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Information to Users INFORMATION TO USERS This manuscrit has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The qnali^ of this reproduction is dependent upon the qnali^ of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins^ and inqnoper alignment can adversefy afreet reproductioiL In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beghming at the upper left-hand comer and continuing from left to right in equal sections with small overlaps. Each original is also photographed in one exposure and is included in reduced form at the back of the book. Photogrsphs included in the original manuscript have been reproduced xerographically in this copy. Higher quali^ 6" x 9" black and white photographic prints are available for aiy photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order. UMI A Bell & Howell Information Company 300 North Zeeb Road. Ann Arbor. Ml 48106-1346 USA 313.'761-4700 800/521-0600 ARISTOHE ON MATHEMATICAL INFINITY DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of the Ohio State University By Theokritos Kouremenos, B.A., M.A ***** The Ohio State University 1995 Dissertation Committee; Approved by D.E. Hahm A.J. Silverman Adviser J.W. Allison Department of Classics H.R. Mendell ÜMI Number: 9612214 UMI Microform 9612214 Copyright 1996, by UMI Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. UMI 300 North Zeeb Road Ann Arbor, HI 48103 Copyright by Theokritos Kouremenos 1995 VTTA June 7, 1967 ..............Bom -Patra, Greece 1989 ......B.A., Aristotle University of Thessaloniki, Greece 1989 - 1990 ..Visiting Fellow, University of Cologne, Germany 1990 - 1992 ... Teaching Associate, Department of Classics, The Ohio State University, Columbus, Ohio 1992 M.A. Classics, The Ohio State University, Columbus, Ohio 1992 -1994...................Teaching Associate, Department of Classics, The Ohio State University, Columbus, Ohio FIELDS OF STUDY Major Field: Classics u TABLE OF CONTENTS VITA.............................................................................................................Ü LIST OF FIGURES......................................................................................iv INTRODUCTION.........................................................................................1 CHAPTER PAGE I. ARISTOTLE’S POTENTIAL INFINITY AND MODERN SCHOLARSHIP......................................... 16 n. DC 271b26-272a7: ARBITRARILY LARGE MAGNITUDES AND EUCLIDEAN GEOMETRY..............44 m. POTENTIAL INFINITY AND ACTUALIZATION................65 IV. ARISTOTLE’S CRITIQUE OF PLATO IN PHYSICS T ........77 V. ACTUAL INFINITY AND GREEK MATHAMATICS.......... 8 8 VI. ARISTOTLE AND EUDOXUS’ PLATONISM.....................120 VH. ON THE ORIGIN OF THE EXHAUSTION METHOD 129 BIBLIOGRAPHY...................................................................147 m LIST OF FIGURES FIGURES PAGES 1. Euclid’s exhaustion proof of El. 12.1.....................23 2. Aristotle’s argument against an actually infinite cosmos {DC 271b26-272a7) ........ .45 3. Graphic representation of the argument in Phys. 266b6-14.................................................. 48 4. The proof of El 1.31 in Aristotle’s Met. 1051a21-29..................................................... 6 8 5. Aristotle’s argument against an actually infinite cosmos {DC 272b25-28) ..............92 IV INTRODUCTION My purpose in this disertation is to take a fresh look at Aristotle’s notion of potential infinity in its relation to mathematics, an issue Aristotle explicitly touches upon only once in Phys. 207b27-34. The upshot of the passage is clear: according to Aristotle his rejection of actual, untraversable infinity does not cause any problem to mathematicians who neither use nor need this kind of infinity but require in their proofs only the existence of arbitrarily small finite magnitudes, i.e. potential infinity. Aristotle’s brief remark is of particular historical importance. One has only to think of Cantor’s rebuttal of Aristotle and his concomitant assertion that actual infinity is indispensible to mathematics; the constructivist reaction which reasserted the sufficiency of potential infinity; Hilbert’s famous "On the Infinite" (1906) which initiated proof theory in an attempt to resolve the issue of mathematical infinity. In the formative period of the history of calculus mathematicians agreed with Aristotle that Greek geometry did not need actual infinity -they pointed out the proofs by means of the exhaustion method which, instead of the actual infinity of a magnitude’s divisions, require only that a finite magnitude less than a given magnitude. This is a classic case of potential infinity in the Aristotelian sense but recent scholarship has responded to Aristotle’s remark in P/iy5.207b27-34 in a rather 2 hostile way. It is useful to give here a detailed hst of all senses of potential infinity Aristotle operates with in order to delineate clearly both my purpose in this dissertation and the received doctrine which I will question. Aristotle operates with various formulations of potential infinity: Pj! Potential infinity of numben VneN Cm (m > n) A -<(3MeN Vm (M > m) Pz: Potential infinity of extended magnitudes, lines areas or volumes, which is adequately captured by Pj if n and m range over extended magnitudes instead of natural numbers; this reduction is home out by DC 271b26-272a7 (to be discussed in ch. 2 ). P3 : Potential infinity by division of an extended magnitude, which can be expressed by means of a function f as follows: f(l, y) = y , f(n, y) > 1/2 f(n-l, y) A y > x 3neN f(n, y) < x Since n ranges over the natural numbers, this formulation captures Aristotle’s reduction of the potential infinity of numbers to that of division of extended magnitudes (Phys. 207bl-15): each n corresponds to the multitude of parts that results in each step of the operation. P 3 also captures the convergence lemma ("Euclid’s lemma") in EL 10.1 which plays a crucial role in exhaustion proofs. 3 P^: Potential infinity by inverse addition, which is the reciprocal of P,. Defining s„ as the sum f(2, y) + f(3, y) +...+ f(n, y), then4 Passerts Vs. s. < y. In Phys. 206b3-b22 Aristotle explicitly commits himself to P 4 and P 3 and, therefore, to Pj, as wül turn out later in 207b 1-15. In Phys. 206b9-12 he also commits himself to a version of P^, namely that VM, m < M 3neN (M = mn), a principle that does not hold universally for numbers. Aristotle commits himself to Pj in DC 271b26-272a7 and Phys. 266b2-4. But there is no reason to assume that by the time he wrote Phys. 206b3-22 he had not commited himself to P%, since in this passage he commits himself to a variation of P^. In Phys. 206b3-22 Aristotle brings m the variation of P^ in contrast to P 3 which means that it is 3P on which he focuses. This is natural, as I argue in ch. 4, because Pj plays a crucial role in Plato’s account of infinity. The thesis usually attributed to Aristotle is exactly the negation of the stipulation in P^-Pz and my main point in this dissertation is that Aristotle does adhere to P^ The received doctrine arises from the fact that Aristotle develops the notion of potential infinity in the context of his belief in the finitude of the universe. Correspondingly, so goes the communis opinio, Aristotle might allow arbitrarily small magnitudes via P 3 , as he does in Phys. 207b27-34, in geometrical proofs but has to reject arbitrarily large extensions (Pj) -otherwise there would be magnitudes potentially greater that the greatest physical extension, the diameter of the universe: but Euclidean geometry does need arbitrarily large extensions for the theory of 4 parallel lines. This interpretation, already found in Chemiss (1935) and Solmsen (1960), became influential with Hintikka (1973) who read rejection of arbitrarily extended geometric magnitudes in Phys. 207b 15-20; he, moreover, argued that this reading is cogent given that, for Aristotle, geometric magnitudes have to be abstracted from corresponding physical magnitudes. This view had, on the one hand, led to some remarkable new conclusions and, on the other, resurrected an old one. Hintikka himself concluded that Aristotle’s physical universe cannot be Euclidean since the parallel postulate, which requires straight lines produced ad infinitum, cannot be satisfied in it. For Knorr (1982) this situation blatantly contradicts Aristotle’s claim in Phys. 207b27-34 and suggests his lack of mathematical sophistication, a charge levelled against Aristotle already by Milhaud (1902). Hussey (1983) and White (1992) tried to defend Aristotle. For Hussey Aristotle demanded a finitist overhaul of Euclidean geometry which would get rid of the problem by substituting for the parallel postulate an equivalent proposition, El 1.32. White went even a step further than Hintikka attributing to Aristotle a conception of physical space that has both Euclidean
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