Aristotle's Mathematical Matter and Eudoxus' Proportion Theory

KbuREMENOS JT, (aq €. le q aa 6 | THEOKRITOS KOUREMENOS / COLUMBUS, OHIO Aristotle’s Mathematical Matter and Eudoxus ci Proportion Theory Aristotle’s mathematical matter, the 6An vont) of mathematical objects, is the starting point of this paper, whose purpose is not only to propose a new interpretation of this notion but also to elucidate certain points in Aristotle’s philosophy of mathematics: whether he thought of mathematics as axiomatizable' in accordance with his theory of science”, and whether ' J. Hintikka, Reply to Dorothea Frede, Synthese 28 (1974), 94 denies the application of Aristotle’s axiomatic model to mathematics because the only medium of deduction is the syllogism; cf. W. Leszl, Mathematics, Axiomatization, Hypotheses, in: E. Berti (ed.), Aristotle on Science. The Posterior Analytics, Padova 1981, 284, who proposes that the axiomatization of mathematics necessitates recourse to technical mathematical proof. Hintikka does not explain his thesis and, although I. Mueller, Greek Mathematics and Greek Logic, in J. Corcoran (ed.), Ancient Logic and Its Modern Interpretations, Synthese, Historical Library 9 (Dordrecht-Boston, 1974), 48 — 57, and J. Barnes, Proof and the Syllogism, in Berti 17— 59, have shown the limitations of categorical syllogism as the logic of mathematical proof, this does not imply that Aristotle did not think of his syllogistic theory as the underlying logic of mathematics in some restricted sense; cf. J. Corcoran, Aristotle's Natural Deduction System, in Corcoran 92, D. Frede, Reply on Hintikka's Paper ‚On the Ingredients of an Aristotelian Science’, Synthese 28 (1974), 88 and Barnes 41/42. A way to justify anachronistically Aristotle’s apparent conviction that syllogistic is the underlying logic of mathematics seems to be provided by Corcoran and Smiley's (What is a Syllogism?, JPhL 2, 1973, 136— 154) interpretation of Aristotle's syllogistic as a natural deduction system. J. Hintikka and U. Remes, Ancient Geometrical = KOUREMENOS, | heokritos. — Aristotle's mathematical matter and Eudoxus’ proportion theory. WS 1996 109: 55-85. * Ausgehend von einer Interpretation der Aristotelischen ÚAn vont) mathematischer Objekte wird der Standpunkt untersucht, den Aristoteles der Mathematik innerhalb der Wissenschaften zuschrieb. Hierfür wird das Verhältnis zwischen der Eudoxischen Proportionentheorie und den Antworten des Aristoteles auf die gegebenen Probleme betrachtet und der Hintergrund für die Aristotelische Lehre vom Kontinuum beleuchtet. [67-00489 en Analysis and Modern Logic, in R.S. Cohen, P. K. Feyerabend and M. W. Wartofsky, Essays in Memory of Imre Lakatos, Dordrecht - Boston 1976, 253 — 275 have shown the close relationship between ancient geometrical analysis and modern natural deduction systems, especially the method of semantic tableaux, developed by E. W. Beth, Semantic Entailment and Formal Derivability, Mededelingen van de Koninklijke Nederlandse Akademie van Wetenschappen Afdeling Letterkunde N. R. 18.13 (1955). Now, the method of semantic tableaux can be used for the logical analysis of mathematical proofs and has certain similarities with Aristotle’s „proof by ex-

such an axiomatization would entail the unification of geometry and arithextensionality which is arrived at by means of mental abstraction and underlies the properties of mathematical objects. Thus indeterminate extension becomes the ultimate product of mathematical abstraction, the ‚material‘ metic into a single science of mathematics’. A considerable part of the paper is devoted to the relationship between Eudoxus’ proportion theory and Aristotle’s answers to the above problems. Related to this topic is the math- 57 substratum upon which mathematical properties are superimposed’. Such ematical background of Aristotle’s concept of ouvex&c* and, last but not an interpretation of (An vont admits three genera of extensionality, inleast, his often questioned mathematical competence’, as it is indicated by determinate extension in one, two and three dimensions’. However, the his Physics. distinction between one-, two- and three-dimensional genera, which are the matter of lines, planes and solids respectively, is problematic. Aristotle considers the 6An von as an object of metaphysical inquiry (Met. 1059b14— 21). If the mathematical matter is the abstract notion of inde- I. ÚAn vontn and ouvexés Aristotle’s references to 6A vonti’, the material substratum of mathematical objects, are usually understood as implying the abstract notion of terminate extension, why should metaphysics deal with it?? No apparent ? For a negative answer to this question see W. Kullmann, Die Funktion der relationship between indeterminate extension in one, two or three dimensions and metaphysics suggests itself easily. Moreover, one of Aristotle’s objection to the Academic mathematical matter is that, if mathematical objects have as their matter ,,the great and the small“, then two absurd consequences mathematischen Beispiele in Aristoteles’ Analytica Posteriora, in Berti (above, n. 1), follow: either each dimension is irrelevant to the others or the distinction position"; see E. W. Beth, The Foundations of Mathematics, New York 1966. 190— 192. 253/254. 3 H.G. Apostle, Aristotle's Philosophy of Mathematics, Chicago 1952, claims His solution, that Gin vont „in its widest conception is the thinkable generic element” an instance of which is extension, is accepted by T. L. Heath, Mathematics in Aristotle, Oxford 1949, 214 and elaborated by I. Muellér, Aristotle on Geometrical that Aristotle conceived of a general science which incorporates proportion theory and the properties of the category of quantity. A similar approach is taken by C. B. Crawley, Universal Mathematics in the Aristotelian-Thomistic Tradition: The Hermeneutics of Aristotelian Texts Relative to Universal Mathematics, Washington 1980; for criticism see H. R. Mendell, Aristotle and the Mathematicians: Some Cross- Objects, AGPh 52 (1970), 156— 171; see also the extensive discussion by H. Happ, Hyle, Berlin- New York 1971, 581ff., J. Annas, Aristotle's Metaphysics Books M and N, Oxford 1976, 30— 34, S. Gaukroger, Aristotle on Intelligible Matter, Phronesis Currents in the Fourth Century (Diss.), Stanford 1985, 229ff. However, it is not 28 (1980), 187— 197, and E. Hussey, Aristotle’s Physics Books III and IV, Oxford easily understandable what scientific value could such a general science of quantity, 1983, 184. For a different interpretation of 6An vont see J. Lear, Aristotle on or ,,posology”, have. As it will be shown below (ch. 10 and 11), Aristotle was familiar Mathematics, PhR 91 (1982), 179— 183, who relates the notion of (An vont to the constructions involved in geometrical proofs. Frère Augustin-Gabriel, S. G., Matière with the mathematical research of his day and it seems quite improbable that he could have entertained a notion like ,,posology”, whose concrete mathematical meane—— ing, if any, is hard to grasp. Intelligible e Mathématique, A, Laval Théologique e Philosophique 17 (1961), 173— 196, esp. 191 — 196, relates the notion of 6An vonti to continuity. * The mathematical background of Aristotle’s ovvexés is examined by H.J. 7 The notion of indeterminate extension is problematic. There is no indication Waschkies, Von Eudoxos zu Aristoteles. Das Fortwirken der Eudoxischen Proporthat Aristotle conceived indeterminate extension as abstract mathematical space, although he lacked the notion of physical space, as Happ (above n. 6) 604/605 assumes; cf. H.G. Zekl, Topos, Paradeigmata 10, Hamburg 1990, 32/33. Phys. 204b20—22 does not support the assumption of an Aristotelian indeterminate extionentheorie in der Aristotelischen Lehre vom Kontinuum, Amsterdam 1978, and M. Caveing, Quelques Remarques sur le Traitement du Continu dans les Eléments d’Euclide et la Physique d’Aristote, in Penser les Mathématiques, Paris 1982, 145— 166. Waschkies and Caveing take the mathematics of Euclid's Elements as the tension in three dimensions; on the contrary, this passage denies the possibility of an infinitely extending body. It is important to note in this context that any assumption about mathematical space is absent from the Euclidean postulates; see A. Seidenberg, Did Euclid’s Elements, Book I, Develop Geometry Axiomatically?, AHES 14 (1974/75), 272/273. background of Aristotle’s cvveyéc, which is thus put in an anachronistic historical context; see below n. 69. * See especially G. Milhaud, Aristote et les Mathématiques, AGPh 16 (1902/1903), 367 — 392. © The term An vont occurs only in Met. 1036a9/10, 1037a4, 1045a34 and ® This interpretation, which depends heavily on Alexander’s understanding of 36. The first two passages deal with the problem in what sense a semicircle is part Din vontí (cf. the comments in Annas [above, n. 6] 30), is proposed by Mueller of a mathematical circle and Ps.-Alexander of Aphrodisias In Met. 510,3—5; 515, 26— 28 identifies 6An vonth with extension. In Met. 1045a34 and 36 HAN vonti is used for the generic element in a definition. For an attempt to connect the two uses of the term see W. D. Ross, Aristotle’s Metaphysics 2, Oxford 1924, 199/200. (above, n. 6). : ? Ross 2 (above, n. 6) 309 explicitly equates the 6An vont with space and WMnA offers to answer to this question; cf. Happ (above, n. 6) 612 and Annas (above, n. 6) 33 who simply state the fact without any explanation.

between dimensions is meaningless (Met. 1085a9 — 20)'°. Aristotle’s objection presupposes that dimensions are species of ,,the great and the small“ of the received interpretation which admits three genera as material substrata but, since according to the received interpretation of 6An vonti the dimensions correspond to genera of indeterminate extension, the question whether Aristotle’s 5Am vonti is immune to his objection against the Ac- 59 genus, that of ovvexfj peyé0n. This modification avoids a further problem of geometrical objects but posits a single genus of geometry!*. For Aristotle the genus of geometry comprises peyé0n (An. Post. 76a42—b2) which are of indeterminate extension in one, two or three dimensions? defined as cuveyî) (Met. 1020a10— 12, DC 268alff.) and it is noteworthy that in Met. 1061b2/3 Aristotle emphasizes the single genus of geometry after an enumeration of geometrical properties (Met. 1061a35 — b2), the rá0n The obvious answer is that all of them encompass ovvexfj peyé0n and this suggests a modification in the received interpretation of 6An vonti: the definition of the ultimate product of mathematical abstraction as nooöv ademic theory arises naturally. What is the relationship between the genera of this genus as rá0n of its objects qua rocév and ouvexéc!*, and the focus must shift from dimensionality to ovvexéc!!. Aristotle talks about the „plane“ and „solid“ as genera of plane and solid figures (Met. 1024436 — b4) and ovveyés (Met. 1061a29— 34). The context of Met. 1061a28— b3 shows but taking these genera as the öAn vonti, the ultimate product of matheof mathematical matter as moodv and ouvexéc. Geometry is the paradigme matical abstraction, would deprive the 64 vonti of its required generality of the science of being qua being which is possible because, although its a close relationship between the single genus of geometry and the definition as a material substratum'”, because, although all of them are ovvexn (DC objects belong to distinct genera, they are subsumed under the single de- 26826 — 10), each is clearly distinct from the others!?. Aristotle’s description scription qua óvta (Met. 1061b3— 17; cf. 1003a33ff.) which effects the single of mathematical abstraction in Met. 1061a28—b3 and 1061b21—25 puts the emphasis on ovveyéc as the product of abstraction while dimensionality, genus necessary for an Aristotelian science'®. The analogy between geometry and the science of being presupposes that geometry’s genus comprises objects especially in Met. 1061a28—b3, appears as a further characterization of belonging to distinct genera, i. e. the dimensional genera of Met. 1061a33/34, Guvexés. However, there is no reason to assume that such a dimensional but nevertheless it is a single genus, and geometry a single science like the characterization plays any major role in the definition of ouveyéc. moody and cvveyéc, the final product of abstraction in Met. 1061a28 — b3, is defined science of being (cf. Met. 1060b33—35), because the generically distinct objects fall under the single description qua ovvex&g which, paralleling qua in Met. 1020a7— 11 without any reference to dimensionality which is treated dv, does not take into account their differentiae, i.e. dimensionality. The in 1020a11/12 as a further characterization of ovveyéc, albeit irrelevant to fact that geometry studies the properties of generically, i.e. dimensionally, its definition. Moreover, the discussion of péye00g in DC 268a1 — b10 shows distinct objects, as the analogy between geometry and the science of being that dimensionality is defined in terms of ovveyéc. Thus ouveyéc seems to requires, is clearly shown by Met. 997a25—30 (cf. 997b14/15) and DC be a notion prior to dimensionality and therefore it is reasonable to assume 268a30—b2. Moreover, the analogy implies that it is ovveyég, the mathethat ovvex&g, without any reference to dimensionality, must have been matical matter, that generates the genus of geometry, since all dimensionally considered by Aristotle as 6An vontn; indeed, in Met. 1061b21—25 the oîkgia 6An of mathematics are one-, two- and three-dimensional ueyé0n distinct objects are examined qua ovveyfj, exactly as the science of being examines generically distinct objects qua Svta. But, if „qua ouvexéc* funcqua ovvexn. 14 See Mueller (above, n. 6) 167/168. Mueller accomodates the dimensionally 2. VA vont and the Genus of Mathematics According to this interpretation, there is no need to distinguish among three genera corresponding to the three dimensions, An vonti is only one distinct geometrical objects within the single genus of geometry by appealing to the general notion of péye00g that encompasses all geometrical objects. However, he fails to account explicitly for the common characterization of all geometrical objects as peyé0n. 12 On this passage see Annas (above, n. 6) 184. '! Cf. Frère Augustin-Gabriel (above, n. 6) 190— 193. ‘2 Happ (above, n. 6) 612 recognizes that 6An vont must be „etwas überindividuelles”. 12 The implication that each dimension has its own »Moetic”, i.e. substratum is clearly stated by Gaukroger (above, n. 6) 189. material, !5 On the qua operator see W. Wieland, Die Aristotelische Physik, Gôttingen 1962, 197— 202, and Lear (above, n. 6), 162— 175. 16 On the single genus of metaphysics effected by means of focal meaning see G.E.L. Owen, Logic and Metaphysics in Some earlier Works of Aristotle, in I. Düring and G. E. L. Owen, Aristotle and Plato in the Mid-Fourth Century, Gôteborg 1960, 163 — 190; see also M.T. Ferejohn, Aristotle on Focal Meaning and the Unity of Science, Phronesis 25 (1980), 117— 128.

61 tions like „qua ôv“, then the mathematical matter has to be „something Particularly interesting is Aristotle’s characterization of the entire mathcommon“ (Ev tt kai xorvòv, Met. 1061b12), apparently to the dimensionally ematics in terms of ovvexéc (Met. 1061b21 — 25). The context of this chardistinct genera of geometrical objects. This is a requirement that the genera acterization is the paradigmatic role of mathematics for the science of being of indeterminate extension in one, two and three dimensions cannot meet. (Met. 1061b17— 33) and, therefore, it implies that mathematics, like geo- Thus, contrary to the received interpretation, the genus of geometry not metry in Met. 1061b2/3, has a single genus despite the distinct genera of its only implies a single mathematical matter but it is also effected by it. objects?!, Aristotle's common characterization of geometrical peyé0n and Why does Aristotle emphasize that geometry has a single genus in relation to the mathematical matter? Apart from the analogy he draws of the genus of mathematics. However, he distinguishes between the primitive numbers as ouvexfj is apparently to be understood in relation to the unity between geometry and the science of being, his comment bears on his conterms and the naßn of geometry and arithmetic respectively (An. Post. ception of geometry and mathematics in general. The question is ultimately 76b1 — 9). Moreover, he emphasizes that the difference between geometrical related to the meaning of ouvexés as the matter of mathematical objects which Aristotle relates to the unity of geometry’s genus. arithmetic; however, he allows for certain points of contact by stipulating Up to this point, the product of mathematical abstraction in Met. 1061a28 — b3 has been referred to almost exclusively as ovvexéc, although Post. 75238—b20). Aristotle’s stipulation implies that there are proofs, in and arithmetical proof results from the different genera of geometry and that, in these limited cases, the ney&ßn are characterized as numbers (An. Aristotle links ouvex&g with zocóv!”. This conjunction appears contradicwhich the primitive terms of geometry and arithmetic are identical, but, tory. In Met. 1020a7 — 12 Aristotle distinguishes ovveyég as a sub-category according to An. Post. 87a38/39, such an identification implies that geometry of rocév thus precluding the synonymy of these terms. It is reasonable to assume that xooév in Met. 1061a33 and 34 means not ovveyés rocóv"*, and arithmetic have the same genus. It seems that Aristotle is ambivalent but the latter holds for numbers whereas ouvexéc applies to geometrical characterization of ney&ßn and numbers in terms of mathematical matter”? peyé0n only and, despite this fundamental distinction, the rá8n in Met. and, by implication, a common science of mathematics effected by it, appears 1061a35—b2 are exclusively geometrical. The same contradiction appears problematic”. reversed in Met. 1061b21—25, where ouveyéc alone characterizes mocà, which include not only geometrical peyé@n but also numbers!?. Moreover, Aristotle’s emphasis on ouveyxés and rooöv is striking; he defines ouveyéc as 5umperóv eig dei Öraıpera” and rocév as simply Sumperóv, but what is the point of reducing geometrical, and mathematical in general, properties about whether there exists a single genus of mathematics. Thus a common 3. ÜAn vontrj and Eudoxus’ Proportion Theory. In both Met. 1061a28—b3 and 1061b21 — 25 Aristotle states that geometry and mathematics examine mathematical properties of objects qua cuveyî. Among these properties he includes proportion-theoretic properties to properties of objects qua ovveyi and mood in the above sense? 2! The passage is not examined by Heath (above, n. 6) 222—224 under the !7 There is no reason to assume with Happ (above, n. 6) 604 n. 217 that xooév does not play any rolein the understanding of this passage. !8 = nAî00g according to Met. 1020a8 — 10; cf. Frère Augustin-Gabriel (above, n. 6) 191. heading „Universal Mathematics‘ and it is passingly referred to by Mendell (above, n. 3) 237, although it is extremely important for the development of any argument about universal mathematics in Aristotle. 2 Frère Augustin-Gabriel (above, n. 6) 193 posits two kinds of intelligible 12 Qua ouvexés must pick out an essential characteristic of both numbers and matter, one for numbers and one for peyé0n. This division corresponds to the division peyé0n, as qua óv does for the objects of metaphysics (the two terms are contrasted in Met. 1061624), and thus it cannot imply the ,,mimetic’ characterization of numbers of mathematics into number theory and geometry and presupposes Frére Augustinin terms of peyé0n referred to in Met. 102063 — 6; on „linear“, „plane“ and „solid“ geometry, would be impossible according to Aristotle's standards for demonstrative numbers see Heath (above, n. 6), 209. science (Matière Intelligible e Mathématique, B, Laval Hong e Philosophique 20 According to Waschkies (above, n. 4) 237— 242 thisis Aristotle's second Gabriel's conviction that a unified mathematics, incorporating both arithmetic and 18 [1962], 207 — 210); see, however, ch. 8 below. characterization of the continuum developedin Phys. T ch. 4—8 and Phys. Z ch. 23 W. Fiedler, Analogiemodelle bei Aristoteles. Untersuchungen zu den Ver- 1/2. For Aristotle's first characterization of continua „als geordnete Structuren“in gleichen zwischen den cinzelnen Wissenschaften und Kiinsten, Amsterdam 1978, Top. A ch. 2 and Phys. E ch. 3 see Waschkies (above, n. 4) 198— 220. On Aristotle's 57ff., and Kullmann (above, n. 2) 253, wrongly understand Met. 1026a26/27 as referring to ,,allgemeine Mathematik". For the correct interpretation of this passage see Mendell (above, n. 3) 237/238. definition of the continuum see D. Bostock, Aristotle on Continuityin Physics VI, in L. Judson (ed.), Aristotle’s Essays: A Collection of Essays, Oxford 1991, 180— 188,

(Met. 1061b1/2), which help understand the above two passages. In An. Post. 74a17—25 a proportion-theoretic theorem is proven for all mathematical objects as a property they have qua todí, irrespectively of their doxus’ removal of any homogeneity restrictions with the pre-Eudoxean ve 63 proof in which dimensions, determining distinct gión of lines, planes, solids and numbers, had to be treated separately. That the pre-Eudoxean approach dimensional genus or whether they are ney&ßn or numbers. Aristotle refers implies a fragmentation of the genus of proportion theory into eiön of lines, to Eudoxus’ reformulation of proportion theory and it is unlikely that his planes, solids and numbers?” is shown by An. Post. 85a37—b1, where proreference in Met. 1061b1/2 to proportion-theoretic properties proven as portion theorems are said to be proved as properties of mathematical objects properties of objects qua cuveyî] does not imply his reference in An. Post. not qua lines, planes, solids and numbers but qua napa tadta ti (An. Post. 74a17— 25 to the same properties proven as properties of objects qua todi, 85b1)%. An. Post. 85a37— bl simply restates An. Post. 74a17— 25 and both especially given that both qua ovveyéc and qua todi do not take dimenpassages clearly state a metamathematical result: the proof of the alternando sionality into account. Indeed, ovveyéc, like todi in An. Post. 74a21, makes theorem for all distinct mathematical objects qua todi is a universal all distinct geometrical peyé0n and numbers @vopacpévov ti tadta navta (xa86%.ov) proof”. But, if in An. Post. 74a17— 25 and 85a37 — bl the unity Ev. Leaving aside for the time being the meaning of ouveyés and toëi?* (see of the genus of proportion theory is due to a metamathematical result about below § 10), the obvious analogy between An. Post. 74a17—25 and Met. universal proofs of properties of objects qua toöi, the unity of the genus of 1061a28 — b3 and 1061b21 — 25 explains two problematic features of the last geometry in Met. 1061a28 — b3 must entail an analogous metamathematical two passages: first, the importance of qua todi consists in the absence of result: the properties of generically distinct mathematical objects are conany dimensional distinction between geometrical peyé0n, and thus it supsidered as properties of these objects qua rocóv and ovvexéc, which neports the above interpretation of ocvveyéc as a non-dimensional notion; cessitates universal proofs of theorems about these properties. second, since qua todi does not distinguish between numbers and geometrical Met. 1005a19ff. and 1061b17ff. substantiate the above interpretation. peyé0n, Met. 1061a35—b2 and 1061b21—25 do not involve any contra- In the former passage the set of common axioms is the kxoıvöv of the distinct diction because ouvexéc, as its conjunction with noodv indicates, covers genera of departmental sciences. The set of commön axioms is the koıvöv both geometrical ney&ßn and numbers”. If this is so, why are arithmetical of these genera qua beings. In the latter Aristotle focuses only on the category properties properties of numbers qua cuvexñ in Met. 1061b21 — 25? In what of rocòv whose genera share the common axiomsTM. Aristotle contrasts the way does An. Post. 74al7—25 explain Aristotle’s emphasis on the single science of being with mathematics: mathematics is concerned with lines, genus of geometry and mathematics in Met. 1061b2/3 and 1061 b21 —25 angles and numbers qua ovvezij, not qua beings. The implication is that respectively? In An. Post. 74a17— 25 the alternando property of proportion is said to have been proved by Eudoxus for lines, planes, solids and numbers as a property of these mathematical objects qua todi. Aristotle contrasts Eu- 26 See I. Mueller, Homogencity in Eudoxus’ Theory of Proportion, AHES 7 (1970), 1-6. 27 Cf. Mueller (above, n. 26) 1, T.L. Heath, A History of Greek Mathematics, Oxford "1965, 384 and (above, n. 6) 223. 28 Which, however, must not be conceived as a Platonic independent entity; see Mueller (above, n. 6) 157 and Lear (above, n. 6) 165. # It is very important to determine the exact meaning of todi. Heath (above, 2 On metamathematical results in Aristotle’s logic see Corcoran (above, n. 1) n. 6) 44 and 223 proposed that todi stands for nooöv or péyedos (cf. Waschkies [above] 143/144). This terminological interpretation of todi goes back to Philoponus who identified it with nooöv; cf. J.P. Zabarella, Opera Logica, Hildesheim 1966, proofs within this certain genus as conditions for the unity of a science. On the 727. However, as W. Knorr, The Evolution of the Euclidean Elements, Synthese relationship between the genus’ homogeneity and indemonstrable/demonstrable sen-- 112-122. 30 In An. Post. 87a38— b4 Aristotle relates explicitly the unity of the genus and Historical Library 15 (Dordrecht-Boston 1975), 291 n. 32 rightly observes, An. Post. tences referring to the same domain of objects see H. Scholz, The Ancient Axiomatic 99a10 implies that Eudoxus’ universal proofs require a common property that char- Theory, in J. Barnes, M. Schofield and R. Sorabji (eds.), Articles on Aristotle 1, acterizes both peyé0n and numbers ; cf. Lear (above, n. 6) 167. The whole issue is London 1975, 53 and 62. not simply terminological but related to Eudoxus’ new formulation of proportion theory and thus todi must imply a basic characteristic of it; cf. Mueller (above, n. 1) 46. a metaprinciple dealing with provability of statements within a genus and thus necessitating the existence of this genus; see Ferejohn (above, n. 16), 124— 127. 25 Cf. Lear (above, n. 6) 167. 31 It is probable that Aristotle’s argument for the unity of metaphysics involved 32 See Ross 2 (above, n. 6) 314.

mathematical proofs are about mathematical objects not qua beings, because in this case mathematical proofs would not be mathematical as An. Post. and ka0” até, lacks necessity, an indispensable characteristic of scientific demonstration (An. Post. 71b9— 16, 73b25— 28), and, therefore, the meta- 75b37 — 76a3 shows”, but qua ovvexij. However, since proofs by means of mathematical importance of Eudoxus’ proportion theory lies in establishing the common axioms are universal proofs for all generically distinct beings the required necessity of proportion-theoretic proofs. But, as EE 1222b23—41 shows, the necessity of mathematical propositions depends on qua beings, the complete analogy between the science of being and mathematical objects qua ouvexij are universal proofs. Moreover, if qua ovvexñ, certain indemonstrable principles, hypotheses, and, therefore, the necessity of proportion-theoretic propositions must imply such a principle, which like qua todi, implies a metamathematical result which necessitates universal characterizes all ueyéOn and numbers”. proofs, since in the science of being it is the set of common axioms that That Aristotle considered the anthyphaeretic proof not necessary but accidental, is clearly born out by another reference to Eudoxus’ proportion theory in An. Post. 99a1 — 10, where, in mathematical terms, Aristotle discusses the question whether two proofs of the same theorem can be given. In such a context a reference to the anthyphaeretic and the Eudoxean proof ematics requires that mathematical proofs for all generically distinct matheffects the universal proofs for objects qua beings, in mathematics the universal proofs for objects qua ouvez must imply some indemonstrable principle(s) that characterize all dimensional genera and numbers. 4, Postulates and the Genus of Mathematics According to the above interpretation, in Met. of the alternando theorem is indeed fitting: the alternando theorem can be 1061a28—b3 and 1061b21—25 Aristotle generalizes a metamathematical result concerning demonstrated separately for lines qua lines and numbers qua numbers. A particularly Eudoxus' proportion theory over the entire genus of geometry property of all dimensionally distinct geometrical ney&ßn and numbers qua and mathematics in general. The proof of the alternando theorem within Eyovta abënoiv toravdi. However, since for Aristotle only the universal proof qualifies as a scientific proof, according to the distinction he makes in An. Post. 99a1 —6 the anthyphaeretic proof must have been considered the pre-Eudoxean anthyphaeretic proportion theory* explains Aristotle’s reasons for such a generalization. Modern reconstructions of the anthyphaeretic proof show that it must have treated the case of areas as primary universal demonstration of the alternando property can also be given as a by him as an accidental proof. and then a proof for the case of lines was derived from the case of areas”. An. Post. 74a17— 25 shows that the S', necessary for Eudoxus’ universal In Aristotle’s terminology, the anthyphaeretic proof was neither xatà mavtés proofs, must be Gv@tepov napa tó xa’ Exactov (An. Post. 74a7—9), i.e. nor xa0" aúró, as these notions are developed in An. Post. 73b25 — 74a3*: superordinate to all dimensional gión of geometrical peyé0n and the genus if S* and S' are the species (An. Post. 74a20—21) of lines and numbers of numbers. But, since tà ka’ Exaota are the eïôn of lines, planes, solids respectively in the Porphyrean tree of proportion theory’s genus, the antand numbers, S' must be the entire genus of proportion theory”. Thus hyphaeretic proof proves the alternando property separately, or derivatively Eudoxus’ universal proofs of proportion theoretic results presuppose a single (An. Post. 73b32—74a3), for S* and S', and not universally for a S' with genus of proportion theory superordinate to the particular genera of mathi<k andi<l. For Aristotle the anthyphaeretic proof, since it is not universal ematical objects. This genus is required for the necessity of mathematical 3 For a comprehensive discussion of Aristotle's reference to Bryson's attempt to square the circle see I. Mueller, Aristotle and the Quadrature of the Circle, in N. Kretzmann (ed.), Infinity and Continuity in Ancient and Medieval Thought, Ithaca 1982, 152— 154 and 160 — 164. # For the general characteristics of the anthyphaeretic proportion theory see Knorr (above, n. 24) 270/271. 3% For the reconstruction of the anthyphaeretic proof see O. Becker, EudoxosStudien I. Eine Voreudoxische Proportionenlehre und ihre Spuren bei Aristoteles und Euklid, OS B.2 (1933), 311—333, B.L. van der Werden, Science Awakening, Groningen 1954, 177/178, and Knorr (above, n. 24) 340 — 342. 26 The following analysis is based on J. Barnes, Aristotle's Posterior Analytics, Oxford 1975, 120/121. 7 On the necessity of mathematical propositions see Leszl (above, n. 1) 293 — 305; cf. the brief comment in Heath (above, n. 6) 280. 35 Pace Mendell (above, n. 3) 229— 238. See also W. D. Ross, Aristotle's Prior and Posterior Analytics, Oxford 1949, 524. Philoponus assumes that the genus of proportion theory was identified as moody; his conjecture is accepted by Heath (above, n. 6) 223 and elaborated by Apostle (above, n. 3) and Crawley (above, n. 3). It is obvious, however, that any relationship between the genus of proportion theory and rocòv requires a formal specification of the meaning of nooöv from a mathematical point of view. Otherwise, xooôv is just an empty term which does not explain anything. For my interpretaion of rocóv and its relationship to Eudoxus' proportion theory see below ch, 7 and 8.

proofs, but one should also take into account the special condition for mathematical necessity laid out in EE 1222b23 — 41. Indeed, qua todi, which, paralleling qua rocôv and ouvexéc, effects S', is said in An. Post. 74a24/25 geometrical peyé0n are characterized as numbers (An. Post. 75b3— 6)" to be posited as a hypothesis (6 xa@6Aov ürotidevrar Úrápyerv). The Aristotelian hypotheses are equivalent to Euclid’s postulates and thus it remains to examine in what way a postulate can effect S', the single genus of Eudoxus’ proportion theory and, consequently, the single genus of geometry and mathematics in general”. In An. Post. 99a7— 10 Aristotle considers all geometrical peyé0n and numbers qua Exovra abënoiv toravéi” as wc év yéver. From a comparison with An, Post. 74a17—25 it is clear that Aristotle refers to S', the single genus of Eudoxus’ proportion theory in contrast with the fragmented genus of anthyphaeretic proportion theory. The same problem is brought up in An. Post, 7565/6, where Aristotle discusses the applicability of arithmetical proofs to geometrical peyé8n. He is aware that the genus of arithmetic is 67 paralleling the characterization of numbers as cuvezij, i.e. peyé0n, in Met. 1061b21 —25. There can be no doubt that the cases Aristotle has in mind are, or at least include, Eudoxus’ universal proportion-theoretic proofs but, since peyé0n and numbers are distinct genera, the applicability is allowed only if peyé0n and numbers are ,,one genus in some sense” (An. Post. 75b8/9), i.e. if there is a S', as in An. Post. 99a7— 10. Since Aristotle stipulates that a demonstrative transition from one genus to another presupposes that the two genera share something xoivév (An. Post. 75b20), which is their proper principles (An. Post. 76a15), it is reasonable to assume that, as An. Post. 87a38 — b4 shows, Si, „one genus in some sense” over distinct genera (= @¢ év yeveı), presupposes that S*, S' and all other genera share proper principles. The proper principles of a science include hypotheses (An. Post. 93b23) and thus a universal proof requires a postulate which establishes its necessity according to EE 1222b23—41. distinct from the genus of geometry (An. Post. 75b3—5 and 7/8); however, The set of principles that necessitate S', a requirement for the demonhe does allow for such an applicability in some cases and in these cases strative transition of a universal proof, supports the above interpretation of mathematical matter as causally related to the single genus of mathematics, its oixeia An, and implying a set of indemonstrable principles that characterize the distinct genera of geometrical ney&ßn and numbers. If qua zocóv and cuveyéc (Met. 1061a34/35; 1061b24) = qua toi (An. Post. 74a24) = qua Exov abEnoiv toiavôi (An. Post. 99a10; see below $ 10), the kowvóv of * For the relationship of Aristotelian hypotheses and the Euclidean postulates sec H. P. D. Lee, Geometrical Method and Aristotle's Account of the First Principles, CQ 29 (1935), 113—124; cf. B. Einarson, On Certain Mathematical Terms in Aristotle’s Logic, AJP 57 (1936), 33— 54 and 151 — 172, and Heath (above, n, 6) 53— 57, On the significance of hypotheses in Aristotle's conception of science see J. Hintikka, On the Ingredients of an Aristotelian Science, Nous 6 (1972), 66 — 69. Foran extensive discussion of hypotheses pertaining to the axiomatization of mathematics see Leszl (above, n. 1) 305-315. Any relationship between the Aristotelian hypotheses and the Euclidean postulates has been denied by Ch. H. Kahn, The Role on nous in the Cognition of First Principles in Posterior Analytics ii 19, in Berti (above, n. I) 391— 393; cf. also Ross (above, n. 38) 59, who complains that Aristotle should have recognized proper principles corresponding to the Euclidean postulates. Kahn thinks that Aristotle's distinction between definitions and hypotheses, which are assertions of existence, does not leave room for postulates of the Euclidean type. K. von Fritz, Die APXAI in der griechischen Mathematik, Archiv für Begriffsgeschichte | (1955), SAT. discusses extensively the relationship between Aristotle's hypotheses and Euclid’s postulates and shows that the required link between the two is the conception the distinct genera of geometrical ney&ßn and numbers according to An. Post. 75b20, imply a postulate which characterizes all these genera, these genera are (bc év yéver in the sense that all are true interpretations of the same postulate*. Thus in Met. 1061a28—b3 and 1061b21 —25 the distinct * See Knorr (above, n, 24) 310. In view of Eudoxus' proportion theory I cannot agree with Barnes (above, n. 36) 129 that Aristotle's characterization of geometrical ucyéôn as numbers is ,,a jocular hypothesis. Heath (above, n. 6) 45 does not relate this passage to Eudoxus’ proportion theory. Kullmann (above, n. 2) 253/254 construes this passage as excluding the possibility of a common characterization of peyé0n and numbers; thus he understands Aristotle's reference to Eudoxus' universal proofs as an indication that for Aristotle mathematics is compatible with his theory of science, which depends on the homogeneity of the genus. of geometrical existence as constructibility; since Euclid’s first three postulates are # On interpretations in Aristotle's logic see Corcoran (above, n. 1) 103— 107. construction postulates that assert the constructibility of basic ney&ßn, whose ex- For numbers and all classes of geometrical peyé8n as isomorphic models of an axiom system in Eudoxus' proportion theory see F. Beckmann, Neue Gesichtspunkte zum 5. Buch Euklids, AHES 4 (1967), 31; Beckmann, however, considers the fifth book of Euclid’s Elements as representing Eudoxus’ proportion theory (see below n. 69). A universal proof of the alternando theorem for all true interpretations of an axiom is a mathematically more palpable notion than the „proof by analogy to which Apostle (above, n. 3) 65 and Mendell (above, n. 3) 232/233 appeal. istence is assumed by Aristotle's hypotheses, an interpretation of hypotheses as postulates is consistent with a basic characteristic of Greek geometry. On Aristotle’s hypotheses as existence assumptions see B. Landor, Definitions and Hypotheses Posterior Analytics 72a19—25 and 76b35—77a4, Phronesis 26 (1981), 308—318. * For the mathematical meaning of this phrase see Knorr (above, n. 24) 291

genera of geometrical and mathematical objects belong to a single genus, the science dealing with „demonstration and knowledge" (Met. 1059b14—20). All references to Eudoxus’ proportion theory, and thus to the mathematical matter, in the Analytics are from a model-theoretic point of view because they are concerned with universal proofs. Although Aristotle vo that of geometry and mathematics respectively, as true interpretations of a postulate®, Moreover, since the single genus of these sciences is causally related to the product of mathematical abstraction, the mathematical matter, the latter must have been conceived by Aristotle as a true interpretation of 69 proofs of theorems“. Thus it turns out that Aristotle’s generalization of the realizes that universal proofs require a model-theoretic structure that subsumes distinct genera under ,,one genus in some sense“, it is in the Metaphysics that this requirement becomes of primary importance for the science of being*’. Moreover, Aristotle's usual example of a common axiom, which belongs to the domain of metaphysical inquiry, is an axiom pertaining only to the ,mathematical* category of quantity (equals subtracted from equals leave equals)*, and, since the mathematical matter implies a postulate common to all branches of mathematics, this notion is an appropriate subject metamathematical property of Eudoxus’ proportion theory over the entire for metaphysics. a postulate: that geometry and mathematics examine lines, planes, solids and numbers qua nocóv and ovveyéc, means that all these genera are true interpretations of a postulate which is implied by rocév and cvveyég. These interpretations are mathematical matter in the sense of ,explanatory’ or ‚demonstrative‘ matter for mathematical theorems, because, as An. Post. 74a17—25 shows, it is these interpretations which necessitate the universal genus of geometry, and especially mathematics, suggests a foundational Although in Met. 1061a28—b33 Aristotle draws an analogy between unification of geometry and arithmetic under an axiomatic system that matter is a fitting subject for the science of being. Because, since the genera the unified genus of mathematics and the unified genus of the science of being, it is impossible that he would have considered a unified genus of mathematics as a completed result“. In An. Post . 7663 — 6 he restricts the demonstrative transition of an arithmetical proof to geometrical peyé0n in some cases only, which definitely include proportion-theoretic results like the alternando theorem, and elsewhere he maintains the formal distinction between ueyéOn and numbers*’. The common characterization of peyé0n of departmental sciences qua beings are characterized by the set of common and numbers in Met. 1061a28—b33 can be explained by assuming that, axioms whose true interpretations are all genera, a model-theoretic structure under the influence of Eudoxus’ reformulation of proportion theory, Arsimilar to that of mathematics underlies the science of being. Aristotle is, istotle envisaged a unification of all branches of mathematics under an axiomatic system that characterizes all their genera in accordance with his theory of science*'. The start had been made by Eudoxus whose proportion necessitates universal proofs of the type Eudoxus introduced into proportion theory“ (see below $ 9). 5. DAN vont and Metaphysics If S' is a model-theoretic structure, then the notion of mathematical therefore, right in saying that the mathematical matter does not belong to # Cf. Scholz (above, n. 30) 59 n. 26. Corcoran (above, n. 1) 103/104 emphasizes the absence of any mention of alternative interpretations in Aristotle’s logical theory but one should also take into account sciences like mathematics and metaphysics whose genus comprises distinct gencra; see also below ch. 6. * On material explanation see Barnes (above, n. 36) 215/216. 4 It is usually assumed that Aristotle's conception of axiomatic science influ- # Cf. Mendell (above, n. 3) 238. * This is Euclid's third Common Notion. The fact that the third Common Notion is more at home in the fifth book of the Elements, a development of Eudoxus’ proportion theory (sce below n. 69), than in the first leads van der Waerden (above, n. 35) 183 and Seidenberg (above, n. 7) 273-279 to the conclusion that the third Common Notion was originally formulated within Eudoxus' proportion theory. This hypothesis is attractive: it provides a further relationship between Eudoxus’ proportion theory and Aristotle’s metaphysics and shows that Eudoxus had an interest enced Euclid’s Elements; see von Fritz (above, nr. 39) 45— 63 and 99 — 103, A. Szabo, Anfange des Euklidischen Axiomen-Systems, AHES 1 (1960), 104/105, K. Berka, Aristoteles und die Axiomatische Methode, Altertum 9 (1963), 200—205, and I. in establishing general principles. Mueller, Euclid’s Elements and the Axiomatic Method, British Journal for the Philosophy of Science 20 (1969), 289— 309. See, however, below n. 90. %9 Cf. Aristotle's awareness that his metascientific argument in favor of the unified genus of metaphysics cannot be unqualifyingly conclusive; see Ferejohn “ The validity of the alternando theorem for all distinct mathematical objects is guaranteed by the ,,Principle of Form“; see the comments in Scholz (above, n. 30) 59 n. 26 and the discussion in Corcoran (above, n. 1) 105/106 (cf. also 107/108). It is interesting to note that Euclid does not employ the Principle of Form; see Mueller (above, n. 1), 43/44. (above, n. 16) 126. 30 Cf. Knorr (above, n. 24) 310. 5! There are, however, indications that Aristotle considered the majority of the sciences complete or almost complete; see J. Barnes, Aristotle’s Theory of Demonstration, in Barnes, Schofield and Sorabji (above, n. 30) 85 n. 88.

71 theory proved theorems for all distinct genera of mathematical objects as istotle’s hints at a single science of mathematics should not be lost sight of. true interpretations of at least one axiom. Moreover, Eudoxus’ proportion The validity of the above interpretation requires that nooöv and ouvex&s theory most probably resulted from his reexamination of theorems from in Met. 1061a28 — b2 and 1061b21 — 25 bear the same meaning with todi in different branches of mathematics? and thus it was a theory with a wide An. Post. 74a24 (see below $ 10); noo6v and oœuveyéc and todi must be range of applications that could have inspired foundational speculations. It is quite possible that Eudoxus himself might have shared, or even initiated, shown to imply a basic characteristic of Eudoxus’ proportion theory, a such speculations*?, But one cannot venture further than that. postulate that necessitates the universal proofs of his theory. According to the Aristotelian theory this postulate has to be an existence postulate and it must also yield a definition of the primitive terms of both geometry and 6. Aristotle’s notion of interpretation arithmetic, peyé0n and numbers respectively (see below $ 8). The modern notion of interpretation should not be foisted upon Aristotle. A single science is one which has a single genus and a single genus 7. rnooôv, cvveyéc and Eudoxus’ ,Bisection Principle’ is defined by the primitive terms (An. Post. 87a38—bl). If the genera of Aristotle defines both nooöv and ovveyéc in terms of division (Met. geometry and arithmetic coalesce into a single genus of mathematics, their 1020a7 — 11). However, whereas rocóv is simply diaipetov, ovvexés is deprimitive terms, peyé0n and numbers respectively (An. Post. 76a43—b5), fined as tó cic Gxeipa diaipetdov (Phys. 200617 — 20) and in Phys. 20667 — 20 have to be identical”*. According to An. Post. 76a31 — 34, this implies the and 266b10—20 Aristotle expresses infinite divisibility by means of a prinidentification of proper geometrical and arithmetical principles that assume ciple that allows to exceed ravrós @picpévon [ueyéBovc] (Phys. 206b19/20; the existence and meaning of ney&ßn and numbers respectively, This is cf. 206b18/19, 266b3/4 and 266b19/20) by successive removal of a propor- Aristotle’s only implicit definition of interpretation and his concept of a tional part, which in Phys. 266b11/12 is the half. The principle Aristotle single genus of mathematics must necessarily conform with it; his stipulation employs to express the infinite divisibility of ovveyéc has been identified as that the demonstrative transition requires a set of proper principles common ‚the Bisection Principle* (henceforth BP) which played a cardinal role in to both genera suggests that this is the case and, therefore, Aristotle’s char- Eudoxus’ proportion theory”. Unlike the Euclidean convergence lemma (El. Prop. 10, 1), Eudoxus’ BP was an unproven assumption” and thus Aristotle's acterization of jey¿0n and numbers qua ovvezñ in Met. 1061b21 — 25 must imply such a set (see below $ 8). A further desideratum is to explain Aristotle's misgivings about the feasibility of a foundational unification of arithmetic mathematical matter implies indeed a postulate,which, according to An. and geometry (see below § 9). Moreover, the historical background of Aruniversal proofs of proportion-theoretic theorems: indeed, it is BP that allows #2 See Knorr (above, n. 24) 261 — 288. * For supporting evidence see above, n. 48. On Eudoxus and the formalization of proportion theory see Knorr (above, n. 24) 275. A drive to axiomatic mathematics Post. 74a17—25, has the significant foundational advantage of effecting Eudoxus to dispense with the anthyphaeretic definition of proportion‘, which, as shown above, not only is problematic from a mathematical point of view but also conflicts with Aristotle’s requirement of formal necessity for scientific propositions”. could have been related to Eudoxus’ stay in Plato’s Academy; see G. E. L. Owen, The Platonism of Aristotle, in Barnes, Schofield and Sorabji (above, n. 30) 26—28. % For a comprehensive discussion of BP and its relation to Eudoxus' proportion % See the analysis of An. Post. 87a38— b4 in Barnes (above, n. 36) 183; cf. J. theory see W. R. Knorr, Archimedes and the Pre-Euclidean Proportion Theory, AIHS van Rijen, Aspects of Aristotle's Logic of Modalities, Dordrecht-Boston-London 28 (1978), 200— 205 (see also 210/211 for Aristotle's use of BP). Aristotle’s consistent use of the verb brepBaAAew and the phrase ravrós dpropévov [ney&ßovg] in his enunciations of BP indicates that he refers to the wording of a particular formulation 1989, 177 — 179. . 5 Scholz (above, n. 30) 62 is mistaken in asserting that ,,we ... have to extend the notion of a domain and characterize it ultimately by the totality of the arguments of the principle. whose substitution for the free variables in a formalized axiom-system makes these axioms into true sentences“. Scholz understands such an extension as a characterization of primitive terms but An. Post. 87a38—b4 shows that Aristotle starts with of Mathematics and Philosophy in Antiquity, in Kretzmann (above, n. 33) 124/125. 5 See Knorr (above, n. 56) 205ff. and Infinity and Continuity: The Interaction the identification of the primitive terms of these domains in which propositions are 58 See Knorr (above, n. 56) 193 and cf. (above, n. 24), 272/273. 39 According to my interpretation of ÜAn vont as a true interpretation of BP true; cf. Corcoran (above, n. 1) 103/104 on the notion of intended interpretation in which necessitates a formal proof, a semicircle is indeed the 6An vontn of a circle, Aristotle's logic. as Aristotle takes it in Met. 1036a9/10 and 1037a4, because certain theorems about

A eee ee AA CIO There is, however, one fundamental difference between the Aristotelian formulation of BP and its use by Eudoxus in his exhaustion method, where BP most probably originated from®: whereas the Eudoxean formulation of BP avoids any reference to infinite continuation of division®', Aristotle's formulation in Phys. 206b7—20 and the other passages of the Physics presuppose the potentially infinite continuation of division. But according to Met. 1020a11 the potentially infinite divisibility of ouvexés characterizes only peyé0n, whereas in Met. 1061b23 —25 it extends to numbers, too. For Aristotle the axiomatization of mathematics depends on this common characterization of peyé0n and numbers. The problem that arises is to explain in what sense ueyé0n and numbers are characterized by BP and can be analyzed into two particular questions: what formal characterization of both ney&ßn and numbers is entailed by BP and what is the foundational importance of potentially infinite divisibility. In Met. 1061b23 — 25 numbers and peyé0n are roca. According to Met. 1020a7/8, this means that both are simply divisible, without any implication of finite or infinite divisibility that distinguishes numbers and peyé0n respectively. Thus rocóv, the sheer assumption of divisibility, seems to have the distinctive mark of Eudoxus’ formulation of BP as is applied in exhaustion proofs; indeed, since ovveyég is based on BP and, as Met. 1020a7— 11 shows, the only difference between nocdév and ovveyég is that the latter implies potentially infinite divisibility, and moreover, ouveyés is rogöv but not necessarily vice versa, rocòv must hark back to the original Aristotle's Mathematical Matter and Eudoxus’ Proportion Theory 73 If both numbers and peyé0n qua mood are characterized by BP, BP must not appeal to infinite divisibility. From Phys. 206b3 — 12 it is clear that finite and infinite divisibility are distinguished by means of a finite and infinite sequence respectively, depending on the common ratio“, Aristotle’s infinite sequence is not an arbitrary sequence but an approximation to a uéye00g given beforehand (Phys. 206b16—20), and thus it exhibits a basic feature of Eudoxus’ exhaustion method”. Now, BP allows Eudoxus to dispense with any reference to the infinity of such a sequence of proportional terms. If Aristotle operates with a formulation of BP that does not refer to infinite divisibility which characterizes only peyé0n, he must follow the exhaustion method in his formal handling of divisibility. Indeed, in Phys. 232b20—233a12, although he sets out to prove that time is cvveyég, i.e. ad infinitum divisible (Phys. 232b23— 26), and his aim calls for the construction of the infinite sequence in Phys. 206b7—9, he performs only a finite number of divisions and then he generalizes by appealing to the proportionality of these divisions (Phys. 233a3—5; for a detailed analysis of this proof see below,$ 11). In Phys. 207b27 — 34 Aristotle refers to proportional division as a recent mathematical development that makes the assumption of actual infinities unnecessary in mathematical proofs: by means of proportional division, given any péye00g divided in a certain ratio, the existence of another, arbitrarily small, péye0oc is guaranteedTM, The indirect handling of infinite sets of proportional terms by means of their proportionality alone, without any direct appeal to their mathematical formulation of BP as a principle of division to which Aristotle’s infinity, is a mark of the exhaustion method and shows that Aristotle operates potential infinity is irrelevant. In order now to solve the problem of the with a formulation of BP which per se does not imply either actual infinity common characterization of numbers and peyé0n by BP, it remains to see if Aristotle operates with the Eudoxean formulation of BP to which the Aristotelian notion of potential infinity is adjoined and, moreover, what is or potentially infinite divisibility‘°. For obvious reasons Aristotle's proof in Phys. 232b20—233a12 cannot imply a finite formulation of BP; the same holds for Eudoxus’ exhaustion method, because in that case his reasoning the link between Eudoxus’ BP and its reformulation by Aristotle as a characterization of ouvexés that covers both numbers and peyé0n. 62 See the discussion of this passage in Heath (above, n. 6) 108— 110. 63 See Mueller (above, n. 59) 235/236. the circle are proved by application of BP, which divides the arc of a circle. It cannot be accidental that, as Archimedes notes in the preface to his Quadrature of the Parabola, Eudoxus was the first to prove formally the theorem that circles are to one another as the squares on their diameters (El. Prop. 12,2) by means of BP, which successively divides arcs of a circle by bisecting the sides of an inscribed polygon; see Knorr (above, n. 56) 200 and I. Mueller, Philosophy of Mathematics and Deductive Structure in Euclid’s Elements, Cambridge, Mass.-London 1981, 200/201. ® See Knorr (above, n. 56) 200-205 and (above, n. 57) 123ff. 6! See Knorr (above, n. 57) 127. Pace J. Hintikka, Aristotelian Infinity, PhR 75 (1976), 197— 218, and Knorr (above, n. 57) 122 n.22, Heath, (above, n. 6) 111/112, is right in relating Aristotle's comment to Eudoxus’ exhaustion method. vüv in Phys. 207630 must refer to a recent mathematical development; cf. viv in An. Post. 74a23 which introduces Eudoxus’ new proof of the alternando theorem. Since Aristotle says that mathematical proofs avoid any assumption of infinity by means of a technique that combines division and proportionality, he can only refer to Eudoxus’ exhaustion method. 65 The case against any philosophical influence on Eudoxus’ introduction of BP and his method of exhaustion has been argued plausibly by Knorr (above, n.57) 123— 135; cf. Mueller (above, n. 59) 123— 135.

would fall victim to the geometrical naiveté of Antiphon’s attempt to square 8, the admission of infinity is required not only for physical applications but also for purely mathematical considerations that pertain directly to the Aristotelian conception of the foundational unity of mathematics. Similarly, in Met. 1061b21 —25 all peyé0n and numbers are rocà, true interpretations of BP in its Eudoxean formulation, but mathematics examines them qua ovvezij, i.e. true interpretations of BP in its Aristotelian formulation applicable to peyé0n only. Leaving aside potential divisibility, the only difference between Eudoxus' and Aristotle’s formulation of BP, for the time being, rocòv and ouvexij must presuppose a common characterization of both peyé0n and numbers. The infinite sequence of pro- 74 the circle by employing an intuitive precursor of BP“. Thus Aristotle seems to admit two formulations of BP: one that avoids any assumption of infinite divisibility by means of proportionality and is related to rocòv (Met. 1020a7/8), and one that implies potentially infinite divisibility, applicable only to pueyé0n and related to ovvezés (Met. 1020a10/11). Aristotle’s conjunction of rocóv and ouveyés in Met. 1061a28 — b3 implies the two formulations of BP which, coupled together as in the proof at Phys. 232b20— 233a12, exemplify two diverse Aristotelian tendencies: on the one hand mathematical finitism®’, which according to the above interpretation finds a parallel in Eudoxus’ exhaustion method and is thus related to formal mathematical proof, and on the other admission of infinity in a certain (potential) sense required for the geometrical analysis of physical processes among other things*. As it will be shown below in $ % For an analysis of Antiphon's argument sce Knorr (above, n. 57) 130— 132 and Mueller (above, n. 33) 154 — 156. A conception of BP that bypasses considerations of finite or infinite continuation of division explains Aristotle's problematic reference to general mathematical proofs in Met. 1077b17—20. According to this passage universal mathematical proofs treat peyé0n and numbers neither as ,,possessing péye080c" nor as „being divisible", As Ross (above, n. 6) 416 observes, it is odd that Aristotle seems to do away with divisibility, although it can be an essential property of all mathematical objects; cf. Heath (above, n. 6) 223. Indeed, Aristotle does characterize both ney&ßn and numbers in terms of divisibility, which, as rooöv and cuvezés, becomes of paramount importance for the axiomatization of mathematics; he simply filters out the specific kind of divisibility that characterizes only numbers or only peyé0n. „Being divisible must be a peculiar characteristic of numbers, in contrast to ,,possessing héyedos”. But, since péyedos implies infinite divisibility, according to Met. 1020a7—11 „being divisible“ must be a lax way of referring to the finite divisibility of numbers. According to this reading, Met. 1077a17— 20 does not undermine the interpretation of universal mathematical proofs as related to divisibility obtained by a formulation of BP that avoids any assumption about finite or infinite divisibility. #7 Aristotle’s mathematical finitism is extensively discussed by Hussey (above, portional terms in Phys. 206b7—9 and Aristotle's proof in Phys. 232b20— 233a12 show that ovvex&g presupposes the notion of proportionality; his brief reference to Eudoxus' exhaustion method in Phys. 207b27 — 34 shows that the Eudoxean application of BP depends on proportionality, too. Since BP plays a central role in Eudoxus' proportion theory, which treats commensurable peyé0n as primary, proving the same result for incommensurable peyé0n by means of an indirect argument that leads to a contradiction”, it seems that the required meeting point of the Aristotelian and the Eudoxean formulation of BP must be Eudoxus' characterization of proportionality. Eudoxus' effective definition of proportionality is Euclid’s parts-conception of proportion among integers (El. 7, Def. 20) and thus it accounts for the homogenous treatment of numbers and geometrical ueyé0n as peyé0n in an extended sense of the term that Aristotle requires in An. Post. 75b5/6. Now if it istaken for granted that, as will be shown in § 10, Aristotle operates with Eudoxus’ arithmetical notion of proportionality, the concomittant application of BP is bound to divide the proportional peyé0n into commensurable parts, so that the peyé6n can be treated as either geometrical ney&@n or numbers (see below $ 10). The above interpretation of nooöv as implying Eudoxus’ formulation of BP that does not commit one to an infinite sequence, shows that, on these premisses, Aristotle’s characterization of numbers and peyé0n as rooôv is justified. n. 6) 93-96 and 178/179. Hussey, however, mistakenly considers Eudoxus' proportion theory and his method of exhaustion as involving assumptions about infinitely large totalities; see Knorr (above, n. 57) 123— 135 and Mueller (above, n. 59) 233/234. It is more plausible that Aristotle’s finitism did not provide an alternative basis for the mathematics of his day but was indebted to the mathematical practice of his day and especially to Eudoxus’ methods which avoid any assumption about infinite totalities. to retain the infinite as potential infinite was the analysis of physical processes; for ouveyés and infinite as a prerequisite of motion sce Wieland (above, n. 15) 278. attempt to save the concept of infinity in the face of a movement among his con- For Aristotle’s mathematical physics see Hussey (above, n. 6) 185—200 and Anstotle’s Mathematical Physics: A Reconstruction, in Judson (above, n. 20) 213-242. ® For these and other characteristics of Eudoxus' proportion theory see Knorr (above, n. 56) 190/191. Based on certain theorems of Archimedes and Pappus, Knorr has plausibly reconstructed Eudoxus' proportion theory and has shown that the proportion theory of Euclid's Elements Book V is an evolution of Eudoxus’ protemporary mathematicians, like Eudoxus, to give up that concept. As Aristotle’s portion theory. “5 Knorr (above, n. 57) 121/122 considers Aristotle’s potential infinity as an mathematized definition of physics in Phys. 202b30ff. shows, the only reason he had 7% See Knorr (above, n. 56) 190 and 218.

The above discussion accounts for Aristotle’s use of nooöv in Phys. 237b23 —238a29, In Phys. 237b30/31 and 238a31 rooûkis rooôv is used for the number of times a pépog measures a péyedog; the same expression is used in Met. 1020b5/6 for plane and solid numbers. In Met. 1023b12— 17 Aristotle's Mathematical Matter and Eudoxus' Proportion Theory 77 § 11) shows that Aristotle employs both the BP and proportionality, exactly as Eudoxus does in his exhaustion method, in order to derive potentially infinite divisibility, required for the analysis of physical processes, by means of a formal mathematical method that avoids any assumption of actual infinity. However, the mathematical method is arithmetical. to his own formulation of BP for the foundational unification of numbers 8. Numbers and ovveyéc In Phys. 207433 —b21 Aristotle also implies a much closer connection between numbers and ouveyés that demonstrates the importance he attaches Aristotle defines pépog”! as the product of the division of nooöv and the arithmetical nature of both rooöv and the notion of measurement is shown proportional relationships”. The term is used in the same sense in Phys, and geometrical ey&ßn into the single genus of mathematics. For Aristotle, by Met. 1052b20ff. and Phys. 221a26ff. (esp. 221b14—20). Moreover, in Phys. 250217, 30, 31 and b2 rooóv is used for peyé0n that enter into 237a30/31 and 238a23, in proofs which presuppose the proportionality of (2)”, (3). und Metaphysik bei Aristoteles, Bern 1987, 209, who offers a stricter formulation for the successor of m. Cf, M, Mignucci, Aristotle's Arithmetic, in A, Graeser (cd.), Mathematik Charlton’s R should be understood as proportionally divided. % This characterization of potential infinite divisibility agrees with claim (5) in W, Charlton, Aristotle’s Potential Infinites, in Judson (above, n. 20) 141. In (1) ™ For this formulation see Mueller (above, n, 59) 140, 7 For the semantics necessary for the introduction of the modal operator see Hussey (above, n. 6) 178/179. Each Y%(pfi+1,y)) and %4(p(i+2,y)) can be conceived as a unit and thus OF (+2) (+2) > (i+1)) VG+1) But for Aristotle (2) is a restatement of(1) in terms ofi+l and i+2, i.e. the aggregates of ¥2(@(i+ 1,y))'s and 'A(@(i+ 2,y))'s: Ym ©3n (n>m) Thus Aristotle’s characterization of ovvex&g can be understood as Eudoxus’ BP, to which the semantics of a modal operator is adjoined. Apparently it is pli+1,y) > pli+2,y) and i+ 1 and i+2 correspond to the multitude of %(p(i+ 1,y))'s and %(p(i+2,y))'s that make up the &yedog y at the two successive divisions. According to Phys. 207433 =b21 a collection of units n greater than the collection m has to be introduced by a modal operator, since its existence depends on the potential divisibility of the péyeQoc y in {1}: operator ©” which accounts for potentially infinite divisibility: (1% Vo(l,y) = y, O3p(it+ Ly) = % (pi,y)) (y is a n&yedog), the Aristotelian formulation of BP requires the modal pll,y) = y and git ly) = “(@(iy)) a number is a finite set of units and the possibility of finding a greater set Eudoxus’ proportion theory”, Besides Aristotle's arithmetical treatment of the uniform motion theorem, most probably established by Eudoxus in his work On Speeds (see below $ 11), and employ techniques compatible with is grounded in the potentially infinite divisibility of peyé0n. If the BP is represented as a function satisfying the conditions’ peyé0n, it is interesting to note that Aristotle simply assumes the existence of a pÉpos that measures a given péysbog. This assumption, the taking of the m-part of a péyeOog, is tacitly used by Euclid in his proof of El. 5,5” and its use by Aristotle is justified, if he presupposes Eudoxus’ formulation of BP, which, as Phys. 207b27 — 34 shows, guarantees the existence of any m-part, however small; indeed, in Eudoxus’ proportion-theoretic proofs BP is used to derive the existence of a required péyebog assumed to be commensurable with a given uéy£0oc”*. A way, albeit derivative, in which Aristotle feels free to characterize — 233a12 (below proportionality. The analysis of the proof in Phys. 232b20 numbers as ovveyi] becomes now apparent: ouveyés depends on the Aristotelian version of BP, which, besides Eudoxus’ BP, captures his arithmetical notion of proportionality and, moreover, introduces potential infinity in lieu of actual infinity that the Eudoxean formulation of BP avoids. As it is, at the root of Aristotle’s ouvvexig lies an arithmetical characterization of 71 wépo¢ can also be used in the arithmetical sense of „an aliquot part“ (cf. El. Def. 7, 3); see Ch. Mugler, Dictionnaire Historique de la Terminologie Géométrique des Grecs, Paris 1959, sub voce HÉpog. On the Euclidean notion ofpépog see Mueller (above, n. 59) 61/62; cf. D.H. Fowler, The Mathematics of Plato's Academy: A New Reconstruction, Oxford 71990, 226—228. 7? On the proportionality in Phys, 249b7—250025 see Hussey (above, n. 68) 215f. Crawley (above, n. 3) 168/169 proposes that noodv might mean „quanta proportional in a proportionality". ™ See Knorr (above, n. 56) 211 n. 80. ™ Sce Mueller (above, n. 59) 122. ™ See Knorr (above, n. 56) 187 and 189; for further applications of this Eudoxean technique in Aristotle's Physics see 233a35 — b3 (cf. Knorr (above, n. 56] 197 n. 57) 237b28, 238a6/7, 238a11/12 and a21, 266b21 — 24.

Aristotle can use the BP to derive the existence of a collection of units n as (3) explains Aristotle's statements that tò 8° Ev ét pétpov onpaiver pavepôv (Met, 1078b33/34) and onpaiver yap zò Ev Gt pétpov nÂndovs tıvög, Kai 6 dpiOpds Sti roc pepetpynpévov Kai rAN0os nörpwv (Met. 1088a4 —6). According to Met. 1020a8/9 rAùdog is defined as &pOuntov rogóv and jtétpov can be understood only in its technical geometrical sense, because any Yılpli,y)) is a measure of y. As it is, Aristotle’s definition presupposes the fundamental undefined notion in Euclid’s arithmetic, measurement”, But, since in Met. 1020a9/10 the ap.Gpytov rooóv is contrasted with petpnrôv noaóv, which is a ouveyès péyeBoc, Aristotle’s characterization of numbers in terms of measurement as a nAfjdog peperpnpévov Kai nÂndos pérpwv offers the link between ovvegég and numbers. ouveyés is grounded in BP, which, as Phys. 207a33—207b2 shows, Aristotle uses in order to deal with a fundamental point in arithmetic: whereas Euclid seems to assume the existence of an infinite number of units, any finite collection ofwhich is a number"', Aristotle uses BP in order to account for the potential existence of any finite collection of units in agreement with his exclusion of actual infinities**, Thus a geometrical postulate accounts for the existence (rö elvat, An. Post. 75b6) of units, the primitive terms of arithmetic (An. Post. 76b3—5). Moreover, Met. 1088a4—6 shows that divisibility in the broad sense accounts for the todi eivat (An. Post. 75b6) of a unit and, therefore, the BP, the axiomatic guarantor of divisibility, fully qualifies as a principle of the genus of arithmetic according to An. Post. 76a31 — 36. Aristotle's Mathematical Matter and Eudoxus’ Proportion Theory 79 Met. 1061b21—25 to numbers and peyéOn qua ouvexnj as the olxeia An of mathematics does not involve any contradiction. 9, Some Historical Comments Such a unification of the principles of arithmetic and geometry respectively into a single set of principies of mathematics agrees with some fundamental characteristics of Euclid’s Elements. Euclid’s treatment of jeyé07 is similar to his treatment of numbers”, although Euclid never formulates a common characterization of peyé0n and numbers like Aristotle does. Moreover, in (1) (i+ 1) and (i+ 2) are metalanguage variables, whereas (3), which Aristotle must consider equivalent to (1), they are object language variables; the same formal situation occurs in Euclid’s Def. 7, 16, his only definition of an arithmetical operation", As shown above, Aristotle, like Euclid, takes measurement as a fundamental notion for both numbers and peyé0n. Thus Eudoxus’ proportion theory could have provided a basis for an Aristotelian axiomatization of mathematics consistent with fundamental characteristics of Greek geometry and arithmetic. Given the fact that Eudoxus’ handling of ratio could have led to a concept of real number®®, such an axiomatization could have made a significant contribution to Greek mathematics. But, although Aristotle realizes the possibility of a unification of geometry and arithmetic consistent with his theory of science, he prefers to propositions (An. Post. 87b1 —4)%. It follows that, if there is a single genus keep the two domains separate. For Aristotle a demonstrated proposition must belong to the same genus ag the indemonstrable propositions, i.e. the same primitive terms must appear in both demonstrable and indemonstrable of mathematics, no subset of its primitive terms? could be a counter-interpretation®® either of an indemonstrable or demonstrable mathematical proposition. However, there are indemonstrable propositions which are true only for peyé0n, like the assumption of the m-part and fourth proportional, both of which are used by Aristotle (see above§ 7 and below § 10 respectively). But BP does the same for the domain of geometrical peyéOn. In Phys. 207027 — 34 it is the BP that account for the elvaı (Phys. 207b30) of any Héyebog required in geometrical exhaustion proofs, Thus the BP is central for the existence of both numbers and some peyé0n. Moreover, the tobi elvar of geometrical ney&ßn, given in Met. 102047— 11 as petpntov zooóv, turns out to be exactly the same as the rodi elvat of numbers given in Met. 1088a4—6. It follows that the formal expression of divisibility is also a principle for the genus of geometrical j1eyé6n and, since the genus of numbers tation and the theory of incommensurable jeyé01n provides numerous de- ** On counter interpretations in Aristotle’s logic see Corcoran (above, n. I) 105. #5 See Knorr (above, n. 56) 219/220. #6 See Scholz (above, n. 30) 62, 5” Cf. Corcoran (above, n. 30) 62. See Mueller (above, n. 59) 59/60. # See Mueller (above, a, 59) 121/122, In these cases, the primitive terms of arithmetic can be a counter-interpreand the genus of peyê0n have the same principles, the primitive terms of both arithmetic and geometry can be unified into the single genus of mathematies as required in An. Post. 87a38/39. Thus ouvex&g encapsulates both BP and potential infinite divisibility which yield common proper principles for the genera of both geometry and arithmetic and Aristotle's reference in * For measurement in Euclid’s arithmetic see Mueller (above, n, 59) 61-64, *! For Euclid’s conception of unit and number see Mueller (above, n. 59) S8IT. #7 See Hussey (above, n. 6) 97 and 178.

tpb1hue0ety6é1pa0bNrln2eosv1opeao—nr2odth5nie.-olunEesmutsbdh,eoerAxoseriuyasntoohtfcaolEbednulscdyailocdiahedbir’sasietvmnfiepitodflntfysthraobofmuonhrkito?fhm!uie,oclEaglteinomenaiontpfrysmoap(rotserhetimnboatenrloitdchwuseconie.rndy1Mi0fen1ot)r.o,- 1ewfamsv0oxhuaoa.elilvanlolsetAtomyruwairiesnraludtitoblpozneyteaiarlrcttuedain’itposrhrnfneaoCy”tcpdh:otfaiGostrntoruihahonnetecdittaheoktarnotsilpmhozyeanlawtapithTstchlihehklaoeasmsyenotacrkbtthreoafiiatfsecutrciiotoPasecsuurcn,rotKaepeiodrousvmeirpr-aztecofnairmrnthtoyeeaislennedtogamoranlesoplistfrotiefmyozchtafepeatdrlahoireipydpnmonoearsEvtnsaueidircclobesnoAli*apredt’lm’hi,AsseetrnIyoiEttsplfltiresoeoimrwtsmceioobnltumotuiiletslavd-h,nde element. rationofKnor’sanalysis. theoryproportionEudoxus’ofdevelopmentaistheoryproportionwhoseVElements GtpMhu1aBMAa)isrxgaeesutliorlcn1aihepo.m1tklf3eeha(m.rnotntaa'dhoob5nseritozi9ugeva)sadAretkcnngho,i”hxda,o-.ifMzIniNPoaun(orA.mhsexasfltyilbihptssooterhinvmohezokeratHp,uftViolsgiIMwerhescIunompmtsenvecIaaiEtwIlcehtA.lohierneniaSs,cmlEcsernuIsieuchtny.oe1knteh7dfnloti7n.aG)etosnw.urdinre(Cv,osdav11.ehfstev99a(gsk2r7ooaenu88fnxbog),midtio,nepEcsvoeDf.tlmpn3oi,ehlasrTromieaty2Shdnoiv5,cKehrsdnz4eioeaot;ddncrtmeawe2ohyicanv-dw4otbetSk(kn0a-xltere—2Bsdounm4soBgplde.os(cuea)vl1Adt,ic9rseoenrh7tAinrEs5agu-7An)xMt1ar.cr,Li0zyoloa6ts2imwn/luhiftdFad1emtoo0uESmitrAl7.nAehctlmaxeMah43tliib2enls1hok)Tytad2meo,,ehsh—nauGroEeP.3ntcbrgudl1vihe(umoy”4irkaemplss.ibintoEaccoidgihvh5soenyet9snd,,re) limitations.spaceofbecausedetailin wEGnDittur.mhhcepel(iHoamc-dirbhy'nootdsivfapniertk,awcohniepaManyouy,troeGdhtflrJiehi.useoascinxtdeurilxsteIopafhsfmlgeo.tcaeuoitsan2niksmd6trzn)ieaougiashntmfis.poaodltrxn(youiaDemihobsipmenomoptpvsdoihaetSsyoepE,lissnrurtciiMoshncbuploshleoiau,zorldng,mtcehioaeisHfotegr6.innh8to.t)serna,tfesmeohoaifenld4t,rE2seinlytH),aocerontamtsfodilie.xmconiAlp6pnnt(reoltsaisnteuHsrb.hotntefoedhv(dtueecaulfc,bneo.tod'flfivB6steoe9hrrbfn),ecsoo;4rttHk8oeihan;uspomke.nnsr4dtFoafnhs6spiebeinf—oyAgtrnrhavat%liie*itfns'?hiotdw(ceonisat4oeonlhtffe ntermsofarithmetic.w2(seaoiemAt2fscptAEBK4x7u.rrhfhmun)Z3iitcoreae7sd5shphhTus6ntrltoyifa3p39%%ba)OmuhswrnAxopcot1allzpterouyhunlf/b1(iqhtAeSSptUas9n3aosmur—rc'eehté'f2b2dntgsiia(tlhvheyqo-s3sraBzipsvptEAet8)bKuKdonkrnehiloA(0ayiiwnrTsEd,etrfbEs0n.iosdlo’eoat2u=pHf,g<ienEtrrfpaBn3oT.itsceIxhnyorufZalTdlp)ts8ntoer5de,hm’itEfn(b'noaD(6utkpocoisehmda=d(2)lws1ati’nBbCxr1BbpMmeri(oDw6fZoAE0odnsau1teaeqpZvi(.—vuCt9hnb2sgTntraueB<seBhida8ot7,hryH,gea1Ao/n(A2vZlcpai3tiom81,xeyl27rnéBap,sanihbd9oc,uA..)nttIyrE=9e3adseB’ch2ilnta,po5yimr’a5(.(h6Zcen—xt62f,siloipdv6ha'5E(rn))gDsMtnye)9ilPfls,ep1oaBr.vao—C3)rbs(1piwt-Mt1eoy4o39Aualsns91r6Bhp)—x,e21Es0t95thof.epTi’.nD8ec)7bu+hrinohts,lHeved.34réoACsrTaoetiewhgbBybpm)u(nwAoDraefxo3hyZa?t2n,erlEviuC0B—asixod,m”den7rvTpttuifc6ib,rZa-he3hs’2sEg)qwoody,)ctaepr(lube7smKohyfniu”ttiad,a3c)1ei=ndro..bflaroimnbos0,cwdeetAinxtf1irh—id(mnapessuectacyAbh1s—uoBngiirslt”pmienA5nidhm3cPt(fe.srcu,on)ufsghri3cdatbe:ouism)eétpArhwnDfBpcmoslqobasgBeuZtevCorpsnwlfda=h)triA.oucrea(ifB'i,te2ysnos)o(c7hZt,.n;a”resnEipP3mf*ToseT.hrtIwpbnas,igdthyos5erai.pnaseoft/(Tclnivreuo,rfso6dta(hDKmeey'ifA.2nlycaonptCusB)é2chtiforeAn.gsPi,Z'0atcr2dsyB=fhntAn6is7arE(eoc0log3ZunbmreiywE(hoq)dpe9r<a.ns2bumBtcÈ,ef(bh—l/ixwtAaoroZag3qDrudeihstvb1$mT)nBsaoAleCo21Mimdyt’n,e-s g5(evKos2abnmr1b6ear3o.tnierfvc)oyhT;ure.,s veHxcD2taeoihp7wrl3remcnaove2yCreci1szfarit,to.n

Thus Aristotle does not state explicitly that BZ is finite because this is already implicit in the definition of proportionality appealed to in (3). (ii) is presupposed by Eudoxus’ treatment of proportionality and, moreover, (ii) amounts to a formal characterization of proportionality that covers both peyé0n and numbers and thus expands the notion of n&yedog to include numbers”, as Aristotle requires in An.Post. 75b5/6 for universal proofs”. (ii) is provable within the proportion theory of Euclid’s Elements V'TM, and, since, according to Aristotle, the homogeneity in Eudoxus’ universal proofs is effected by a hypothesis that characterizes both peyé0n and numbers, there are strong reasons to believe that (ii) was axiomatically posited by Eudoxus as a definition of proportionality for the extended notion of uéyedos!". todi in An. Post. 74a24 presupposes this extended notion of péyedos and éyov abEnotv roravöi, the equivalent of todi in An.Post. 99a10, turns out to be an accurate description of proportional terms, as they are defined in (ii), that can be both peyé@n and numbers. The fact that in Met. 1061a32— b3 proportion-theoretic properties are proven for all geometrical pey¿0n qua rocé and ovveyzij, which imply BP, does not contradict this result; because once it is realized that (ii) offers a definition ofproportionality to think that Aristotle implies Euclid’s Def. 5,4; on the axiom of Archimedes and Euclid's Def. 5,4 see Mueller (above, n. 59) 139 — 145, 98 (ji) explains why for Aristotle the ouveyés can be divided only in those ratios which are specified by the natural numbers. Pace Bostock (above, n. 20) 186/187. this restriction should not be considered as an intrinsic problem of the Aristotelian characterization of ouveyxés, nor should Aristotle’s ovveyéc be criticized from a modern mathematical point of view. For an interpretation of physical continuum not as „Punktmenge“ but as „Teilbarkeitseigenschaften“ in the Aristotelian sense see C.F. von Weizsäcker, Das Kontinuum, in E. Rudolph (ed.), Zeit, Bewegung Handlung. Studien zur Zeitabhandlung des Aristoteles, Stuttgart 1988, 144— 167 esp. 146/147. % See Mueller (above, n. 59) 137/138. = For the proof see Mueller (above, n. 59) 137/138. : Mueller has proposed (ii) in order to account for the notable absence of a definition of proportionality for both ney&ßn and numbers from Euclid's Elements, although it is necessary for Def. 10,5. According to my interpretation, Aristotle presupposes such a definition of proportionality, which agrees with Eudoxus' proportion-theoretic proofs and accounts for his removal of homogeneity assumptions. dI for both peyé0n and numbers, BP provides not only a way out of the anthyphaeretic definition of proportionality but the necessary condition that ueyé0n in the extended sense have to satisfy in order to enter into proportion relationships'”. Moreover, the term aö&nv occurs in Phys. 206628 as successive addition of proportional terms that complements successive division by means of BP. This complementarity is formally expressed in Phys. 26662 —4 as the equivalence of BP and Archimedes’ axiom’. If abEnow toravéi in An. Post. 99a10 recalls the corresponding term abEnv from Phys. 206b28 and the axiom of Archimedes, then Exov avena touavôi and moody and ouvexés pick out two equivalent postulates, the Archimedean axiom and BP respectively, as properties that ney&ßn have to satisfy in order to enter into proportion relationships as defined in (ii) ‘*. In Phys. 207b27—32 divisibility, obtained by means of BP, is a property of ney&ßn that can enter into proportionalities; in the context of proportion theory, the emphasis on divisibility can be historically explained by the fact that Eudoxus’ proportion theory evolved from his method of exhaustion'®’. On the other hand, (ii) requires that the proportional ney&ßn be Archimedean. As a final point, one notices that, although todi = Exov adEnow toravdi = rooôv and ouvexés presuppose a definition of the mathematical entities, both ney&ßn and numbers, of proportion theory’s genus, they actually amount to existence postulates. This situation is symptomatic of the close relationship between definitions and postulates that has been observed in both Aristotle and Greek mathematics’. : 11. Aristotle’s ouvey&g and Eudoxus’ Exhaustion Method Phys. 232b20—233a12 contains Aristotle’s only proof that a péyedos, time, is ouveyéc. Since ovvexés is mathematically defined in Phys. 206b7 —9 102 Cf. the comments in Knorr (above, n. 56) 235. However, his thesis that a complete theory of proportion on the Eudoxean basis was never worked out completely is not convincing. Aristotle's references to Eudoxus' proportion theory and proofs for both peyéôn and numbers imply the existence of certain postulates, which suggest a complete, or almost complete, formalization of Eudoxus’ proportion theory. 103 For this equivalence see Mueller (above, n. 59) 142/143. 10% For a formulation of the axiom of Archimedes compatible with (ii) see D. If this is so, Becker (above, n. 35) 311 — 333 is right in speculating about an original Bostock, Logic and Arithmetic, ii. Rational and Irrational Numbers, Oxford 1979, of Euclid’s tenth book which made use of a treatment of proportionality not found 130— 134. in the Elements. Given the essential relationship between Eudoxus' proportion theory 105 See Knorr (above, n. 56) 235. book can very well go back to Eudoxus’ work on incommensurable peyé0n which 106 See von Fritz (above, n. 39) 391 — 393 and 412/413, and J. L. Ackrill, Aristotle’s Theory of Definition and Some Questions on Posterior Analytics ii 8-10, did contain a definition of proportionality for both peyé0n and numbers. in Berti (above, n. 1) 395— 384. and his work on incommensurability (see Knorr [above n. 24] 261ff.), Euclid’s tenth

by means of the BP as an infinite decreasing sequence of proportional terms, one expects that Aristotle’s proof should aim at the construction of such an infinite sequence and Aristotle's definition of ouveyéc in Phys. 232b24/25 seems to justify this expectation'”. His proof, however, seems to follow a quite different route, He considers a péye00g PA, a temporal interval ZH and an object A moving faster than an object B which during the time ZH traverses the péye8o0c TA. According to the result established in Phys. 232b5—20 A traverses TA in a time ZO<ZH and correspondingly B in ZO traverses a péyeboc TK <TA; again, A traverses FK in a time ([ZAJ< ZO. This divisions, which can be represented by the inequalities TA<TK and ZH>ZO> [ZA], must be shown to continue ad infinitum, if Aristotle is going to prove that time is infinitely divisible. How then can he claim in Phys. 233a5— 7 that any temporal interval ZH can yield a subinterval ZO? Before this generalization he rounds off the reciprocal divisions of FA and ZH with the remark that the divisions of TA and ZH are proportional (Phys. 233a3/4). The proportionality of time and péye@0g has not been established before it is implied in Phys. 220b15—221a9, where it is said that time measures motion and motion measures time (Phys. 220623/24; cf. 220b14— 18) and, moreover, that time determines a motion that measures the entire motion (Phys. 220b32—221a4). Aristotle’s comments clearly pre- 85 Thus Aristotle's proof runs as follows: by appealing to the result established in Phys. 232b5 — 20, he divides ZH proportionally to PA. But for any submultiple of TA, however small, the proportionality established by the theorem of uniform motion guarantees the existence of a corresponding submultiple of ZH. Thus the combined use of division and proportionality as established by a formally proven theorem eliminates any need for the non-constructive introduction of an actually infinite set. This technique is a distinctive characteristic of Eudoxus’ exhaustion method and Aristotle’s — 233a12 he presents remark in Phys. 233a13— 16 shows that in Phys. 232620 a formal proof and not a commonsensical argument that time is OUVEZÉS because half the distance is always traversed in half the time and so on'"!. The problem with such a Zenonian argument, which Aristotle immediately proceeds to refute in Phys. 233a21 -b5, is obviously the introduction of actual infinity. But it would be mistaken to assume that it is the mathematics involved that derive the potentially infinite divisibility implicit in ovveyéc'””. Nowhere does Aristotle imply that the mathematics of his age was in any way concerned with the notion of infinity and Phys. 207b27 — 34 answers negatively the question in Phys. 204a34 —b1. As the discussion of divisibility in GC 1,2 shows, potential division is a matter not of mathematics per se ". but of the semantics of the modal operator adjoined to (1)! suppose the definition of uniform motion which is referred to in Phys. The implied commensurability of time and péye00c means that TA=nTK !!! For the formal character of book Z of the Physics see J. Jope, Subordinate Demonstrative Science in the Sixth Book of Aristotle's Physics, CQ 22 (1972), and ZH=nZO; if one operates with (ii) as a definition of proportionality, 279 — 292. the proportionality of time and péye0oc results immediately. Eudoxus must who claims that Physics Z does not know the potential infinity of Physics T. Besides 233b4—6 and, most probably, comes from Eudoxus’ work On Speeds'®, have established as a theorem that in uniform motion the distance is pro- = pi 112 This is not the place for a detailed refutation of Bostock (above, n. 20) 180, the évééyetar in Phys. 232b21, which clearly implies potentiality, against Bostock's proportional divisions of TA and ZH must appeal to the uniform motion thesis speaks the fact that, although he is willing to recognize similar mistakes in the handling of infinity in both Z and T, he separates the two books as far as the theorem, which is the drodederypévov in Phys. 233a7!!°. Moreover, (ii) notion of potential infinity is concerned. explains Aristotle's conclusion in Phys. 233a11/12 that time and uéye0oc the ¿vóéyerar in Phys. 232b21. As far as I know, the modalities in Aristotle's Physics is a topic still waiting detailed investigation. On the notion of chance, which figures twice in the important passage in Phys. 249b27— 250b7, see M. Mignucci, "O ext portional to time’ and, since (ii) was his definition of proportionality, the have the same and equal divisions. 107 Cf. Heath (above, n. 6) 130. 108 See Knorr (above, n. 57) 120 n. 19. ' See Knorr (above, n. 57) 120 n. 19; for fragments of Eudoxos' work see F. Lasserre, Die Fragmente des Eudoxos von Knidos, Berlin 1966, 67 — 74 and 198 — 212. © This word cannot refer to the result established in Phys. 232b5— 20, which is clearly referred to in Phys. 233a5 — 7; it remains that the dnodederypévov can only be the proportionality in Phys. 233a3/4, which has nothing to do with Phys. 232b5 — 20 and thus must imply an independent result. 113 See the discussion of GC 1,2 in Charlton (above, n. 78) 135ff. Note also tó mob et Nécessaire dans la Conception Aristotélicienne de la Science, in Berti (above, n. 1) 173—203, and L. Judson, Chance and Always or for the Most Part, in Judson (above, n. 20) 73-99.