, at least not in this sense. In Euclid the point is not guaranteed through construction (something like a formation of intersection), but rather is assumed as given in arder to guarantee the geometric formations through construction.1s2

In contrast, there is the variously criticized pedantry-factually

152. Euclid, Elements, Book 1, postulates 1 and 2.

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unjustified-with which Euclid constructs the irrational quantities in Book X of his Elements, quantities that are conceived to be the relationships of a line segment to a given initialline segment. The explicit and detailed work that Euclid dedicates to every newly introduced irrational relationship impressively shows that he does not simply accept the existence of all irrational (quadratic) points or numbers. Rather, he sees the necessity of securing their differently comprised mode of being-megethe and not arithmoi-by a unique way of construction. This construction of irrational numbers must have been important for Euclid, since the obviously specific mode of existence of these quantities could come to light only in it. For Euclid, they "exist" only to the extent that they can be established constructively, yet they do exist in this mode even if they are given as unlimited (apeiron) from the standpoint of rational number relationships.

It must not be overlooked that the irrationalities existing in Euclidean geometry remain limited to the quadratic. Since for Euclid the only existent formations are those that are attainable constructively through circles and lines, then geometrically speak­ing, only such points are acceptable as can be composed from quadratic roots and the irrationalities consisting of them, and no others. And yet there are these points in the sense that geometrical existence had for antiquity. They are acquired from constructive generation in precisely the prescribed way.153 The limitation of the constructive means of antiquity contains the fact that antiquity's

153. A point of division tripartitioning a circle's are thus do es not exist for antiquity, even if intuitively it may be evident that there must be one. It comes into existence later when a new means of construction was intro­duced by means of conic sections (circles and lines in space). There is nothing in Euclid concerning their application. The only exception to the quadratic irrationalities in the geometry of antiquity is the number pi, which is applied for cubings and rectifications. (All the problems of this kind at that time were solely dependent upon pi). Yet pi here assumes a unique place to the extent that this transcendent number is "introduced" as a relationship between circle surface and square of radius, i.e., it is not constructed with the pre-given means of construction, since pi cannot be constructed with the means of antiquity. If a convergent approximate construction were to be allowed as validation for the proof of existence, it would have contradicted antiquity's basic conception of the essence of mathematics since the existential proofs of antiquity are always concerned with finite constructions. Strictly speaking, the number pi does not exist in the sense of antiquity's mathematics.

300 Mathematical Space

definitions of irrationalities are not limited solely to quadratic roots, but also that such definitions are finite in principie. The existence of irrational quantities is ascertained through finite construction and their entire sense of existence is exhausted by geometric construction with the given implements of construction (compass and straightedge) in a finite number of steps. Today one might reject antiquity's conception of the essence of what is mathematical as primitive. Yet it has a well founded sense in that for the Greeks there are simple forms. For them the circle is a simple form to the extent that it can be produced with an instrument of construction, and thus it can serve as an element of proof for the mathematical existence of composite formations. To determine it through an endless nesting of connected series of line [Polygonzugen]-despite the affinity of such a conception to pictorial-symbolic intuition­would have contradicted antiquity's conception of the essence and uniqueness of mathematical construction.

The fundamental difference between modern mathematics and that of antiquity consists, as .was already shown, precisely in the extension of the concept of construction in such a way that it is transposed to certain operations of thought (algorithms). It is only through this that an "infinite construction" is acceptable as a legitimate mode of generation of mathematical formations. Thus for modern times, the quadratic irrational numbers are no longer the results of finite geometric constructions, but limit formations of infinitely converging processes. Moreover, the admission of endless processes as a justifiable principie of construction of numbers leads not only to quadratic, but also to other algebraic as well as transcen­dent irrationalities.

In geometric terms this obviously means that the existence of mathematical points is ascertained only when a procedure can be given on the basis of which they are discoverable as limits of number sequences oras interval nestings. Their specific problem of existente depends on the "sequence" character of their construction-more precisely, on the endlessness of the converging processes that ought to lead to them.

First, let us briefly consider those sequences that are, in the narrower sense, regular sequences capable of being constructed by a so-called closed analytical expression.154 And yet these are the sale

154. It is to be observed that the concept of the regularity of a sequence is always relative to specific characteristics. Even sequences whose numbers are determined by a general number, which thus cannot be presented as

Riemann's Geometry 301

sequences that in an extended sense can be regarded as phenome­nologically accessible at all. The closedness of the expression allows such a sequence to be in a certain sense a really "admittable" sequence. The accessibility of the formation defined by it will here be guaranteed at least in relation to its foundations of construction, i.e., if the beginning member of the sequence and thus the initial interval of nesting are "themselves given," then due to the all­pervasive regularity of the sequence, they are also constructed, in a certain sense.

To be sure, there are always only a finite number of discrete intervals capable of fulfillable postulation, i.e., symbolically intui­tive fulfillment. It is thereby essential that similar individuals and discrete postulations be taken one after the other and must be actually accomplished in time. The factually fulfillable positing intention of such nesting is thus a sequence of acts whose structure is finite in principie. The open endlessness of the sequence will not be attained in it. Yet these acts, which appear necessary in accor­dance with the idea of nesting, need not be individually fulfillable acts. The "and so forth," in which the actually positing intention exhausts itself after a few steps, is not completely indeterminate, due to the pre-given law of formation of such a succession. Rather, such a regularity makes it possible to survey the entire succession. Phenomenologically speaking, such an overview is of a most unique kind, and in its total structure it differs essentially from all previ­ously considered "viewing" intentions. In itself it is neither one of the positing acts of nesting, nor is it a synthetic act comprising its potential totality. Directed "toward" the open-endless succession, such an overview is an intention of a second level. More precisely, the infinite sequence is surveyed with one glance: a singular finite act of the mathematizing consciousness. It does not intend the infinite progressively-in accordance with the Hegelian "bad" in­finity-but, dueto the power of the "and so forth ... ," there is the categorial fulfillment of the intended as it is ruled and secured through the law of formation of the sequence. This is the mode in which the infinite can become a problem for a being who is finite, but whose consciousness is at the same time such that its variously

merely recurrent, are not simply regular, as if they were determined vis-a-vis all characteristics. The problematic of the endless is contained in this partial indeterminateness. Since the endless cannot be surveyed to the end, there is everywhere a partial indeterminateness of sequences and thus an arithmetic problem as such.

302 Mathematical Space

stratified structure of acts is capable of constructing and mastering the infinite in finite thought.

Certainly, in this lies the fact that the infinite can never become a phenomenon, can never be an object in the strict sense. Thus the point to which the convergence of nesting "leads" cannot be a really given and designatable phenomenon. The iterative construction of sequences-and thus of (sorne) mathematical points-reveals the fact that these sequences move in a continually open horizon and remain endless processes. The construction of such points can never be closed and can only be a constant becoming; in its construction, the point can be grasped in perpetua! becoming.

This character of becoming in time shows up more impressively and radically in the so-called freely chosen sequences. After all, not every point is constructable through a regular or through a determi­nate recurrent sequence. In the effort to reach "any" chosen point (but not "all" points!), modern mathematics thought it necessary to allow sequences in which the succession of the members is not determined by any pre-given regularity, but rather in which each member is posited by an act of free choice. Naturally, here only a finite number of members is factually positable. What is meant by the "and so forth" in this freely chosen sequence is obviously quite different in its intentional sense from that of the regular sequence. While in the latter case the "and so forth" is a kind of direction to break off further procedure, since in accordance with the rEigularity nothing "new" can occur, the idea of the freely chosen sequence contains the notion of a factual progression in infinitum, since its becoming constantly contains new theses. Yet this idea essentially abolishes the horizon in which any further postulation should occur. A mere requirement to posit "something" further is purely formal and completely abstract-an empty intention that can be fulfilled by any arbitrary ad hoc postulation; no finite chain of postulations insufficient. Moreover, there is no positing consciousness having a capacity for an overview of a higher arder.

The question as to whether such sequences can still be seen as constructible is essentially a terminological question. If we do not wish to designate a series of arbitrary postulations without pre-given rule of procedure as "construction," then the constructive character explicated up to now must be denied to them. Yet the existence of points that are to be built through the freely chosen sequences becomes extremely controversia!- and controversia! not because of the indefiniteness of the process leading them, but above all due to the nature and mode of access to them. As was stressed, however, the

Riemann's Geometry 303

question of the existence of the mathematical point can be sensibly posed only as a question of its access.155

What the freely chosen sequences offer for the discussion of this question-and offer not differently from, but more clearly than, regular sequences-consists above all in clarifying the particularly mathematical view of infinity in its specific potential character. Just as a real number is not, but becomes through its generation, so does its corresponding point. The sequence is not merely a method for grasping a preexistent point, but rather the manner and mode of generating it in the first place. And it generates the point only as long as it is a sequence in process-the point is not just generated within it. The manner and mode of its generation allows that it will be generated in infinitum without being generated.

Thus a mathematical point does not exist. At any rate, its exis­tential claim is not united with any such claim defensible in the geometric sense; it is not "given," since no construction can actually "give" it. Even in the regular sequences, a constructible point is, in a strict sense, "not given." If one insists, its mode of being lies in the paradox of being a constructively generatable formation only to the extent that this construction never reaches an end. Not only is all ontology fooled by it, but it also remains the l'enfant terrible of pure mathematics "any" point.

The paradox of its existence finally becomes an aporia, however, "all" points are meant. Mathematics is completely clear about the risks of this kind of massive postulation. Behind it lies the question of the mathematical continuum. The problem of what is a genuine mathematical continuum, and how does it relate to real numbers, is confronted by mathematics with astounding reservation, and is enveloped in cautious and not completely clear exposition.15B This must be considered less an expression of perplexity than the positive

155. The mathematical problems of such sequences and the critical objections raised against them from the side of pure mathematics cannot be discussed here. The concern here is with what they accomplish for the mathematical conception of the intuited continuum.

156. H. Weyl, who in many ways has made the continuum into a theme of mathematical investigation by seeing and clearly formulating the prob­lematic contained in it, raises the decisive question: "Why do we postulate under the continuum the concept of real number?" - only to discard it immediately ("This is not the task here ... ") and to turn for information to the concept of real number as "the abstract schema of the continuum." See (1), p. 70-71.

304 Mathematical Space

insight that here the concern is with an "object" of mathematics whose conception, in contrast to any other mathematical concep­tion, must receive a strongly modified sense.

If one wanted to see the continuum intended in the sense of approximation to ever greater exactness the longer the process goes on, then it would mean that one is imputing a false sense to the infinite sequence of numbers and respectively to the processes in consciousness correlated to them. That taken singly these processes possess precisely definable limits, which are interpretable as points, does not mean that the continuum consists of points or that it is a (not enumerable) multitude of points; rather, it allows us to recog­nize what can occur with the continuum in mathematical treat­ment-and above all,. what cannot occur. The character of the sequence as a process running into infinity is mathematically a precise expression of the possible in infinitum division of the continuum, which remains a pure possibility. If the point sought in it were to be actually attained, the continuum would be disrupted. The condition, that the point never is but always becomes, saves the continuum. The linguistic expression stating that the continuum can be "divided" into points has a defensible sense only when one avoids the paradox of the point and assumes it in the total actuality of its becoming.

What is most significant for the freely chosen sequence is that this becoming, as well as its incompletability, becomes directly obvi­ous.157 They themselves appear only through their becoming, pos­ited in acts of free choice-and they are only insofar and to the extent that they are posited. Any member in them is a discrete and continually new postulation. lt is predetermined and predisposed by nothing; it is an act of freedom which, in accordance with its idea, is unlimited in its possibilities. And yet it is an action in time, finite, factually limited to a finite number of really possible steps in the framework of duration possessed by a factual subject.

157. Since within the framework of our inquiry the concern is only with the mathematical conception of the intuitable continuum, we do not explicate the question any further with respect to what the theory of freely chosen sequences achieves for the solution of the purely mathematical problem of the contiunuum. For details see O. Becker (2), pp. 600 ff.

Concluding Observations

The problems treated last were especially concerned with the constitutive relationship of mathematical objectivity to the mathe­matizing subject-more clearly: to the subject as a temporal, finite being. It is an old insight. Aristotle had already recognized the temporal moment of mathematics in the phenomenon of the se­quence. Kant stressed the relationship of number to time which for him was a form of intuition of a finite and indeed specific human being.

Finally, O. Becker has convincingly treated, in minute phenome­nological analyses, the modern problematics of pure mathematics and the decisive role of temporality for the being-characteristic of mathematical objects. Committed to the transcendental-constitutive mode of research of Husserl as well asto Heidegger's hermeneutical phenomenology, his effort is led by the intention of illuminating the ontological meaning of the mathema, of the mathematical, in an encompassing sense from the process of "mathematizing," of mathematikheiesthai, as a mode of the living existence of the human being.158 The anthropological founding of mathematics, as seen by Becker, is not to be understood in a sense of "anthropologism." The basic structures of mathematics cannot be dissolved into empirically determined processes of thought, into the psychic acts of the factual human being. Rather, he intends to show that there are essential relationships between the sense-structure of the mathematical and the ontological meaning of the human essence or the idea of a finite being.159

The reason why the finitude of the mathematizing subject appears decisively in this anthropological relationship will be comprehen­sible directly from the problem of modern mathematics, understood

158. O. Becker (2), pp. 441, 637 ff,; correspondingly (7), especially pp. 157 ff.

159. O. Becker (3), pp. 279-83.

306 Concluding Observations

as a method for the mastery of infinity: only a finite being can make sense of the problem of the infinite, only he can want to master it, and thereby find himself confronted by the abyss of the unsurvey­able, innumerable, and undecidable.

Our preceding investigation immerses itself in this problematic. It does so primarily in the sense in which even the geometric domain ultimately obtains its existence from the mastery of the infinite in pure mathematics. Just as the latter is related to the being of the subject in time, geometry too appears to be essentially related to a being in space: only a spatial and spatially bound being can pose problems for himself of the kind that lead to geometry as a science of space.

This relationship is not exhausted in its depth and in its specific nature by such an analogy alone, as the preceding investigation, particular! y in the field of the mathematics of space, believed to have been able to show. Immersed in the specific question of the basis of being of geometric phenomena, the investigation did not first dis­cover it in the domain of mathematical syntheses and the accom­plishment of certain acts of the mathematizing consciousness, but rather attempted to understand it from preceding achievements of the subject in his corporeity. In our retrogression to these founda­tions, we found ourselves placed befare two limits that a phenome­nologically attained understanding has to respect. On the one side, there was the doxic-thetic character of consciousness with its "positing" activities as a contingency not to be inquired behind, which could not be deduced from the corporeal activities of the subject. What appeared as uniquely demonstrable was an intercon­nection of the achievements of the lived body and consciousness such that in the structure of the latter-contingent in itself-and specifically in the domain of geometrically constitutive acts, impli­cations were encountered on every side that could be deciphered only by retrogression to the functioning of the lived body as such. On the other side, our phenomenological observations discovered an impassable limit in the facticity of the corporeal being-more concretely, in the corporeal constitution of the human. While it was conceived as a specific concretization of the general principie of corporeity, in this specification it could not be made further under­standable, but had to be left in its impenetrable facticity.

If we recall the function that had to be attributed to the lived body for the constitution of space per se, then the result for us is that the question of the existence of mathematical spaces-as an ontological question concerning the mode of their being-will not be answered

Concluding Observations 307

merely in relationship to sorne finite being or other. Rather, it must be answered more concretely and fundamentally from the mathe­matizing process of a corporeal human being who is designed in this and no other way. It is first of all "on the basis" of its corporeity that this being is a temporal, finite being-it is "on the basis" of this, its lived body, that it is a spatial being at all. To discover in this the ultimately founding condition of the geometrical sense of being was the intent of the preceding investigation.

It has nevertheless served the understanding of this sense only one-sidedly, because it has not brought to light what occurs through the multi-stratified constitution of the geometric by the corporeal human being, and what happens to him himself. The essential correlative relationship between the full sense-structure of the geometrical and the total ontological structure of the space­constituting subject should not be permitted to rigidify into a statically conceived relationship of arder. Rather, it must be pre­sented in the dynamics of its reciprocally implicative becoming. This task was hardly mastered by the means of the phenomenolog­ically descriptive method. Ultimately, however, we remain indebted to the subject for illuminating the meaning that must be attributed not only to the subject's "way" through the diverse spaces, but finally to his reflection on this way as his own.

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Register

W. Ahlmann, p. 128.64

Aristotle, p. 157, 161 ff. W. Arnold, p. 96.36

H.U. Asemissen, p. 100.41

J.J. Bachofen, p. 34.14

O. Becker, p. 11, 184,92 193, 211,109 220,116 246,126 261 ff., 266,1 38

297,151 305. G. Berkeley, p. 95, 98, 101 f., 123. E.W. Beth, p. 197.101

O.F. Bollnow, p. 20.3

F.J.J. Buytendijk, p. 16,2 31,9 33, 35.19

R. Carnap, p. 182 f. E. Cassirer, p. 35.40

J. Cohn, p. 162.81

H. Conrad-Martius, p. 7,2 162.81

R. Descartes, p. 201, 203, 243. K. v. Dürckheim, p. 15,1 28.2

Euclid, p. 211,109 213, 217 ff., 227, 231 ff., 240, 265 ff., 270, 286, 298 ff.

F. Fettweis, p. 74.29

E. Fink, p. 7,2 175.87

P. Finsler, p. 219.115

K. v. Fritz, p. 211,109 267.138

W. Fuchs, p. 133.68

A. Gehlen, p. 34.14

W. Gent, 7. 2

G. Gentzen, p. 220.117

K. Godel, p. 220. J.W.v. Goethe, p. 22. K. Goldstein and A. Gelb, p. 78,32 127 ff.

319

320

C.F. Graumann, p. 83.33

A.A. Grünbaum, p. 71.27 N. Günther, p. 96.36

A. Gurwitsch, p. 107.46

Register

N. Hartmann, p. 166,84 171 f., 256.131

M. Heidegger, p. 2, 4, 8 f., 16,2 41 f., 51 f., 53,21 385. H. Heimsoeth, p. 7. 2

H. Hermes, p. 219.115

D. Hilbert, p. 200, 207 ff., 213, 220, 227 f., 269. O. Holder, p. 197,101 213.11 2

H. Hopf, p. 259.132

E. Husserl, p. 4 f., 8, 11 f., 38,17 42, 83,33 88,34 98 f., 104 ff., 123, 150, 166,84 171 f., 175, 180, 184, 187,93 191 ff., 196,99 197, 202,212,219 ff., 245,125 305.

E.R. Jaensch, p. 96.36

M. Jammer, p. 7,2 162.81

W. Kaiser, p. 21.4

l. Kant, p. 1, 2, 105, 126, 165 f. D. Katz, p. 12458 , 126. W. Killing, p. 260,134 261,135

L. Klages, p. 22. F. Klein, p. 259, 265, 275 ff. J. Konig, p. 22.5

A. Koyre, p. 7. 2

J.H. Lambert, p. 268 f., 274 f. L. Landgrebe, p. 107.48

H. Lassen, p. 94,35 131, 153 ff. K. Lewin, p. 72. J. Locke, p. 123, 188,94 196. F. Lobell, p. 259.132

G. Martin, p. 220.118

M. Merleau-Ponty, p. 2, 16,2 58, 99 ff., 158 ff. S. Monat Grundland, p. 128.63

L. Nelson, p. 213.11 2

O. Neugebauer, p. 211.109

M. Palágyi, p. 37, 42, 126. R. Pauli, p. 126.60

A. Podlech, p. 17.2

A. Portmann, p. 31.9

H. Plessner, p. 16,2 58.23

Proklos, p. 267 f., 275. K. Reidemeister, p. 202,104 209,108 212.

E. Rothacker, p. 16.2

F.S. Rothschild, p. 65.25

Register

B. Riemann, p. 269, 274, 285 ff. Sacceri, p. 268. F. Sander, p. 96.36

J.P. Sartre, p. 17,2 58.23

W. Schapp, p. 124.58

M. Scheler, p. 16,1 44, 116 ff., 158 ff., 166,84 181 ff. P. Schilder, p. 65,25 71,27 78,32 127. F.v. Schiller, p. 33. H. Scholz, p. 211.109

P. Schroder, p. 20.3

A.v. Senden, p. 128.64

Th. Skolem, p. 220.U8

A. Speiser, p. 211.109

P. Stackel and F. Engel, p. 272.142

A.D. Steele, p. 217.U4

S. Strasser, p. 20.3

E. Straus, p. 21,4 29, 36. E. Stroker, p. 22,4 125.59

C. Stumpf, p. 101,42 153. H. Tellenbach, p. 44.19

J.v. Uexküll, p. 31.9

P. Valery, p. 35.13

W. Voss, p. 135.69

V.v. Weizsacker, p. 31.9

H. Weyl, p. 242,123 288, 303.156

W. Wieland, p. 158.79

J. Wittmann, p. 128.64

H.G. Zeuthen, p. 217.114

J. Zutt, p. 78.32

321