SPACE.

Summary. 1. Introduction. 2. Cosmologists and theologians. 3. The age of Newton. 4. The mathematical revo lution of the 19th century. a. Non-Euclid ian geometry. b. Riemann’s conception of physical space. c. Geometries and groups. 5. Relativistic kinematics and Minkowski spacetime.

1. Introduction

Any normal speaker of a classical or modern European lan guage—and, presumably, of other languages too—is able to discern in his en vironment certain properties and rela-tions we call ‘spatial’, such as direction and dis tance, containment and contiguity, size and shape. This ability alone neither presupposes nor yields the physico-mathematical idea of space. The lat ter does, indeed, bestow systematic unity on our use of spatial predicates. But the experi ences motivating such use do not imply or even suggest some of the more salient features of that idea, viz. (α) that space is the seat of all bodies and the site of all intervening spans; (β) that it is—at least no tionally—distinct from the things in space; and (γ) that the aggregate of all spatial lo cations and relations is an instance—a “model”—of a definite math ematical structure (identified before Einstein with so-called Euclidean 3-space, in which, characteristi cally, different volumes can have the same shape). Clas sical Greek did not have a word for such an idea; and the Latin word ‘spatium’ employed by Newton to convey it had for the Romans an es sential connotation of betweenness, which is still alive in the modern descen dants of Latin, but is inimical to the notion of all-en compassing space. The said idea of space has belonged to mathematical phys-ics throughout its history and must have pre ceded it, though only in the thoughts of a few unconven tional minds. Due to the big cultural success of physics it has become an indispens able component of the intellectual stock-in-trade of the more edu cated seg ment of western ized mankind. The history of this idea will be sketched in sections 2-5, from its foreshadowings among Greek cosmolo gists and me dieval theologians (2), through its mature articulation and dis cussion by the great philosopher-physicists of the 17th and 18th century (3), to its analysis and generalization by 19th-century mathematicians (4), and the premier application of their work to physics in Minkowski’s chronogeometric reading of Einstein’s relati vistic kinematics (5). On the subsequent use of the gener alized idea of space in rela tivistic cosmology: v. space-time models.

Commissioned by Enciclopedia Italiana. Published in Italian translation in Dizionariodelle Scienze Fisiche. Roma: Istituto della Enciclopedia Italiana. Vol. 5, pp. 427–433. The more technical passages and the scholarly notes were omitted in the translation.

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2. Cosmologists and theologians

In Hesiod’s Theogony (ca. 700 B.C.), “Chaos (Xãow) was born first of all”, followed by Earth, Love, Night, Day, the starry Heaven, etc.1 The noun xãow is etymologically related to xãskv—‘gape’, ‘yawn’—and xãsma—‘gaping mouth’, ‘chasm’—and may have meant the gap between heaven and earth.2 In the beginning, before these two came into being, Xãow stood alone, a bor der less opening, a jawless yawn, sheer betweenness with nothingto sepa rate. To us, this strongly suggests pure, empty space; but in Greek intellec tual history, Hesiod’s Xãow comes out rather as a forerunner of tÚ êpeiron, the unlimited indefinite stuff which the philosopher Anaximander (ca. 550 B.C.) conceived as the source of everything.3

Primordial stuff is also designated by a word connoting extension in Plato’s Timæus (ca. 360 B.C.). Here, “the whole heaven” and all things in side it are molded out of the pre-exist ing x≈ra by a divine craftsman who fash ions them after the changeless, perfect Platonic Forms. Now, x≈ra is a very common Greek word, meaning ‘room’, ‘place’, ‘land’, ‘country’ (the cog nate verb xvr°v means ‘to move, to go forward’, but also ‘to contain, to hold’). Plato says that the primary x≈ra is just what we have in mind when we claim—mistak enly—that all being is in some place (tÒpow) somewhere (pou).4 In his story, this “reservoir of becoming” seethes with confused physical powers (dÊna-meiw)—watery and fiery, earthy and airy—until the god puts order and balance into it.5 He does it by forming elementary par ticles of reg ular shapes germane to the powers of each of the four classical elements: cubes for stable, tightly packed earth, icosahedra for smoothly flowing water, tetrahedra for stinging fire, octahedra for air. (The availabil ity of still one more regular shape, the dodecahedron, does not raise the question of a “miss-ing element”, for Plato simply assigns it to heaven it self). Significantly, the god does not knead the elementary particles out of the x≈ra, but con structs them geometrically by apposing regular polygons made from plane figures of two kinds, viz. right-angled tri-angles with their angles in the sim ple proportions 1:1:2 and 1:2:3.6 These triangles are thus the true elements of Plato’s cosmos, which, however, are confined to the faces of the elemen tary particles.

A boundless expanse, deliberately purged of even the suggestion of ma te riality, is afforded by tÚ kenÒn, the void surrounding the atoms of Leu cippus (ca. 440 B.C.) and Democritus (v.: atom). Cornford7 hailed these thinkers as the inventors of space (with all three features α, β, γ, noted in 1). In his opinion, tÚ kenÒn was not only intended to provide room for the in cessant motion of atoms, but also to be a suitable domain for the construc tions of geome try. There is, however, a big difference between the space of modern physics and the atom ists’ void, namely, that bodies do not fill the latter but move through it. The atomists counter the Eleatic denial of plu ral ity and change by adding nothingness (tÚ mhd°n) to being (tÚ ˆn, tÚ d°n), but these complementary realities are, of course, mutually exclusive.8 The void outside the atoms enables at once the multiplication

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and the mo tion of being, while at the same time securing—by its very alienness—the unity and invariability of each existent.

There is no evidence that the atomists had the needs of geometry in mind when they conceived tÚ kenÒn. As a matter of fact, in Antiquity working ge ometricians concentrated on figures, their properties and rela tions, and did not explicitly consider the structure of the domain in which the figures are imbedded. On the other hand, the demands or “postulates” stated by Euclid at the beginning of his Elements certainly entail the avail-ability of points standing farther apart than any assignable distance. By Postulate 3, one can construct a circle taking any point for its center and any distance for its ra dius. Hence, there is always a seg ment twice as long as any given segment σ, viz. the diameter of a circle with radius σ. By Postulate 5, two coplanar straights a and b which form with a third one c, angles α and β, on one side of c, such that α + β < π, must meet somewhere on that same side of c. Obvi ously, as π – α – β decreases to 0, the distance from c to the intersec tion of a and b increases beyond all bounds. Many propo sitions in Euclid, begin-ning with I.29, depend on Postulate 5, and the un boundedness of dis tances is al ready implicit in the proof of I.16. So, even though Euclid and his predeces sors possibly had no inkling of infinite Eu clidian space, this idea lurks in their mathematical practice. As Koyré9 rightly saw, it was bound to come out into the open when geometry was taken seriously as the “language” of physics by Galilei and his suc cessors.

Insight into the implications and physical significance of Greek geometry may have been hindered by Aristotle (384-322 B.C.), the most articulate and influential of Greek cosmolo gists, who expressly denied that physics had any proper use for mathematics, and accounted for what we call spatial proper ties and relations in a way that is deeply at variance with mod ern physico-mathematical conceptions. The key notions of his account are body (s≈ma) and place (tÒpow). Bodies are either simple or compound. There are five simple bodies, viz. earth, water, air, fire and aether. The first four go into ev erything that lies under the moon, while aether is the peculiar stuff of the 55 concentric spheres which Aristotle judged necessary to explain the regular motions in the sky.10 In the Categories, Aristotle wrote that each part of a body holds (kat°xei) a part of the body’s place, thus suggesting that the place of a body is none other than the extension (diãsthma) within its boundary. In the Physics, however, he expressly rejects this, on the ground that if this ex tension were something on its own, differ ent from the body it self, it would in turn have a place, which, in turn, would hold another one, etc. Moreover, if the place of a body had length, width and depth it would also be a body, so there would be two bodies in the same place.11 Aristotle therefore defined the place of a body as the first immobile limit of its con tainer (tÚ toË peri°xontow p°raw ék¤neton pr«ton).12 It follows that the innermost limit of heaven is the place of the aether and, in a sense, of ev erything, while heaven itself is not in something else.13 This takes care of the question: What lies beyond heaven? “There is nothing outside the whole”.14 Inside heaven Aristotle distinguishes five

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concentric regions which consti tute the proper places of the sim ple bodies (ascending from the center of the world in the order in which they were listed above). By nature the sublunar bodies tend to rest at their respective proper place or to move directly to it if they happen to be somewhere else. The aether, on the other hand, cannot be dragged away from its place but moves in it eternally with uniform cir cular motion. For Aristotle the void is a physical absurdity:15 a body located in such an uniform environment would have no ground for moving in a par ticular direction or for resting at a particular point, and yet, not meeting any resistance, it would travel through it with infinite speed. He also rejects the possibility of an infinite body.16 On the other hand, bod ies are infinitely di visible. But, in his view, and contrary to what Zeno of Elea had argued ca. 450 B.C, this does not imply that a fast runner cannot catch a slow one or reach the goal.17 To achieve it the run ner does not have to traverse the racetrack’s infinitely many parts in a finite time, for, though the track is po tentially divisible in infinitum, it is not actually divided.18 Besides, for each piece of track, no matter how small, which the runner must cover, there is a corresponding time interval in which to cover it.19 This matching of distances with times in the analysis of motion was perhaps Aristotle’s greatest contri bution to modern scientific thought.

Aristotle’s influence in Antiquity was strong, but not decisive. Epicurus and the Ro-man poet Lucretius (†55 B.C.) upheld atomism and the infinite void. The Stoics saw the world as a plenum standing in a boundless vac uum. The Aristotelian Strato of Lampsacus (†269 B.C.) understood that a body filled its place, which he conceived as an empty extension. The Christian commentator John Philopon revived this notion in the 6th cent. A.D. “Place—he wrote—is not the limiting surface of the surrounding body… It is a certain interval, measur able in three dimensions, incorporeal in its very nature and different from the body con tained in it.”20

Western Christendom received the Aristotelian corpus from Muslim Spain ca. 1200. Aristotelian physics, favored by the Dominicans, was re sisted by the Franciscans, but even tually prevailed. In the 16th century it domi nated European universities and tinged the wording of Catholic dogma at the Council of Trent. In 1276, however, the Church had for bidden, under pain of excommunication, the teaching of 219 largely Aristotelian proposi tions, in cluding this one: “That God could not move the heaven in a straight line, the reason being that He would then leave a vacuum.”21 Though ortho doxy seemed thus to require a vacuum extramundanum, most medieval authors shunned it, for it raised a grave theological dilemma: it would either be an infinite being outside God or an attribute of God Himself. Among those who grappled with it, Bradwardine (†1349) came up with the notion of an “infinite imaginary vacuum” or “place” (situs),22 which presumably God might realize—inso far as it is required—should He will to move the world. Oresme (†1382) also refers to the extracosmic void as an imagined place (lieu),23 or space (espace).24 But at times he plainly as serts its actual exis tence: “Hors le ciel est

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une espace vide incorporelle d’autre manière que n’est quelconque espace pleine et cor-porelle… Cette espace…est infinie et indivisi ble et est l’immensité de Dieu et est Dieu même”.25 Uncannily antici pating Newton, he defines mo tion as a change in a body’s intrinsic stance “with re gard to the motionless imagined space”.26

3. The age of Newton

In the preface to the Principia (1687), Newton stated the task of mathe matical physics thus: “To find out the forces of nature by studying the phe nomena of motion, and then, from those forces, to infer the other phenom ena”.27 In Newton’s analysis, the motion or translation of a body from one place to another is preserved by the body’s intrinsic force (vis insita) until an extrinsic force (vis impressa) acting on the body causes it to change.28 The place (locus) of a body is the part of space (spatium) which it occupies (occupat) at the mo ment, not its position, nor the circumambient surface. Newton distinguishes ab-solute space, which “by nature, and without any re lation to something external, always remains similar and immobile”, from relative space, which is “a mobile measure or di-mension of the former, …defined by its position with respect to bodies”.29 The forces of nature must, of course, ac count for the continuation and change of true motions, i.e. the displacement of bodies in ab solute space. Newton seeks for observable dif ferences between true or absolute, and merely relative motion, and finds them—he believes—in some phenomena of rotation. The very idea of an ab solute space could not be suggested by such phenomena. But it is not en tirely Newton’s creation.

By 1600, ‘spatium’ and its modern renderings generally meant a 3-di mensional expanse which can either be taken up by a body placed in it, or else be empty. Theology prob-ably contributed to this meaning, for the very room which God must have at hand to move the world should that be His will can be readily viewed as providing, at creation, a site for the world it self. Space, in some such sense, received much attention from the great Ital ian philosophers of nature. Telesio (1509-1588) conceived it as the incorpo real, immobile, ho mogeneous receptacle of bodies, and declared that it was neither a substance nor an accident. According to Patrizi (1529-1597) space was created before anyuthing else. It is an infinite continuum containing in finitely many lines, surfaces and solids which are the subject of ge ometry. Space is not a worldly entity (de mundanis), and hence does not fall under the Aristotelian categories. For Bruno (1548-1660) space is “a continuous physical quantity with three dimensions, in which the magnitude of bodies is comprised”. It precedes bodies, stays motionless while they succeed each other, and will remain when all bodies are gone. It indif ferently receives ev erything, and neither acts nor suffers. It is neither matter nor form nor a composite of both, and must therefore be judged prae-ternatural.30 Like views were voiced by Campanella (1568-1639) and the French atomist Gassendi (1592-1655).

Descartes (1596-1650), who had little sympathy for the philosophers of nature, took

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dis tance from them by outrightly denying the possibility of a vacuum. On the other hand, by analyzing geometrical figures into sets of points meeting specific conditions and labelling each point with its oriented distances to three unbounded mutually orthogo-nal planes, he contributed decisively to the conversion of geometry into the science of space. He main tained, moreover, that the substance of bodies was none other than their ge ometrical exten sion. Thus, as Huygens (1629-1695) adroitly noted, body ac cording to Descartes was pretty much the same as the void.31 Yet Henry More (1614-1687) chided Descartes for equating body and extension, be cause the infinity of the latter must then lead to the divinization of the ma terial world.32 More and the other Platonists at Cam-bridge stressed the dis tinction be tween space as the field of God’s action, and the bodies He cre ates in it. Like the Italians, they maintained that space was neither a sub stance nor an attribute of a substance, in the familiar Aristotelian sense.

Newton grew in this tradition. “It may be expected—he wrote in the 1660’s—that I should define extension as substance or accident or else noth ing at all. But by no means, for it has its own manner of existence… It is not a substance, for it does not subsist absolutely by itself, but as an emanative effect of God and an affection of all being; nor does it underlie such affec tions of its own as would signal a substance, i.e. actions (such as a mind’s thoughts or a body’s motions). …Moreover, since we can clearly conceive extension ex isting without any subject, as when we may imagine spaces outside the world or places empty of body, and we believe it to exist wher ever we imagine there are no bodies, and we cannot believe that, if God should anni hilate a body, [extension] would perish with it, it follows that [extension] does not exist in the manner of an accident inherent in some subject.”33

But Newton went further and deeper than any of his predecessors in elu cidating the pe culiar ontology of space: “Just as the parts of duration are in dividuated by their order, so that (for example) if yesterday could change places with today and become the later of the two, it would lose its individ u ality and would no longer be yesterday, but today; so the parts of space are individuated by their positions, so that if any two could exchange their posi tions, they would also exchange their identities, and would be converted into each other qua individuals. It is only through their recip rocal order and po sitions (propter solum ordinem et positiones inter se) that the parts of du ra tion and space are un derstood to be the very ones that they truly are; and they do not have any other principle of individuation besides this order and posi tion.”34

Newton’s description of space not as a substance or a relation between substances, but a structure, forestalled the objections that Leibniz (1646-1715) would raise against his sup posed substantivalism. Unfortunately, it was not printed until 1962. In his lifetime he en trusted his defense to Samuel Clarke, who argued that if—as Leibniz claimed—“space was nothing but the order of coexisting things”, then, “if God should remove in a straight line the whole material world entire,…it would still…continue in the same

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place”.35 But this is also true if the points of space get their individual iden tity solely from their reciprocal posi tions and order, for, if space is Euclidian, that order is invari-ant under translation. So the theologian’s vi sion of world transport is nonsense also in Newton’s own terms.

Ironically, in Newton’s physics, his concept of absolute space stands idle. By Cor. V to the Laws of Motion, “the motions of bodies included in a given space are the same among themselves, whether that space is at rest, or moves uniformly in a right line with-out rota tion”.36 So the job assigned to the admittedly inaccessible absolute space can be performed by any relative space tied to an inertial frame of reference. (By Cor. VI, the same can be said of a frame falling freely in a uniform gravitational field.)37 As a matter of fact, the absolute structure underlying Newton’s relative spaces is not 3-di mensional space, but 4-dimensional spacetime (v. space-time models). But this could only be un-derstood in the light of 19th-century mathematics (v. 4,5).

In the 1770’s Kant (1724-1804) sought to detheologize Newton’s overt views on space while at the same time can onizing them as eternal necessi ties of human reason. He had advo cated Leib nizian relational ism, but in 1768 he satisfied himself that bod-ies presuppose space. (He noted that the incon gruence of, say, a right shoe with the matching left shoe cannot be de scribed in terms of the mutual relations between the parts of each, but only by re ferring them both to the surrounding space.)38 Thus, space was no at tribute of bodies. Since, in his view, it could not be pronounced a substance without running into antinomies, he declared it an attribute of the mind: the “form” of our external sense. (Whence the bodies, which pre suppose it, were de moted to appear-ances.) This thesis, however, was turned into a sort of philo sophical joke by the critical reflections he mounted on it. For they im ply that the mind-substance is unknowable; indeed, the substance–attribute relation would lack “objective va lidity” if applied to it. The philosophy of space must therefore rest upon the analysis of hu man experience and its presup positions as disclosed from inside. That analysis shows—ac cording to Kant—that ordinary self-awareness pressupposes the perception of objects in space.39 Space does not therefore depend on the human psyche as we know it, for it is indeed the latter which requires the prior availability of space. Kant revolutionized philosophy, and his conception of the human un der standing as Nature’s legislator probably contributed to the uninhibited in tellectual creativity displayed by the founders of 20th century physics.40 But his tortuous thinking on space does not appear to have had any direct influ ence on them.

4. The mathematical revolution of the 19th century

Standing on the shoulders of Newton and Leibniz, of Euler and Lagrange, 19th century mathematicians quietly carried through a revolution in thought whose import has not yet been fully appreciated. The next three subsections refer to those aspects of it which directly concern physical ge ometry.

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4a. Non-Euclidean geometry

Euclid’s Postulate 5 never appeared self-evident to all. Of course, if two coplanar straights a and b form with a straight c internal angles on the same side of c adding to less than π, a and b must indefinitely approach each other on that side of c. But must a and b therefore meet? After all, a hyperbola approaches its asymptotes without ever intersecting them.

Several Greek mathematicians sought to derive Postulate 5 from more obvious state-ments. In modern times, Wallis (1616-1703) showed that it holds if and only if there exist figures of different size with equal shape, and Saccheri (1667-1733) contended that the exis tence of a single rectangle war rants its validity. But then, there is no way of certifying that a given quadri lateral is exactly rectangular.

Saccheri41 considered a so-called Saccheri quadrilateral, i.e. a quadrilat eral with two equal sides perpendicular to the base. By symmetry, the other two angles are equal. They could ei ther be (a) larger than, (b) equal to, or (c) smaller than a right angle. Saccheri dismissed (a) offhand, for it implies that there can be a polygon with 2 sides. From (c) he inferred a good many consequences without coming across any blatant contradiction. One of them is that some pairs of coplanar straights which form with another straight and on the same side of it internal angles adding to less than π share a per pen dicular at their meeting point at infinity. This would contradict the fact that all right angles are equal, so that (iii) has to be dismissed too. Thus, only (b) remains, which is provably equivalent to Postulate 5.

Less than a century later Lobachevski (1793-1856) founded the first system of non-Euclidian geometry on assumptions equivalent to Saccheri’s alternative (c). Lobachevski de fined a straight as the locus of all points which remain unmoved if space is rotated about two fixed points.42 This definition is compatible with both the assertion and the negation of Postulate 5. Lobachevski’s geometry rests on its negation. In a Loba chevskian space there are no rectangles (and hence no cubes) and the 3 internal angles of a triangle add up to less than π. More surprisingly per haps, through the inte rior of any right angle α there runs a straight line λ parallel to both sides of α. (The distance κ from λ to α’s vertex is a charac teristic of the space, so Lobachevskian spaces form a one-parameter family; if κ = ∞, the space is Eu clidian.) A system of geometry equivalent to Lobachevski’s was indepen dently developed by Bolyai Janos (1802-1860). Gauss (1777-1855) had worked much earlier on such a geometry, but re frained from publishing for fear of “the clamor of Bœotians”.43

Lobachevski showed that the formulae of Lobachevkian trigonometry yield those of stan dard spherical trigonometry if for each segment-length x occurring in the former one substi tutes x√(–1).44 The algebraic correspon dence between both sets of formulae ensures that any contradiction arising in Lobachevskian geometry would be matched by a contradiction in Euclid ian geometry.

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Is physical space Euclidian or Lobachevskian? The question was raised in Schweikart’s correspondence with Gauss in 1818-19,45 and later, pub licly, by Lobachevski himself, who tried to solve it by calculating the sum of the in ternal angles of a triangle formed by three stars. He concluded that contem porary astronomical data were compatible with the Euclidian value π.46 Some philosophers were unnerved by the idea that Euclidian geome-try—hitherto the epitome of reason—could thus be handled as an empirical hypothesis, con firmed to within the margin of experimental error. Lotze (1817-1881) argued that if the 3 angles of an astronomical triangle added to a value dif ferent from π, this would show, not that physical space is non-Euclidian, but rather that light rays are not straight.47 By thus drawing at tention to the un derdetermination of geometrical theory by experience, Lotze un wittingly contributed to a much more radical reassessment of ge ometry—and of reason it self—than Gauss or Lobachevski had in mind.

4b. Riemann’s conception of physical space

In 1854 Riemann (1826-1866), at Gauss’s behest, delivered in Göttingen an inaugural lecture on “The hypotheses that lie at the foundation of geome try”.48 This is perhaps the most original essay ever written on space, and cer tainly the most fruitful one for our current understanding of physical ge om etry. The author’s interest is focussed on physical space, or rather on the mathematical structure instantiated by it. He conceives this structure as a species of the fairly broad genus of “n-fold extended quantities”. This genus is narrower than that of n-dimensional (topological) spaces, and is presum ably coextensive with the class of real n-dimensional differentiable mani folds. Riemann took it for granted that physical space is 3-dimensional. He thought that physicists did well in assigning it a metric, the nature of which, however, he regarded as an open question, subject to theoretical hy pothesizing and experimental corroboration. The lecture’s main effort goes to introducing some mathemati cal concepts which should be useful in the sci entific handling of this question. To avoid lengthy explanations they will be sketched here in today’s mathematical vocabulary.

It may seem that, through its use of coordinates, physics assumes that space is homeo-morphic (topologically equivalent) to Â3. Riemann saw that such an assumption would be needlessly strong. The physicist never coordi natizes more than a finite region surrounding the objects under study. Hence, what he tacitly presupposes is that each point of space has a neigh borhood homeomorphic to Â3. But such neighborhoods may well be pieced together to form a global structure which is very different from Â3. (As the several patches mapped by geographers onto the flat pages of an atlas jointly form the—topologically spherical—surface of the earth.) In order to use mathe matical analysis physics requires that coordinate trans formations be ex pressible by differentiable functions. Now, a real n-dimensional dif fer en tiable manifold—or n-manifold, for short—is a topological space M, ev ery point of which has a neighborhood homeomorphic with Ân which is mapped

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bijectively and bicontinuously onto an open region of Ân, so that (i) every point of M lies in the domain of one of these maps or “charts”, and (ii) if two such charts, g and h, have overlapping domains, the compos ite map pings (“coordinate transformations”) g ∗ h−1 and h ∗ g−1 are C∞–differ entiable. Thus, through its use of coordinates and analysis, physics had in effect pre supposed that space is a 3-manifold.

Physicists also assume that any path joining two points in space has a definite length. If the path is straight, its length obeys the Theorem of Pythagoras, so that, if a, b and c are the the lengths of the three sides of a triangle, and ω is the internal angle formed by a and b,

c2 = a2 + b2 — ab cos ω (1)

If the path is curved, its length is, by definition, equal to the (demonstrably unique) limit as n → ∞ of the lengths λ(P

1), λ(P

2),… of any sequence of non-intersecting po-

lygonal arcs P1, P

2,… inscribed in the path, and such that the length of the longest side

in Pn converges to 0 as n → ∞. However, in a Lobachevskian space, lengths do not

satisfy eqn. (1). In Riemann’s view, the length of a path in physical space depends on the interplay of forces acting on and about the path. The successful employment of the Pythagorean met ric in physics bespeaks its approximate validity on the human scale. But on a much larger or a much smaller scale it might well break down. Riemann is inter-ested in developing a more general concept of a metric, which would enable a physicist to formulate—whenever experience makes it advis able—a more accurate definition of the metric of physical space. Such a concept can be spec ified at different levels of generality. Since the Pythagorean metric had up to then proved suitable, Riemann thought that, for the time being, one ought to abide by a type of metric which agrees to first order with the Pythagorean metric on a neighborhood of each point. This is now known as a Riemannian metric. Riemann conceived it as a generalization to n-mani folds of the intrinsic metric of surfaces—i.e., 2-manifolds—developed by Gauss.

Let M be an m-manifold and N an n-manifold (m,n ≥ 1). Consider a mapping ϕ: M Æ N by P Å ϕP. ϕ is smooth at P ∈ M if and only if there is a chart g defined at P and a chart h defined at ϕP such that the composite mapping h ∗ ϕ ∗ g−1 (of Âm into Ân) is C∞–differ entiable. ϕ is smooth if it is ev erywhere smooth. A scalar field on M is a smooth mapping of M into Â. Scalar fields on M form a ring F(M), with addition given by (ϕ + ψ)(P) = ϕ(P) + ψ(P), and multiplication given by (ϕψ)(P) = ϕ(P)ψ(P). A curve in M is a smooth mapping of an open interval I ⊂  into M. Consider a curve γ: (a,b) Æ M; u Å γu. The tangent to γ at γτ, denoted by γ∗τ (τ ∈ (a,b)), is the linear map ping of F(M) into  defined, for each ƒ ∈ F(M), by:

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γ ∗t (ƒ) =

dƒ⋅γdu u = t (2)

If P ∈ M lies in the range of a curve γ, γ is said to go through P. The tan gents at P of all curves through P form an m-dimensional real vector space at tached to P: the tangent space T

PM. The tangent spaces attached to all points of M constitute the tangent bundle

TM. TM is a 2m-manifold which is mapped onto M by the projection π defined by the condition: for each w ∈ T

PM, πw = P. A vector field on M is a section of TM, i.e.

a smooth mapping W: M Æ TM, by P Å WP ∈T

PM. The vector fields on M form a

module v(M) over the ring (M). In particular, we define for each curve γ in M a vec-tor field γ∗ by setting γ∗P

= γ∗τ if P = γτ for any τ in the domain of γ; letting γ∗ fade smoothly to 0 in a narrow neighborhood of the range of γ, and putting γ∗ equal to 0 everywhere else.

Consider now a mapping g: v(M)

× v(M) → (M). If V,W ∈ v(M), we denote by g

P(V,W) the value of g(V,W) at P ∈ M. g is a Riemannian metric if and only if it is:

(i) bilinear, so that ∀V,W,Y ∈ v(M) and ∀ƒ ∈ (M), ƒg(V,W) = g(ƒV,W) = g(V,ƒW); g(V,W) + g(Y,W) = g(V+Y,W), and g(V,W) + g(V,Y) = g(V,W+Y);

(ii) symmetric, so that ∀V,W ∈ v(M), g(V,W) = g(W,V);(iii) non-degenerate, so that ∀W ∈ v(M) and ∀P ∈ M, g

P(W,W) ≠ 0, un less W

P =

0; and(iv) positive definite, so that ∀W ∈ v(M) and ∀P ∈ M, g

P(W,W) ≥ 0.

In that case, ⟨M,g⟩ is said to be a Riemannian manifold, and the length of a curve γ in M between two points γa and γb is given by the definite integral:

(gγτ(γ ∗,γ ∗)1 21 2)d τ

a

b

g determines a 4-linear mapping ℜ of v(M) × v(M)

× v(M)

× v(M) into F(M) called

the curvature of ⟨M,g⟩, and a scalar field R ∈ F(M) known as the curvature scalar. Defi-nitions of ℜ and R can be found in any good text book of differential geometry. (If g meets all the above conditions except (iv) it is said to be a semi-Riemannian metric. ⟨M,g⟩ is then a semi-Rieman nian manifold. The concept of curvature remains, but the concept of length needs refurbishing, as g no longer is positive-definite.)

Riemann’s view of geometry lies at the heart of Einstein’s theory of grav ity (v. rela tivity, general). This theory imposes a semi-Riemannian metric on a 4-manifold repre senting, not physical space, but spacetime. The metric agrees to first order with the

Torretti, Space 12

metric h of Minkowski spacetime (5). In agree ment with Riemann’s conception of the dependence of geometry on physical forces, the theory’s field equations link the met-ric to the distribu tion of matter and non-gravitational energy. On the other hand, the metric alone fixes the spacetime trajectories of freely falling (uncharged, non-spinning) test particles.

4c. Geometries and groups

The 19th century witnessed numerous innovations in geometry, besides those pre-sented in 4a and 4b. The center of the stage was held by projec tive geometry, which was generally regarded as a fairly innocuous im provement on Euclidian geometry, although it breaks with it at the topolog ical level, and thus ultimately poses a more serious challenge to intuition than Lobachevs kian geometry. Though projective geometry can be traced back to Kepler (1571-1630) and two of its basic theorems were proved by Desargues (1591-1661) and Pascal (1623-1662), its flourishing began with Poncelet (1788-1867), in the wake of the re vival of synthetic geometry by Monge (1746-1818). In projective geometry every straight has a “point at infinity”, where it intersects its parallels. Since this point can be approached from ei ther direction, the points on the projective line lie in cyclical order. The pro jective plane contains a straight line at infinity; projective space, a plane at infinity. Projective space is topologically compact. The projective plane is one-sided and non-orientable.

Analytic methods based on the use of real-valued homogeneous coordi nates were intro-duced into projective geometry by Möbius (1790-1868) and Plücker (1801-1868). There-upon Staudt (1798-1867) enriched projec tive space with “imaginary points” labelled with complex-valued homoge neous coordinates and constituted, under some natural—mostly tacit—topo logical assumptions, the complex projective space.49 This new creation of the Euro pean mind is often the referent of late 19th-century mathemati cians’ state ments on “space”. In particular, it is tacitly presupposed by the group-theo retical investigations of Felix Klein (1849-1925) and Sophus Lie (1842-1899) on the foundations of geometry.

In his so-called Erlangen Programme (1872),50 Klein presented the al ge braic theory of groups as a means for unifying the variegated de velopments in geometry. Let P stand for (3-dimensional) complex pro jective space and consider all the bijective mappings of P onto it self. Such mappings form a group (with the identity on P as the neutral element and com position of mappings as the group product): the most general group of “space” trans forma tions. Under such mappings P transforms as an un structured set. Car dinality is the only property they all preserve. Bicon tinuous bijections con stitute the subgroup of topological transformations, which preserve neig bor hood and connectivity relations. Consider now the projective transfor ma tions, which map sets of collinear points bijectively and bicontinously onto sets of collinear points. These form a subgroup of the topologi-cal transfor mations. The properties they preserve constitute the full scope of projective

Torretti, Space 13

geometry. A subgroup of pro jective transformations permutes imaginary points among themselves, thus justifying the standard treatment of real projective space as an independent subject. Among the latter trans forma tions, one may distinguish the subgroup of affine transformations, which map the plane at infinity onto itself and therefore preserve the Euclidian relations of incidence, collinearity, linear order and parallellism. Euclidian similari-ties, which preserve the ratios between Euclidian dis tances form a subgroup of the affine group. It contains as a proper subgroup the distance-preserving Euclidian isometries. The congruence-preserving Euclidian mo tions are, in turn, a subgroup of the latter.

Klein, however, did not approach the subgroup of Euclidian motions in this way, but ac cessed it directly from the projective transformations of P.51 Any such transformation will preserve the cross-ratio between 4 collinear points. Following an idea of Cayley (1821-1895), Klein defined the distance between 2 points A and B relative to a quadric surface S as a func tion of the cross-ratio between A, B and the two points where the line AB meets S. The pro jective transformations that map S onto itself constitute a subgroup which plainly preserves this function. Klein showed that if S consists exclu sively of real points, the points in the inte rior of S, as related among them selves by the said distance function, satisfy the theorems of Lobachevs kian geometry—which Klein called hyperbolic. If S consists only of imaginary points, the real points of P, as related by the said distance function, comply with a different non-Euclidian geometry discovered by Klein—which he called elliptic. Finally, if S is the de generate conic consisting of the line at infinity taken twice, the real points of P, not on the plane at infinity, as re lated by the said function, satisfy the theorems of Euclidian geome try—called parabolic by Klein. In this very specific and somewhat contrived sense, ome may therefore regard the same underlying space as a relization of Euclidian geometry or of one of the said non-Euclidian geometries—de-pending on one’s free choice of a transformation group. This discovery prompted the geomet ric conventionalism of Poincaré (1854-1912) for, as he noted, “the existence of a group is not incompatible with that of another group”,52 and the mathe matical physicist may choose the group he feels most comfort able with.

A characterization of Euclidian geometry by the congruence-preserving motions had been attempted by the philosopher Ueberweg in 1851. Later, Helmholtz (1821-1894) lighted on the same idea, but he soon realized that not just Euclidian, but also Lobachevskian geometry could be characterized in this way.53 Are these the only geometries which can be character ized by a group of motions? Lie tackled this question with the powerful means of this theory of differentiable transformation groups.54 A Lie group is a math em atical structure which sports the algebraic properties of a group and the ge ometric properties of an n-manifold. The dimension number n is usually referred to as the number of pa-rameters of the group. Take an m-manifold M endowed with a Riemannian metric g. It can be shown that ⟨M, g⟩ ad mits at most an (n(n +1)/2)-parameter Lie group of g-pre-serving trans for mations. Indeed, ⟨M, g⟩ does admit such an (n(n +1)/2)-parameter group

Torretti, Space 14

if and only if the curvature scalar R deter mined by g is constant. ⟨M, g⟩ is Euclidian if and only if R = 0. If the constant R is less than 0, ⟨M, g⟩ is Lobachevskian, and if it is greater than 0, ⟨M, g⟩ is a realization either of Klein’s elliptic geometry or of the related but topologically distinct system known as spherical ge ometry.

5. Relativistic kinematics and Minkowski spacetime

In the last third of the 19th century the foundations of Newtonian me chanics were closely scrutinized. Mach (1838-1916) argued that Newton’s rotational thought experi-ments did not prove the existence of absolute space, for their effects might be due to the cosmic dis tribution of matter.55 This idea, dubbed Mach’s Principle by Einstein, played a heuristic role in the develop ment of General Relativity.

James Thomson and Ludwig Lange would not question the key Newtonian idea of real forces causing absolute acceleration, but they rightly saw that its effectiveness does not re quire the existence of an absolute space. In Newto nian kinematics absolute accel-eration is ac celeration with respect to the rel ative spaces determined by inertial frames of reference. By 1885 Thomson and Lange had independently worked out precise defini-tions of an inertial frame and an inertial time scale attached to it.56 Lange’s ap proach is akin to Einstein’s in the original formulation of relativistic kine matics (1905), and indeed Einstein’s famous defini tion of simultaneity in an inertial frame may be regarded as a necessary supplement to Lange’s defi nitions.

Lange presupposes the Newtonian distinction between free particles and particles acted on by an external force. He bids us consider three arbitrarily chosen free particles α, β and γ, moving in three non-collinear directions from a point P fixed in the relative space SF de termined by a frame of refer ence F. If α, β and γ describe straight lines in SF, F is by defini tion an inertial frame. Also, any time scale T by which the particle α traverses equal dis tances in equal times is by definition an inertial time scale. If these defini tions are adopted, then the following are testable laws of nature:

I. Any free particle describes a straight line in SF.II. Any free particle traverses equal distances (in SF) in equal times (as measured

by T).

Lange’s definitions, together with these laws, pick out an infinite family of inertial frames and inertial time scales. What they do not do, however, is to assign a definite value to a free particle’s constant speed (or to a non-free particle’s acceleration). As Einstein remarked in 1905, such a valuation pre supposes the synchronization of clocks at distant places, viz. the places suc cessively traversed by the particle in question.57

This had already been noted by Thomson,58 but he apparently thought that any plausi ble signalling method would satisfy the stated requirement in a manner com patible

Torretti, Space 15

with the admissible margin of experimental error. However, as Ein stein would show, the level of accu racy achieved in optics with Michelson’s inter ferometer demanded a more discerning defini tion of synchronism. The one proposed by Einstein can be stated in Lange-like fashion as follows.59 Let F be an inertial frame in Lange’s sense and P a point fixed in its relative space SF. Use any established metod for measuring time at P—say, a frictionless top. Consider a pulse of light λ issuing at a given moment from P and propa gating in vacuo in every direc tion. If in time t the light pulse reaches a point Q ≠ P, is instantaneously reflected at Q, and comes back to P, then by definition it takes ex actly Ht to reach Q. As Q ranges over SF, this condi tion defines a universal time for the inertial frame F. It follows that the wave front of λ forms in SF at any given mo-ment a sphere centered at P. If the said condition is substituted for Lange’s defi nition of an inertial time scale, the testable laws I and II still hold. Moreover, the following is also a testable law of nature:

III. Any light-signal in vacuo propagates in a straight line (in SF) with constant speed (according to the stated definition of time).

As Einstein showed, under the proposed definitions, this Principle of the Constancy of the Velocity of Light does not conflict with the Principle of Rel ativity, according to which the laws of nature afford no grounds for dis tin guishing any particular inertial frame from the others. As is well known, the compatibility between these two principles is secured if the laws of nature, when formulated in terms of Einstein time and Cartesian space co ordinates adapted to an inertial frame, take the shape of equations invariant under transformations of the Lorentz group. To satisfy this requirement, the basic concepts and laws of kinematics and dynamics were subjected by Einstein to a substantial revision.

One of the baffling features of the new relativistic kinematics is that the spatial dis-tance between two separate points and the time interval between two successive events depend on the inertial frame to which the points or events in question are referred. Both magnitudes are greatest relatively to the rest frame of the object bearing the points and the clock marking the events, and they both tend to 0 as the relative speed of that object in the chosen frame of ref erence approaches the speed of light c. On the other hand, if the time interval between two events A and B relatively to an iner tial frame F is denoted by τF and the distance relative to F between the lo cations of A and B is denoted by σF, the quantity ∆(A,B) = c2τF

2 – σF2 re mains unchanged as F ranges over

the inertial frames.Pursuing an idea first adumbrated by Poincaré, Minkowski (1864-1909) interpreted

the quantity designated above by ∆(A,B) as a measure of the separation or interval be-tween A and B in the 4-dimensional cosmic contin uum of events which he dubbed “the world”, but is currently called space time.60 One sees at once that ∆(A,B) = 0 if

Torretti, Space 16

and only if A and B respec tively coincide—at least potentially—with the emission and the reception of a light signal propagated in vacuo. All other pairs of events fall into two classes: those separated by a “timelike” interval (∆(A,B) > 0), and those separated by a “spacelike” interval (∆(A,B) < 0). Clearly, the intervals between two event pairs are properly comparable only if they both be long to the same class. The cosmic continuum of events—or instantaneous event-sites—can be read ily conceived within the classical tradition of physics as the Cartesian prod uct of the Euclidian space SF determined by any given inertial frame F and the domain TF of an Einstein time coordinate for F. The differentiable struc tures of TF and SF induce in TF

× SF the topology—indeed, the differ-entiable structure—of Â4. Let the 4-manifold thus defined be denoted by M. Minkowski spacetime consists of M endowed with some additional struc ture. This can be presented in two seemingly disparate yet logically equiva lent ways, either in the spirit of Riemann (4b), or in that of Klein(4c).

Given the arbitrary choice of a Euclidian space SF and an Einstein time TF attached to an inertial frame F, every spacetime point E ∈ M projects onto a unique spatial point P

E ∈ SF and a unique instant T

E ∈ TF. (P

E and T

E are the place and the time—relative

to F—at which E occurs). If x, y, z are Cartesian coordinates for SF and t is an Einstein time coordi nate for TF, one can define on M a global chart u which assigns to each spacetime point E a real-num ber quadruple ⟨u0(E),u1(E),u2(E),u3(E)⟩ given by:

u0(E) = t(TE), u1(E) = x(P

E), u2(E) = y(P

E), u3(E) = z(P

E)

A semi-Riemannian metric h is everywhere defined on M by specifying its action on the vec tor fields ∂/∂uk determined by the coordinate functions uk (k = 0,1,2,3), viz.:

(∂/∂u h,∂/∂u k) =

c if h =k = 0–1 if h = k >00 if h ≠k

⟨M,h⟩ is Minkowski spacetime. [Note that, for any P ∈ M, (∂/∂uk)P

is the tan gent at P of the one curve γ through P which is such that, for any t ∈ Â, uk(γt) = t, and uh(γt) = uh(P) if h ≠ k.] If A and B are two events —as above— and γ: (a–ε,b+ε) Æ M is a geodesic such that γ(a) is the spacetime site of A, and γ(b) that of B,

((γ ∗ τ,γ ∗ τ)

1 21 2)d τa

b= ∆(Α,Β)

This approach, though straightforward, is somewhat anachronistic, for the concept of

Torretti, Space 17

a geodesic of zero length—such as γ would be if it tracks the spacetime path of a light signal in vacuo—made little sense until Levi-Cività (1873-1941) introduced in 1917 the notion of parallelism along a curve in an arbitrary n-manifold.61 By means of this no-tion, it became possible to characterize a geodesic not by its (extremal) length, but by its (steady) direction, viz. as a curve whose tan gents are mutually parallel along it.

But Minkowski spacetime can also be introduced as follows. Let M be just the set of spacetime points. (Forget its topology and differentiable structure.) Let the additive group of the vector space Â4 act transitively and effectively on M; i.e., let there be a mapping ƒ: Â4

× M → M by ⟨,E⟩ Å E, such that (i) ∀, ∈ Â4, ∀E ∈ M, (+)E = (E);

(ii) ∀E1,E2 ∈ M there is some ∈ Â4 such that E1 = E2; and (iii) E = 0 for every E ∈ M if and only if ∈ 0. Endow Â4 with the inner product ∧ defined as follows: For every = ⟨α0,α1,α2,α3⟩ ∈ Â4 and every = ⟨β0,β1,β2,β3⟩ ∈ Â4,

∧ = Σ

h = 0

3η hkαh β kΣ

k = 0

3

(3),

where

η hk =

c if h =k = 0–1 if h = k >00 if h ≠ k

Recalling that ∀P,Q ∈ M there is some ∈ Â4 such that P = Q, one readily defines the interval ∆(P,Q) between two spacetime points P and Q:

∆(P,Q) = ∆(P,P) = ∧ (4).

The vector a and the interval ∆(P,P) are said to be spacelike, timelike, or null if ∧ is, respectively, positive, negative or equal to 0. A null or timelike vector = ⟨α0,α1,α2,α3⟩ is said to be future-directed if α0 > 0. The structure ⟨M,Â4,ƒ,∧⟩ is equivalent to Minkowski spacetime (as presented earlier) if the action ƒ satisfies the following physical requirements: ∀P,Q ∈ M such that P = Q ( ∈ Â4), (i) if is a future-directed null vector, P and Q may stand for the spacetime sites of the emission and reception, respec tively, of a light pulse in vacuo; (ii) if is a timelike vector such that ∧ = 1, P and Q may be the sites of two ticks, separated by unit time, of a clock at rest in an inertial frame; (iii) if ∧ = 1 and R = P = Q (, ∈ Â4; R ∈ M), where = ∧ = 0, P, Q and R may be the sites of three distinct events such that the first two occur at the same end of an unstressed rod of unit length at rest in an inertial frame, while the third oc-

Torretti, Space 18

curs at the other end of it. The inner product (1) and the partition of Â4 into timelike, spacelike and null vectors are, of course, invariant under Lorentz transformations. Such trans formations can be understood actively, as hyperbolic rotations of the vector space, or passively, as changes in the decomposition of each vector into 4 components. Either way, the structure signalled on M by (4) may be char acter ized, in the style of the Er-langen Programme, as the geometry of the Lorentz group. (Needless to say, the action ƒ induces in M the topology and differentiable structure of Â4—which must hold at least on a neighbor hood of each spacetime point to ensure the applica bility of mathematical analysis).

Minkowski’s restatement of relativistic kinematics has often been de scribed, somewhat disparagingly, as a “formalism”. This usage is all right, provided that it does not suggest an unfavorable comparison with earlier modes of thought. Though less familiar, Minkowski’s spacetime is not more formal, and certainly not a whit more artificial than Newton’s space–and–time. Indeed, the impressive successes of General Relativity in the very large and of Quantum Field Theory in the very small have ratified Minkowski’s forecast: “From now on space by it self and time by itself shall be wholly re duced to shadows, and only a sort of union of them both shall subsist inde pendently.”

ROBERTO TORRETTIUniversidad de Puerto Rico

NOTES

1 Theog., 116-125. Aristotle quotes line 116 pãntvn m¢n pr≈tista xãow g°net', aÈtår ¶peita ga›' eÈrÊsternow, and adds this comment: …w d°on pr«ton Ípãrjai x≈ran to›w oÔsi, diå tÚ nom¤zein, Àsper ofl pollo¤, pãnta e‰nai pou ka‹ §n tÒpƒ (Phys., IV, i, 208b30-33).

2 Other scholars prefer to think of xãow as the gap between earth and Tartarus; see Hesiod, Theogony, edited by M.L. West, Oxford 1966, ad 116.

3 Diels-Kranz, Vorsokratiker, 12A9-11.4 prÚw ˘ dØ ka‹ ÙneiropoloËmen bl°pontew ka¤ famen énagka›on e‰nai pou tÚ ¯n ëpan ¶n

tini tÒpƒ ka‹ kat°xon x≈ran tinã—Tim., 52b.5 Tim., 52d-53b.6 Tim., 53c-55c.7 F.M. Cornford, “The invention of space”, in Essays in Honour of Gilbert Murray, London 1936, pp.

215-235.8 See, for instance, Diels-Kranz, Vorsokratiker, 67A6, 68A37.9 A. Koyré, From the closed world to the infinite universe, Baltimore 1957.

Torretti, Space 19

10 On the simple bodies, see De Cælo, I, i-iv. On the 55 spheres which Aristotle be lieved were necessary and sufficient to make mechanical sense of of the planetary the ory of Eudoxus and Calippus, see Metaphysics, Λ, viii.

11 ÉAdÊnaton d¢ s«ma e‰nai tÚn tÒpon: §n taÈt“ går ín e‡h dÊo s≈mata—Phys., IV, i, 209a6-7.

12 Phys., IV, i, 212a20-21.13 ı d' oÈranÚw oÈk°ti §n êllƒ —Phys., IV, v, 212b22.14 parå d¢ tÚ pçn ka‹ ˜lon oÈd¢n §stin ¶jv toË pantÒw —Phys., IV, v, 212b16.15 The impossibility of the void (kenÒn) is the subject of Phys., IV, vi-ix.16 The impossibility of an infinite body (s«ma êpeiron) is argued in De Cælo, I, v-vii. The infinite

(tÚ êpeiron) is discussed in Phys., III, iv-viii.17 On Zeno’s arguments against motion, see Phys., VI, ii, ix; VIII, viii.18 §n d¢ t“ sunexe› ¶sti m¢n êpeira ≤m¤sh, éll' oÈk §ntelexe¤& éllå dÊnamei —Phys., VIII,

viii,19 t«n m¢n oÔn katå posÚn épe¤rvn oÈk §nd°xetai ëcasyai §n peperas m°nƒ xrÒnƒ, t«n d°

katå dia¤resin §nd°xetai: ka‹ går aÈtÚw ı xrÒnow oÏtvw êpeirow. Àste §n t“ épe¤rƒ ka‹ oÈk §n t“ peperas m°nƒ sumba¤nei dii°nai tÚ êpeiron, ka‹ ëptesyai t«n épe¤rvn to›w épe¤roiw, oÈ to›w peperasm°noiw. Phys., VI, ii, 233a26-31.

20 Ioannis Philoponi in Aristotelis physicorum libros quinque posteriora com mentaria, ed. H. Vitelli; Berlin 1888, p. 567.

21 “Quod Deus non possit movere celum motu recto: Et ratio est, quia tunc relin queret vacuum.” H. Denifle and E. Chatelain, Chartularium universitatis parisiensis, 1200–1452, Paris 1889 ff. Vol. I, p. 546.

22 “Deus essentialiter et præsentialiter necessario est ubique, ne dum in mundo et in eius partibus universis; …verumetiam extra mundum in situ seu vacuo imaginario in finito.” Thomas Bradwardine, De causa Dei contra Pelagium, ed. H. Savile; London 1618, p, 177; my italics.

23 Nicole Oresme, Le Livre du ciel et du monde, ed. by A.D. Menut and A.J. Denomy; Madison 1968, p. 724.

24 Oresme, l.c., p. 368: “Hors le monde est une espace ymaginee infinie et im mobile, …et est pos-sible sanz contradiction que tout le monde fust meu en celle espace de mou vement droit. Et dire le contraire est un article condampné a Paris.” (Cf. supra, n. 21.)

25 Oresme, l.c., p. 176. (French spelling modernized by R.T.)26 “Estre meu selon lieu est soy avoir autrement en soy meisme ou resgart de l’espace ymaginee

immobile, car ou resgart de celle espace ou selon elle est mesuree le isneleté [speed—R.T.] du mouvement et de ses parties.” Oresme, l.c., p. 372.

27 “Omnis enim philosophiæ difficultas in eo versari videtur, ut a phænomenis motuum investige-mus vires naturæ, deinde ab his viribus demonstremus phænomena reliqua.” Newton, Philosophiæ Naturalis Principia Mathematica, ed. by Koyré and Cohen, Cambridge MA, 1972, p. 16.

28 “Definitio III. Materiæ vis insita est potentia resistendi, qua corpus unumquodque, quantum in se est, perseverat in statu suo vel quiescendi vel movendi uniformiter in directum. […] Definitio IV. Vis impressa est actio in corpus excercita, ad mutandum ejus statum vel quiescendi vel movendi uiniformiter in directum.” Newton, Principia, ed. cit., pp. 40f.

29 Newton, Principia, ed. cit., p. 46.30 “Est ergo spacium, quantitas quædam continua physica triplici dimensione con stans, in qua corporum

magnitudo capiatur. […] Ipsum etenim locandis corporibus præesse oportet, et cum locatis esse, et

Torretti, Space 20

mutuo iisdem succedentibus immobiliem consis tere, et omnibus demum recendentibus remanere. […] Pauperrima illa naturæ et natu ralium distributio in materiam, formam, atque compsitum non satisfaciat; inde enim spacium præternaturale, et antenaturale judicare oporteret. […] Indifferenter omnia re cipiat […]. Nil agat, nilque patiatur.” Bruno, De immenso et innumerabilibus, I.viii, in Opera Latina Conscripta, Vol. I.1, Napoli 1879, p. 231. On Telesio and Patrizi I am indebted to Grant, Much ado about nothing, Cambridge 1981, Chapter 8.

31 “Videtur corpus secundum Cartesium non differre a vacuo philosophorum”—Huygens, Œuvres complètes, La Haye 1888-1950, vol. 16, p. 221.

32 This drastic step away from the Judæo-Christian tradition would indeed soon be taken by the Cartesian Spinoza (1632-1677). On More, see Koyré, From the Closed World to the Infinite Universe, Baltimore 1957, chapter V and VI.

33 “De extensione jam forte expectatio est ut definiam esse vel substantiam vel ac cidens aut omnino nihil. At neutiquam sane, nam habet quendam sibi proprium exis tendi modum qui neque substan-tiis neque accidentibus competit. Non est substantia tum quia non absolute per se, sed tanquam Dei effectus emanativus, et omnis entis affectio quædam subsistit; tum quia non substat ejusmodi propriis affectionibus quæ substan tiam denominant, hoc est actionibus, quales sunt cogitationes in mente et motus in cor pore. …Præterea cum extensionem tanquam sine aliquo subjecto existentem possumus clare concipere, ut cum imaginamur extramundana spatia aut loca quælibet corpori-bus vacua; et credimus existere ubicunque imaginamur nulla esse corpora, nec possumus credere periturum esse cum corpore si modo Deus aliquod annihilaret, sequitur eam non per modum accidentis inhærendo alicui subjecto existere.” “De Gravitatione et equipondio fluidorum”, in Hall & Hall, eds. Unpublished Scientific Papers of Isaac Newton, Cambridge 1962, p. 99.

34 “Quemadmodum enim durationis partes per ordinem individuantur, ita ut (instantiæ gratia) dies hesternus si ordinem cum hodierno die commutare posset et evadere posterior, individuationem amitteret et non amplius esset hesternus dies sed hodiernus: Sic spatii partes per earum positiones individuantur ita ut si duæ quævis possent positiones commutare, individuationem simul commu-tarent, et utraque in al teram numerice converteretur. Propter solum ordinem et positiones inter se partes du rationis et spatii intelliguntur esse eædem ipsæ quæ revera sunt; nec habent aliud in-dividuationis principium præter ordinem et positiones istas”. “De Gravitatione…”, l.c., p. 103.

35 The Leibniz–Clarke Correspondence, ed. by H. G. Alexander, Manchester 1956, p. 32. (Dr. Clarke’s Third Reply, §4.)

36 “Corporum dato spatio inclusorum iidem sunt motus inter se, sive spatium illud quiescat, sive moveatur idem uniformiter in directum sine motum circulari.” Newton, Principia, ed. cit., p. 63.

37 “Si corpora moveantur quomodocunque inter se, et a viribus acceleratricibus æqualibus secundum lineas parallelas urgeantur; pergeant omnia eodem modo moveri inter se, ac si viribus illis non essent incitata.” Newton, Principia, ed. cit., p. 64.

38 Kant, “Von dem ersten Grunde des Unterschiedes der Gegenden im Raume”, Wochentliche Königs-bergsche Frag– und Anzeigungs–Nachrichten, 6-8, February 1768.

39 Kant, Kritik der reinen Vernunft, 2. Aufl., Riga 1787, pp. 274-279.40 Remember the strong presence of Kantianism in the better universities of Ger many, Britain and

France at the turn of the century.41 G. Saccheri, Euclides ab omne nævo vindicatus, Milan 1733. Chelsea Publications has recently issued

a reprint, with facing English translation.42 “During the revolution of the surface containing it the straight line does not change its place

if it goes through two unmoving points in the surface (i.e. if we turn the surface containing it about two points of the line, the line does not move).” Lobachevski, Geometrische Untersuchungen zur Theorie der Parallellinien, Berlin 1840, fact #1.

Torretti, Space 21

43 Gauss to Bessel, January 27, 1829. Gauss, Werke, VIII, Leipzig 1900, p. 200.44 Lobachevski, Zwei geometrische Abhandlungen, German translation by F. En gel, Leipzig 1898-99, p.

65.45 Gauss, Werke, VIII, Leipzig 1900, pp. 180f.46 Lobachevski, Zwei geometrische Abhandlungen, l.c., p. 22.47 H. Lotze, Metaphysik, Leipzig 1879, pp. 248f.48 B. Riemann, “Über die Hypothesen, welche der Geometrie zu Grunde liegen”, Abhandlungen

der Kgl. Gesellschaft der Wissenschaften zu Göttingen, vol. 17, 1867. See the English translation with mathematical commentary in Spivak, A Compre hensive Introduction to Differential Geometry, Berkeley 1979, vol. 2, pp. 135ff.

49 On Staudt, see H. Freudenthal, “The impact of von Staudt’s Foundations of Geom etry”, in R.S. Cohen et al., eds., For Dirk Struik, Dordrecht 1974, pp. 189-200.

50 F. Klein, Vergleichende Betrachtungen über neuere geometrische For schungen, Erlangen 1872. Revised text in Mathematische Annalen, 43: 63-100 (1893).

51 F. Klein, “Über die sogenannte Nicht-Euklidische Geometrie”, Mathematische Annalen, 4: 573-625 (1871); 6: 112-145 (1873); 7: 531-537 (1874).

52 H. Poincaré, Œuvres, Paris 1916-56, vol. XI, p. 90.53 H. von Helmholtz, “Über die Tatsachen, die der Geometrie zum Grunde liegen”, Nachrichten der

Kgl. Gesellschaft der Wissenschaften zu Göttingen, 18: 193-221 (1868).54 S. Lie and F. Engel, Theorie der Transformationsgruppen, Leipzig 1888-93, vol. 3; cf. H. Freudenthal,

“Neuere Fassungen des Riemann-Helmholtz-Lieschen Raum problems”, Mathematische Zeitschrift, 63: 374-405 (1956).

55 E. Mach, Die Mechanik in ihrer Entwicklung, 9e. Aufl., Leipzig 1933, pp. 224-26. (The first edition appeared in 1883.)

56 J. Thomson, “On the law of inertia, the principle of chronometry and the prin ciple of absolute clinural reat, and of absolute rotation”, and “A problem on point-mo tions for which a reference-frame can so exist as to have the motions of the points rel ative to it, rectilinear and mutually proportional”, R. Society of Edinburgh Proc., 12: 568-578, 730-742 (1884). L. Lange, “Über das Beharrungsgesetz”, K. Sächsische Akad. der Wissenschaften zu Leipzig, Berichte der Math.-phys. Kl., 37: 333-351 (1885).

57 A. Einstein, “Zur Elektrodynamik bewegter Körper”, Annalen der Physik, (4) 17: 891-921 (1905), pp. 892f.

58 J. Thomson, “On the law of inertia,…”, l.c., p. 569. 59 A. Einstein, “Zur Elektrodynamik…”, §1.60 Cf. H. Poincaré, “La dynamique de l’électron”, Rendiconti del circolo matematico di Palermo, 21: 129-

175 (1906); H. Minkowski, “Raum und Zeit”, Physikalische Zeit schrift, 10: 104-111 (1909).61 T. Levi-Cività, “Nozione di parallelismo in una varietà qualunque”, Rendiconti del circolo matematico

di Palermo, 42: 173-205 (1917).