1SyntacticreductioninHusserl'searlyphenomenologyofarithmeticMirjaHartimo,UniversityofJyvskyl,FinlandandMitsuhiroOkada,KeioUniversity,JapanAccepted for publication in Synthese

Introduction

In this paper we will discuss what we call syntactic reduction in the early writings ofEdmund Husserl (18561938). Clarifying Husserls notion(s) of reduction sheds light on therole of a constructive aspect in Husserls early conception of mathematics. The concept is alsointeresting because it shows how and in what context various ideas commonplace to thepresent-day proof-theory and logical theory of (equational) computation, such as termrewriting and equational proof-reduction, were conceived already in the 19th century. Ouraim in this paper is to identify various kinds of syntactical reduction, to examine how Husserldevelops them from his Philosophy of Arithmetic (1891) to the notorious Double Lecture heldin Gttingen in 1901, and finally to explain the role of the reduction in Husserls conceptionof mathematics.

Husserl developed various kinds of reductions in his attempts to clarify Hermann Hankels(18391873) principle of permanence under which algorithms are extended. He foundHankels principle problematic in two respects: first Hankel equated the sign and the signifiedand second, Husserl held that Hankel had not properly justified the coherence of the extendedalgorithm. To ensure the correctness of the extended technique of calculation, he held, thetechnique had to be reducible to the identities between the signs. The present paper detailsHusserls indebtedness to Hankel in detail for the first time.1 The earliest occurrence of theterm reduction can be found in Husserls text in a manuscript dated to around 188990.2 Theend of Husserls Philosophy of Arithmetic (1891) can then be read as offering Husserlsputative improvement of Hankels principle. In it Husserl distinguished between the systemsof signs and systems of concepts, a distinction he had already made in his Habilitationsschrift(see p. 14 below), but which in the Philosophy of Arithmetic is ultimately conceived as two

1 Previously it has been discussed in Hartimo 2007. In addition to that discussion, the present paper elaboratesHankels view in more detail and shows in particular that where Hankel used mathematical induction to definevarious number systems, Husserls strategy was to examine their correctness with a reduction-basedcomputational approach.2 He uses the term reduzierbar in Die wahren Theorien dated to around 18891890 (Husserl 1983, 35, see alsoArithmetik der Reihen dated to around 18891891, Husserl 1983, 167168). Note that Husserl also sometimesuses the word reduction for another sense, such as for a reduction from n-ary addition to binary addition sothat a + b + c is reduced to (a +b) + c (e.g., Husserl 2003, 279). Such a notion is not a topic of primary interestfor us here. To be sure, the reduction under examination has nothing to do with Husserls notorioustranscendental phenomenological reduction either.

2ways of understanding the signs. Husserl defines systematic numbers with the decimalsystem. To ensure the coherence of the use of complex expressions, the notion of reduction ofany complex expression to a (unique) canonical or normal number in the system isintroduced in the last section of the book Philosophy of Arithmetic, Psychological and LogicalInvestigations (1891). When doing this, Husserl uses what is now called term-rewritingreduction (or proof reduction) of the underlying equational proof system. Husserl returns tothis topic in his Double Lecture in 1901. In addition to most of the existing literature onHusserls notion of Definitheit, the present paper draws attention to the notion of syntacticreduction used in Husserls discussion of mathematical manifolds.3 While not all manifoldsare mathematical, mathematical manifolds are definite when their elements are syntacticallyreducible to the elementary individuals.4

In section 5 below, we will show how in his Logical Investigations (19001901) Husserlexplains the philosophical motivation for such reduction: the task is to bring mediatefulfilment to (algebraic) formal mathematics, which could not otherwise receive suchfulfilment.5 Its role is thus to show how (at least part of) formal mathematics is accessiblefrom our immediate intuition of simple definitions.

1. Hermann Hankel and the principle of permanenceHermann Hankel (18391873) presented the principle of permanence in his Theorie dercomplexen Zahlensysteme. Insbesondere der gemeinen imaginren Zahlen und derHamiltonschen Quaternionen nebst ihrer geometrischen Darstellung (1867). As the titlesuggests, Hankels treatise attempts to present a theory of functions of a complex variable.The topic was at the time relatively new; it had been founded by Gauss and Cauchy, and thenfurther developed by Riemann. Hankels aim was to give it a complete, rigorous, andscientific presentation (Hankel 1867, v). In the first part of his Theorie der complexenZahlensysteme, Hankel intended to give a rigorous treatment of complex number systems. Inthe second part he then meant to discuss the theory of functions of a complex variable, but thesecond part was never published (Monna 1973, 66).

The work is famous for having made Hermann Grassmanns approach known to the generalpublic (Monna 1973, 71). In it Hankel also discusses at length Hamiltons approach toquaternions, which Hankel claims to be rather unknown in Germany (1867, vi). Hankelswork thus may have been instrumental in introducing also Hamiltons work to a Germanaudience. Conceptually, Hankels work is particularly important for having presented a purelyformal, symbolic conception of mathematics. In it, logic and mathematics are conceived to beindependent of the content of the objects, such as pictures or quantities, or, according toHankel, at least so they can be (1867, 1).

3 With the exception of Okada 2013. Centrones discussion of definite manifolds comes very close to thepresent discussion. In distinction from her very detailed and elaborate discussion of Husserls central concepts,the present approach explains the connection to the equational proof reduction not discussed in those termsby Centrone, as well as Husserls indebtedness to Hankel in detail (Centrone 2010, 149192; 2011). See also daSilva 2013a; 2013b. For a detailed account of the development of Husserls early mathematics see (Ierna 2005,2006).4 These elementary individuals are what are called in the PA normal numbers [Normalzahlen] (Husserl 1970,261); i.e., the canonical elements of the domain.5 To be specific, in Chapter 3 of the sixth logical investigation.

3To introduce us to such a purely formal discipline, Hankel starts with an investigation of whathe calls the vulgar foundation of the concept of number that is tied to ordinary intuition(1867, vii). The combinations in it are either thetic or lytic. The thetic combinations areaddition, multiplication, and the power operation. The lytic combinations are the reverseoperations, such as subtraction and division. Hankel considers the combinations one afteranother and discusses their properties. For example, he shows that addition is governedby two main laws [Hauptgesetze]: associativity and commutativity (1867, 2).

Further, addition is a well-defined operation [eindeutige Operation], so that when one of thesummands changes, the result of the operation also changes. In the modern terms, Hankelexplains that addition is a function. According to him, these properties define the operationformally:

The properties of addition given here are sufficient for deriving all the furtherconsequences about the construction of sums, without a need to be remindedabout the real meaning of addition. In this respect, they build the system ofconditions that are necessary and sufficient for defining the operation formally.(1867, 2)6

Hankels discussion of addition captures his general approach: the aim is to capture the lawsthat govern the arithmetical operations so that the operations become defined by means oflaws such as associativity and commutativity. In other words, the calculation rules are givenby combining laws that govern the rules of operation. Multiplication is subsequently defined,according to Hankel, by means of addition, commutativity, associativity, distributativity, anda postulation that 1a = a (1867, 3).

The lytic combinations are defined by means of the thetic operations. For example, consider asummation x + b = c. In accordance with the properties of addition, x has a value, which canbe described as x = b c. When b > c, the value of x is not among natural numbers, and isthus impossible [unmglich]. Such impossible results necessitate extending the numberconcept, in this case with negative numbers (1867, 5). Hankel distinguishes this sense ofimpossibility from the logical impossibility. Impossible numbers are not contradictory, butimpossible only in the sense that they cannot be represented intuitively (1867, 67). By meansof this distinction Hankel then distinguishes between purely formal numbers that are thosethat cannot be constructed by intuition, and actual numbers that represent actual magnitudesand their relationships. Between the two types of numbers, there are those numbers that canbe given a complete definition, but about which we do not know whether they can berepresented intuitively. They can be called potential numbers, insofar as they can becomeactual numbers, or else they may remain abstract, insofar as they can be only thought and notintuited. Or they can be simply called formal, in so far as they merely express certain formalrelationships. The division between the transcendent and actual numbers is thus not rigorousbut fluctuating (1867, 78).

6 Die hier angegebenen Eigenschaften der Addition sind ausreichend, um aus ihnen alle weiteren Folgerungenber Summenbildung abzuleiten, ohne dass man sich jemals dabei der realen Bedeutung der Addition erinnernmsste. Sie bilden insofern das System der Bedingungen, welche nthig und ausreichend sind, um dieOperation formal zu definiren (1867, 2). The translations are by the authors, unless otherwise indicatedand/or the text refers directly to the English translation. If the original text includes s p a c e d words they arehere written with italics.

4The consideration of the lytic or the reverse operations and the need to extend the numberdomain beyond what can be intuited leads Hankel eventually to conclude that generalarithmetic is purely formal, completely removed from all intuition:

The presupposition for setting up a general arithmetic is thus a purelyintellectual mathematics, removed from all intuition, a pure theory of forms,which has for its objects not the combination of quantities or their images, thenumbers, but intellectual objects, thought-objects, which could correspond toactual objects or relations, even though such a correspondence is not necessary.(1867, 910)7

Furthermore, the purely formal mathematics is not a generalization of the usual mathematics.According to Hankel, it is an entirely new discipline, which does not prove the rules ofordinary arithmetic, but exemplifies them [Regeln nicht bewiesen, sondern nurexemplificirt, werden] (1867, 12). To be sure, Hankel does not intend to restrict such a purelyformal account of mathematics to the realm of ordinary arithmetic, but intends it to cover theentire Organismus der Mathematik (1867, 12).

The generality of the formal mathematics is achieved by the reliance on the principle ofpermanence. Hankels formulation of the principle is as follows:

The entailed introductory basic law can be described as the principle ofpermanence of the formal laws, and it consists in the following: When twoforms expressed in general signs of the arithmetica universalis are equal, theymust remain equal also when the signs cease to describe simple magnitudes andthe operations receive some other content. (1867, 11)8

Here, by two forms expressed in general signs of the arithmetica universalis, Hankel seemsto refer to laws such as associativity and commutativity; e.g.,

a + b = b + a,

which hold also when we move from positive whole numbers to negative, irrational andeventually real numbers. Hankel also remarks that these laws define the operations so thatthey cannot yield any contradictions. To ensure this, he claims, the laws have to beindependent from each other (1867, 1011).The principle of permanence thus requires the algebraic laws to be permanent. Thearithmetical operations are permanent insofar as they are defined by algebraic laws. Theprinciple allows the use of impossible numbers (i.e., negative, imaginary, and generallycomplex numbers) in the purely formal domain and the operations defined therein. As Hankellater puts it:

7 Die Bedingung zur Aufstellung einer allgemeinen Arithmetik ist daher eine von aller Anschauung losgelste,rein intellectuelle Mathematik, eine reine Formenlehre, in welcher nicht Quanta oder ihre Bilder, die Zahlenverknpft werden, sondern intellectuelle Objecte, Gedankendinge, denen actuelle Objecte oder Relationensolcher entsprechen knnen, aber nicht mssen. (1867, 910)8 Der hierin enthaltene hodegetische Grundsatz kann als das Princip der Permanenz der formalen Gesetzebezeichnet werden und besteht darin: Wenn zwei in allgemeinen Zeichen der arithmetica universalisausgedrckte Formen einander gleich sind, so sollen sie einander auch gleich bleiben, wenn die Zeichenaufhren, einfache Grssen zu bezeichnen, und daher auch die Operationen einen irgend welchen anderenInhalt bekommen (1867, 11).

5By means of this principle, it was possible to replace the initial concept ofnumber as an expression of actual relationships of objects and their operations with a more general concept of formal operations that move only in thedomain of logical thought, as well as (replace it) with numbers that result fromthe mental linking of objects, which are first thought without content, withpurely abstract forms of the combining thought about the discrete. (Hankel1867, 47)9

According to Hankel, this principle is used everywhere to define the necessary and sufficientlaws for arithmetical operations. In order not to be too restricted, however, the commutativityof the operations is not absolutely presupposed (1867, 11). With such an admission Hankelincludes in his treatment Hamiltons theory of quaternions, in which the multiplication is notcommunitative (cf. Hankel 1867, 105).

The principle of permanence, to which Hankel above refers to as introductory orpedagogical [hodegetische], is a metaphysical principle tied to all our intuitions (Husserllater disagreed with this, see below). Therefore, formal mathematics is a fundamentaldiscipline that the abstract combinations of magnitudes, as much as those of spatial intuition,as well as the mechanical magnitudes, all fall under (1867, 1213). Hankels description ofhis intentions suggests that he has in mind something like a theory of rings or different kindsof algebraic structures that have two binary operations (+ and ) that satisfy a set of axiomsthat define the properties of the binary operations.

However, for Hankel this is not enough. Following Hermann Grassmanns Lehrbuch derArithmetik Hankel also constructs a number system from simple elements, and proves thatthey have desired properties such as associativity. Hankel thus wants to work on both fronts atthe same time: he sets up a formal theory of arithmetic and then generates an actual numbersystem and shows that it has the desired properties.

Motivation for such an approach can be deciphered from the following explanation:

We will then in general proceed as follows: After a domain of objects is given,we will next ask whether there is an applicable operation that has the propertiesof addition. There is no exact method to answer this question, rather it has to besolved with a creative invention; the principle of permanence serves us wellhere. If, however, an operation with the properties of addition is found, then onewill further ask whether there is a corresponding operation for multiplication; toanswer this question one will use the principles of multiplication in more or lessspecial cases, and so one arrives at an actual definition of multiplication. Whenthis has taken place, then there remains a task to show in a synthetic way that, infact, all fundamental principles of operation that have been taught in thisparagraph are fulfilled, and that only then can one strictly consider the operationto be multiplication. The principle of permanence is everywhere only in a

9 Mittels dieses Principes war es mglich, an Stelle des zunchst liegenden Begriffs einer Zahl, als desAusdrucks der actuellen Relationen von Objecten und deren Operationen, den allgemeineren Begriff formaler,bloss im Gebiete des logischen Denkens sich bewegender Operationen und aus der mentalen Verknpfung vonObjecten hervorgehender Zahlen zu setzen, welche zunchst inhaltsleer, rein die abstracten Formen deszu[s]ammenfassenden Denkens des Unstetigen sind. (1867, 47).

6methodological sense of this word analytic; a series of arbitrary presuppositionsneeds to be made, which it does not prove but only guides. That thesepresuppositions are arbitrary is made sufficiently clear by the fact that differentactual operations can be given that all satisfy the general formal rules. (1867,3334)10

In other words, even though Hankel claims above that the general laws give the necessary andsufficient conditions for defining an operation formally, this is not enough for him in general.In addition, we have to exemplify the operations in question and then show that they satisfytheir formal definition. While the formal definition ensures the permanence of the operations,it will not guarantee the existence of the operation. Hankels explanation suggests that sincethe formal definition can be exemplified in many different domains, it does not define theoperation uniquely. Hankel thus distinguishes between the system of algebraic laws and itsmodels.

Hankels project is thus a combination of an analytic definition of the operations by means ofalgebraic laws and a synthetic generation of the system of calculation. For example, by meansof a basic law such as

a + (b + 1) = (a + b) +1,

Hankel defines the calculation rule of addition.11 However, he then also verifies the inverse,namely that the operation of addition implies the basic law as a theorem, provable bymathematical induction. Hankels synthesis (i.e., the generation of the system of calculation)initially aims at an actual construction of the system. However, the actual execution of thecalculations is soon impossible for the results of operations may be impossible objects andthus they cannot be intuited. Hankel thus suggests building a number system that contains allpossible results of operations carried out on certain elements, signified by new signs. Theoperations are then applied to new signs so that yet new signs will be added to the system.The process will be continued until no further signs can thus be reached:

Such a system can only be created by starting from certain elements, the units,and combining them in every possible way through certain operations andsignifying the results of these operations with new signs. These new signs will

10 Wir werden dabei im Allgemeinen so verfahren: Wenn ein Gebiet von Objecten gegeben ist, so wird manzunchst fragen, ob es eine auf sie anwendbare Operation gebe, welcher die Eigenschaften der Additionzukommen. Eine stricte Methode zur Beantwortung dieser Frage gibt es allerdings nicht, vielmehr wird dieproductive Erfindung sie lsen mssen; das Princip der Permanenz leistet dabei gute Dienste. Ist aber eineOperation gefunden, welche die Eigenschaften der Addition hat, so wird man weiter fragen, ob es eineentsprechende Multiplication gebe; um dies zu beantworten, wird man die Principien der Multiplicationwiederum in mehr oder minder speciellen Fllen benutzen, und so dazu gelangen die Multiplication actuell zudefiniren. Ist dies geschehen, so bleibt es dann noch brig, in synthetischem Gange nachzuweisen, dass in derThat alle fundamentalen Principien der Operation, wie sie in diesem . gelehrt sind, erfllt sind, und erst dannwird man die Operation streng genommen als Multiplication bezeichnen knnen. Das Princip der Permanenz isthiebei berall nur ein im methodologischen Sinne dieses Wortes, analytisches; es mssen stets eine Reihe vonarbitrren Annahmen gemacht werden, welche es nicht beweist, sondern nur leitet. Dass jene Annahmenarbitrr sind, geht gengend daraus hervor, dass verschiedene actuelle Operationen gegeben werden knnen,welche smmtlich den allgemeinen formalen Regeln gengen. (1867, 3334)11 Hankel makes a shortcut here. The basic law should be a + (b + c) = (a + b) + c for which this is an instance c =1.

7then be further combined according to the aforementioned rules, thus originatingin new signs, etc. If one continues so far that one does not reach any new signs,that is, the results of the new operations can always be expressed with theexisting ones, then one calls the constructed series of signs a closed system ordomain, whose order I name according to the number of units that have beenused in its construction. (1867, 35)12

Hankel calls the signs of such a system numbers, thus identifying the numbers with the signsfor them. He then defines a formal number as follows:

A number is the expression of certain formal relationships between arbitraryobjects; a number system represents a systematically ordered series of suchrelationships or combinations, whose essence determines the character of thenumber system. (Hankel 1867, 36)13

Actual numbers can be subsumed under these formal numbers.

To build a number system of individual elements, he first defines (setzt)

1 + 1 = 2, 2 + 1 = 3, 3 + 1 = 4,

He calls such numbers absolute, and 1 the numerical unit [numerische Einheit]. Todefine addition he considers an application of the associative principle:

A + (B + 1) = (A + B) + 1 (1)

which holds for any sum.14 It is important to note that Hankel uses the associative laws toobtain the primitive recursive calculation rules, whereas in the usual modern primitiverecursive addition, they are just rules; i.e., rules governed by the primitive recursion format.The proof of (1) thus seems redundant from the modern point of view.

Hankel does not use the term reduction, but that is essentially what he does with thecalculation rules induced from the algebraic laws; namely the algebraic equation (1) is takenas the calculation rule from the right hand side to the left hand side. He reduces more complexformations into a definition in his sequence. Hankel thus seems to try to show that the results

12 Ein solches System kann nur geschaffen werden, indem man von gewissen Elementen, den Einheitenausgeht, diese auf alle mgliche Weise durch gewisse Operationen verbindet und die Resultate dieserOperationen mit neuen Zeichen signirt. Diese neue Zeichen werden dann nach vorstehenden Regeln wiederumzu verknpfen sein und zu neuen Zeichen Veranlassung geben u.s.f. Fhrt man so fort, bis man zu neuenZeichen nicht mehr gelangt, also die Resultate der neuen Operationen durch die schon vorhandenen jedesmalausgedrckt werden knnen, so nennt man die gebildete Zeichenreiche ein abgeschlossenes System oderGebiet, dessen Ordnung ich nach der Zahl von Einheiten benenne, welche seiner Bildung verwandt wordensind. (1867, 35).13 Eine Zahl ist der Ausdruck gewisser formaler Beziehungen beliebiger Objecte zu einander; ein Zahlensystemstellt eine systematisch geordnete Reihe solcher Beziehungen oder Verknpfungen dar, deren Wesen denCharacter des Zahlensystems ausmacht. (Hankel 1967, 36).14 Hankel proves it for the case where B = 1, by showing that the outcome is a number in the sequenceaccording to its definition. He then shows that this can be done also when B = 2. If B = 2 then (1) holds; i.e., thatA + (2 + 1) = (A + 2) + 1 or A + 3 = (A + 2) + 1, where (A + 2) and thus also (A + 2) + 1, or A + 3, are also numbersof the sequence. (1867, 37).

8of addition operation belong to the natural number sequence. Hankel then points out that onecan prove a similar result for any two numbers without any intuition and purely mechanicallyusing the property that each number is defined as the sum of its predecessor and the unit(Hankel 1867, 37). He then proves the associativity in a very modern fashion bymathematical induction. He then uses associativity to prove commutativity.15

Multiplication, according to Hankel, is an operation for which

A 1 = A,

which shows the significance of the unit used for the construction of the number system [denhierin liegt die Bedeutung der zur Bildung des Zahlensystems verwendeten Einheit] (1867,38). In general multiplication can be defined recursively [recurrirend definirt werden] bymeans of the equation

A (B + 1) = AB + A,

which can be taken as a special case of the distributive law. He then proves the general formof the distributive law, and then associativity and also commutativity of multiplicationreferring to Grassmanns Lehrbuch der Arithmetik (Hankel 1867, 3840).

Finally he defines what he calls the modulus of the operations:

A + 0 = Aand

A 0 = 0.

With these equations he thus defines a system of natural numbers, as opposed to the earlier,formal definition that created a more general, algebraic domain. Hankel next moves on toconsider equations that extend the domain to include the negative whole numbers, rationalnumbers and complex numbers.

Hankel thus proves that the calculation rules exemplify the formal conditions set up earlier bymeans of formal laws. Above he claimed that whereas the formal arithmetic does not consistin a generalization of ordinary arithmetic; it is a completely new science, whose rules are notproven, but only exemplified by the latter [ordinary arithmetic] [Regeln nicht bewiesen,sondern nur exemplificirt, werden] (1867, 12). He further claims that in the ordinaryarithmetic the definitions of operations determine their Regeln [presumably arithmeticallaws], and in the formal mathematics the Regeln define the sense of the operations (ibid.). Inaccordance to this claim, Hankel thus first gives a formal, analytic definition of the operationsby means of algebraic laws and then constructs desired systems and shows that in them the

15 Hankels proof of associativity of addition is as follows. Assuming the equation A + (B + ) = (A + B) + (2) andusing (1) twice, we get: A + {B + ( + 1)} = A + {(B + ) + 1} = {A + (B + )} + 1. According to (2), this is {(A + B) + }+ 1, and according to (1) {A + B} + ( + 1); thus, A + {B + ( + 1)} = {A + B} + ( + 1). So if (2) holds, it also holdswhen is replaced by + 1. Where (2) holds for = 1, according to (1), it holds for any case, so (2) holdsaccording to the known general ways of demonstration [Schlussweise]. He then derives commutativity ofaddition from associativity. Let 1 + A = A + 1 (3); then according to (1) and (3), 1 + (a + 1) = (1 + A) + 1 = (A + 1) +1. Since (3) holds for A = 1, and by replacing A with A + 1, (3) holds in general. Let A + B = B + A (4); then A + (B +1) = (A + B) + 1 = (B + A) + 1 = B + (A + 1) = B + (1 + A). According to (1), (4), (1), (3) and also (2), A + (B + 1) = (B +1) + A.

9same laws are exemplified. This also explains why Hankel writes in the introduction that thedistinction between transcendent and actual numbers, mediated by the formal numbers, is notstark, but fluctuating, which will be in the following clearly presented [wird sich himFolgenden klar genug herausstellen] (1867, 8).

Hankel claims to be everywhere guided by the principle of permanence, which enables themove from the realm of actual numbers to more general formal concepts:

By means of this principle, it was possible to replace the initial concept ofnumber as an expression of actual relationships of objects and their operations with a more general concept of formal operations that move only in thedomain of logical thought, as well as (replace it) with numbers that result fromthe mental linking of objects, which are first thought without content, withpurely abstract forms of the combining thought about the discrete. (Hankel1867, 47).16

The principle of permanence is also crucial for reaching the pure theory of the complexnumbers (ibid., 100). The complex number system eventually defined by Hankel iscategorical in being maximal:

A higher complex number system, whose formal operations of calculating fixedby the conditions given in 28, and whose products of unity are particular linearfunctions of the original unities, and in which no product can vanish without oneof its factors being zero, contains within itself a contradiction and cannot exist.(Hankel 1867, 107, translation from Detlefsen 2005, 286)17

Detlefsen calls the theorem Hankels theorem and puts it in more modern terms to say thatthe field C is, up to isomorphism, the only commutative field obtainable by adding roots ofpolynomials with complex coefficients to the field C. The theorem thus claims that theprinciple of permanence does not apply to number concepts beyond complex numbers(Detlefsen 2005, 286). Nevertheless, Hankels principle of permanence preserves the validityof arithmetical laws as purely formal conditions for the operations.

2. Husserls clarification of Hankels principle and the origins of syntactic reduction

In his Formal and Transcendental Logic (1929), Husserl claims that with his attempts toformulate a concept of a definite manifold, he tried to clarify Hankels principle of thepermanence of formal laws. He writes,

16 Mittels dieses Principes war es mglich, an Stelle des zunchst liegenden Begriffs einer Zahl, als desAusdrucks der actuellen Relationen von Objecten und deren Operationen, den allgemeineren Begriff formaler,bloss im Gebiete des logischen Denkens sich bewegender Operationen und aus der mentalen Verknpfung vonObjecten hervorgehender Zahlen zu setzen, welche zunchst inhaltsleer, rein die abstracten Formen deszuammenfassenden [sic.] Denkens des Unstetigen sind (Hankel 1867, 47).17 Ein hheres complexes Zahlensystem, dessen formale Rechnungsoperationen nach den Bedingungen des 28 bestimmt sind, und dessen Einheitsproducte ins Besondere lineare Functionen der ursprnglichen Einheitensind, und in welchem kein Product verschwinden kann, ohne dass einer seiner Factoren Null wrde, enthlt insich einen Widerspruch und kann nicht existiren. (Hankel 1867, 107).

10

The concept of the definite manifold served me originally to a different purpose,namely to clarify the logical sense of the computational transition through theimaginary and, in connexion with that, to bring out the sound core ofHermann Hankels renowned, but logically unsubstantiated and unclear,principle of the permanence of formal laws. My questions were: Under whatconditions can one operate freely, in a formally defined deductive system (aformally defined multiplicity), with concepts that, according to the definitionof the system, are imaginary? When can one be sure that deductions that involvesuch an operating, but yield propositions free from the imaginary, are indeedcorrect that is to say, correct consequences of the defining forms of axioms?How far does the possibility extend of enlarging a multiplicity, a well-defined deductive system, to make a new one that contains the old one as apart? (Husserl 1969, 97)18

Husserl notoriously discussed the notion of a definite manifold in his two Gttingen lecturesin 1901. However, it seems that Husserl engaged in clarifying Hankels principle already inhis Philosophie der Arithmetik (1891). Indeed, it seems Husserl was troubled by Hankelsprinciple already when writing his Habilitationsschrift, at least by July 1887. To habilitate atthe University of Halle, Husserl had prepared to defend publicly eight claims on July 1, 1887.Two of the theses relate to Hankel, namely thesis VI Hankels Principle of the permanenceof formal laws in arithmetic is neither a metaphysical nor an introductory[hodegetisches] principle and thesis VII [l]ogical justification of the use of irrational andimaginary numbers in all mathematical domains has not been demonstrated up to now(Husserl 2003, 357). Clearly Husserl was not satisfied by Hankels principle of permanence,but as we will see, despite his critical attitude, Husserl eventually adopts what he above callsthe sound core of the principle.

Husserls Habilitationsschrift On the Concept of Number: Psychological Analyses (1887),19

and subsequently the first chapters of Husserls first properly published book Philosophy ofArithmetic, Psychological and Logical Investigations (1891) (hereafter PA), representHusserls initial approach applying Brentanian methodology to Weierstrassian problems.Husserl met the limits of this approach already within the PA, which makes the PAparticularly difficult to assess, as Husserl changes his approach in the middle of the work. Inthe secondary literature this has given rise to a variety of stories about the stages Husserl

18 Der Begriff der definiten Mannigfaltigkeit diente mir ursprnglich zu einem anderen Zwecke, nmlich zurKlrung des logischen Sinnes des rechnerischen Durchgangs durch Imaginres und im Zusammenhang damitzur Herausstellung des gesunden Kernes des vielgerhmten, aber logisch unbegrndeten und unklaren H.Hankelschen Prinzips der Permanenz der formalen Gesetze. Meine Fragen waren: an welchen Bedingungenhngt die Mglichkeit, in einem formal definierten deduktiven System (in einer formal definiertenMannigfaltigkeit) mit Begriffen frei zu operieren, die gem seiner Definition imaginr sind? Wann kan mansicher sein, da Deduktionen, die bei solchem Operieren von dem Imaginren freie Stze liefern, in der Tatrichtig sind, das ist korrekte Konsequenzen der definierenden Axiomenformen? Wie weit reicht dieMglichkeit, eine -Mannigfaltigkeit, ein wohldefiniertes deduktives System zu erweitern in ein neues, das dasalte als Teil enthlt? (Husserl 1974, 85).19 On the Concept of Number: Psychological Analysis has been published as Husserls Habilitationsschrift inHalle a. S.: Heynemannsche Buchdruckerey (F. Beyer), 1887; second edition in Husserliana XII, (Husserl 1970,289338), and translated by Dallas Willard in (Husserl 2003, 305356). Carlo Ierna has argued that the originalHabilitationsschrift went significantly farther than On the Concept of Number (see Ierna 2005, 2330 ).

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went through before 1891 (e.g., Miller 1982; Willard 1984; and in most historical detail Ierna2005). The end of the Philosophy of Arithmetic is an outcome of considerable struggle onHusserls part. Several manuscripts from around 188991 give clues about the nature ofHusserls problems and the kinds of solutions he considered. As will become clear, thesemanuscripts seem to revolve around Hankels principle of permanence in one way or another.

The most important indication of Husserls struggle is a letter he sent to his mentor CarlStumpf in February 1890. In it, Husserl writes about a shift in his views as follows:

The results which I have obtained are striking enough. The opinion by which Iwas still guided in the elaboration of my Habilitationsschrift, to the effect thatthe concept of cardinal number forms the foundation of general arithmetic, soonproved to be false. (The analysis of the ordinal number already made this clearto me.) By no clever devices, by no inauthentic representing, can one derivenegative, rational, irrational, and the various sorts of complex numbers from theconcept of the cardinal number. The same is true of the ordinal concepts, of theconcepts of magnitude, and so on. And these concepts themselves are not logicalparticularizations of the cardinal concept. The fact is that general arithmetic(including analysis, theory of functions, etc.) finds application to the cardinals(in number theory), as well as to the ordinals, to continuous quantities, and ton-dimensional Ausgedehntheiten (time, space, color, force [Kraftkontinua], etc.)(1994, 13).20

What he thus realizes is that general arithmetic is not tied to any conceptual basis (See alsoDie wahren Theorien written around 188990). Husserl was obviously struggling with theproblem of extending the number field. Husserl had realized that it should be possible toconsider calculation entirely devoid of its conceptual basis. However, to Husserl, this meansthat instead of extending the number domain, one only extends the arithmetical technique.

No negative, imaginary, fractional numbers can be proved to be generated asstages of development or combination forms of number concepts. The numberconcept permits no extensions; what will be extended and allows extension isonly the arithmetical technique. (Husserl 1983, 4243).21

As mentioned earlier (in 1887), Husserl had prepared to defend the thesis Hankels Principleof the permanence of formal laws in arithmetic is neither a metaphysical nor an introductory

20 Die Resultate, zu denen ich gelangt bin, sind merkwrdig genug. Die Meinung, von der ich noch bei derAusarbeitung der Habilitationsschrift geleitet wurde, da der Anzahlbegriff das Fundament der allgemeinenArithmetik bilde, erwies sich bald als falsch. (Schon die Analyse der Ordnungszahl fhrte mich darauf.) Durchkeinerlei Kunststcke, durch kein uneigentliches Vorstellen kann man die negativen, rationalen, irrationalenund die mannigfachen komplexen Zahlen aus dem Anzahlbegriff herleiten. Dasselbe gilt vomOrdnungszahlbegriffe, dasselbe vom Grenbegriffe usw. Und diese Begriffe selbst sind keine logischenSpezialisierungen der Anzahlbegriffe. Tatsache ist, da die allgemeine Arithmetik (inkl. Analysis,Funktionentheorie etc.) Anwendung findet auf Anzahlen (Zahlentheorie), desgleichen auf Ord[inal]z[ahlen],auf stetige Quantitten, auf n-fache Ausgedehntheiten (Zeit, Raum, Farbe, kraftkontin[uum] etc.).(1983, 245)21 Keine negativen, imaginren, gebrochenen Zahlen lassen sich nachweisen, die als Entwicklungsstufen oderKombinationsformen der Anzahlbegriffe entstehen knnten. Der Anzahlbegriff lt keinerlei Erweiterungen zu;was erweitert wird und Erweiterung zult, ist nur die arithmetische Technik (Husserl 1983, 4243).

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[hodegetisches] principle. Obviously we do not know what would have been the moredetailed content of Husserls thesis. However, we are now in a position to guess: Husserlprobably thought of Hankel as a formalist who relied only on symbolic calculations, and thatthe principle of permanence is hence not a metaphysical principle. On the other hand, itcertainly was not a pedagogical or introductory principle either owing to its central role inHankels approach. In any case, for such a principle the most important property is that ityields correct results. Consideration of this takes Husserl to emphasize the need to prove theconsistency of the extended calculations. Husserls own formulation of the principle ofpermanence in Die wahren Theorien (around 18891890) is the following:

Principle of Permanence: When, by virtue of the peculiarity of the concepts thealgorithm is founded on, certain algorithmic operations cannot be executed infull generality without arriving at contradictory constructions of concepts, thealgorithm is extended after detaching it from the conceptual foundation andregarding it as a conventional one. This is done by experimentally adding everysuch construction to the algorithmic domain and by adding the convention thatthe old laws remain valid also for the objects that they symbolize (signs). Theold laws should thus be executable in full generality. In any case, one must thenprove the consistency of the extended algorithm. (Husserl 1983, 33)22

In other words, according to Husserls formulation, the principle allows extending thealgorithm so that one can use the operations by stipulating that the old laws (presumablyassociativity, commutativity and so forth) remain valid. Contrary to Hankel, Husserlemphasizes that the extended algorithm [erweiterten Algorithmus] has to be shown to beconsistent. While Hankel first gives a formal, analytic definition of the operations by meansof algebraic laws and then constructs desired systems by means of mathematical induction,Husserl investigates the correctness of algorithms by means of term reductions of anequational proof system. Indeed, Husserl claims that the algorithms produce correct resultswhen every equation for relations between the signs can be, using the definitions of the signs,reduced to an identity (Husserl 1983, 35). He writes,

We have so far considered only pure algorithms, if you will, pure game systems.The importance of this consideration lies in the possibility to control scientificdomains through algorithms. And the usefulness of it is immediately clear:When a scientific domain is controlled through a restricted algorithm so thatbetween the two there is a thoroughly investigated and characterized fullparallelism, then, for the scientific purposes of this domain, the limitedalgorithm can be substituted salva veritate with an extended and unlimitedalgorithm, and it will as such in even higher measure control that domain,through the greater completeness of its mechanism. All the concepts andconventions of the extension lack a conceptual grounding that can be verified inthe scientific domain, they are meaningless and, insofar as they provide a formal

22 Princip der Permanez: Wenn vermge der Besonderheit der einen Algorithmus begrndenden Begriffegewisse der algorithmischen Operationen nicht in voller Allgemeinheit ausfhrbar sind, ohne da man aufwiderstreitende Begriffsbildungen kommt, so erweitert man den Algorithmus, nachdem man ihm von derbegrifflichen Grundlage losgelst und als einen konventionellen gedacht hat, dadurch, da man jede solcheBildung versuchsweise dem algorithmischen Gebiete adjungiert und die Konvention hinzufgt, da auch fr diedurch sie symbolisierten Gegenstnde (Zeichen) die alten Gesetze gltig bleiben, also die alten Gesetzeunbeschrnkt ausfhrbar sein sollen. Man mu dann in jedem Fall die Konsistenz des erweiterten Algorithmusnachweisen. (1983, 33).

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solution to previously unsolvable problems, indications of contradictoryconcepts. The calculation with the help of the technique of the extended domainmust, under all circumstances, produce correct results, because according to ourgeneral investigation, every sign equivalence, which contains and presupposesonly signs and conventions of the more restricted domain, is in this sense correct[im Sinne dieser eine richtige], thus reducible to one identity. Everysignequivalence of this sort is, however, the necessary expression of a correctjudgment for the considered scientific domain. (1983, 3435)23

Husserl is troubled by the algorithm as a pure game system, whose signs do not correspond toanything conceptual in the scientific domain. Elsewhere Husserl attributes such a view toHelmholtz and complains that Helmholtz is not able to explain how such games can haveworking applications (Husserl 1994, 14). However, Husserl admits that the calculation ishelpful if it yields correct results. To ensure this, Husserl claims above, every equation of ofsigns should be reducible to an identity. This idea seems to have been inspired by the resultsof many mathematicians, and in particular Grassmann. Husserl continues his above mentionedletter to Stumpf:

Arithmeticians who now with hesitation, and now decisively explainnumbers as signs, allow themselves to be guided merely by the study ofalgebraic formalisms. These mathematicians (Grassmann above all) havebrought to Evidence the possibility of deriving the whole algorithm of arithmeticand analysis by means of mere sign definitions (1 + 1 = 2, 2 + 1 = 3, etc.; a a =a2, ( a)2 = a, etc, all understood in the sense of mere equivalences of signs uponpaper). This occasioned their identification of number and sign. (Husserl 1994,14)24

Husserl then proceeds to say that this is quite acceptable: there should be nothing to wonder atin the fact that a system of signs and operations with signs can replace a system of conceptsand operations with judgments, where the two systems run rigorously parallel (ibid. our

23 Wir haben bisher nur reine Algorithmen, wenn man will, reine Spielsysteme betrachtet. Die Bedeutungdieser Betrachtungen liegt in der Mglichkeit wissenschaftliche Gebiete durch Algorithmen zu beherrschen.Und der Nutzen daran ist sofort klar: Wenn ein wissenschaftliches Gebiet durch einen begrenzten Algorithmusdadurch beherrscht wird, da zwischen beiden jener ausfhrlich untersuchte und charakterisierte volleParallelismus besteht, dann kann fr die wissenschaftlichen Zwecke dieses Gebiets dem begrenzten auchirgendein erweiterter und unbegrenzter Algorithmus salva veritate substituiert werden, und er wird als solcherin noch hherem Mae jenes Gebiet beherrschen, durch die grere Vollkommenheit seines Mechanismus.Alle Begriffe und Konventionen der Erweiterung ermangeln einer in dem wissenschaftlichen Gebietnachweisbaren begrifflichen Fundierung, sie sind sinnlos und, sofern sie sich formell als Lsungen frherunlsbarer Aufgaben prsentieren, Anzeigen widersprechender Begriffe. Das rechnen mit Hilfe der Technik deserweiterten Gebiets mu unter allen Umstnden zu richtigen Resultaten fhren, weil nach unserer allgemeinenUntersuchung jede Zeichen quivalenz, welche nur die Zeichen und Konventionen des engeren Gebietseinschliet und voraussetzt, im Sinne dieser eine richtige, also auf eine Identitt reduzierbar ist. JedeZeichenquivalenz dieser Art ist aber der notwendige Ausdruck eines richtigen Urteils fr das betrachtetewissenschaffliche Gebiet (1983, 34-35).24 Die Arithmetiker, welche bald schwankend, bald entschieden die Zahlen als Zeichen erklrten, lieen sichblo leiten vom Studium des algebr[aischen] Formalismus. Die Mglichkeit, durch bloe Zeichendefinitionen(1+1=2, 2+1=3, etc.; aa=a2, ( a)2=a etc., alles verstanden im Sinne bloer quivalenzen von Zeichen auf demPapiere) den ganzen Algorithmus der Arithmetik und Analysis herzuleiten, haben jene Mathematiker, vor allemGrassmann, zur Evidenz gebracht. Dies veranlate sie, Zahl und Zeichen zu identifizieren, (Husserl 1983,245246).

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italics).25 Contrary to Hankel and others, Husserl does not want to identify the number and thesign. Indeed, already in his Habilitationsschrift, Husserl criticized Helmholtz for conceivingnumbers as mere signs, as Stumpf pointed out in his summary of Husserls work (Gerlach andSepp 1994, 173). Husserls problem is thus how to establish the existence of the conceptualsystem parallel to the symbolic one:

Upon this fact syllogistic and the logical calculus rest, as does the much moresubtle system of ordinary arithmetic. To calculate is not to think (infer), butrather is to derive sign from sign systematically, in conformity with set rules.The sign which is obtained at the end is interpreted, and thus the desired thoughtresults. So we have a method of deriving a judgment from given judgments, notby means of actual inferring, but rather though a rule-governed procedure inwhich, from the arithmetical symbols for the data, the symbol for the result isobtained in a mechanical fashion.(1994, 14)26

In a letter to Stumpf he accordingly sketches his view of arithmetica universalis at the time.

The sign system of arithmetica universalis divides into a certain sequence oflevels, comparable to that of a system of concentric circles. The lowest level (theinnermost circle) is occupied by the signs 1, 2 = 1 + 1, 3 = 2 + 1, etc.; the nextby fractional signs; and so on. The signs of the lowest level, and they only, areindependent. Those of the higher levels are formally dependent upon those ofthe lower levels, and ultimately upon the lowest. To each domain there belongrules of calculation (formal laws). Those of the higher domains are dependentupon those of the lower, and include them formally. The rules of calculation are,then, so formed that each equation (in whatever way it may be set up, i.e., bymeans of whatever domain levels) is satisfied as an identity with reference to thesigns and the domain of rules which it actually involves. Thus, for example, ifan equation between whole numbers (or their signs) is proven with the aid of allthe parts of the arithmetica universalis, it nonetheless incorporates into itselfnothing of this route of proof: it is an identity with reference to the signs whichit contains. It is an identity in the sense of the signs and sign rules of the lowestlevel. (1994, 16)27

25 Da ein System von Zeichen und Zeichenoperationen ein System von Begriffen und Urteilsoperationen mitdenselben zu ersetzen vermag, wenn beide Systeme streng parallel laufen, hat nichts Verwunderliches(Husserl 1983, 246).26 Darauf beruhen Syllogistik und Logikkalkl, darauf das viel feinere System der gemeinen Rechenkunst.Rechnen ist nicht Denken (Schlieen), sondern systematisches Herleiten von Zeichen aus Zeichen, nach festenRegeln. Das result[ierende] Zeichen wird gedeutet und so resultiert der gewnschte Gedanke. Also eineMethode: ein Urteil aus gegebenen Urteilen herzuleiten nicht durch ein wirkliches Schlieensondern durch einregelmiges Verfahren, bei dem aus den arithmetischen Signaturen der Daten diejenige des Resultats reinmechanisch gewonnen ist. (Husserl 1983, 246)27 Das Zeichensystem der arithmetica universalis gliedert sich in eine gewisse Stufenfolge, vergleichbarderjenigen eines Systems konzentrischer Kreise. Die tiefste Stufe (den innersten Kreis) fllen die Zeichen 1,2=1+1, 3=2+1 usw., die nchste die Bruchzeichen usw. Die Zeichen der untersten Stufe und nur sie allein sindindependent; die der hheren sind von den der tiefern und schlielich den der untersten Stufe formalabhngig. Jedem Kreise kommen Rechenregeln (formale Gesetze) zu, die des hhern sind abhngig von dendes tiefern, schlieen sie formell ein. Die Rechenregeln sind nun so formiert, da jede Gleichung, auf welchemWege, d.h. mittelst welcher Stufenkreise sie auch gewonnen sei, identisch erfllt ist mit Beziehung auf dieZeichen und das Regelgebiet, die sie wirklich impliziert. Ist also z.B. eine Gleichung zw[ischen] ganzen Zahlen(sc. Zeichen) bewiesen durch Hilfsmittel aller Teile der arithmetica universalis, so haftet ihr doch von diesem

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Husserl claims that this view is formally consistent with Helmholtzs view, but isconceptually importantly different for Husserl did not want to adopt Helmholtzs signtheory (ibid., 16). Arithmetik der Reihen dated to around 18891891 seems to be Husserlsattempt at constructing some such system (Husserl 1983, 154214). In any case, according toHusserl, arithmetica universalis is a segment of formal logic, which he defines as a symbolictechnique. In Die wahren Theorien Husserl was confident about his approach. He claimsthat the principle of permanence is an undoubtedly correct principle that removes allDunkelheiten der Arithmetik and fully justifies arithmetical procedures (1983, 35).However, around the same time Husserl also points out that the solution based on calculation,even though interesting and useful, does not yield any insight into the essence of arithmetic(1983, 2223). To this effect, at the end of Die wahren Theorien, Husserl demands furtherinvestigation in the conceptual realms. He thus concludes his manuscript, with a formulationof the next task:

The clarity that we brought to authentic arithmetic illuminates our way forwarda little. If the algorithm is applicable also outside the number domain, then it cantake place only due to the fact that it either works because the conceptualdomain, even though different from the number concepts, has a formallyanalogous structure with the number concepts, or since it simply involvesconcrete numbers. The conceptual domains in question must be researched andthe true state of affairs set forth.Our task will thus be to study the different domains of application of thenumbers and the relations between the numbers, and carefully examine theconcepts which one has generally wanted to use as the basis for arithmetic.(Husserl 1983, 4344)28

The question with which Husserl proceeds is thus to examine different kinds of domains tosee whether they share the same structure with the number theory. Most of the othermanuscripts from 18891891 accordingly discuss different kinds of domains.

Having written the above mentioned manuscripts (the letter to Stumpf, Die wahren Theorien),Husserl finished the PA. At this point Husserls clarification of Hankels principle meansthat he introduces a distinction between the signs and the objects signified and that hespecifies the condition of consistency of the extended system to rest on the reducibility toidentities.

3. PA and the symbolic reduction

Beweiswege nichts an: sie ist identisch mit Beziehung auf die Zeichen, die sie enthlt. Sie ist eine Identitt indem Sinne der Zeichen und Zeichenregeln der untersten Stufe (1983, 247248).28 Die Klarheit, welche wir in die eigentliche Anzahlenarithmetik brachten, leuchtet unserem weiteren Wegnun schon ein Stck voraus. Lt der Algorithmus auch auer dem Anzahlgebiet noch Anwendungen zu, dannkann es nur dadurch geschehen, da er entweder dies leistet, weil die bezglichen Anwendungsbegriffe,obwohl von den Zahlbegriffen verschieden, doch ein formell analoges System bilden wie diese; oder weil essich einfach um konkrete Anzahlen handelt. Es mssen die betreffenden Begriffsgebiete demgemdurchforscht und das wahre Sachverhltnis dargelegt werden.Unsere Aufgabe wird also sein, die verschiedenen Anwendungsgebiete der Zahlen und Zahlverhltnisse zuuntersuchen und diejenigen Begriffe, welche man sonst der Arithmetik hat zugrunde legen wollen, sorgfltig zustudieren. (Husserl 1983, 4344)

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In the last chapter of Philosophy of Arithmetic, Husserl moves beyond the view advocated inan earlier chapter of the same book. In the final chapter, Husserl defines arithmetic as thesystematic treatment of the techniques of calculation (1970, 257; 2003, 272). In accordancewith his above distinction between the sign and the signified, Husserl distinguishes betweentwo kinds of construction: one takes place through the formation of concepts, the other bymeans of mechanical-exterior sign formation (ibid., 271272).29 Accordingly Husserldistinguishes between two kinds of method of derivation: as a conceptual operation, or else asa sense perceptible operation that, utilizing the system of number signs, derives sign fromsign according to fixed rules, only claiming the final result as the designation of a certainconcept, the one sought (ibid., 272).30 Whereas the former method is highly abstract,limited, and even with the most extensive practice, laborious, the latter is concrete, sense-perceptible, all-inclusive, and it is, already with a modest degree of practice, convenient towork with (ibid., 272).31 Husserls sympathies rest first on the symbolic methods, and hewrites: The method of sensible signs is, therefore, the logical method of arithmetic. (2003,272).32 Indeed, the main aim of what follows is to guarantee the reliability of the method ofsigns, that is, to give foundations for the symbolic calculations.

Husserl then defines calculation as any rule-governed mode of derivation of signs from signswithin any algorithmic sign-system according to the laws or better: the conventions forcombination, separation, and transformation peculiar to that system (2003, 273).33 Bylaws in this definition, Husserl refers to calculation rules rather than algebraic basic laws.According to the definition, calculation consists of three separable stages: Conversion of theinitial thoughts into signs calculation and conversion of the resulting signs back intothoughts (2003, 273).34 To ensure the reliability of the method of signs, the method has tobe given a logical foundation (2003, 274).35 This means working out the structure of thesystem of concepts:

Only the systematic combination of the concepts and their interrelationships,which underlie the calculation, can account for the fact that the correspondingdesignations interlock to form a coherently developed system, and that therebywe have certainty that to any derivation of signs and sign-relations from givenones, which is valid in the sense prescribed by the rules for the symbolism, theremust correspond a derivation of concepts and conceptual relations fromconcepts given, valid in the sense that thoughts are. Accordingly, for thegrounding of the calculational methods in arithmetic we will also have to goback to the number concepts and to their forms of combination. (2003, 274)36

29 mechanisch-uerlichen Zeichenbildung (1970, 257)30 welche aufgrund des Zahlzeichensystems nach festen Regeln Zeichen aus Zeichen herleitet, um erst dasResultat als die Bezeichnung eines gewissen, des gesuchten Begriffes zu reklamieren (Husserl 1970, 257).31 Die Methode der Begriffe ist hochst abstrakt, beschrnkt und selbst bei grter bung mhsam; die derZeichen konkret-sinnlich, allumfassend und schon bei miger bung bequem zu handhaben (1970, 257).32 Die Methode der sinnlichen Zeichen ist also die logische Methode der Arithmetik (1970, 257)33 jede geregelte Art der Herleitung von Zeichen aus Zeichen innerhalb irgendeines algorithmischenZeichensystems nach den diesem System eigentmlichen Gesetzen - oder besser: Konventionen - derVerknpfung, Sonderung und Umsetzung (1970, 258).34 Umsetzung der Ausgangsgedanken in Zeichen - Rechnung-, Umsetzung der resultierenden Zeichen inGedanken (1970, 258).35 logisch zu fundieren (1970, 259).36 Nur an der systematischen Verknpfung der ihr zugrunde liegenden Begriffe und deren Beziehungen kannes ja liegen, da die korrespondierenden Bezeichnungen sich zu einem konsequent gebildeten System

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The number system for Husserl consists first of all of a series of normative numbers, normalnumbers [Normalzahlen] fixed standards, as it were which all other number forms arereferred back to (2003, 276). The first basic task of arithmetic is to identify the types offorms (e.g., as an additive, multiplicative or a more complex type), and then to findcalculation rules for each type with which to reduce the given number form into the normalnumbers. He writes,

Accordingly there arises, as the first basic task of Arithmetic, to separate allconceivable symbolic modes of formation of numbers into their distinct types,and to discover for each type the methods that are reliable and as simple aspossible for carrying out that reduction. (2003, 277)37

The systematic numbers are systematically formed number series. The systematic formationof the series could be given by, for example, the series 1, 1 + 1, 1 + 1 + 1, , or perhapssomewhat more elegantly by a successor function, but at this point Husserl chooses to capturethem with a decimal (base-10) system or representation. The reason for this is that his aim isto show that the operation on concepts is strictly parallel to the external method of signs andthat thereby the latter is reliable (cf. ibid., 282283). The first sense of reduction Husserldiscusses is the one in which any complex symbolic formation (i.e., any closed term,Husserls example is 18 + 48) is to be reduced to one of the systematic numbers (66 inHusserls example). Husserl understands the arithmetical operations to be methods forcarrying out the reduction (cf. ibid., 284).

For each systematic number and the operation on it, there corresponds a univocal sign, and aparallel reduction on signs that gives the same result:

Thereby is proven, all steps taken one by one, the rigorously univocalcorrespondence between the method of addition by thinking in concepts and themethod of addition by calculating in signs; and we can place our complete trustin the latter. (2003, 282283)38

Husserl then discusses similarly multiplication, subtraction and division. He thus obtains thelogical foundation for mechanical calculation, the logical soundness of which is guaranteedby means of the rigorous parallelism between the systematic of the numbers and numberrelations, on the one hand, and that of the number signs and relations of number signs

zusammenschlieen und dabei die Sicherheit besteht, da jeder im Sinne der Zeichen regeln folgerichtigenAbleitung von Zeichen und Zeichenbeziehungen aus den gegebenen eine im Sinne der Gedanken folgerichtigeAbleitung von Begriffen und Begriffsbeziehungen aus den hier gegebenen entsprechen msse. Demgemwerden wir auch zur Begrndung der arithmetischen Rechenmethoden zurckgehen mssen auf dieZahlbegriffe und deren Verknpfungsformen. (1970, 259).37 Demgem erwchst als die erste Grundaufgabe der Arithmetik, alle erdenklichen symbolischenBildungsweisen von Zahlen in ihre verschiedenen Typen zu sondern und fr einen jeden sichere und mglichsteinfache Methoden jener Reduktion aufzufinden (1970, 262).38 Damit ist allen einzelnen Schritten nach die streng eindeutige Korrespondenz zwischen der in Begriffendenkenden und der in Zeichen rechnenden Additionsmethode nachgewiesen, und wir drfen der letzterenvolles Vertrauen schenken. (1970, 267).

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(equivalences of symbols), on the other (2003, 287).39 In this way, the progression along thesequence of concepts there corresponds, in rigorous parallelism, a progression along thesequence of names. And the system of names taken by itself is every bit as coherent as that ofthe concepts (2003, 265).40

As a consequence of this parallelism, Husserl thinks, each arithmetical operation can beviewed either as a conceptual operation or as an operation on signs. He thus explains thataddition, for example, can be carried out

by a determinate process of computation, or perhaps by referring to the table oftruths for addition. But the quasi-additions of the signs corresponding to themare also univocal as to their result, whether carried out through the parallelexternal process of enumeration, or through referring to the table of signequivalences for addition. Thereby is proven, all steps taken one by one, therigorously univocal correspondence between the method of addition by thinkingin concepts and the method of addition by calculating in signs; and we canplace our complete trust in the latter. (2003, 282283)41

Thus, it seems, the systematic numbers defined above can be viewed either as numberconcepts or as signs for them. Likewise, the systematic reduction can be regarded as aconceptual reduction or else as a symbolic operation on signs. Husserl refers to suchreductions as calculational/formal as opposed to conceptual in a footnote in which Husserlpoints out that

the negative, imaginary, fractional, and irrational numbers have not yetbeen introduced. Through them there occurs in our number domain acalculational/formal although by no means a conceptual reduction of theinverse number forms to the direct ones. (2003, 298)42

Husserl thus thinks that the conceptual calculation cannot be extended to cover the imaginarynumbers. However, one can use them in the calculations on the basis of the system of signs,which is purely calculational/formal.

39 dessen logische Triftigkeit durch den strengen Parallelismus zwischen der Systematik der Zahlen undZahlbeziehungen auf der einen und derjenigen der Zahlzeichen und Zahlzeichenbeziehungen(Zeichenquivalenzen) auf der anderen Zeite gewhrleistet ist (1970, 271).40 Dem Fortschritt entlang der Reihe der Begriffe entspricht in strengem Parallelismus ein Fortschritt entlangder Reihe der Namen, und das System der Namen ist in sich genau so konsequent als das der Begriffe (1970,250).41 die Elementaradditionen sind ihrem Resultat nach eindeutig; sie werden ausgefhrt, sei es durch einenbestimmten Zhlungsproze, sei es durch den Hinweis auf die Tabelle von Wahrheiten der Eins und Eins. Aberauch die ihnen korrespondierenden Quasi-Additionen der Zeichen sind ihrem Resultat nach eindeutig, sei esdurch den parallellaufenden uerlichen Zhlungsproze oder durch den Hinweis auf die Tabelle vonZeichenquivalenzen der Eins und Eins. Damit ist allen einzelnen Schritten nach die streng eindeutigeKorrespondenz zwischen der in Begriffen denkenden und der in Zeichen rechnenden Additionsmethodenachgewiesen, und wir drfen der letzteren volles Vertrauen schenken (1970, 267).42 die negativen, imaginren, gebrochenen und irrationalen Zahlen noch nicht eingefhrt sind. Durch siefindet auf unserem Anzahlengebiete eine rechnerisch-formelle - obschon keineswegs begriffliche - Reduktionder inversen Zahlformen auf die direkten statt. (1970, 282n).

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In sum, Husserl thus presents systematic numbers as decimal numbers. These systematicnumbers form simultaneously a sign system and a conceptual number system. The reductionsof the complex expressions to the canonical definitions can likewise be regarded asconceptual or symbolic reductions, or the latter as calculational/formal in terms that Husserluses. In the above quote, Husserl also distinguishes between a reduction, which could becalled direct and in which we reduce from a complex form to a canonical natural (or what hecalls normal) number/symbol, and an indirect reduction, in which the reduction takes usfrom the inverse number forms to the direct ones.

The first sense of the reduction is the usual reduction as computation/calculation.43 It is areduction inside direct terms or complexities. It seems that Husserl considers only the so-called innermost reduction, which avoids so-called confluence arguments. By the inner-most reduction strategy, we mean the reduction strategy, in which an innermost position ofpossible reduction is taken first if there is more than one position in the underlying term, forwhich reductions are possible. We denote a term or a complex term, say t with free (oralgebraic) variable occurrences, x and y, as t[x,y]. Consider two reduction rules,

t[x,y]s[x,y], p[x,y]r[x,y],

and consider a term or a complex term t[n,p[m,k]]. Here, t[n,p[m,k]] stands for the termobtained from t[x,y] by substituting term n with x and substituting term p[m,k] with y. Thereare then two possible reducible subterms (called redeces) of this term, namely, the termt[n,p[m,k]] itself and the subterm p[m,k]. By applying the first rule for the first redex, thewhole term is reduced to s[n,p[m,k]], while by applying the second rule for the second redex,the whole term is reduced to t[n,r[m,k]].

The outermost reduction strategy means that if there are alternative possible redeces (orpossibly reducible subterms) with the underlying reduction rules, the innermost redex ischosen fo reduction. In the above example, t[n,p[m,k]] is reduced to t[n,r[m,k]], instead of tos[n,p[m,k]].44

Husserl considers the reduction rules for the decimal number system or generalized X-expansion-based number system. For him, the reduction rules are set to reduce any (closed)term or complex term to a normal number (of a decimal or X-based number system), anirreducible term of the underlying reduction rules. Now, we denote such normal numbers, n,

43 From the point of view of equational arithmetic based on equational logic, the equation 18 + 48 = 66 hasmany different types of proofs including the one in which 66 is traced back to 18 + 48. Instead, the reduction-based proof starts with a complex (compounded term), which reduces to a normal (irreducible) term, 66 in thiscase. In the term-rewrite reduction proof-strategy, for a query, say 18 + 48 = 31 + 35, the left-hand term 18 +48 and the right-hand term 31 + 35 are independently reduced to normal terms, say, n and m, and when n andm are identically the same term, the whole chain of reductions of both sides is understood as the term-rewritereduction proof of the original query (which is now a proved theorem).44 An example of the inner reduction is a reduction in which the term (18 + 4) x 3 is reduced first with the rulesof addition to 22 x 3 and eventually to a normal term 66. The outer reduction would be one in which there isreduction with rules of multiplication to (18 + 4) + (18 + 4) + (18 + 4) and eventually to the normal term 66. Thediscussion of why different reduction strategies employed to a single term always result in a same normal termis called the confluence issue. Husserl seems to avoid this issue by considering only the fixed innermostreduction strategy.

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m, l, etc. Then, keeping this innermost reduction strategy for the application of rules, after thefirst step of reduction resulting in t[n,r[m,k]], where we now assume n, m, and k are normalnumbers (i.e., irreducible), one does not move to apply the first rule (although the first rulecould be applied for reduction to s, [n,r[m,k]]), but keeps reducing the part r[m,k] repeatedlyso that this part eventually terminates with a normal number, say h. Then, the whole term ist[n,h]. Now, since both n and h are irreducible, there is no inner redex and the first reductionrule can be applied to t[n,h], which results in s[n,h], and this term is eventually reduced to anormal term.

Towards the end of his discussion, Husserl discusses the reduction from the indirect to thedirect. At this point Husserl sketches two groups of problems that call for general arithmetic:

The first has to do with an indirect determination of number by means of anequivalent complex of given conjunctions of known numbers, and the task hereconsists in reducing to a minimum the difficulties and complications involved inthe actual execution. The second has to do with a number determination whichis indirect to a yet much higher degree, by means of a complex of operationsthat are only incompletely given, inasmuch as the unknown number itselffunctions as one term in the conjunctions. (2003, 298299)45

Here, what he called the indirect way of the number determination is the equational definition(with existential quantifier in the modern logical sense); in this way, the inverse/reversefunction can be defined. It is incompletely given in the sense that the determination is givenonly when the equation is solvable. Husserl remarks that in this way the inverse (i.e., the lyticnumbers in Hankels terms), negative, fractional, and irrational numbers can be defined, justas Hankel did.

Husserl was well aware of the development of set theory and the existence of infinitecardinals. This enlargement of the conceptual domain should have forced Husserl to give upthe strict parallelism between the system of signs and system of concepts. Admittedly, he doesnot consider the X-basis number symbol representation after Philosophy of Arithmetic, butstill in 1905 he insists on strict parallelism between some system of signs and system ofconcepts (Daubert 2004, 295).

However, Husserl criticizes the reliance on the mere calculation technique on moreconceptual grounds, as can be read already in his review of Schrders Vorlesungen ber dieAlgebra der Logik in 1891. As Bernhard Rang remarks in his introduction to the Husserlianavolume XXII where Husserls review is reprinted, Husserl held that Schrders algorithmlacks a theoretical basis, the theory of the algorithm (Husserl 1979, xv). Husserl seeks for atheory of deduction and holds that calculation is not deduction (1994, 55). One could perhapssee in Husserls criticism increasing appreciation for Hankels idea that there should be anunderlying formal theory that secures the calculations with operations. For Husserl, more thanfor Hankel, however, the role of such formal theories is to provide a conceptual foundation for

45 Die erste geht auf eine indirekte Zahlbestimmung durch einen quivalenten Komplex gegebenerVerknpfungen von bekannten Zahlen, und die Aufgabe besteht darin, die wirkliche Ausfhrung auf einMinimum von Schwierigkeiten und Verwicklungen zu reduzieren; die zweite geht auf eine in noch viel hheremMae indirekte Zahlbestimmung durch einen Komplex nur unvollkommen gegebener Operationen, sofern dieunbekannte Zahl selbst als das eine Fundament der Verknpfungen fungiert, (1970, 283).

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the calculations. Ultimately, the idea develops into Husserls critique of psychologism inProlegomena.

In any case, Husserl was, at least initially, happy with what he achieved in Philosophy ofArithmetic:

All of the difficulties and doubts encountered in Chapter X with regard to theunderstanding of the calculational operations and the arithmetic which treats ofthem, we may already at this point regard as resolved. With the modified sensewhich the operations acquire in the domain of symbolic number formations, ithas become fully intelligible why scientifically elaborated methods for carryingout the operations are here required, which seemed pointless there. (2003, 287288.)46

However, he never came to a final resolution about the general arithmetic. He meant todiscuss it in the second volume of PA, which never appeared. In his Foreword to thePhilosophy of Arithmetic, Husserl explains briefly what he plans to discuss in the secondvolume of the book: the first part will consist of a logical investigation of the arithmeticalalgorithm, and justification of utilizing, for example, negative and imaginary numbers incalculations. In its second part, Husserl will discuss the fact that identically the samealgorithm, the same arithmetica universalis, governs a series of conceptual domains that haveto be carefully distinguished (2003, 7). Husserl refers to it also in his Selbstanzeige forPhilosophy of Arithmetic: The higher level symbolic methods, quite different in nature,which constitute the essence of the general arithmetic of cardinal numbers, are reserved to thesecond volume, where that arithmetic will appear as one member of a whole class ofarithmetic, unified in virtue of the homogeneous character of identically the same algorithm(2003, 300).

4. Double LectureAbout a decade later, in 1901, Felix Klein and David Hilbert invited Husserl to attend themeetings of the Gttingen Mathematical Society. Husserl complied with the invitation andsoon gave at the Society two lectures known as the Double Lecture [Doppelvortrag], entitledDer Durchgang durch das Unmgliche und die Vollstndigkeit eines Axiomensystems. Theexact composition of the lectures is not known. There exist several fragments that address theissues discussed in the lectures. On the basis of these fragments, it can be conjectured whatHusserl said in the lectures. Yet, Elizabeth Schuhmann and Karl Schuhmann, the editors ofthe most recent edition for the text of the lectures, assume that the greater part of themanuscript must have been lost (Schuhmann and Schuhmann 2001, 88).

Husserl begins the lectures with a discussion of the formal nature of mathematics as thescience of theoretical systems in general. It is a study of theory forms defined by a

46 Alle Schwierigkeiten und Zweifel, die wir im X. Kap in dem Verstndnis der Rechnungsoperationen und dersie behandelnden Arithmetik fanden, drfen wir schon jetzt als gelst ansehen. Bei dem vernderten Sinn,welchen die Operationen auf dem Gebiet der symbolischen Zahlbildungen erlangen, ist es vllig begreiflichgeworden, warum hier wissenschaftlich ausgebildete Methoden der Operationsvollziehung ntig sind, die dortgegenstandslos schienen (1970, 272).

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totality of formal axioms, i.e., by a limited number of purely formal basicpropositions, mutually consistent and independent of one another. Systematicdeduction supplies in a purely logical manner, i.e., purely according to theprinciple of contradiction, the dependent propositions, and therewith the entiretotality of propositions that belong to the theory defined. But the object domainis defined through the axioms in the sense that it is delimited as a certain sphereof objects in general, irrespective of whether real or Ideal, for which basicpropositions of such and such forms hold true. An object domain thus definedwe call a determinate, but formally defined, manifold (2003, 410).47

The theory forms are defined by the axioms, and the axioms also define the domain of objectsthat satisfy the theory in question. These theory forms can be set in relation to one another:

they can be systematically classified; one can broaden or narrow such forms;one can bring a certain previously given theory form into systematicinterconnection with other forms of determinately defined classes and drawimportant conclusions concerning their interrelationship (2003, 410).48

According to Husserl, the theory forms are abstracted from concrete theories of differentsciences; e.g., Euclidean geometry. An example of an abstracted theory form is a three-dimensional Euclidean manifold, which in turn is one among many interconnected manifoldsof varying degrees of curvature (1970, 431; 2003, 410). Such formal mathematics aims to bethe instrument of concrete mathematical discoveries. More importantly, formal mathematicsprovides the calculations with a theoretical basis, as what Husserl demanded from Schrder.Husserl now seems to have adopted the Hankelian approach that there are formal algebras thatare exemplified by more concrete theories. In contrast to the thinking of Hankel, the formalalgebras define formal domains and thus are not understood in a mere symbolic sense as howHankel seemed to understand them.

However, according to Husserl, the exact relationship between formal mathematics and itsemployment in substantive mathematics or in particular domains of knowledge remainsproblematic. In particular, the problem Husserl intends to solve in these lectures is stated asfollows:

Problem: Suppose a domain of objects given in which, through the peculiarnature of the objects, forms of combination and relationship are determined thatare expressed in a certain axiom system A. On the basis of this system, and thuson the basis of the particular nature of the objects, certain forms of combinationhave no signification for reality, i.e., they are absurd forms of combination. Withwhat justification can the absurd be assimilated into calculation with what

47 durch einen Inbegriff von formalen Axiomen, d.h. durch eine begrenzte Zahl rein formaler, miteinanderkonsistenter und voneinander independenter Grundstze. Die systematische Deduktion liefert rein logisch, d.i.rein nach dem Prinzip vom Widerspruch, die abhngigen Stze und damit den Gesamtinbegriff von Stzen, diezu der definierten Theorie gehren. Das Objektgebiet aber ist durch die Axiome in dem Sinn definiert, da esumgrenzt ist als irgendeine Sphre von Objekten berhaupt, gleichgltig ob realen oder idealen, fr welcheGrundstze solcher und solcher Formen gelten. Ein so definiertes Objekt-Gebiet nennen wir eine bestimmte,aber formal definierte Mannigfaltigkeit (1970, 431).48 sie lassen sich systematisch klassifizieren, man kann solche Formen erweitern und verengern, man kannirgendeine vorgegebene Form in systematischen Zusammenhang mit anderen Formen bestimmt definierterKlassen bringen und ber ihr Verhltnis wichtige Schlsse ziehen (1970, 431)

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justification, therefore, can the absurd be utilized in deductive thinking as if itwere meaningful? How is it to be explained that one can operate with the absurdaccording to rules, and that, if the absurd is then eliminated from thepropositions, the propositions obtained are correct? (Husserl 2003, 412)49

Husserls task is thus to tackle, once again, Hankels principle of permanence. In Hankelspicture, a theory form would justify the use of concrete operations even when the results of anoperation are impossible (e.g., negative or rational). Given that Husserl thinks above that theproblem is in the exact relationship between formal mathematics and its concrete applications,Husserl seems to be thinking about how the theory forms justify the calculations in theconcrete theories, even when the calculations use signs that have no representation in theconcrete theory. Using the example of Euclidean geometry that Husserl frequently mentions,the question would be, for example, how the three-dimensional Euclidean manifold justifiesthe calculations with imaginary elements that are carried out in the Euclidean geometry.

Husserl starts by discussing several approaches to solve the problem, the fifth one being theone he defends. The solution explicitly uses the principle of permanence. It, he claims,enables the shift from the specific domain to the formal domain in which one can freelyoperate with imaginary numbers. We rise, according to the principle of permanence, abovethe particular domain, pass over into the sphere of the formal, and there can freely operatewith 1 (Schuhmann and Schuhmann 2001, 96, Husserl 2004, 417418). Immediatelybefore this claim Husserl discussed the ideas of manifolds that are defined by means of aseries of stipulations, such as the associative law. According to Husserl, the real domains ofthis same form are governed by what he calls an algorithm of the manifold. In Husserlsproposed solution the idea is not to discuss real domains but the formal domain that generatesa formal algorithm:

Now the algorithm of the formal operation is indeed broader than the algorithmof the narrower operations, which alone are really presupposed in a givenconceptual domain. But if the formal arithmetic is internally consistent, then thebroader operating can exhibit no contradiction with the narrower. Thereforewhat I have formally deduced in such a way that it contains only signs of thenarrower domain must also be true for the narrower domain. (Husserl 2003,418)50

Husserls task is thus to show that the formal arithmetic is internally consistent.

He starts by defining a formal domain called D obtained through abstraction from the realdomain. The formal domain is then extended so that when new axioms are added, the old ones

49 Problem: Es sei ein Gebiet von Objekten gegeben, in welchem durch die besondere Natur der ObjekteVerknpfungs- und Beziehungsformen bestimmt sind, die sich in einem gewisse Axiomensystem Aaussprechen. Aufgrund dieses Systems, also aufgrund der besonderen natur der Objekte, haben gewisseVerknpfungsformen kein reale Bedeutung, d.h. es sind widersinnige Verknpfungsformen. Mit welchem Rechtdarf das Widersinnige rechnerisch verwertet, mit welchem Rechte kann also das Widersinnige im deduktivenDenken verwendet werden, als ob es Einstimmiges wre? (Schuhmann and Schuhmann 2001, 93).50 Nun ist der Algorithmus der formalen Operation zwar weiter als der Algorithmus der engeren Operationen,die allein realen unterlegt sind in einem gegebenen Begriffsgebiet. Ist aber die formale Arithmetik in sichkonsistent, so kann das erweiterte Operieren keinen Widerspruch zeigen mit dem engeren; also was ich formalso abgeleitet habe, da es nur Zeichen des engeren Gebietes enthlt, mu fr das engere Gebiet auch wahrsein (1970, 438)

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are conserved. In other words, the extended system is a conservative extension of the originalsystem. The axiom system thus defines a domain that Husserl also calls a manifold. Themanifold, as a totality of objects, is therefore defined by means of the relational formalproperties expressed in the axioms (2003, 420). According to Husserl, any axiom systemincludes existence axioms. For example, if there is a combination +, then it means that givena and b,

x (x = a + b),

certain laws (e.g., association and commutativity) remain valid. (Husserl writes that for thiscombination such and such laws are valid (2003, 420).) These existence axioms can beunivocal or equivocal. The equivocation can be eliminated by the joint force of the axioms(2003, 421).

Husserl then continues [i]n the domain belong all univocally determining object forms out ofwhich constructively arise, through singularization, univocally defined totalities of objects(2003, 421). The objects of the formal domain are thus object forms. The proper,univocally produced objects have to be generated constructively. A special case of the formaldomain is one where the entire domain is constructed from a finite number of objects of thedomain (2003, 422). Like Hankel, Husserl considers general axiom systems that createoperation systems in which the number sequence is generated. Accordingly, in one fragmentdated to 1901, he writes Operations: That literally means to generate (2003, 486).51

In the fragment that Dallas Willard, the translator and the editor of Philosophy of Arithmetic,has placed next, Husserl discusses the definiteness of manifolds. He discusses three cases, ofwhich the first two phrases are as follows.

a) A definite manifold is ruled out by the inessential closure axiom.b) Can a purely algebraic manifold, which defines no individual of the domain

whatever can such a manifold have the character of a definite manifolds? (2003,422)52

The first case establishes the definiteness by adding to it a closure axiom such as Hilbertscompleteness axiom. The second case considers a possibility of definiteness, when theconstruction of the manifold is not taken into consideration. According to Husserl, if only oneoperation is defined, the associative and commutative laws form a definite combination. Ifmore operations are added, and the same laws remain valid, and

any sentence which contains only the +, and regardless of how I have derivedit, is decided as to truth and falsehood. Likewise, the well-known laws ofaddition and multiplication are definite in this sense, under presupposition of thesaid supplementary axiom. (2003, 4223)53

51 Operationen: Das sagt dem Wortlaut nach Erzeugungen. (1970, 482)52 a) Definite Mannigfaltigkeit durch das unwesentliche Schlieungsaxiom wird ausgeschlossen. b) Kann einerein algebraische Mannigfaltigkeit, welche keinerlei Individuen des Gebietes definiert, kann eine solche denCharakter einer definiten Mannigfaltigkeit haben? (Schuhmann and Schuhmann 2001, 99-100)53 jeder Satz, der nur das + enthlt und den ich wie immer abgeleitet habe, ist entschieden in Wahrheit undFalschheit. Ebenso sind die bekannten Gesetze der Adition und Multiplikation definit in diesem Sinn unterVoraussetzung des genannten Zusatzaxioms (Schuhmann and Schuhmann 2001, 100). (Schuhmann andSchuhmann 2001, 100).

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By the said supplementary axiom, Husserl presumably refers to the inessential closureaxiom mentioned under a). Husserl discusses elsewhere (or in another fragment; namely,VIII Abhandlung dated likewise to 1901) closure axioms, saying, for example, that In theformal definition of a manifold possibilities remain open under all circumstances, if noclosure axioms of any kind are added (2003, 475).54 Hilbert had, according to Husserlsnotes, lectured on his completeness axiom at the Gttingen Mathematical Society onNovember 5, 1901, whereas Husserls lectures took place on November 26 and December 10,1901 (Gutzmer 1902, 147). Hence, Husserl was aware of the completeness axiom and he hadto consider it. However, he does not seem too happy with Hilberts solution. In anotherfragment Husserl refers to the closure axioms as producing spurious completeness since wecan make any axiom system quasi-complete with such a closure axiom. He then explains hisnext solution that refers to the operational formations of natural numbers and the executabilityof the operations so that the identity a = a is produced (2003, 429), to which we will now turn.

The third possibility that Husserl discusses under c) takes up the operation systems (2003,423; Schuhmann and Schuhmann 2001, 100). There are two subcases. In the first, not everygenerally defined and existing operational result belongs in the sphere of the operationallyproducible and distinguished individuals. Husserl points out that this would be the case if theordering axioms were absent from arithmetic (2003, 424).55

The other is the consideration of the mathematical system for which he also uses the termconstructible system [konstruierbar Mannigfaltigkeit] (1970, 452; 2003, 433). Suchsystems are those in which the objects can be determined only through operations andoperationally characterizing concepts (2003, 423)56, and such objects can be determinedoperationally so that no further individual that is similarly determined can be added (2003,424). According to Husserl, the mathematical system is one in which

any individual existing on the basis of the axioms admits of an operationaldetermination and must belong within the sphere of specific operational results(which are obtained on the basis of a certain finite number of objects, whetheroriginally assumed as given in the definition of the manifold, or whether to bearbitrarily selected and given).(2003, 424)57

Such a system would be defined recursively in Hankels (and others) fashion by generatingthe number system from a finite number of objects so that it has certain properties, where thecomplex expressions are reducible to the canonical definitions, namely the simplestirreducible expressions in the number series. Such a constructible reduction system formationis called a constructor-based rewrite system, in the modern terms of rewriting theory, wherethe irreducible or normal terms are p