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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 and non-Euclidean characteristics -necessarily, if the parallel postulate is not satisfied. Nevertheless, for White Aristotle can account for the needs of
Euclidean geometry, as far as arbitrary extensions are concerned, because of his instrumentalism: geometric magnitudes are formally correct but meaningless and are to be accepted only insofar as they do not involve absurd physical consequences, i.e. they do not violate the finitude of the Aristotelian cosmos. 5 Both attempts fail. There is no evidence to support White’s anachronistic attribution of instrumentalism to Aristotle, Hilbert’s way of adherring to strictly finitary mathematics without foregoing its infinitaiy extension. Hussey, on the other hand, is wrong because El. 1.32 is equivalent to the parallel postulate only on the assumption of the Archimedean axiom which does allow for arbitrary geometrical extensions. But the real problem is that the charges against Aristotle to which
Hussey and White responded are unsubstantiated to begin with, Aristotle not only explicitly invokes the Archimedean axiom in Phys. 266a33-b3, he also clearly implies it in his discussion of potential infinity in Physics r (206b9-12). In DC 271b30-272a3, moreover, he explicitly admits that it is always possible to obtain a magnitude greater than a given magnitude, exactly as there is always a munber greater than a given number; he makes this assertion in the most unlikely context, an argument designed to show the finitude of the universe. This is a clear proof that for Aristotle this finitude does not interfere with the needs of geometry and, therefore, in Phys.
207b 15-20 and 207b27-34 Aristotle cannot contradict what he explicitly admits elsewhere, as Hintikka’s thesis implies.
In ch. I I set out to see what went wrong with the standard interpretation of
Phys. 207b27-34. If Aristotle does not contradict himself, then this passage must have nothing to do with arbitrarily extended magnitudes. The dialectic sXxixcim&ot Physics r is of particular help here. The tone of Aristotle’s discourse is set by the five aporiai at Phys. 203b 15-30 which are thought to support actual infinity. Aristotle’s aim is to show that theseaporiai do not necessitate the acceptemce of actual infinity 6 which the secondaporia justifies because mathematicians use the actual infinity of a magnitude’s divisions {Phys. 203bl7-18). Consequently, when in Phys. 207b27-34
Aristotle contends that mathematicians neither use nor need acmal, "untraversible" infinity, he must answer the second aporia and, therefore, the proofs he refers to in
Phys. 207b27-34 involve not arbitrarily extended magnitudes but rather the infinite divisibility of magnitudes. The characterization "untraversible" causes no problem here: Aristotle also calls "untraversable" the inverse addition of a magnitude’s divisions, i.e. (l/2)m + (l/4)m + ...+ (l/2")m {Phys. 206b9), a way by which he explicitly calls a magnitude infinitely divisible {Phys. 207a22-23), in the potential sense of course, i.e. that the difference between the inverse addition and the original magnitude m can be chosen less than any assigned magnitude, no matter how small
{Phys. 206b 16-20). If this is the only interpretation of Phys. 207b27-34 that agrees with the context of the passage, then exhaustion proofs are the only proofs in Greek geometry that, instead of the actually infinite inverse addition of a magnitude’s divisions demand a finite magnitude chosen in the above said maimer. But exhaustion proofs do bear out Aristotle’s claim in Phys. 207b27-34.
A similar problem appears in Hintikka’s interpretation of Phys. 207b 15-20.
This peissage is clearly parallel to Phys. 206b20-27: Aristotle contends that infinity by
"addition" cannot exceed all assigned magnitudes, even potentially, because, if it
could, there would be actual physical infinity since there can be potentially as great
a magnitude as there can be actually. Taken out of context Aristotle’s claim supports
Hintikka’s reading. But in both passages "addition" has nothing to do with addition 7 that produces arbitrarily extended magnitudes: the broader context makes clear that
Aristotle talks about inverse addition and, as Ross (1936) 557 had realized, Aristotle’s stipulation that this sum cannot exceed all assigned magnitudes even potentially amounts to the condition that the variable never reaches its hmit. Consequently these passages by no means restrict the range of extended magnitudes available to geometry within the confines of the universe.
There are of course interesting questions raised here:
(a) What prevents the actualization of potential infinity, if for Aristotle all potentialities are bound to actualize?
(b) If Aristotle admits arbitrarily large geometrical magnitudes abstracted from physical magnitudes, does this contravene the finitude of the universe?
(c) Why does Aristotle stipulate in Phys. 206b20-27 and 207b 15-20 that infinity by inverse addition cannot exceed all assigned magnitudes even potentially? Why does not this modal restriction extend to the potential infinity of arbitrarily large extensions as well? Such an extention would be natural since Aristotle supports this restriction on account of the modal postulate "there can be potentially as great a magnitude as there can be actually".
(d) If in Phys. 206b20-27 and 207bl5-20 Aristotle does not rule out arbitrarily extended magnitudes, why does he employ a cosmological reductio ad absurdum designed to protect the finitude of the universe?
The answers to (a)-(d) are necessary to round off the refutation of the received interpetation. Before that, however, I discuss in ch. 2 Aristotle’s argument 8 in DC 271b30-272a3, which has been seriously misinterpreted in the literature, and shows that an even stronger version of his claim in Phys. 207b27-34 can be borne out mathematically: because in Euclidean geometry it can be shown that, if one straight line is never actually infinite for any choice of the unit line, then all straight lines are finite, i.e. there are no actually infinite straight lines (Baldus (1930)). This result shows that Aristotle’s rejection of actual infinity does not pose a threat to Euclidean geometry and, in view of the above, one has to conclude that, contrary to Knorr,
Aristotle’s theory of infinity does not show remarkable insensitivity to the issues that must have occupied his contemporary geometers. Not only is a reference to exhaustion proofs right to the point in 207b27-34, but also, generally speaking, potential infinity does not run counter to the needs of Euclidean geometry.
Before I examine further the relevance of Aristotle’s comment in Phys.
207b27-34 to his contemporary mathematics, I show the internal coherence of his notion of potential infinity. The answer to (a) is given by Aristotle himself in Met.
1048b 14-17. There he denies that infinity is potential in the sense that it will have actual independent existence: it is potential in the sense that it will be actualized for the sake of mathematical knowledge, a thesis that harks back to Met. 1051a21-33 where the noetic activity of a mathematician amounts to the actualization of potential mathematical entities required for a proof. Thus potential infinityis actualized but, since it depends on a mathematician’s thought, it does not necessarily have ontological import (see Phys. 208al4-22) which would violate the finitude of the universe, (b) does not present any problem either for two reasons. First, it need not 9 be assumed that there is an exact correspondence between the magnitudes in a geometric proof and the physical magnitudes they are abstracted from: Aristotle emphasizes that the dimensions of a magnitude do not affect the validity of a proof.
Second, since xa. Meteor. 340a7-10 Aristotle implies a characterization of the universe as actually infinite for practical purposes, it seems that in his finite universe there is enough physical space to satisfy the most demanding needs of abstraction.
As far as (c) is concerned, the modal postulate applies not to physical reality but to the mathematician’s thought. Were there potentially a magnitude or number greater than all potential magnitudes or numbers, the modal postulate would nercessarily lead to actual infinity. But Aristotle avoids the problem because, as will turn out below in ch. 5, he has valid reasons to reject the notion of an actually infinite number or magnitude on account of the mathematical incoherence of such a notion. Nevertheless, in the case of inverse addition there is by definition an actual magnitude greater than all sums (l/2)m + (l/4)m + ...+ (l/2“)m so that, on the modal postulate, there would potentially be a sum (l/2)m + (l/4)m + ...+ (l/2“)m...
= m, i.e. m would be the actually infinite sum of its parts. But for Aristotle this situation amounts to the actual, completed infinity of numbers {Phys. 207a33-bl5) which, as will be shown below in ch. 5, he has valid reasons to reject. To avoid exactly this problem Aristotle stipulates that in inverse addition the variable cannot potentially exceed all magnitudes: in modem terms it never reaches its limit, a stipulation obviously justified by the modal postulate.
Why Aristotle further bolsters this stipulation by means of an cosmological 10 reductio ad absurdum, (d), is explained in ch. 4. The crucial fact is that in both relevant passages, 206b20-27 and 207bl5-20, Aristotle couples the cosmological reductio ad absurdum with a critique of Plato’s notion of arithmetic infinity. Does this notion commit Plato to actual physical infinity by Aristotle’s lights? If it does, the cosmological reductio ad absurdum fits in naturally with Aristotle’s defence of the modal restriction on potential infinity by inverse addition which is designed to exclude the actual infinity of numbers. That Plato is committed to this construal of arithmetic infinity is shown not only by scattered remarks in his dialogues but also by Aristotle himself in Phys. 204b7-10, 207bl0-15 and Met. 1073al8-22. Aristotle, moreover, must have thought that Plato is somehow committed to actual physical infinity because of his construal of arithmetic infinity. He does, of course, dissociate
Plato from actual physical infinity in his historical survey of older views on infinity
{Phys 202b30-203al6): Plato posited infinity, "the great and small," as a principle of perceptibles and forms, but, unlike the Pythagoreans, did not posit actual physical infini^ because he denied that forms are spatially located. But then why does
Aristotle take into account Plato’s views in a discussion explicitly devoted to actual physical infinity, irrespective of whether such infinity exists in intelligibles and mathematicals{Phys. 204a34-b4) and, on top of that, criticize solely Plato’s construal of arithmetic infinity? These incongruities show that Aristotle’s critique of Plato in
Physics r is rather allusive. But a clear picture emerges if one takes into account other passages firom the Physics which show that for Aristotle it is exactly Plato’s denial of spatially located forms that commits him to actual physical infinity, la Phys. 11 209b33-210a2 Aristotle argues that Plato cannot deny easily the spatial location of forms and numbers if "the great and small," principle of both forms and numbers, is also interpreted as "place," i.e. a principle of perceptibles. But if this is so, as
Aristotle thinks, then Plato’s adherence to the actual infinity of numbers commits him to actual physical infinity as well.
The above discussion has shown how Aristotle’s claim in Phys. 207b27-34 about actual infinity in mathematics fits in with the programmatic statement in Phys.
204a34-b4. The focus in Physics r is indeed on actual physical infinity; but for the
Pythagoreans and, according to Aristotle, Plato too actual physical infinity coincides with actual mathematical infinity. Aristotle, consequently, has to refute actual mathematical infinity along with actual physical infinity, all the more since the second aporia inPhys. 203b 15-30 justifies actual infinity on account of mathematical practice.
As I show in ch. 5, this does not imply that Aristotle’s objections to actual infinity in mathematics are motivated by physical or cosmological considerations. In view of the above discussion it is easy to see why in Phys. 204b7-22 a refutation of the notion of an actually infinite number is coupled with the refutation of an actually infinite physical body. But Aristotle’s argument is entirely mathematical, contending that an actually infinite number contravenes the characterization of number as a numbered set. In Met. 1083b36-1084a7 he also argues that, since all numbers are either odd or
even, and an actually infinite number can be neither, the notion of an actually infinite number turns out arithmetically incoherent. Aristotle’s attitude to the issue
at hand bears a clear affinity to Cantor’s attempt to mathematically legitimize his 12 transfinite numbers by establishing their arithmetic coherence. Aristotle’s arguments against the notion of an actually infinite number eo ipso extend to actually infinite magnitudes as well, although, as I show, his argument in DC 272b25-28 against the infinity of the cosmos suggests that he could have also refuted the existence of such magnitudes because they entail a violation of what came to be the third Euclidean postulate.
To further appreciate Aristotle’s position on mathematical infinity it is essential to view it vis-à-vis his contemporary mathematicians’ attitude to infinity,
Knorr (1981) has claimed that it was Eudoxus who abandoned actual infinity for strictly mathematical considerations without any philosophical influence: according to Knorr, Eudoxus’ exhaustion method developed out of an older method, employed by Hippocrates of Chios, which applied EL 5.12 to an actually infinite set of terms, a step Eudoxus would have found unacceptable, given that the "formal organization of the Elements", as Knorr puts it, was well under way in his time. Following in
Knoix’s footsteps White (1992) 144 describes Aristotle’s characterization of potential infinity in Physics r as "a general conceptual account -a Scientific American account for the educated and interested layperson, if you like- of the "scientific" notion of infinity that was applied technically, with increasing sophistication, by ancient mathematicians." White here refers to Eudoxus’ exhaustion method and, in view of my interperetation of Phys. 207b27-34, Aristotle appears to have been critically influenced by his contemporary mathematical practice. Viewing the exhaustion method as the rigorous method par excellence for handling mathematical infinity 13 comes directly from the eighteenth-century discussions of the foundations of the
calculus. This of course does not imply that Eudoxus conceived of the exhaustion
method as a rigorous means for handling mathematical infinity. Indeed, if one is to judge from Euclid’s Elements, the culmination of "the formal organization of the
Elements", Aristotle’s contemporary mathematicians do not seem to have abandoned
actual infinity as Knorr has it. First, actual infinity appears clearly in El. 3.16 and
9.20. Second, the application of El. 5.12 to an actually infinite set of terms appears
in El. 12.4, a theorem crucial to the exhaustion proof of El. 12.5; and it would have
been bizarre if Euclid had not known that Eudoxus developed the exhaustion method
in order exactly to avoid such a step in Hippocrates’s technique. Consequently, on
Knorr’s account of the origins of the exhaustion method, one has to admit that Euclid
either lost sight of, or did not care about, the standards of rigor set by Eudoxus, one
of the most prominent mathematicians of the pre-Euclidean era.
Further evidence against any possible worry over actual infinity on Eudoxus’s
part is provided by the mathematician’s own philosophical affiliations. The Knidian
had adopted a version of Plato’s theory of forms and, if he had also adherred to a
Platonist philosophy of mathematics, actual infinity would bave been philosophically
acceptable to him. Aristotle seems to confirm this hypothesis. As I argue in ch. 6 ,
in Metaphysics M he criticizes an anonymous Platonist theory which posits
mathematicals in perceptibles, a view Aristotle puts in the context of a theory that
posits form in perceptibles. Since elsewhere Aristotle attributes this theory explicitly
to Eudoxus, this version of mathematical Platonism must have been put forth by 14 Eudoxus as well. But the way in which Aristotle leads Eudoxus’ Platonism to absurdity hinges on the particular assumption of the actual infinity of a magnitude’s divisions.
Neither the philosophical nor the mathematical evidence suggests that
Eudoxus would have perceived a problem with the appeal to an actual infinity of terms in the Hippocratean proof: in ch. 7 I attempt to show how the exhaustion method could have arisen in the context of Eudoxus’ problem solving activity. The upshot of this discussion is that Aristotle’s rejection of actual infinity and his insistence on potential infinity did not exert any influence on his contemporary mathematics (only much later Pappus did adopt an axiom of infinity Aristotle would wholeheartedly subscribe to). On the other hand, Aristotle’s views on infinity cannot be plausibly said to have been molded by his contemporary mathematics either.
When in Phys. 207b27-34 he brings up exhaustion proofs in order to illustrate the sufficiency of potential infinity in proofs, he carmot imply that the exhaustion method originated out of a concern over actual infinity. On the contrary he implies that his contemporary mathematicians had not abandoned actual infinity despite the fact that in their proofs, i.e. exhaustion proofs, they relied solely on potential infinity. He argues that, if mathematicians do not use actual infinity in their proofs, they do not need it exactly because actual infinity enters the mathematical discourse, as is shown by Euclid’s Elements, but without contributing anything to proofs: this is indeed characteristic of actual infinity in E/. 3.16,9.20 and 12.4. Aristotle’s subtle point can only testify to his deep respect for, and insight into, the mathematical practice of his 15 day. To no small measure, the importance of his theory of infinity lies in its independence from his contemporary mathematicians’ attitude to infinity: he manages to delineate an original, well-argued and coherent position, which takes into account the needs of proofs as they were done in his day, without demanding any reforms on philosophical grounds. CHAPTER I
ARISTOTLE’S POTENTIAL INFINITY AND MODERN SCHOLARSHIP
The Berlin Academy prize proposal for the year 1784 marks a landmark in the history of mathematics as an official recognition of the foundational problem in the calculus/ According to the prize committee, the problem amounted to the need for a precise characterization of mathematical infinity:
It is well known that higher mathematics continually uses infinitely large and infinitely small quantities. Nevertheless, geometers, and even the ancient analysts, have carefully avoided everything which approaches the infinite; and some great modem analysts hold that the terms of the expression "infinite magnitude" contradict one another. The Academy hopes, therefore, that it can be explained how so many true theorems have been deduced from a contradictory supposition, and that a principle can be delineated which is sure, clear -in a word, truly mathematical- which can appropriately be substituted for "the infinite". This is to be done without making the researches which had been expedited by using the concept of the infinite too difficult or tedious. We require that the matter be treated with all rigor, clarity, and simplicity.^
It is indeed all but ironical that the solution to the problem climaxed in the second part of the nineteenth century with a characterization of real numbers which
‘On the historical context of the 1784 Berlin Academy prize proposal see Grabiner (1981) 40ff. ^For the entire proposal see Lagrange (1784) 12-13 and Dugac (1980) 12.
16 17 introduced completed infinities in mathematics and, via Cantor’s set theory, gave rise to a new foundational crisis. As Russell put it wiyly:
The mathematical theory of infinity may also be said to begin with Cantor. The Infinitesimal Calculus, though it cannot wholly dispense with infinity, has as few dealings with it as possible, and contrives to hide it away before facing the world. Cantor has abandoned this cowardly policy, and has brought the skeleton out of its cupboard. He has been emboldened in this course by denying that it is a skeleton. Indeed, like many skeletons, it was wholly dependent on its cupboard and vanished in the light of day.^
But the story is much older than Cantor and even the Berlin Academy prize proposal. It seems that the skeleton and its cupboard were put away in the crypt already in classical antiquity by Aristotle himself:
My theory does not deprive mathematicians of their study by refuting an actual, untraversable infinity towards increase. Because now they do not need [this kind oQ infinity (for they do not use it) but require only the existence of arbitrary finite magnitudes; and it is possible to divide an arbitrary [i.e. arbitrarily small] magnitude in the same ratio as the greatest magnitude. Therefore, it makes no difference in proofs whether a magnitude is among the magnitudes that exist (Phys. 207b27-34).
To put Aristotle’s statement in the appropriate historical context, Gauss had vigorously objected to actual infinity in mathematics;'* but, as Cantor had realized,
Aristotle was the first to banish it from mathematics, whose demonstrative needs are weU served by potential infinity.* Cantor, on the contrary, asserted that the
mathematically rigorous application of potential infinity requires an underlying actual
^Russell (1903) 304.
‘‘See Gauss (1860) 269. *See Cantor (1883) 173-175 and (1887) 395-396. 18 infinity which thus gains conceptual and ontological priority/ like Aristotle, however, Weyl insisted that mathematics is the science of the infinite, not the actual but the potential,’ and the controversy is still at the heart of the debates between the successors of Kronecker on the one hand and Cantor on the other -in 1975 the constructivist Bishop declared a "crisis in contemporary mathematics-.ifue to our neglect o f philosophical issues", foremost among them that of infinity.* Since, moreover, it was the same issue that motivated Hilbert’s famous program, it is evident that with his brief comment in. Phys. 207b27-34 Aristotle has secured a place, for some dubious, for others honorable, in the history of mathematics and its philosophy.
Indeed, as will be seen later in this and the following section, Aristotle was the first to delineate precisely mathematical infinity in its potential and actual sense.
His rejection of actual infinity in Physics F, moreover, leaves no doubt that he breaks new ground in the philosophy of his time. But of particular importance to the history of the philosophy of mathematics are Aristotle’s reasons for the rejection of actual infinity and, in view of the intuitionistic opposition to mathematical practice, his claim that Greek mathematicians neither use nor need actual infinity. The same view was obviously held by Lagrange and his fellow judges for the 1784 Berlin
®Cantor (1886a) 9. ’See Weyl (1949) ch. 2.8; cf. (1931).
*Bishop (1975) 507. 19 Academy prize® who credited Greek geometers with scrupulous avoidance of actual infinity, i.e. infinitely large and small quantities. Aristotle does come very close to the attitude of eighteenth-century mathematicians towards infinity. Lagrange’s contemporary analysts were not the first who held the expression "(actually) infinite magnitude" to be a contradiction in terms, as remarked in the prize proposal.^” It is well known that, in spite of their sharp formal difference, number and magnitude shared certain interesting aspects for Aristotle" and thus he refutes the notion of
®The committee also included Johann (U) Bernoulli and Johann Karl Gottlieb Schultze. But Lagrange’s concern over the foundations of the calculus long preceded the 1784 prize proposal (see Grabiner (1981) 38ff.) and strongly suggests that the topic of the competition was his own idea; see Hofinann (1953-1957) 6 8 and lushkevich (1971) 155. “See GerdU (1760-1761) 1. The contradiction lies in the fact that an actually infinite magnitude violates the Archimedean axiom.
"This is home out by the fact that inMet. 1020a7-ll Aristotle defines "magnitude" as "measurable quantity" and "number" as a countable quantity (i.e. multitude) whereas in Met. lG88a4-6 he defines number as a "multitude of measures". These definitions suggest that Aristotle understands numbers as commensurable magnitudes, although this is not his official definition of number, a situation that would plausibly account for Euclid’s failure to provide a common characterization of proportionality for both magnitudes and numbers (cf. Mueller (1981) 138). The most plausible explanation for such an attitude can be found in the notion of measurement, a fundamental undefined notion in Euclidean arithmetic (see Mueller (1981) 61), which is naturally understood both arithmetically and geometrically. A good index of the situation is Aristotle’s argument in£>C 273a21-b29 where, assuming the proportionality of (physical) magnitudes and their weight and given a weight P, Aristotle takes a weight E (
The exhaustion method is the major characteristic of the twelfth book of
Euclid’s Elements and was later improved upon by Archimedes for the calculation of the area of curvilinear plane figures and volumes of solids. In eighteenth-century discussions of the foundations of calculus the method came to be considered the paradigm of mathematical rigor:" most notably Maclaurin attempted to show that the exhaustion method, though long-winded and difficult to handle, could prove all results obtained by the new "analytical" means of the calculus without, however.
are not valid for numbers. When later on in the course of the same argument he considers a weight E incommensurable with F (DC 273b 10-15) he clearly presuposses the parts-characterization of proportionality for integers in El. 7 Def.20. "Cf. Grabiner (1981) 42 n.78. "See Whiteside (1961) 333-348. 21 appealing to infinitary entities that violate the Archimedean axiom.*'* Incidentally, it is Archimedes who implicitly credits Eudoxus with the development of the exhaustion method in a historical justification of the axiom, or lemma as Archimedes calls it, that bears his name today: "the excess by which the greater of (two) unequal areas exceeds the lesser can, by being added to itself, he made to exceed any given finite area."'^ Archimedes claims that thislemma is similar to that by means of which earlier geometers proved El. 12,2 and 10, theorems whose proof he explicitly attributes to Eudoxus in the preface to On the Sphere and Cylinder: Since the
Euclidean proofs of these theorems employ the method of exhaustion, it follows firom
Archimedes’ historical note that the method originated with Eudoxus*^ and thus the lemma to which Archimedes compares his own must be the so-called 'bisection principle"."
^^Maclaurin (1742). For a concise account of the foundations of the calculus according to Maclaurin see Guicciardini (1989) 47-51. Ironically, Maclaurin himself abandoned the exhaustion method after the first book of his treatise. '^Preface to the Quadrature o f the Parabola', formulated for lines and solids as well the axiom appears in a slightly different form among the lambanomena in the first book of the On the Sphere and Cylinder. In Euclid’s Elements the Archimedean axiom figures as an unstated assumption (see Mueller (1981) 139ff.), although Aristotle formulates it explicitly raPhys. 266b2-3 (cf. Heath (1956) 3,16; this passage wUl be discussed below). The most detailed account of the origins of this much discussed axiom is Knorr (1978); previous literature is found in Knorr (1978) 206 n.70 and in his bibliographical essay in Dijksterhuis (1987) 433. Cf. Heath (1956) 3, 365ff.
"The term 'bisection principle" seems to have been coined by Knorr (1978) in order to distinguish between the convergence principle in Euclid’s exhaustion proofs 22 This principle allows for the crucial formal step in an exhaustion proof that made Greek geometry notorious for avoiding actual infinity’* and the application of the principle in the proof of the theorem that "circles are to each other in the ratio of the square of their diameters" (E/. 12.2) shows how incisively Aristotle characterized for the first time the mathematical notion of potential infinity.’’ and in El. 10.1 with which the convergence principle of the exhaustion proofs has been usually identified (see e.g. Heath (1956) 3, 15-16, Mueller (1981) 233, Dijksterhuis (1987) 54 and Jesseph (1993) 125-126); the term "bisection principle" is obviously parallel to the term "dichotomy" employed by Dijksterhuis (1987) 54 as a shorthand for E/. 10.1 in his discussions of Archimedes’ exhaustion proofs. As Knorr showed, there are cogent reasons to assume that in the exhaustion proofs convergence is guaranteed only by an unstated assumption, the bisection principle, which later gave rise to the theorem in E/. 10.1, most probably in the context of the revision of proportion theoiy that culminated in the fifth book of Euclid’s Elements. In theElements the bisection principle appears as a special case of E/. 10.1. For the equivalence of Archimedes’ axiom and the bisection principle in the exhaustion proofs see Mueller (1981) 142-143. White (1992) 152n.25 objected to Knorr’s distinction between El. 10.1 and its special case of bisection m the context of exhaustion proofs but his objections are not very convincing: El. 10.1, the so-called "Euclid’s lemma", does cover the special case of the bisection principle and it might "represent a considerable conceptual advance over any possible bisection principle", as White puts it; this, however, does not give to El. 10.1 historical precedence.
18Cf., Maclaurin (1742) 37.
”There are many general accounts of the exhaustion method in both its Euclidean and Archimedean formulations; see e.g. Pedersen (1980) 31-32, Grabiner (1981) 34-35, Dijksterhuis (1987) 130-133 and Jesseph (1993) 124-129. An excellent account of Euclid’s exhaustion proofs is found in Mueller (1981) 230-236 along with perceptive comments on the crucial differences between the exhaustion method and integral calculus. Euclidean, i.e. Eudoxean, exhaustion proofs seek to establish by reductio ad absurdum that two areas or solids x,y are to each other in a certain ratio
(a, 0). If (x, y) (a, 8 j), then there is a fourth proportional z such that (x, z) = (a, j8 ). First let z < y. The idea then is to show that within y there is ay* > z so that. 23
Fig. 1 Euclid’s exhaustion proof of EL 12.1
Euclid proceeds by reductio ad absurdum assuming that two circles ABCD, EFGH
(Fig. 1) are not to each other as the squares of their diameters BD, FH; thus there must be an area S
if X* is another entity similar to y* and inscribed inx, then via an already established result it is (x*, y*) = (a, 0). Consequently, (x, z) = (x*, y*) whence it follows x > X* and z > y*, which is contrary to y* > z. If now z > y, this hypothesis is easily reduced to the previous one and the reductio ad absurdum is successfully completed. The critical point is of course how one obtains the required y* > z. To do so Euclid inscribes in y an entity yo and then another one y^ so that y^ > 1/2 (y - yo) and y^ = yo + yi. Each y„ obviously divides further y - yo, continually approaching y, and by such successive divisions, Euclid argues, one obtains a y^ < y - z which is the required y* > z. This step, the application of the bisection principle, as Knorr has dubbed it, obviously resembles the modem definition of limit (for any small e > 0 there is a large N |f(x) - k| < e for all x > N) and explains the historical relevance of the exhaustion method to discussions of infinitesimals and limiting processses.
^°For a formulation of the exhaustion method without the assumption of a fourth proportional see Stolz (1891) 517 n.3 and Hasse & Scholz (1928) 27. 24 is as BD^ is to FI?. Euclid then inscribes a square EFGH in the homonymous circle and argues that it is greater than half the circle EFGH; next he bisects the circumferences EF, FG, GH, HE at the points K, L» M, N and joins EK, KF, FL»
LG, GM, MH, HN, NE. Now each of the resulting triangles EKF, FLG, GMH,
HNE is greater that half of the circle’s segment about it and the bisection principle is appealed to in the next step:
bisecting the remaining circumferences and joining the straight lines and doing this continually we will leave some segments of the circle that are less than the difference by which the circle EFGH exceeds the area S.“
The intuitive idea behind the application of the bisection principle is that the circle can be approached as closely as one desires by a series of inscribed polygons,
square, octagon, 16-gon, 2 “-gon... so that the difference between the circle and a given 2“-gon is less than any assigned magnitude, in this case EFGH - S.
D’ Alembert and de la Chapelle appealed to this situation in order to illustrate the notion of limit:
One magnitude is said to be the limit of another magnitude, when the second can approach nearer to the fust than a given magnitude, as small as that magnitude may be supposed; nevertheless without the magnitude which is approaching ever being able to surpass the magnitude which it approaches...Properly speaking, the limit never coincides, or never becomes equal, to the quantity of which it is a limit, but the latter can always approach
^*For a collection of Euclid’s and Archimedes’ applications of the bisection principle see Knorr (1978) 235ff. 25 doser and doser, and can differ from it by as little as desired.^
In the proof oiE l. 12.2 each bisection of circumferences, i.e, each 2“-gon, generates increments s„ satisfying the conditions (1) s^+i ^ 1/2 (EFGH - s j and (2) Sn+j = s„
+ Sn+i; the bisection prindple guarantees the existence of an s^ so that EFGH - s^ <
EFGH - S.® It is exactly this step that brings forth the inappropriateness of the term "exhaustion method":^ unlike the sophist Antiphon, Euclid (or Eudoxus) does not imply that the circle will be reached after an infinite number of bisections so that it coincides with an ultimate polygon.^ Had he done so, the proof of £/. 12.2 would have obtained far more easily than by the double reductio ad absurdum that follows
“ D’ Alembert & de la Chapelle (1789) s.v. Limite; for a brief discussion of this definition see Grabiner (1981) 8-9. ^ o complete the proof, assume that the requisite s^ gives rise to the polygon EKFLGMHN (Fig. 1). Let a polygon AOBPCQDR similar to EKFLGMHN be inscribed in the circle ABCD so that (BD^, F If) = (AOBPCQDR, EKFLGMHN) (E l 12.1). But ex hypothesi (BD^ Fif) = (ABCD, S) so that (ABCD, S) = (AOBPCQDR, EKFLGMHN) and by cdtemando (ABCD, AOBPCQDR) = (S, EKFLGMHN). It follows that S is greater than the polygon EKFLGMHN but it was assumed that S is less than the polygon EKFLGMHN. Euclid then proceeds to refute the assumption that BD* is to FHP as the circle ABCD is to an area greater than the circle EFGH by showing that this assumption leads to the previous one whose absurdity has been demonstrated. *^Cf. Dijksterhuis (1987) 130; for the origin of the term "exhaustion method" see Knorr (1982) 123 n.23. “The argument was part of Antiphon's attempt at the quadrature of the circle; see Simplicius InArist. Phys. 54f. (Diels) and Themistius In Arist. Phys. 4 (Schenkl). For a discussion of Antiphon’s quadrature see Mueller (1982). 26 in the exhaustion proof.“ Instead, Euclid treats the polygon as a finite variable
that, satisfying (1 ) and ( 2 ), can be chosen less than anygn/en assigned magnitude as in D’ Alembert and de la Chapelle’s definition of limit,” Thus, to put Lagrange’s comment on Greek geometry in more precise terms, what is avoided here is not infinity per se but rather actual infinity. The appeal to the bisection principle in an exhaustion proof is a prime example of potential infinity which, in Cantor’s words.
is mostly witnessed when one has an undetermined variable finite quantity which either increases beyond all limits (here we can take as an example that so-called time which is counted from a definite initial moment) or which decreases beneath any finite small limit (as, for example, in the correct presentation of a so-called differential).^
“One need only appeal to E/,12,1. A similar type of argument was most probably used by Hippocrates of Chios in his proof oiE l. 12,2 before Eudoxus developed the exhaustion method; see the discussion below in § 5. ”The situation is nicely illustrated by Maclaurin when he contrasts the "rigorous" Archimedean method, i.e, the exhaustion method, with the indivisibilist methods current in the eighteenth century: "In what Archimedes had demonstrated of the limits of figures and progressions, there were valuable hints towards a general method of considering curvilinear figures...But, that his method might be more easily extended, its old foundation was abandoned and suppositions were proposed which he had avoided. It was thought unnecessary to conceive the figures circumscribed or inscribed in the curvilinear area, or solid, as been always assignable and finite; and the precaution of Archimedes came to be considered as a check upon Geometricians, that served only to retard their progress. Therefore, instead of his assignable finite figures, indivisibles or infinitely small elements were substituted; and these being imagined indefinite, or infinite, in order, their sum was supposed to coincide with the curvilinear area, or solid" (Maclaurin (1742) 37),
28 Cantor, (1887-1888) 400; cf. Cantor (1886b) 371. 27 Differentials aside, the second case in Cantor’s account reflects not only the use of the bisection principle in an exhaustion proof but also the two complementary senses of mathematical infinity according to Aristotle (Phys. 206b3-33):
In a way the infinite by addition is potential in the same sense as the infinite by division; for it will always be possible to obtain something beyond but it will not exceed any magnitude, as it does in the division where it will always be less than any assigned magnitude (Phys. 206bl6-20)
To emphasize first the affinities with Cantor’s account, Aristotle stresses explicitly that these variable quantities are finite (phys. 206a27-29, 33-206bl) and later on
(Phys. 207bl0-15) he parallels mathematical potential infinity with time. What is more important, however, the two senses of potential infinity he recognizes, by division or inverse addition (phys. 206b4-7, 207a23), seem to be more closely interrelated than scholars have assumed. Aristotle begins with the complementarity of these two senses (phys. 206b3-6) but does not give a precise characterization of potential infinity by division as he does for potential infinity by inverse addition in
“ On the difference between Aristotle’s two kinds of potential infinity see below § 3. It is important to emphasize that the two complementary senses of potential infinity in this passage are not potential infinity by division and addition simpliciter but potential infinity by division and inverse addition (whose formal characterization follows below in this paragraph). Although this obvious point is duly recognized by Ross (1936) 557, it is totally missed by Hussey (1983) 85 who understands the passage as referring to "the impossibility of a potential infinite in respect of unbounded addition"; as will turn out below, the failure to distinguish between addition simpliciter and inverse addition in Physics T has led to a serious misunderstanding of Aristotle’s potential infinity in its relation to mathematics. 28 Phys. 206b7-9:
If in a finite magnitude one sets an assigned part [of it] and adds proportionate but not equal parts of the whole magnitude, one does not exhaust the [initial] finite magnitude.^
In modem notation Aristotle simply states here that for a magnitude AB, if f(l, AB)
= AB , f(n, AB) ^ 1/2 f(n-l, AB) A Vx, AB > x 3nGN f(n, AB) < x (P 3 ), then, defining s„ as the sum f(2, AB) + f(3, AB) +...+ f(n, AB), s„ is always less than AB
for any n, i.e. Vs^ s„ < AB (P 4 ). Given the term "inverse addition", it is tempting to conclude that Aristotle implicitly constmes the potential infinity by division as e.g. the series of divisions AB/2" which can yield a magnitude less than any assigned magnitude/^ That this is so is shown by Phys. 266b2-4, where Aristotle formulates
^hp yap T(p Tevepaapepcp peyéBei &p \a0wp n ç utpiapepop TcpoaXap.^a.PTQ t
\oy "‘a . Ross (1936) 556-557, Heath (1949) 108-109 and Hussey (1983) 84. 29 a version of the bisection principle in terms of potential infinity,“ but there is cogent textual evidence that cannot be accomodated solely on this view. Immediately after Phys. 206b7-9 Aristotle concludes that the infinite "is only in this sense and in no other, i.e. potential and by division" (ôXXtoç pikv ovv o v k lanv, o v t u s ç b’ eari t o Sticeipov, bvpdtfjL€i T€ KttikvX KaOaipêaei, Phys, 206bl2-13).^^ Thus, unless the infinite by inverse addition is somehow related to the infinite by division, Aristotle here passes abruptly to the latter, although he gives the impression that he simply sums up Phys. 206b7-9, where he has detailed potential infinity by inverse addition.^ ^^For a discussion of this passage see below. ^^KaOaîpeaiç is explained by Bonitz ^.v. as oi4>aipeaiq but in this context there is no formal difference between subtraction and division. Subtraction instead of division appears in El. 10.1 as well in the application of the bisection principle by Archimedes in Quadrature of the Parabola 20 (corollary) and Theon in his Commentary on Ptolemy I 361 (Rome); cf. z\s,o Phys. 207a22-23 (biouperov b' kvi re T^v KaOaCpemp). ^Ross (1936) 556 does not seem to realize the problem posed by the adverb ovTosç immediately after the characterization of potential infinity byinverse addition in.Ph.ys. 206b 12-13. He remarks only that KaOaîpeoiç "refers to the same process that has hitherto been called biatpeoiç, but refers to it in a slightly different aspect. In the one the whole is regarded as divided into parts; in the other as diminished by the removal of parts". Ross’ interpretation of KuBaipeaiç could certainly fit the successive removal of parts in Aristotle’s inverse addition and, if it were so, Phys. 206bl2-13 would be absolutely straightforward. This interpretation, however, is not home out by Phys. 207a22-23 where Aristotle couplesKaOaCpeaiç and inverse addition (fa n yap TO &Treipop...biaipeTop b ’ kvi re t^p KoOaCpeaip Koi tt}p àpTeaTpap,p,épiiP vpôaBeaip). On Ross’ interpretation of KaOaCpeaiq, Aristotle would here refer twice to the same operation, which is implausible; it is plausible, however, to think that Aristotle simply refers to the two senses of potential infinity and, therefore, that KaOaipeaiç is a mere synonym of biaipeaiç. 30 One might argue of course that here bonus dormitat Aristoteles; in Phys. 207a22-23, however, he claims that theinfinite is potential and divisible by both division and inverse addition (e Kai Ti)p àprearpappéprip vpoaBeaip)?^ Since it is difficult to posit an inadvertent slip in this case too, the blurring of the neat distinction between the two senses of potential infinity can be naturally accounted for if Aristotle does not omit a characterization of potential infinity by division but simply expresses it by means of inverse addition: given the characterization of potential infinity by inverse addition, he can do that only by allowing f(n, AB) to become less than any assigned magnitude for a sufficient n 206b 16-20). This is the only way to account for the text and, in its obvious affinity with the application of the bisection principle in exhaustion proofs, it bears out not only the complementarity of the two senses of Aristotle’s potential infinity but also Heath’s interpretation of Phys. 207b27-34: The principle or axiom on which Eudoxus founded his "method of exhaustion" for measuring the content of curvilinear plane and solid figures was framed precisely in accordance with the principle enunciated by Aristotle in this passage.^ Heath did not support his view with textual evidence, probably because he did not live to put the final touches on his book, but the structure oiPhys. T fully justifies ^% e passage was excised by Stolzle but, as Ross (1936) 559 notes, it is sufficiently supported by the commentaries of Simplicius and Philoponus and the paraphrase of Themistius, % ath (1949) 111. 31 his brief remark. In the aporematic section of Phys. 203bl5-30 Aristotle lists five reasons that are usually thought to support the existence of infinity -obviously in the non-Aristotelian sense of actual infinity: (1) the infinity of time, (2) the division of magnitudes in mathematics (xpâvrai yap k u I oi iia6i]p,anKol rÿàiretp(p, Phys. 203bl7- 18), (3) constant generation and destruction, (4) the fact that there is always something beyond a limit and, most importantly, (5) the belief that numbers, geometrical magnitudes and the space outside the cosmos are infinite because they never give out in thought. These qporiai provide the dialectical subject matter that informs Aristotle’s discussion of infinity and are answered at the end of Phys. F (207bl0-208a23). But in Phys. 207b27-34 Aristotle clearly contravenes the claim in aporia (2 ) (xp&VTai yap koù oi /ladrjpanxoi Tÿ aireiptfi ~ ovSk yap vvv béovrai [the mathematicians] t o v aireipov (ov yap xpûi'Tai),Phys. 207b29-30) and therefore his answer must refer to proofs that appeal to an arbitrarily small, finite magnitude instead of the actual infinity of a magnitude’s divisions. Since these proofs agree with Aristotle’s rejection of an actual, untraversable infinity (&Treipov...èvepyei ®’On the meaning of "untraversable" see below ch. 2 . ®®Contraiy to Ross (1936) 560 according to whom Aristotle’s point in Phys. 207b27-34 is that "mathematics does not need very great magnitudes, let alone infinite ones" and Hussey (1983) 93 according to whom "Aristotle claims that geometers do not use, and therefore do not need, infinitely extended lines (or planes, or solids, a fortiori). He implies that such lines did occur in the geometry of his age; his claim is that they can always be replaced, without damage to the reasoning by 32 magnitude is untraversable by way of inverse addition (ov ôiê^eiai to Tevepaaiiepov, Phys. 206b9) which fits naturally the division of magnitudes, as said above, all the more since Aristotle has explicitly denied in Phys. 207b 15-18 that the potential infinity by inverse addition can be actual too/^ Potential infinity by inverse addition also explains well how any arbitrarily small magnitude (AB - s j can be divided in the same ratio as another far greater magnitude (AB) since the terms in inverse addition are in constant ratio (Phys. 206b7-9);‘*‘’ but, since for Aristotle this is how the arbitrary finite magnitude required for the proof comes about, the type of proof he suitably long finite ones". The evidence from Physics P, however, speaks clearly against the assumption that in Phys. 207b27-34 Aristotle has in mind actually infinite extended magnitudes. This interpretation unfortunately became popular with Hintikka (1973) on whose views see below in this section. ^’See below ch. 3. ‘*“On the traditional reading of Aristotle’s theory of infinity to néyioTov néyedoç in. Phys. 207b31 is the greatest extended magnitude available in the finite universe. Obviously, however, to peyioTov /leyeOog is the original magnitude AB, which, in an inverse addition of its parts, is the greatest of all AB - s^. Hussey (1983) 93ff. does not e^lain clearly how Aristotle’s statement to the effect that "it is possible to divide an arbitrary magnitude in the same ratio as the greatest magnitude" fits in with his interpretation of Phys. 207b27-34. Ross (1936) 560 understood this statement as a reference to Euclid’s El. 6.10 (to cut a given uncut straight line similarly to a given cut straight line) arguing that, according to Aristotle, "any property that could be proved by the use of a great magnitude can equally be proved by the use of a small one". Hussey (1983) 95 also claims that "for the purposes of proof any geometrical diagram may be scaled down arbitrarily small" and thus he must understand Phys. 207b27-34 in the same way as Ross. El. 6.10 cannot have the important implications Ross reads into it. But even if the notion of "scaling down" were mathematically applicable, it is hard to see how it could make sense within Phys. 207b27-34 and its broader context; cf. also below n.l07. 33 has in mind can only be an exhaustion proof where the divisions produced by the bisection principle are in constant ratio/^ Given the dialectic structure of Physics F, Aristotle’s characterization of potential infinity by division/inverse addition and the nature of the exhaustion method itself, Phys. 207b27-34 can only refer to exhaustion proofs -there are indeed no other proofs in Greek mathematics that demand only an arbitrary finite magnitude instead of the actual infinity of a magnitude’s divisions, as Aristotle implies in this passage. Nevertheless, Aristotle’s claim that Greek geometry does not need actual infinity has been violently attacked, despite the fact that his claim is historically justified by the exhaustion method itself and agrees with the attitude to Greek geometry during the formative period in the history of calculus. Knorr ““Cf. Mueller (1981) 235. Recently White also associated Aristotle’s theory of potential infinity with the exhaustion method: "It is certainly the case that Aristotle’s account of the sort of infinite-by-addition process that is the inverse of an infinite-by- division process captures a fundamental feature of the geometrical method of exhaustion: the geometrical process of division, and hence the inverse process of addition, does not terminate in any finite number of steps of division/addition. However, I have also made the stronger claim that Aristotle’s refusal to consider the inverse processes as completed (or completable) at all -even in a so-called "infinite" number of steps- accurately reflects a similar refusal on the part of ancient mathematicians" (White (1992) 146). As I argue below in ch. 5-7, there are reasons to doubt whether White’s stronger claim reflects the reason that motivated the development of the exhaustion method. White’s discussion, moreover, hinges on the undeniable conceptual similarities between Aristotle’s potential infinity and the exhaustion method -he has not attempted to elucidate the text of Physics F, in particular Phys. 207b27-34, by means of these similarities. A fundamental difference between my approach and White’s is exactly the interpretationoîPhys. 207b27-34 -he accepts Hintikka’s reading of this passage; see the discussion below. 34 declared that Aristotle’s theory of the infinite shows remarkable insensitivity to the issues that might have occupied the geometers of his generation. If Euclid and his predecessors knew of his theory, they chose wisely to disregard it.“^ Whether Euclid ignored, wisely or not, Aristotle’s theory of infinity, is a different, albeit very interesting, question that wiU be addressed below (ch. 5-7). But such damning criticism is not home out by the evidence -one has to look only at Aristotle’s and Cantor’s characterization of potential infinity in order to realize that the former was as insensitive to the needs of his contemporary geometers as the latter to the needs of nineteenth century analysts. Unfortunately Knorr did not try to substantiate his thesis; he brushed aside Heath and fell back on Hintikka’s interpretation of Phys. 207b27-34.‘*^ According to Hintikka Aristotle’s claim does not square with the facts because his theory of potential infinity vitiates a fundamental assumption of Euclidean geometry, the indefinite extendibility of straight lines: Aristotle is not here saying merely that a mathematician does not need an infinite "Knorr (1982) 122. "See Knorr (1982) 122 n.22: "At Phys. IH 7, 207b27 Aristotle argues that geometers do not actually require this sense of the infinite [i.e. the indefinite extendibility of straight lines]; but Jaako Hintikka has clearly shown his error here (Hintikka 1966). T.L. Heath misses this aspect of the issue, however, relating it instead to the Eudoxean analysis". 35 magnitude all of whose parts are simultaneously actualized. If this were all that he was saying, he would have a plausible argument. The quoted passage shows him doing much more, however; he is also arguing that a geometer does not even need arbitrarily large potential atensions...All that he needs according to Aristotle is that there be arbitrarVy small potential magnitudes.^ EKntikka’s first point is certainly correct, as will be shown below (ch. 5): Aristotle has not simply plausible but truly compelling arguments against actual infinity in the mathematics of his day. But the context ofPhys. 207b27-34 cannot bear out Hintikka’s reading as far as arbitrarily large potential extensions are concerned; these are out of place in a reference to geometrical proofs which, on the premisses of Aristotle’s discussion, involve the divisibility of magnitudes. For Hintikka,^* however, arbitrarily large potential extensions are explicitly ruled out in Phys. 207bl5-20: The continuum is potentially] divisible ad infinitum but there is no infinite in the direction of increase. Because a magnitude can be potentially of such extent that might actually exist. Thus, since there is no infinite physical magnitude, it is not possible to exceed any assigned magnitude. From this passage Hintikka infers that, since for Aristotle a geometer’s choice of ‘“Hintikka (1973) 119. There is an obvious jump in Hintikka’s reasoning: even if the actual, untraversable infinity Aristotle denies iaPhys. 207b27-34 referred to an actually infinite extended magnitude, one could not self-evidently conclude that Aristotle refuses to admit arbitrarily large potential extensions as well! This conclusion would be correct only if potential infinity presupposed an actual infinity, as Cantor held, but this is not necessarily so. ^^Hintikka (1973) 117. 36 arbitrarily large potential extensions must be restricted within the finite boundaries of the cosmos, Aristotle’s theory of infinity cannot account for the arbitrary extendibility of straight tines required by Euclid’s Def. 123 and Post. 1.5, the famous parallel postulate. This reasoning hinges on a presumed isomorphism between the, undoubtedly finite, Aristotelian universe and geometrical magnitudes, an assumption that prima fade seems to be supported by Aristotle’s abstractionist construal of geometrical objects. As I will argue below (ch. 3), this is not so. Here it suffices to put Hintikka’s reading of Phys. 207b 15-20 in historical perspective, to question the validity of its consequences and whether it is ultimately borne out by the immediate context of this passage m Physics P. By Hintikka’s lights Knorr is of course partially right to claim that Aristotle’s theory of potential infinity is out of touch with the concerns of his contemporary geometers, especially since the theory of parallel lines appears to have been of particular interest to this generation of Greek mathematicians.'*^ Hintildca has traced his view back to Hero who, according to Proclus, reformulated certain Euclidean proofs so as to avoid the possible objection that there might not be enough available space for certain constructions requiring arbitrary production of strai^t lines.^’ This is exactly what one would e)q)ect if Hero, or the putative detractors of Euclid, shared the compunctions Hintikka reads '**See below n.53. '‘’Hero gave such alternative proofe tor El. 1.11 and 20 (cf. also Proclus’ note on El. 1.16). Van Pesch (1900) has argued that Proclus’ alternative proofs otE l. 1.5,17 and 32 which avoid the production of straight lines should also be attributed to Hero. 37 into Aristotle’s conception of infinity.*** Aristotle’s notion of potential infinity, moreover, drew the attention of at least one of Euclid’s successors. Pappus, who added the following to the Euclidean axioms: "the infinite in magnitudes is both by addition and division and in both cases only potentially."’*^ Here the Aristotelian influence is unmistakable, although there is no indication, and it is not necessary to assume, that Pappus shared Hero’s concerns or read Phys. 207b 15-20 like Hintikka.®° This reading is by no means novel,®* but it has influenced recent scholarly discussions primarily through Hintikka’s detailed exposition. Efis main conclusion, though, is problematic. For Efintikka Aristotle’s physical universe is non- Euclidean since the parallel postulate, which requires arbitrarily extendible lines, is "*Cf. Hintikka (1973) 121. aveipop ev t o î ç fieyédeaip kanv k<ù rpoaQéaei Kal rÿèriKaOaipêaei, ôvvâpei Se eKarepov (Proclus In Pr. EL Comnu 198.12-14 (Friedlein)). ®“Hintikka (1973) does not mention Pappus’ axiom. ®*Already Chemiss (1935) 34 had claimed that Aristotle’s "belief in a finite universe has a curious influence on his own theory of infinity...Infinity by division does exist potentially, but in magnitudes, though infinity by addition does exist potentially, it does so only within a finite magnitude as an infinitely converging series. That is, infinity by addition, in the sense that any given magnitude may be surpassed, does not exist even potentially". Furthermore, according to Chemiss (1935) 35 n.l29, in Phys. 207b27-34 Aristotle simply reassures the mathematicians that "they do not need the kind of infinity of which his argument deprives them". Notably Heath (1981) 344 put forth the same thesis but Ross (1936) 52 correctly argued that mPhys. 266b2-3 Aristotle does admit potentially infinite sums that can exceed any preassigned magnitude (the Archimedean axiom) without implying the existence of actual physical infinity. The reasons why arbitrarily extendible geometrical magnitudes do not compromise Aristotle’s denial of actual physical infinity wül be discussed below in ch. 3. 38 not satisfied in it, although Aristotle naturally could not have been aware of this disturbing consequence.® If, however, Hintikka were right, it is hard to beUeve that Aristotle would not have realized the problem posed by the parallel postulate, given that he is aware of the theory of parallel lines as is shown byAnPost. 77b22-24.^^ “ Hintikka (1973) 120. Phys. 207b27-34 is related to non-EucUdean geometry already in Heath (1981) 343; cf. Solmsen (1960) 173 n.57 and Knorr (1982) 122. Such associations clearly have no more validity than the claim that Aristotle’s Physics is a veritable introduction to a treatise on differential calculus (Cantor (1901) 6 ). Conceptual anachronisms can be historically and mathematically justified as is the case with Euclid’s Def. 5.5 and 7 and Dedeldnd’s definition of real numbers; see Heath (1956) 2,124-126 and Mueller (1981) 125-127 who rightly also lays emphasis on the crucial difference between the ancient and modem treatment of irrationality (cf. Simon (1901) 110 who claimed that the Greeks did possess the modem notion of number and were not restricted only to natural numbers). But asserting that space in Aristotle’s universe is non-Euclidean, albeit incidentally, is by no means comparable to this situation because such an argument lacks any solid basis, like El. Def. 5.5 and 7: the concept of non-Euclidean physical space presupposes a formidable technical apparatus that finds no parallel in ancient mathematics (cf. also below n.76 and 78). “ to 8e Toiç TTapaWrjXovç avpTVKTeiv oieoBai yewfierpiKov twç m l ayeapeTpijTov &\\ov rpoTcov. One might not agree with T 6 th (1967) 370ff. who sees in this passage a reference to non-Euclidean geometry but Aristotle clearly operates with a definition of parallels similar to Euclid’s; cf. Heath (1956) 190 and (1949) 28. It is implausible that Aristotle would define parallels as straight lines which do not meet each other without the stipulation of production ad infinitum. As will be shown below, this stipulation does not contravene Aristotle’s insistence that infinity be only potential. There is, moreover, no reason to think, along with Hintikka (1973) 120, that Aristotle himself or the geometers of his time had not realized the indispensibility of the parallel postulate. As Heath (1956) 202 plausibly suggested, Aristotle’s mention of a petitio principii current in his contemporary theory of parallels (AtiPr. 65a3-9) most probably refers to the problem that was solved when Euclid laid down his parallel postulate. If T 6 th (1967) is right, Aristotle’s brief remark iaAnPr. 66all-15 reflects a failed attempt to solve the problem indirectly. 39 Two attempts to show how Aristotle could have accounted successfully for the needs of Euclidean geometry in a finite universe do not measure up to the task. For Hussey in Physics F Aristotle urges for a finitist overhaul of mathematics that explaias the unlimited sequence of numbers but precludes arbitrary extendible magnitudes. As I will show in ch. 2, Hussey is wrong in assuming that one can do away with arbitrary extensions in the way he proposes and still operate with Euclidean geometry. But, apart firom that, would Aristotle have subscribed to such a double-pronged finitism? Given his characterization of number ia Met. 1087b33- 1088a6 and magnitude in Met. 1020a8-10,^ this seems implausible and, indeed, in DC 271b26-272a7 he does allow for both arbitrary numbers and magnitudes.^ by trying to lead to absurdity the proposition that Euclid later on declared a postulate. In this light Aristotle and his contemporary mathematicians did know what became Euclid’s parallel postulate, albeit not as a postulate. The history of Euclid’s fifth postulate is of course impossible to reconstruct with any degree of certainty; what matters is that Aristotle most probably operates with the Euclidean characterization of parallelism. ^See above n.ll. “Hussey (1983) 178-179 contends that Aristotle’s finitism is adequately captured by a system of mutually accessible possible worlds whose underlying logic is S 5 -to be more Aristotelian, Hussey substitutes "possible states of the world" for "possible worlds". But, if Aristotle’s potential infinity of numbers is to be modelled in this way, how could one avoid actual infinity since for Aristotle aU possible states of the world are bound to actualize? The argument that they caimot be actualized at the same time does not work in this case because Hussey (1983) 97 ascribes to Aristotle the thesis (x) C (3y) (y>x), where C means "conceivable" and quantification is over existing numbers: but, if Cp -* Op, as Hussey admits, what prevents the comprehension of all bearers of that property so that Hussey’s Aristotelian finitism turns out parallel to Helhnan’s modal-structural interpretation of mathematics which 40 In a similar attempt to save the phenomena White^* imputed to Aristotle a compromise branding him an instrumentalist who in Hilbert’s fashion accepts formal, i.e. meaningless, geometrical statements involving arbitrary extendibilily as long as they allow correct geometrical results which do not have absurd physical implications; this is part of a broader thesis (it will be discussed in the following section) that attempts to account for Aristotle’s "non-Euclidean" physical universe in Hintikka’s conclusion. But, even if Aristotle’s philosophy of mathematics admits of an instrumentalist interpretation, which is doubtful,^ White cannot evade the difficully raised by Hintikka. White fails to e)q)lain how instrumentalism could furnish arbitrarily large extensions on Hintikka’s reading of Phys. 207bl5-20; if arbitrarily large potential extensions depend ontologically on actual physical extensions, and allows for a-sequences? (See Heilman (1989) 24-33.) “ White (1992) 160-161. ” ln attributing instrumentalism to Aristotle White follows Lear (1982) 187-188. Starting from Aristotle’s doctrine of abstraction Lear concludes that "the key to explaining the truth of a mathematical statement lies in explaining how it can be useful". I do not see how Aristotle’s attempt to account for geometry and arithmetic without assuming the existence oîseparate mathematical objects in Plato’s fashion can be read as a version of instrumentalism; for the variations of this philosophical doctrine see Jesseph (1993) 76-77. It seems that what makes instrumentalism attractive to White in this context is the fact that Hilbert’s formalism posited only fînitary mathematics as truly meaningful but attempted to save infinitary mathematics by treating it instrumentally: meaningless statements about the infinite is a useful tool in order to derive meaningful statements about the finite (see esp. Hilbert (1926)). Projecting, however, Hilbert’s formalism, which is part of a metamathematical program, on Aristotle’s philosophy of mathematics is an unjustified anachronism - there is nothing in common between the two. 41 White himself admits that much/^ instrumentalism is of no help at all. K the charge Hiutikka himself brought against Aristotle as a result of his interpretation and the attempts to defend Aristotle on the face of this interpretation sound unconvinciug, there must be something wrong with the interpretation itself. Suggestively, already Milhaud found Aristotle’s concept of infinity mathematically boorish on other erroneous grounds and especially because Aristotle admits arbitrarily small magnitudes while rejecting arbitrarily large ones: Qu’ Aristote n’ ait pas senti 1’ étrangeté de ses afdrmations, et qu’ il ait adopté pour le problème de l’ infini deux attitudes aussi contraires, cela montre suffisamment qu’ aucune de ses deux conclusions n’ est le fait d’ un esprit profondément pénétré de la pensée géométrique du V® et du IV® siècle.^® Milhaud’s verdict has apparently survived unchallenged to date. Its direct descendant, Hintikka’s thesis, has also broader historical implications because, to a large extent, it informs Knorr’s negative evaluation of Aristotle’s theory of potential **White (1992) 160 (i-iii). ^®Milhaud (1903) 392 (for his discussion of Aristotle’s theory of infinity see 381- 386). Milhaud has also claimed that "ce n’ est pas seulement en supprimant l’ infini de grandeur qu’ Aristote risque de s’ éloigner de 1’ attitude naturelle du géomètre; c’ est aussi et plus encore peut-être en refusant 1 ’ entéléchie à l’ infini additif révélé par la divisibilité" (385). As an example of this "attitude naturelle du géomètre" Milhaud gives the exhaustion method which, he claims, rests on the actual exhaustion of a magnitude, e.g. of a circle via an actually infinite sequence of inscribed polygons, irrespective of Eudoxus’ rigorous procedure that masks this crucial step. This is a misconstrual of the point in an exhaustion proof and Aristotle cannot be criticized on this account. On Milhaud’s critique of Aristotle’s argument in DC 272b25-28 see below n.l34. 42 infinity in relation to mathematics -this is part of Knorr’s broader view that poses an autonomous evolution of Greek mathematics independently of any philosophical influence.® But if one turns to Physics P, Hintikka’s interpretation of Phys. 207b 15- 20 is mistaken simply because this passage, like Phys. 207b27-34, has nothing to do with arbitrarily large potential magnitudes, although it can be so interpreted if taken out of context. Phys. 207b 15-20 comes firom a discussion that picks up the distinction between potential infinity by division, which exceeds all assigned magnitudes, and by addition, which lacks this property (JPhys. 207a33-35), a distinction I discuss in detail below in ch. 3. This addition, however, is not the addition that would produce arbitrarily large potential extensions and thus substantiate Hintikka’s interpretation but rather theinverse addition whose modal contrast with potential infinity by division Aristotle justifies by the same cosmological reductio ad absurdum as xnPhys. 207b 15- 20 : Therefore it is impossible to go beyond any assigned magnitude by addition even potentially unless there is accidentally an actual infinity, as the physicists claim that the body outside the cosmos, whose essence is air or something like that, is infinite. But if there cannot be an actually infinite perceptible bocfy in this sense, it is clear that there cannot be infinite by addition, not even potentially, except, as said, inversely to division... {Phys. 206b20-27). Hintikka does not mention this parallel passage but, as Ross had already realized, Aristotle’s modal restriction on potential infinity by inverse addition simply implies “ See the discussion in ch. 5-7. «Ross (1936) 557. 43 that the variable Sn never reaches the limit AB, a stipulation that played a role in mathematics at least until the time of d’ Alemhert and de la Chapelle.® Although justified by a cosmological reductio ad absurdum, this stipulation does not commit Aristotle to the rejection of arbitrarily large potential extensions, magnitudes of a distinctly different type from inverse additions, as Hintikka takes it. Why Aristotle, quite une^ectedly one might say, resorts to a cosmological argument in this context will be addressed below;® it is clear though that Physics F supports neither Hintikka’s claim that Aristotle barred arbitrarily large potential extensions nor Knorr’s verdict that Aristotle’s theory of infinity is irrelevant to the needs of his contemporary geometry. Elsewhere, moreover, Aristotle couches explicitly the availability of arbitrarily large extensions in terms of potential infinity: Hintikka himself realizes thatDC 271b26-272a7 is a counterexample to his interpretation but he tries to explain it away by labelling it merely "apparent".® Far firom being so, however, this passage subverts his thesis firom the ground up. ®See Grabiner (1981) 84-85. ®See the discussion in ch. 3. “Hintikka (1973) 121. CHAPTER n DC 271b26-272a7: ARBITRARILY LARGE MAGNITUDES AND EUCLIDEAN GEOMETRY Ironically, in DC 271b26-272a7 Aristotle carries on an argument against the infinitude of the universe by reducing to absurdity the supposition that an infinite body rotates: From the following it is clear that a body rotating in a circle is necessarily finite because (1), if such a body is infinite, then the lines produced firom its center will be infinite too. (2) But the distance between infinite lines is infinite. (3) By "distance between [infinite] lines" I mean the distance beyond which no magnitude can be taken so as to touch the lines [0iaaTtiiia...\&Y^ rwy ypannâv où iiijSev êanp ê^a \afielv néyedoç à-irronevop t û p ypamiûp]; this magnitude is therefore necessarily infinite. (4) For the distance between finite lines is always finite. (5) And, moreover, it is always possible to take a magnitude greater than an assigned magnitude [âei êan t o v ôodévroç pà^op XajSetv] so that, as we say that numbers are infinite in the sense that there is no greatest number, the same holds for distance too. ( 6 ) Thus if it is not possible to traverse the infinite [To...ôir«poy pq Ian ôicKdeîp] and, since the rotating body is infinite, the distance between the lines is infinite, it is not possible for the infinite b o ^ to rotate; but we see that the heaven rotates and we have concluded that this is the rotation of a body. Aristotle’s argument is not hard to follow. The tacit implication in ( 6 ) is that the rotation of a sphere (in this case the cosmos) is represented by the displacement of its radius as it sweeps any sector of the sphere’s great circles (cf. DC 272al3-14). 44 45 Fig. 2 Aristotle’s argument against an actually infinite cosmos (DC 271b26-272a7) This displacement is measured along the maximal distance AB between any two fixed radii (Fig.2), which serve as a frame of reference, as the radius sweeps along and passes through it ((l)-(2)). To lead the initial assumption to absurdity Aristotle argues in ( 6 ) that, if AB is infinite, it cannot be traversed (To...&ir«/)oy iiri can ôtékdeîp), so that no displacement and, therefore, no rotation can take place contrary to the observed phenomena.*® In Physics F (204al4, b8-10,207b29) "untraversable" *®It is indeed diffîcult to visualize the situation in Aristotle’s argument, i.e. an actually infinite sphere in which any chord subtending the arc that bounds any sector is equal to any other chord. Albeit tacit, this assumption is necessary for the generality of the argument and, as far as the sphere’s radii are concerned, it does have an intuitive appeal since, in the case of actually infinite diverging straight lines. 46 is the characterization of actual infinity and picks out an important formal property as is indicated by the argument in Phys. 266b6-14. But it should be noted first that this characterization of an actually infinite magnitude is synonymous with the one in (3), i.e. a magnitude 'beyond which no magnitude can be taken" [ou fii]ôep eonv XajSeîu néyeOoç]. For this is again the infinite in the sense Aristotle refutes in Physics T: It turns out that the infinite is the opposite of what is usually claimed to be: the infinite is not that beyond which there is nothing but that beyond which there is always something [où y a p ov fiJiSep e|co, ôXX’ ou àei n kari, t o u t o cnceipop kanp]. Hoopless rings are called "infinite" in the sense that it is always possible to take something beyond [5ti aei n e^a tan Xa/ijSdmu]... Thus the infinite is that beyond which it is always possible to take some quantity [&neipop...kanp ov k u t o i t o iroaop Xafi^apovaip àeî n Xap^âpeip kanp that beyond which [it is not possible to take a quantity, où be prjSkp l| Since for Aristotle "whole" is synonymous with "complete" (Met. A.26 and 16) which is defined as "that beyond which no part can be taken" (rekeiop \éyeTai...ov pri kanp ë|a) n XajSetu prjSe kp popiop Met. 1021b 12-13), his characterization of actual infinity is essentially identical with Cantor’s: i.e. those exceeding any assigned straight line, there is a distance between them that similarly exceeds any assigned distance. Given that Aristotle conceives of the rotation of such a sphere in terms of angular displacement, his argument presupposes non-Archimedean angles which are easily, and of course anachronisticaUy, implied since the Archimedean axiom for straight lines is equivalent to its formulation for angles (see FeigI (1926) and Baldus (1927); Aristotle’s "untraversable" characterizes actual infinity as non-Archimedean; see the discussion below). 47 By an actually infinite is to be understood a quantum which on the one hand is not variable but rather is fixed and determined in all its parts -a genuine constant- but which at the same time surpasses in magnitude every finite quantity of the same kind.^ The second condition amounts to the absence of the Archimedean property which is picked out by Aristotle’s "untraversable", as is shown by the proof mPhys. 266b6-14 that an infinite magnitude cannot have a finite power: Let AB be an infinite magnitude, (a) This magnitude has some force BF which in some time, let it be EZ, moves A. If I a take the double of BF [t^ç BF bmkaaiav \an^&vav\, [it will move A] in half the time EZ, i.e. Z0. (b) Thus taking always [the double of BF] in this manner I will never traverse AB [t^i> pev AB ovSeiroTe ôié^api] and I will always take a time less than the given time [roO xpovov be t o v boOéPToç aiel ekarrui “ Cantor (1887-1888) 400. ®^The mathematical part of Aristotle’s argument presents no problem (it is easily eq)lained by the principles in. Phys. 266b2-4 (to be discussed below) and the second definition of the quicker in Phys. 232a25-27 formulated for powers and their effects (cf. Phys. 266a26-28)), but Aristotle does not state e)qplicitly the physical considerations that necessitate it: to show that an infinite magnitude cannnot have a finite power Aristotle seems to assume that magnitude and power are coextensive so that the power cannot be finite as the hypothesis assumes. Since weight is proportionate to motive power (Phys. 250a8-9) and is, moreover, defined as quantity of matter per volume (DC 299b7-9), this is indeed so if Aristotle has in mind primarily weightqua motive power. In this case the increments 2“BF stand for both motive power and quantity of matter per volume so that power and magnitude are coextensive. 48 ~bF ze7^E Z /2) 2“Br EZ/2“ Fig. 3 Graphic representation of the argument in Phys. 266b6-14 (b) clearly presupposes the second property in Cantor’s definition of actual infinity since AB is infinite in the sense that it is greater than any 2°Br. Indeed, this characterization of actual infinity occurs eq)licitly twice in the De Caelo, once for an infinite distance (diaanina as in DC 271b30) which is greater than any assigned distance ( t o i 't o ç t o v TrpoTeOêvroç fieî^ov...ôiâaT7](ia, DC 301b 14-15) and once for an infinite time which is always greater than any assigned temporal interval and less than none (àel vK&xàv t o v TrporedévToç Kal o v k eoTiv ov eXoTTov, DC 281a33-34). It is important to note that on Aristotle’s conception of time (Phys. 219b2-5) this characterization of an actually infinite time holds for numbers too. Returning now to DC 271b26-272a7, it is clear, both philologically and conceptually, that Aristotle tacitly assumes Phys. 266b6-14(b) in ( 6 ). It also turns out that (4)-(5)do not belong to the premisses of the argument: because (4)-(5) take up what would be the case were the fixed radii finite and not actually infinite as in the supposition Aristotle reduces to absurdity. According to (l)-(3), since the fixed radii are actually infinite, they are cut by an actually infinite transversal AB greater than 49 any given transversal as actually infinite magnitudes are characterized iuDC 301b 14- 15 and 281a33-34. In (4)-(5) Aristotle simply illustrates this situation by way of contrast, pointing out that, were the fixed radii, and thus their transversal, finite, then it would always be possible to obtain a transversal greater than the assigned one. Much more importantly, (5) makes clear that this transversal is infinite too, not, however, in the same sense as the transversal in (3) but in the sense that numbers are infinite, i.e. because there is no greatest number, so that it is always possible to take a number, or transversal for that matter, greater than any assigned one. But, since the contrast between these two kinds of infinity is obviously the contrast between actual and potential infinity in P/iyj. 206b33-207al0, the parenthetic "as we say " in (5) should be read literally: (5) picks out infinity in Aristotle’s own sense, potential infinity qua arbitrary variability of both magnitudes and numbers. Thus there is no evidence that Aristotle shared Hero’s scruples about arbitrarily extended lines in geometrical proofs. Although Hintikka himself realizes that (5) subverts his reading oiPhys. 207b 15-21 by uiuestrictedly allowing arbitrary variability of magnitudes, he simply remarks that in this context Aristotle’s apparent argument for it is fallacious, however. Moreover, Aristotle is in any case conducting there a reductio ad absurdum against the alleged infinity of the world and hence may have appealed to the principle in question merely because he thought that his opponents were committed to it.® But not only does not Aristotle argue in support of (5) in any, much less fallacious. 6 8 'Hintikka (1973) 122. 50 way, he also can by no means attribute his own conception of infinity to any putative opponent. (5), moreover, forcefully brings to the fore a patently absurd implication of Hintikka’s thesis since Aristotle clearly accounts for the infinity of numbers in the same way as for arbitrary geometrical extensions: were Hintikka right, Aristotle should impose on numbers the same constraints as on arbitrary geometrical extensions, but he clearly does the exact opposite so that he can fully accoimt for the needs of both arithmetic and geometry. Aristotle’s argument in DC 271b26-272a7 did not fare much better in the hands of White who understood it as follows: There apparently is areductio implicitly present here. If there were such a maximal distance, that fact would entail that the radii could not be extended beyond the end points of the interval. But then the radii would not be infinite.'^ White goes on to argue that Aristotle employs the "formal correctness" of the principle in (5) to show that, because of the physical impossibility of an infinité cosmos, arbitrarily large magnitudes do not actually exist: Of course, Aristotle is employing the principle within the context of a reductio argument; his purpose seems to be to use the formal correctness of the principle to argue that, because of physical facts connected with the impossibility of infinite linear displacement entailed by any angular displacement of a cosmos of infinite radius, arbitrarily large triangles do not actualfy exist.™ ®White (1992) 157. ’•White (1992) 158. 51 As the reference to "formal correctness" indicates, it is here that White sees a basis for Aristotle’s instrumentalist attitude towards arbitrarily large potential extensions in geometrical proofs; and, since, moreover, according to White the proof in DC 271b26-272a7 relies "on assumptions closely related to the parallel postulate",’* he feels confident to conclude that Aristotle does not face Hintikka’s fallacious charges. Apart from the inappropriateness of instrumentalism. White falls in the same pitfall as Hintikka and in others of his own. As shown above, the principle in question, i.e. (5), captures Aristotle’s own potential infinity, which has nothing to do with the reductio ad absurdum despite Hintikka’s and White’s contention. Consequently, the upshot is not that "arbitrarily large triangles do not actually exist" on cosmological grounds, as White takes it, but that actually infinite triangles do not exist on such grounds, a claim distinctly difierent firom White’s. White also misinterprets Aristotle’s point inDC 271b26-272a7. A maximal transversal AB would contradict the assumed infinity of the radii, as White thinks, only if Aristotle operated with the potential infinity of (4)-(5). This is not so, however, because the text leaves no doubt that the radii, and thus their transversal, are actually infinite; as shown above, potential infinity enters the argument only as a footnote to illustrate the situation by way of contrast.” ” White (1992) 156. ”It is probable that Proclus made the same mistake as White in the interpretation of Aristotle’s argument in DC 271b26-272a7. After stating the Aristotelian axiom, on which his own proof of Euclid’s fifth postulate depends, Proclus gives the following account of Aristotle’s argument: éôei^e '^ovv cKeîpoç on 52 The only reason, moreover, for which White thinks that DC 271b26-272a7 relies on assumptions "closely related to the parallel postulate" seems to be the fact that, after pointing out the flaw in Ptolemaeus’ attempt to prove Euclid’s notorious fifth postulate, Proclus offers his own proof in which he formulates (5) as an axiom Aristotle referred to in the De Caelo: If from a single point two straight lines making an angle are produced indefinitely, the interval between them, when produced indefinitely, exceeds any finite magnitude{In Pr. EL Com. 371.14-17).” Thus the relationship between (5) and Euclid’s parallel postulate is purely contextual, because White himself admits that Proclus’ "Aristotelian axiom", i.e. (5), is not equivalent to Euclid’s parallel postulate; he claims, instead, that it is weaker in the sense that àiceCpav ovawp rwvàxo tov Kévrpov irpoç t^v vepi^épeiap èK^e^Xtip.évœp aireipop ro pera^v. vevepaapepov yap hvroq av^ijaai t^v ôiaoraoip wcrre àôvvaTop ovk areipoi ai evBeîai (In Pr. EL Com. 371.17-19). Proclus interprets Aristotle’s argument in the same way as White (1992) 157 (quoted above) but it is important to note that the Latin translation of Barocius (1560) presupposes the reading Svparop in 371.20 for aSvparop in Friedlein’s edition (Morrow (1970) reads àSvparop). If Barocius found ôvparôp in one of the five manuscripts on which he based his Latin translation (he never published his Greek text), it is fairly possible that Proclus did not misunderstand Aristotle’s argument. In this case the ya p clause at the end of the quote refers to infinity in the potential sense whereas aveipoi, like aveipop in the previous sentence, picks out infinity as actual. It is of course equally possible that Barocius himself read ôvparôp for àôvparop in an attempt to streamline Proclus’ account with Aristotle’s argument. 73 For Proclus’ proof of Euclid’s fifth postulate see Gray (1989) 36-38. 53 it rules out the elliptic counterexample to Euclidean geometry (in which there is an upper bound to the distance between straight lines) but not the hyperbolic counterexample (in which there is no upper bound to the perpendicular distance between two given parallel straight lines).’"* By adhering to (5), however, Aristotle could have never ruled out the "elliptic counterexample to Euclidean geometiy", unless in a vacuous sense. (5) seemingly contradicts the fact that there is an upper bound (tt /2 ) for the distance between any two points in the elliptic plane; but Aristotle cannot be said to have implied such a contradiction, since he did not have any notion of the elliptic plane. White’s further claim, that (5) leaves open the hyperbolic counterexample, is equally meaningless, first, because (5) amounts to a theorem of absolute geometry, as will appear below, so that by definition it can neither leave open nor rule out hyperbolic properties and, second, because the properties of hyperparallel lines in hyperbolic geometry can in no way be analogous to a property of intersecting lines in absolute geometry.” Finally White can defend Aristotle from Hintikka’s charges only by imputing to him a conception of space as bounded, i.e. non-Euclidean, but in other respects essentially ^^White (1992) 158-159. ”l assume that, when White explains the "hyperbolic counterexample" as the "absence of an upper bound to the perpendicular distance between two given parallel straight lines", he refers to the fact that two hyperparallel lines converge towards their common perpendicular and then diverge so that the distance between them increases without limit; see Martin (1982) 342. That the proof of this fact appeals to Proclus’ "Aristotelian axiom" does not substantiate White’s claim unless Aristotle operates with hyperparallel hnes, even incidentally and unconsciously. But this is clearly not the case. 54 Euclidean.’* This is White’s formulation of the problem diagnosed by Hintikka,” a mathematical chimera made even more bizarre by the introduction of instrumentalism which, by White’s lights, turns the Euclidean and the non-Euclidean elements into a coherent whole.’® To interpret "bounded" as "non-Euclidean" in an obvious reference to the Gnitude of Aristotle’s cosmos is outright unwarranted: "bounded" makes sense mathematically only in relation to the intrinsic geometry of a surface, a notion to which Aristotle’s fînitude of the cosmos bears no analogy, even if one is willing to accept conceptual anachronisms.’® Aristotle could not have held that space in his finite universe is bounded in this sense. The closest parallel to ’“White (1992) 160. ” Curiously, Hintikka (1973) 119 claimed that on Aristotle’s principles Euclid’s fifth postulate fails: what can be justified is "merely the statement that, given the situation described by Euclid (line AB falling on the staight lines AC and BD, angle CAB and angle ABD being less than two right angles), there is a point A’ on AB sufficiently near A such that a parallel to BD through A’ meets AC on the side Euclid specifies. This does not, however, guarantee that the resulting geometiy is Euclidean". If I understand Hintikka correctly, he imputes to Aristotle a simple trick: he scales down the configuration described in the parallel postulate so that the parallel to BD through A’ meets AC very soon without the need for the stipulation that they meet when producedad infinitum. But even if this were so, i.e. if Aristotle did have compunctions about arbitrary production of straight lines, the resulting geometry would still be Euclidean because Hintikka’s hypothetical alternative to Euclid’s fifth postulate involves assumptions equivalent to the fifth postulate: in positing a (unique) line through A’ parallel to BD he evokes Playfair’s axiom and at the same time he assumes that any three lines have commona transversal. ’®White (1992) 160-161. ’®On the notion of the intrinsic geometry of a surface see the accounts in Lanczos (1970) 75ff. and Gray (1989) IBSff. 55 White’s "Aristotelian" conception of space seems to be the fact that the non- Euclidean intrinsic geometry of a curved surface becomes Euclidean in infinitesimal domains as the Gaussian curvature vanishes. But, since Aristotle could not have integrated boundedness and Euclideanness in this manner, it is difficult to see how he could have reached and upheld the conception of space White attributes to him, a conception that obviously cannot be justiGed by instrumentalism or any other philosophical stance, no matter how persuasively articulated in itself. Ultimately White’s attempt stems from the mistaken view that Aristotle’s finite universe necessarily compromises the availability of arbitrary extensions, a fundamental property of Euclidean geometry. It will be shown in ch. 3 that Aristotle’s demonstrated acceptance of arbitrarily produced straight tines (to be supported by further evidence below) is not affected by a finite universe. But, since scholarly exasperation at Aristotle’s theory of infinity stems from a concern about the Euclidean nature of geometiy, it is worthwhile to see first whether potential infinity is at odds with this geometry, as has often been assumed;^" the question is also ®“Hussey (1983) 93-96 seems to assume that infinity in Euclid’s Elements, and in Euclidean geometry in general, is actual -this is evident, among other things, in Hussey’s mistaken assertion that in exhaustion proofs there appear "infinitely large totalities" (96). Knorr has also explicitly denied that potential infinity is indispensible for the purposes of geometiy (see above n.69) and, in his critique of Ross (1936) 52, Hintikka (1973) 118 n.10 distinguishes the Archimedean axiom firom the kind of infinity required for the development of the theory of parallels, obviously assuming that the latter involves actual infinity. Nevertheless, as will be seen below (and this bears directly on a thesis in Hussey’s intrepretation of Aristotle’s views on mathematical infinity), non-Archimedean geometries need not be Euclidean and, therefore. Euclidean geometry must be Archimedean, i.e. it cannot accomodate 56 raised by Aristotle himself, who in Phys. 207b27-34 claims that his contemporary mathematicians not only do not use but also do not need actual inhnity. Although this claim concerns a particular type of proof, it will turn out that Aristotle’s insistence on potential infinity can by no means create problems to Euclidean geometry in general. Surprisingly, (5) played a minor role in the history of mathematics. In his own attempt to prove Euclid’s parallel postulate Omar Khayyam (c.l050-1123)®‘ appealed to (5) in a way similar to Proclus’ and only in 1574 Clavius pomted out that (5) cannot be taken as an axiom but requires demonstration.®^ Saccheri, who eventually proved the proposition in 1733, showed that (5) holds under the hypothesis of both the right and the acute angle, i.e. it is a theorem of absolute geometry.®® Thus, by endorsing (5), Aristotle implicitly accepts a proposition of absolute geometry but, since, as said above, (5) exemplifies his notion of potential infinity, there is philological evidence that, like Proclus, he construed (5) not as a theorem but as an application of a particular axiom. In Phys. 266a24-b6, the converse of Phys. 266b6-14 which has been applied above to the interpretation of DC 271b26-272a7, Aristotle shows that there carmot actually infinite magnitudes. ®®For Omar Khayyam’s proof of Euclid’s fifth postulate see Gray (1989) 47-49. ®^For Clavius’ reasons see Heath (1956) 208. ®®Saccheri (1733) proved (5) as his proposition 21; a proof of this proposition can also be found in Martin (1982) 265-266. 57 be am infinite power in a finite magnitude. Assuming tacitly that power is coextensive with the magnitude of the body possessing it, Aristotle proceeds by reductio ad absurdum: If A is the [finite] time in which the infinite power [F] warmed or moved a given body and in [the finite magnitude] AB there is a finite power (i), by adding continually a greater finite power to this finite power [irpbç tuvttju hxn^âpav àeî ireirepaff/tépijj»] I will eventually reach a power which performs its effect in time A; (ii) because by continually adding to a finite magnitude I wiU exceed any assigned magnitude[irpbg ireirepuafiépop yap àéi vpoanBeïç vTrepjSaXû icavToç ùpiapévov] and (iii) in the same manner by [continually] subtracting [from a finite magnitude] I will obtain a magnitude less than any assigned magnitude [/tai à<]>aipûv éXXei^f'W àaavraç]. (iv) Thus a finite power will move [a body] in the same time [A] as the infinite power. But this is impossible. Thus no finite magnitude can have an infinite power. It is clear that the operation in (i) is couched in the terminology of (5) (cf. especially pei^o) \ap^àvav àii ~ àei ean...peî^ov Xa/Seîv in Phys. 266bl and DC 271b33-272al). As a result, whatever principle is evoked in (ii) to justify this operation, must be implicit in (5) too, aU the more since in both (5) and in Phys. 266a6-bl4 Aristotle operates with potential infinity. This is manifest not only in the locution virep^aXS) t o l v t o ç èpiapêpov (ii), which in Physics F occurs in characterizations of potential infinity of both geometrical magnitudes and numbers (206bl9-20,207a33-b5,207bll-20), but also in the operation itself (peifw Xap^àvm àel Treirepaapévrip) which picks out a crucial point in the characterization of potential infinity at P/iys.206a27-29 (koù to Xap^apôpepop pep àel ehai veicepaapepop àXX’ àei ye erepop Kal erepop). The nature of this principle is obvious firom its function withiu Aristotle’s proo^ which speaks against Heath’s charge that the assumption of a finite 58 power in (1) is risky.®^ Ross®* has accounted for this assumption in a prima fade acceptable way: since theinfinite power moves or warms a given body in time A, the finite power AB does the same in a greater time nA by the principle of virtual velocities {Phys. 249b27-250a9), so that by the same principle a finite power nAB moves or warms the same body in time A®* Aristotle could have argued in this manner; but Ross fails to account for the presence of (ii) and (iii), although Aristotle makes clear that they justify the operation in (i) which results in the absurdity in (iv).®’ Nevertheless, Ross is right to supply the easily implied assumption that AB moves the same body as F in time nA andiViy^. 266b6-149(b), the converse oîPhys. 266a6-bl4 quoted above, allows to complete the operation in (i) thus accounting fuUy for the presence of (ii) and (iii): in Phys. 266b6-149(b) Aristotle assumes that, by continually doubling a finite power BF, 2”Br moves the same body in continually less time EZ/2° and (i) should be taken as a compressed reference to this complementary ®^Heath (1949) 153. ®*Ross (1936) 723. ^O n Phys. 249b27-250a9 see Heath (1949) 143-146, Hussey (1983) 194-196 and (1992) ; for a study of Aristotle’s principle of virtual velocities see Vailati (1897). It should be noted that the term "virtual velocities" is used with respect to Aristotle’s principle not m J. Bernoulli’s sense but in that of the thirteenth century mathematician Jordanus de Nemore: what suffices to lift a weight W through a vertical distance H will lift a weight kW through a vertical distance H /k or a weight W/k through a vertical distance kH (cf. Clagett (1959) 78). ®’Heath (1949) 153 remarks that (ii) does not help Aristotle; the text, however, leaves no doubt that (ii) does play an important role in the argument. Incidentally, it should be noted that a major desideratum in Aristotelian scholarship is a detailed study of Aristotle’s arguments involving infinity in both Physics and De Caelo. 59 operation on AB and nA respectively.** Because, as (ii) accounts for the possibility to extend AB arbirtarily beyond any assigned magnitude, the presence of (iii) can only be justified here if it accounts formally for the possibiUly to obtain by continuous subtraction (from nA) a magnitude less than or equal to an assigned magnitude (A).*® It is easy to see that (ii) and (iii) are the Archimedean axiom and the bisection principle respectively, the latter of course in a formulation different from the one considered above.®" The application of the Archimedean axiom in this **Implicit here is the second définition of "the quicker" in Physics Z.2 ("the quicker transverses the same space as the slower in less time" (see Clagett (1959) 178) -"same space" of course here translates into "same effect" of a power as Aristotle explicitly remarks in Phys. 266a26-28). The three definitions of "the quicker" in Physics Z.2 are historically interesting (see Clagett (1959) 178-179) but Aristotle attributes them to anonymous riveç and surprisingly proves them, although they have been first introduced as definitions. If these definitions come from Eudoxus’ astronomical work On Speeds, as Knorr (1982) 120 n.l9 suggests, it is probable that the proofs originated with Aristotle. *®The particular absurdity Aristotle derives in his argument (that a finite power performs a given effect in the same time as the infinite power) does presuppose that the time in which the finite power performs the given effect is an integral multiple of the time in which the infinite power performs the same effect, (ii) and (iii) give Aristotle a plausible argument for the case of inconunensurable times: he could argue that a finite power performs the same effect as the assumed infinite power in less time, a patently absurd conclusion. ®"See Knorr (1978) 210, although he unnecessarily relates (ii) to Euclid’s E/. Def. 5.4. This definition is a characterization of magnitudes that can have a ratio to each other but Aristotle’s argument does not employ ratios; cf. Mueller (1981) 143-145 who distinguishes this Euclidean definition from the Archimedean axiom, a tacit assumption in Elements. 60 proof merits more than the minimal attention it usually receives in the secondary literature.®* In spite of Hintikka’s thesis, Aristotle obviously evokes a geometrical principle, framed in terms of potential infinity, which allows magnitudes of arbitrary extension and he does that in what would be the most unlikely place, were Hintikka right, a physical argument about powers of natural bodies. Not only does not Aristotle worry lest the Archimedean axiom lead beyond the confines of the cosmos,” there is, moreover, no indication that he applies this axiom in White’s ®*Interestingly Heiberg (1904) 23 saw in (ii) and (iii) "die Grundlage des Exhaustionsbeweises". As far as I know, this is the only appreciative reference to Aristotle’s undoubtedly mathematical argument. “This problem follows easily if Aristotle conceives of motive power as directly proportionate to the quantity of matter per volume so that power and magnitude are coextensive (cf. above n.67): if A is sufficiently small then the nAB required for Aristotle’s argument can be greater than the finite universe. This problem, however, cannot arise because, as will be argued below in ch. 3, for Aristotle the sphere of the universe could be conceived as actualfy infinite for practical purposes so that one can appeal to as great an nAB as possible. If one wants to consider the extreme case, since at any rate the universe is not actually infinite in the literal sense, the problem is again avoided inasmuch as the natural bodies in Aristotle’s argument are the elements: because the total volumes of the elements that are paired in the process of elemental transformation are to each other in certain ratios (Meteor. 340al2-14), i.e. are finite, a point made by Aristotle in order to show that the space between the earth and the stars, actually infinite for practical purposes, cannot be filled by a single element (Meteor. 339b30-340al4). In defense of Hintikka’s view one might claim that Aristotle appeals to the Archimedean axiom exactly because the finite volume of each element available in the universe eo ipso prohibits the unrestricted application of the axiom that might run counter to the fînitude of the universe. This cannot be so, however, because physics is "subordinate" to geometry (cf. AnPost. 76a9-15) so that the Archimedean axiom belongs to the latter where there are no physical facts to restrict its application in proofs and, of course, there are no restriction of a geometrical sort. 61 instrumentalist fashion, i.e. as a formal, meaningless principle acceptable only inasmuch as it does not involve physical absurdities. But the role of the Archimedean axiom in Euclidean geometry helps bear out a stronger version of Aristotle’s thesis that actual infinity is not needed in geometry. As already remarked, it turns out from Phys. 266a6-bl4 that Aristotle would not have distinguished (5), Proclus’ "Aristotelian axiom", firom the Archimedean axiom on which surprisingly the proof of (5) as a theorem of absolute geometry hinges.” Thus by endorsing (5) Aristotle not only accepts intuitively a proposition of absolute geometry but also implies that ruling out actually infinite extensions firom geometry amounts to the characterization of geometrical space as Archimedean. In his attempt to absolve Aristotle of mathematical incompetence on Hintikka’s fallacious premisses, Hussey suggested that Aristotle demanded a finitist reworking of Euclidean geometry which would substitute El. 1.32, i.e. the theorem that the triangle’s interior angles are equal to 180 o , for Euclid’s problematic fifth postulate.’* Hussey obviously attempts to retain the Euclidean nature of geometry ” See the proof of Proclus’ "Aristotelian axiom" (= Saccheri’s proposition 21) in Martin (1982) 265-266. ’^Hussey (1983) 95. I have already commented on two attempts to attribute a version of finitism to Aristotle. A third attempt by Welti (1986) bears directly on the issue discussed here and illustrates forcefully once more the dangers of conceptual anachronism in historical studies. According to Welti (1986) 234 "Erst 2.300 Jahre spater wird Kunstaanheimo die Aristotehsche intuitive Überlegungen streng rechtfertigen. In seinen Finiten Geometrien FGp (wo p ein Primzahl) sind alle Wünsche von Aristoteles restlos erfullt." Welti subscribes unquestionably to Hintikka’s views but thinks that Aristotle’s finite universe does not clash with Euclidean geometry because in Kunstaanheimo’s finite geometries FGp there are 62 in a finite Aristotelian universe that does not allow for arbitrary extendibüity of strai^t lines; but El. 1.32 is equivalent to Euclid’s fifth postulate orify on the additional assumption of the Archimedean axiom, the formal expression of such arbitrary extendibüity. A non-Archimedean geometry can be readüy constructed which lacks arbitrary extensions, satisfies El. 1.32 but is non-Euclidean, i.e. the parallel axiom does not hold in it.” The importance of the Archimedean axiom to Euclidean geometry guarantees the validity of Aristotle’s remark in FAys. 207b27-34 finite points (p^) and straight lines (p(p+1)), the Archimedean axiom does not hold, and the parallel postulate does. Inasmuch as finite geometries are applicable to physics (see Jamefelt (1951)) Welti makes a point which might be correctperse but is irrelevant to Aristotle. K Hintikka is right, a justification of Aristotle in view of a mathematical theory based on Galois groups is beside the point. FGp, moreover, are not Euclidean in the usual sense of the word: one can recover arbitrarüy large Euclidean planes or spaces with any desired degree of accuracy when p tends towards 0 0 (see Jam efelt (1949) 168-169). Not only the infinity, be it actual or potential, of prime numbers is obviously involved here, but, along with Euclidean geometry, its Archimedean property is also recovered. To account for potential infinity in Euclidean geometry in this maimer, as Welti (1986) 504-505 seems to do, is unnecessarily complicated, despite the intrinsic mathematical value of FGp which cannot preserve the Euclidean nature of geometry without arbitrarily large magnitudes: these are indeed absent in a FGp model but appear as soon as Euclidean geometry is approximated in the above said manner. ” See Hübert (1971) 43. The relationship between the Archimedean axiom and the sum of the angles of a triangle is investigated in Dehn (1900). Without the Archimedean axiom the resulting geometry need not be Euclidean; Dehn described the "semi-Euclidean" geometry in which the Archimedean axiom does not hold, and, although every triangle has interior angles equal to 180 o , the parallel postulate does not hold (there are infinitely many parallels through a given point). Although Hussey (1983) 96 acknowledges that Aristotle knew and accepted the Archimedean axiom, he surprisingly does not realize that this contradicts his firm belief in Aristotle’s rejection of arbitrary geometrical extensions. 63 for the entire Euclidean geometry. As far as extended magnitudes are concerned, Phys. 207527-34 amounts to the thesis in Phys. 206511-12 and 26652-3, that owy magnitude can he measured, or, more generally, exceeded, by a multiple of one of its parts, which follows from its weaker version: in both absolute and Euclidean geometry it can be shown that, if one straight line relates to any of its parts according to the Archimedean axiom, then any two straight hnes relate to each other in the same way, i.e. the entire space is Archimedean; alternatively, if one straight line is never actualfy infinite for any choice of the unit line, then it follows that all straight lines are finite, i.e. there is no actually infinitely great or smaU line (Baldus (1930)).* Baldus’ theorem justifies, anachronisticaUy of course, the extension of Aristotle’s note in Phys. 207527-34 to the entire EucUdean geometiy, all the more since Aristotle himself expUcitly characterizes actual infinity as non-Archimedean in *The fact that Baldus’ theorem also holds in hyperboUc geometiy does not undermine my point, i.e. that Aristotle’s potential infinity is not an Hi-conceived philosophical idea that encroaches on geometry. There is of course no underlying claim that Baldus’ results can shed Ught on Aristotle’s theory of potential infinity. They do, however, dispel one’s impression that Aristotle’s potential infinity is an essentially ungeometrical notion which EucUd and his predecessors wisely chose to disregard as Knorr thinks. Leaving aside the historical issue whether EucUd took into account Aristotle’s theory of the infinite (see ch. 5-6), potential infinity makes perfect sense geometrically, especiaUy since in actually infinite, i.e. non-Archimedean, geometrical spaces the resulting geometiy need not be EucUdean. The indispensibiUty of the Archimedean axiom to the development of EucUdean geometry is shown by the fact that, if the paraUel axiom holds but the Archimedean axiom does not, then El. 1.5, 1.20 and the Pythagorean theorem cannot be proved; see BHlbert (1971) Appendix H. 64 both DC 281a33-34/301bl4-15 and Phys. 266bl0-13 (and thus inDC 271b26-272a7(3) too). Euclidean geometry can be non-Archimedean inasmuch as it has a model in real numbers which are elementarily equivalent to the set of non-standard real numbers.” Nevertheless, it is obviously a mistake to regard potential infinity in its Aristotelian characterization as a notion foreign to geometry. As will be shown later (ch. 5), Pappus’ "infinity" axiom would have been a welcome addition to the axiomatic foundation of the Elements -this bears directly on Aristotle’s claim in Phys. 207b27-34 that his contemporary geometers do not use actual infinity: Aristotle himself most probably deemed such an axiom necessary because, by the standards of his contemporary geometiy, potential infinity is the only appropriate construal of geometric infinity. First, though, let us deal with some interpretive debts accumulated so far. ”That R, the non-standard extension of R, is non-Archimedean means that there exists AE*R so that |A | >n for every n€"N, where "N is the external set, i.e. it contains the standard subsets of N, the superstructure that contains all mathematical objects under study in the given theory. The fact that Euclidean geometry can have such a model does not contravene Dehn’s results (above, n.96) for the same reason that *R has the same properties as R without any paradox: "the statement "*R has the same properties as R refers to a specified collection of properties of R, including the Archimedean property, that is formulated in a certain formal language. The statements of this language have specific interpretation in R as well as in 'R and the reinterpretation on higher-order properties like the Archimedean property do not retain their full metamathematical strength" (Stroyan & Luxemburg (1976) 5). The remterpretation of the Archimedean property is that for every A in *R there exists a K in *N such that | A| POTENTEAL INFINITY AND ACTUALIZATION To sum up, by evoking (ii) in Phys. 266a24-b6 and (5) in DC 271b26-272a7 Aristotle certainly cannot contradict what he has already denied mPhys. 207b 15-21, as he does on Hintikka’s reading of this passage. It is unfortunate that Hintikka relied onPhys. 207b 15-21 because this passage is parallel to Phys. 206b 16-24 and the text leaves no doubt that Aristotle refers to potential infinity by inverse addition, not to arbitrary extendibüity of magnitudes or addition simpliciter as is the casein. Phys. 266a24-b6 (ii) and DC 271b26-272a7 (5). In this light the cosmological reductio ad absurdum in Phys. 206b 16-24 and 207b 15-21 appears more startling than Hintikka thinks. As already said, the modal restriction on potential infinityby inverse addition simply amounts to the stipulation that a variable never reaches its limit. But why does Aristotle conclude self-evidently that, if it did, there would exist an actual cosmological infinity? A further problem arises firom the modal postulate he evokes mPhys. 207b 17-18 to justify this reasoning, namely that there can actuaUy be as much 65 66 as there can be potentially.’® One would expect that this principle bounds not only potential infinity by inverse addition but potential infinity in every sense which thus emerges as a problematic exception to Aristotle’s theory of potentiality.” Otherwise according to the modal postulate any arbitrary magnitude obtained by (ü) in Phys. 266a6-bl4 would be bound to actualize, even if it exceeded the boundaries of the cosmos. But such an actualization obviously runs counter to the finitude of the Aristotelian universe.**® Nevertheless, to see a crack here in the coherence of Aristotle’s theory of potential infinity hinges on too narrow a reading of certain notions. Aristotle himself addresses the actualization of potential arbitrary magnitudes in Met. 1048b 14-17: The infinite is potential not in the sense that it will actually have separate existence [kvepye.u^...xapiaTÔv\ but only in the intellect, because the fact that the division of a magnitude does not come to an end [Tb...pri viroXeiTreip t^ v diccipeaip] is due to this potential actuality, not to its being separate [To...xcoptfea0a(.].“* ^Aristotle here seems to imply his definition of potentiality as potentiality for a maximum (DC 281al0-12) measured on a scale of what is to be actualized; cf. van Rijen (1989) 89-90. 99,Cf. Knorr (1992) 121-122. **®For this view see also Chemiss (1935) 34, Solmsen (1960) 168, Hintikka (1973) 117, Hussey (1983) 94 and Knorr (1982) 121-122. Although Phys. 206bl6-24 and 207bl5-21 (the passages these scholars rely on) cannot be considered as Aristotle’s acknowledgment of that fact, the problem is unavoidable in view of Aristotle’s conception of potentiality. **’*The "potential actuality" picks out the "knowledge", which in turn refers to the specific mode in which infinity is actualized, in contrast to its enjoying a separate existence, obviously in the Platonist fashion, as is suggested by the verb 67 This passage fleshes out the remark in Phys. 207bl0-15, that the number of a magnitude’s divisions is not ontologically separate (xwpiffToç) and complete, by denying a thesis parallel to aporia (5) (Phys. 203bl5-30), which justifies the actual infinity of geometrical magnitudes and numbers by their not giving out in thought (pLoi...To €P Tg porjaeL ptj wTroXetxe»'),”® Since there is no reason to assume that Aristotle’s account of the availability of small magnitudes in Met. 1048bl4-17 should not be applicable to large magnitudes, he does not lose sight of the modal postulate iaPhys. 207b 17-18. like potential arbitrarily small magnitudes, potential arbitrarily large ones must be actualized, not as separate existents, however, which could contravene the finitude of the cosmos, but only for knowledge. The relationship between actualization and knowledge Aristotle implies here is made more explicit in Met. 1051a21-33, where the actualization of potential geometric entities required for a proof is said to be the p&qavç of a mathematician who obtains knowledge, i.e. ‘“Ross (1924) 2,252-253 also associates aporia (5) with Met. 1048bl4-17 in order to support Alexander’s reading of this passage. Following the commentator Ross takes TO pri vroXeiveip rijp ÔLaipeaip as the subject and accordingly translates Met. 1048bl4-17 as follows: "For the fact that the process of dividing never comes to an end ensures that this activity always exists potentially but not that the infinite exists as a finished given fact". This translation makes good sense both philologically and philosophically (it reflects a crucial point in Aristotle’s theory of infinity) but it is not borne out by aporia (5). As I will argue below, this aporia records a Platonist thesis which grounds the conceivabiUty of always greater numbers and geometrical magnitudes in the actual infinity of these mathematical entities. The adjective xapiarôç suggests that in Met. 1048b 14-17 Aristotle denies the same Platonic thesis for the availability of always further divisions of a magnitude. In this light it is, I think, preferable to take t o e î m t bvp&pei ravnjp rrfp kpepyeiap and t o be xapi^eaQai as subjects in Met. 1048b 15-17. 68 B r A Fig. 4 The proof oiE l. 1.31 in Aristotle's Met. 105la21-29 the proof in question: It is by actualization that geometrical proofs involving figures come through because [mathematicians] obtain the proof by performing constructions on the figure. Had the constructions been alreacfy available, the proof would have been obvious; but they exist in potentiality. Why are the angles of a triangle equal to two right angles? (i) Because the angles around a point are equal to two right angles, (ii) If a line parallel to one of the triangle’s sides had been drawn, the proof would have been evident to anyone looking at the figure...Therefore it is evident that what is potential is discovered by being brought to actuality; for thinking is an actuality and thus potentiality comes from actuality and this is why they obtain knowledge by action, though a particular actuality is posterior in genesis (Met. 1051a21-26). Aristotle’s (ii) obviously refers not to the proof of El. 1.32* proper but to that of El. 1.32 (Fig. 4): 69 (EL 1.32) Let ABF be a triangle and let its side BF be produced to A...For let FE be drawn parallel to AB through the point F. Since AB is parallel to FE and AB falls on them, the alternate angles BAF, AFE are equal to one another. Again since AB is parallel to FE and BA falls on them, the exterior angle EFA is equal to the interior and opposite angle ABF. But the angle AFE has been shown equal to the angle BAF. Thus the whole angle AFA is equal to the two interior and opposite angles BAF, ABF. (EL 132*) Let the angle AFB be added to each. Thus the angles AFA and AFB are equal to the three angles ABF, BFA, FAB. But the angles AFA and AFB are equal to two right angles. Therefore the angles ABF, BFA, FAB are equal to two right angles as well. Aristotle emphasizes the constructive dependence of the proof of El. 1.32* on that of El. 1.32 which validates the crucial equalityArA+ATB=ABr+BrA+rAB (to derive the demonstrandum, i.e. ABr+BrA+FAB=2R, one also needs the additional equality ATA+ATB=2R (via El. 1.13) which Aristotle refers to in (i)). But this step depends on producing BF to A, an operation allowed by the second Euclidean postulate which guarantees the production of a straight line to an arbitrary point. This postulate can be easily conceived as an intuitive substitute, applicable only to straight lines, for the full blown Archimedean axiom and, since the very similar operation in (ii) is said to bring about the noetic actualizitation of a potential geometric entity, potential arbitrarily large magnitudes, like those guaranteed by the Archimedean axiom or the second Euclidean postulate, must also actualize noetically. Thus, in view of Met. 1051a21-23, there must be two theses at play inMet. 1048bl4-17: first, that potential arbitrarily small magnitudes, and, consequently, arbitrarily large ones, are indispensible to mathematical knowledge, a thesis parallel to Godel’s "indispensibüity argument" in favor of certain axioms; second, that the POVÇ in actuality is thevoovnevop (DA 429a22-24), a view which explicitly extends to 70 mathematical entities {DA 429bl8-19, 431bl2-17, 20-432a6).^“ On this construal of actuality, when Euclid in El. 5.8 asserts "let A [ ‘“^See Godel (1983) 477. Ross (1924) 1, 272 denies that the second thesis is implied mMet. 1051a30-33 because mathematical objects contain "intelligible" matter whereas vovç is identical with the voovnevop only when the latter lacks matter(DA 430a2-4). But in DA 431bl2-27 Aristotle states explicitly that vovç voeî èvepyei(f mathematical objects separated from physical ones by abstraction and concludes: okaç 8e b vovç ëanv ô kut’ evêpyeiav toc vpâypaTot. At any rate, as Hamlyn (1993) 135 remarks, "Aristotle vasdllates on the question whether all things or only pure forms or essences are the objects of the intellect". \a.DA 429b 10-21 Aristotle draws an analogy between mathematical entities and composites arguing that responsible for judgments about the latter is either a faculty other than vovç or vovç differently disposed. If "the latter alternative tends to suggest that the intellect by which one judges essences, "what is to be F", is not after all utterly distinct from the senses" (Hamlyn (1993) 138), then the intelligible matter would be an object of the common sense responsible for péyeBoç {DA 425al4-16; the pêyeBoç of course is not the péyedoçof a perceptible object but that of the 4>àvTaap.a which the geometric proof requires; see DM 449b33-450a8). It should be added here that Aristotle’s conception of conceivability as noetic actualization plays an important role in Hintikka (1973) 124ff. Actually the very point of Hintikka’s paper is to show that Aristotle’s theory of infinity is by no means an exception to "the principle of plenitude" (for a critique of Hintikka’s thesis see Lear (1980)). Ironically, Hintikka does not seem to realize that Aristotle must have been singularly blinded not to see that in this scheme arbitrary potential extensions fit very well without compromising the finitude of the universe. A curious feature in Hintikka’s interpretation of Met. 1051a21-23 should also be pointed out: evpiaKerai be m l roc biaypap.p.otToc evepyeiqc- ôiaipovvreç yap evpioKovai. el 5* riv bngpripeva, It is absurd to rely on thought. Because exceeding or falling short [of a given magnitude] is not in the actual thing but only in thought. For one can conceive of each of us as multiplied to infinity; but there exists someone who exceeds the size we have, not because one thinks so but because such a person exists. Thought is only an accident. Time, motion and thought are infinite in the sense that what is taken does not persist. But a magnitude is infinite neither because of division not because of multiplication in thought (208al4- 22 )."* '®*It is this potential actualization that guarantees the availability of always greater magnitudes as in Met. 1048bl4-17 it is the potentiality of mathematical knowledge that allows for always further divisions of a magnitude. Contrary to Hintikka (1973) 134 the "principle of plenitude" does not seem to provide a self- evident explanation of Aristotle’s point in Met. 1048bl4-17, although the passage undeniably suggests that Aristotle’s theory of potential infinity is not a counterexample to the principle. More plausible is to assume that Aristotle would justify both arbitrary divisibilify and arbitrary extendibility on account of their crucial role in geometrical proofs, i.e. geometrical knowledge, and their internal consistency; for Aristotle’s emphasis on the incoherence of notions like an actually infinite number (and magnitude) see below ch 5. '®Ross (1924) 2, 273 on Afef. 1051a32-33 where Aristotle talks about auxüliary constructions. '“Hintikka (1973) 129 sees in this passage Aristotle’s "idea that although in thinking of x one’s mind assumes the form of x (or, alternatively, makes use of an image having the form of x), it need not assume this form in the same size as the original. The replicas of outside forms that one has in one’s mind are merely scale 72 One might of course object, as Hintikka did, that in view of DA 431bl2-17 Aristotle runs into another problem: because, since e.g. the lines actualized by a geometer’s noetic activity are abstracted from physical lines. now the real problem here is that some of the lines that the geometer needs do not seem to be forthcoming at all and of course this existential problem is not alleviated by the possibli^ of abstracting horn certain attributes of lines. If the requisite lines do not exist, there is nothing to abstract from.^” But this objection is misplaced in its emphasis on the exact correspondence between the mathematical magnitudes in a proof and physical magnitudes. It is clear from DM 449b30-450a7 that in a geometrical proof the actual dimensions of geometrical magnitudes, although certainly products of abstraction from physical ones (DA 432a3- 6), do not come into question without compromising the validity of the proof, as models of these forms, as it were". Hintikka here relies on genuine Aristotelian doctrine (see DM 452b7-22) which is, however, irrelevant to Phys. 208al4-22: in this passage Aristotle simply denies the ontological import of thought, a thesis familiar from Plato. At any rate, DM 452b7-22 does not seem particularly relevant to Aristotle’s theory of infinity. This passage addresses the difference in size between perceived objects and their appearance in the soul. According to Aristotle this appearance is in proportion to other appearances so that there is an impression of relative size. In the case of perception there is indeed a "scaling down" of size which, however, does not bear on geometrical magnitudes contrary to Hintikka and Hussey (1983) 95. In DM 449b33-450a8 Aristotle has already related thought and geometrical demonstration by means of diagrams without assuming any scaling down, which of course might, appear in applied mechanics and astronomy but is obviously irrelevant to pure geometry. “"Hintikka (1973) 122. 73 Aristotie repeatedly insists (<4nPoj^76b39-77a2, Me/. 1078al8-21,1089al9-25). Thus the notion of abstraction need not occasion any problem to the use of (ii) either by Aristotle in Phys. 266a6-bl4 or by Euclid in El. 5.8. Aristotle, moreover, does not seem to worry about (ii) even in the course of a physical argument, as already noted, not only because his theory of science can account for mathematical physics (AnPost. 76a4-15, 78b32-79al6), but for another reason too which, as far as I know, has not commanded any attention.^®* It seems that the extent of physical reality in Aristotle’s universe has been largely underestimated because inMeteor. 340a7-10 Aristotle refers to the size of the universe relative to the earth’s in a rather unexpected way: The volume of the earth, where all water is found too, is to the surrounding magnitude [i.e. the rest of the universe] as no part, so to speak [oû5èyixp &ç eiréiy nôpioy b rqç yrjç èanv ôyKoç...Trpoç to Trepiêxov péyedoç]. As the preposition xpoç indicates, Aristotle here measures the relative size of the earth’s volume and the surrounding universe by forming their ratio (cf. Phys. 215b6-7, 216a6-7, 9-11); but, since the earth’s volume is to the surrounding volume as "no part", the earth is to the universe as a point to a solid (pvOev ya p [nopiov, supplied from DC. 296al4] anyn^ tu>v aanarap kariv, DC. 296al6-17). A point, however, does not have a ratio to a geometrical magnitude (Phys. 215b 12-22) and thus Aristotle’s mathematically incongruous comment on the size of the cosmos clearly “®Cf. the views referred to above in n. 100. 74 initiates what became among later astronomers a common analogy for the negligibility of the earth’s size m comparison to larger spheres: perhaps the best example is Aristarchus’ second hypothesis in his treatiseOn the Sizes and Distances of the Sun and the Moon that the earth has the ratio of a point and center to the sphere in which the moon moves.“® Mathematically this would mean that for Aristotle the universe is actually infinite in relation to the earth; but, as his qualification "so to speak" makes clear, this is only a metaphorical way of speaking - the volume of the universe is so incomparably greater than the earth that it can be thought of as actually infinite for practical purposes.”” This is the only way to interpret Meteor. 340a7-10 consistently with Aristotle’s firm belief m the finitude of the universe and is fully substantiated by the context of this passage, an argument ‘””For further examples see Heath (1913) 309-310; the best known example is perhaps Aristarchus’ assumption that the sphere of the universe is so great that the earth’s orbit stands to it in the ratio of a sphere’s center to the surface of the sphere. This was one of the assumptions in Aristarchus’s heliocentric system according to the testimony of Archimedes (Sandreckoner 244 (Heiberg)). We know that Eudoxus had calculated the relative sizes of the sun and the moon {Sandreckoner 288 (Heiberg)) and if Aristotle’s comparison of various cosmic distances mMeteor. 345b 1-9 relies on Eudoxus, as Heath (1913) 331-332 seems to suggest, it is fairly possible that the comparison of the volumes of the earth and the universe in Meteor. 340a7-10 goes back to Eudoxus. Since, moreover, there is a clear fourth century precedent for the assumption in Aristarchus’ heliocentric hypothesis, there is no need to follow White (1992) 163fiL who thinks that Aristarchus put forth his assumption in order to account for the arbitrary extendibility of potential magnitudes, not forthcoming in the finite Aristotelian universe. This is a fictitious problem and it is preferable to adopt Heath’s explanation of Aristarchus’ assumption as a means of accounting for the absence of observed stellar parallax. ””Cf. Heath (1913) 309 on Aristarchus’ similar assumption. 75 that the space between the earth and the sphere of the fixed stars cannot be filled with a single element. Aristotle evokes the size of the universe as characterized in Meteor. 340a7-10 to argue that, were this so, the volume of the single element would have absorbed all other elements during the process of elemental transformation. This is the only physical argument against actual infinity in i*%sics T (204b 10-19) and thus Aristotle indeed treats the volume of the universe in Meteor. 340a7-10 as if it were actually infinite. Thus there is enough physical reality in Aristotle’s universe, despite its finitude, to satisfy even the most demanding needs of abstraction which, however, do not arise at all as shown above. Neither physical actualization poses a threat to his theory of potential infinity nor does this theory violate his conception of potentiality. Nor, moreover, can the modal postulate in Phys. 207b 17-18 introduce actual infinity via the way in which potential infinity is actualized according to Met. 1048bl4-17. Actual infinity has been characterized in Phys. 206b33-207al0 as a whole beyond which no part can be taken, a characterization of a non-Archhnedean magnitude or number inDC 271b33-272b2, i.e. one greater than all potentially assigned magnitudes or numbers and less than none (DC 281a33-34, 301bl4-15). Inasmuch as Aristotle can plausibly deny the potential existence of such a magnitude or number (see below ch. 5), actualized potential infinity in the sense of Met. 1048b 14-17 does not result to actual infinity. The same obviously holds for potential infinity by division since Aristotle rejects the geometrically untenable notion of an indivisible magnitude. But at this point the parity between potential infinity in the above senses and 76 potential infinity by inverse addition breaks down. Aristotle posits that in inverse addition there is always s„+i > s^ to take but not all AB - s„ can be potentially exceeded in this manner. As already remarked, the cosmological reductio ad absurdum in support of this stipulation appears paradoxical at first sight (why Aristotle resorts to such an argument wiU be discussed below in ch. 4). Without this stipulation, however, there would ensue actual infinity on account of the modal postulate in. Phys. 207b 17-18. Indeed, in contrast to potential infinity by division and adédiionsimpliciter, inverse addition does presuppose an actual magnitude AB greater than all potential partial sums s„. But Aristotle here must avoid construing the actual magnitude AB as an infinite partial sum s^ which, on the characterization of actual infinity in Phys. 206b33-207al0, is actually infinite in the sense that beyond this sum there is no part to take. To preclude this intuitively attractive construal (a magnitude can be plausibly thought of as the totality of its parts, i.e. half, quarter, eight...) it suffices to stipulate that no Sn (n= o o ) potentially exceed all AB - S; so that s„ = AB; because, if there were such a partial sum, it could be actualized according to the modal postulate resulting in actual infinity. Thus only in the case of inverse addition could Aristotle’s concept of potentiality introduce actual infinity, but he steers clear from the problem by not allowing the variable to reach its limit. Denying, moreover, that by inverse addition one can exceed all assigned magnitudes does not compromise his theory -he could very easily add the temporal operator "at the same time" as he does in GC 316bl9-27, where he avoids the same problematic implication of the modal postulate for the divisibility of magnitudes. CHAPTER IV ARISTOTLE’S CRITIQUE OF PLATO IN PHYSICS T It seems, moreover, that Aristotle bars not actual infinily per se in its formal characterization at Phys. 206b33-207al0 but a particular kind of actual inbnity. This is suggested by his full justification of the modal restriction. Not only does Aristotle rely on the cosmological reductio ad absurdum but he also includes a critique of Plato’s view of arithmetical infinily: But if it is impossible for a perceptible body to be actually infinite in this sense, it is dear that it cannot be potentially [infinite] by addition but only, as already said, inversely to division. Because this is why Plato himself posited two infinites, i.e. because [the variable] seems to exceed all assigned magnitudes and go ad infinitum by both [inverse] addition and division. Nevertheless, despite his positing two [infinites], he does not use them. For neither the infinite by division exists in numbers (because the unit is the minimum) nor the infinite by addition (because he admits numbers only up to the decad) (Pt^s. 206b24-33). In this particular context, the potential infinity by division and inverse addition, Aristotle’s critique of Plato should be read along with Simplicius’ report of Plato’s "unwritten doctrines": Let there be posited a finite magnitude, e g. a cubit length; after its division in two, if we let the one half undivided and, dividing the other half, we added to the undivided, the original cubit length would have two parts, one decreasing and the other increasing indefinitely. We would never reach an indivisible magnitude by constant division, since the 77 78 cubit length is continuous and the continuum is divided into always divisible parts. And this kind of constant division implies a sort of infinity within the cubit or rather more than one, one increasing, the other decreasing. In these two the "indefinite (fyad" can be seen consisting of both the increasing and the decreasing component.'" According to Simplicius this is Porphyrins' verbatim report of the various treatises On the Good, one of them by Aristotle himself,"^ and there is no reason to doubt his testimony. The passage illustrates Aristotle’s complementary processes of division and inverse addition. What Aristotle objects to in Phys. 206b25-33 is that both processes are similarly infinite because it seems that one can exceed all assigned magnitudes in both directions. This objection obviously concerns Aristotle’s own claim that not all assigned magnitudes can be potentially exceeded by way of inverse addition and, consequently, the absurdity he sees in Plato’s position cannot plausibly follow from his own claim. But in fleshing out his objection Aristotle surprisingly turns to an internal inconsistency in Plato’s view of arithmetical infinity: for Plato numbers are not infinite in either direction despite the fact that he poses the two pronged infinity of "great and small". Now Aristotle’s critique is certainly misguided because it reads Platonic theories literally and out of context: in positing the "great and small" as principles of ‘“This is part of a more extended report (Simpl. In Arist. Phys. 453.22-455.11 (Diels) = Testimonia Platonica 23B in Gaiser (1963)). ‘"De Bono fr.2 (Ross). 79 numbers Plato does not imply an infinity of ever decreasing numbers'" nor could he have stopped the number series at the decad."" Just a few pages above, moreover, Aristotle clearly implied that Plato held the actual infinity of numbers because in Phys. 204b7-10 he refutes the notion of an infinite "separate" (K€x<>3piffiiévoç) number, a term he reserves solely for the ontological independence of Plato’s ideal or mathematical numbers, qua untraversable, i.e. non-Archimedean. But Aristotle need not contradict himself in Phys. 206b2S-33 -he simply attacks the actual infinity of Platonic numbers from a different angle, internal coherence, pushing a step further the disparaging, and equally unfounded, charge in Met. 1073al8-22 that '"For an attempt to explain in what sense the "great and small", or the "indefinite two", are principles of numbers see Annas (1976) 44ff. Annas seems to account for Aristotle’s critique on the grounds that "Plato is not so clear about the difference between using the number series and giving a theoretical account of it" (Annas (1976) 42-43). Her analogy between Plato and Brouwer (1984) 69 is particularly illuminating. When Aristotle says that Plato posited infinity, "the great and small", as ovaùx (Phys. 203a4-5) or àpxij (Met. 987b 17-27) of numbers (and forms) he can only mean that Plato held numbers to be actually infinite (how the principles of numbers can also be principles of forms is an issue that does not afiect the present discussion; see Annas (1976) 62-73): in Met. 1081al4-17 Aristotle makes clear that the Platonic principles of numbers do not differ from the number series and, according to his critique of the Platonic principles in Met. 1089a5-6, the second principle, "the great and small", accounted for the plurality of numbers, which must have been actually infinite for Plato (cf. below n.l23). Therefore, when Aristotle criticizes Plato’s "great and small" or "two infinites", he criticizes Plato’s realist actual infinity of numbers in terms of Academic jargon. ""On Aristotle’s report that the Platonists generated numbers only as far as the decad see the comments in Annas (1976) 54-55. Since the Platonic generation of numbers is obscure (like many points m Plato’s, or in general Academic, philosophy of mathematics), it seems more plausible to associate the decad with the number- theoretic properties of numbers up to ten. 80 Plato refers to numbers sometimes as actually infinite and sometimes as ending with the decad. Within its context Aristotle’s critique makes sense only if Plato inferred the actual infinity of numbers because inverse addition seems to exceed all assigned magnitudes, a pitfall Aristotle avoids by means of the modal restriction on potential infinity by inverse addition: because, given Aristotle’s characterization of number as a "numbered set" (Phys. 223a24-25),“^ the complete infinity of a magnitude’s parts to which potential infinity by inverse addition would give rise, if not modally restricted, would amount to an actually infinite number. This interpretation explains well Aristotle’s abrupt transition from infinity by inverse addition to the distinctly different infinity of numbers in his critique of Plato. Aristotle, moreover, offers sufficient evidence to piece together the Platonic argument he implies in Phys. 207a24-33. In Phys. 207bl0-15 he grounds the conceivability of ever increasing numbers not in the actually infinite, completed and separate (xcopurroç) number of a magnitude’s divisions but in these divisions’ merely "becoming", potential infinity (cf. Phys. 207b 13-15 and 208a33-b3). As indicated by the term "separate", Aristotle here denies a Platonic thesis and, since only here in Physics F does he reject the conceivability of ever increasing numbers justified ontologlcally in terms of an actually infinite number, this is his answer to the argument in aporia (5) that also grounds the actual infinity of numbers in the conceivability of ever-increasing numbers. The Platonic origin of this argument is “®This is the "traditional" characterization of numbers in Greek arithmetic; see Klein (1968) 46ff. 81 also suggested by two further reasons. First, it agrees with Plato’s broader epistemological claim that thought requires the existence of certain objects;”® second, it obviously reflects the Platonic thesis denied in Met. 1048bl5-17, namely that the availability of always further divisions of a magnitude is due to their actual, ontologically independent infinity.”’ Actually Aristotle’s answer to the conceivability argument suggests that this thesis was part of Plato’s argument: Aristotle denies that conceivability of ever increasing numbers hinges on the actually irdSnite number of a magnitude’s divisions, not an actually infinite number per se, as one m i^t have expected. Indeed, the thesis in Met. 1048b 15-17 amounts to the existence of an actually infinite number, given Plato’s characterization of number as the numbered collection of the parts of a whole in Theaet. 204dl-e7. This Platonic argument infers the actual infinity of numbers firom the availability of always further divisions of a magnitude and, given the unconditional complementarity of division and inverse addition in Plato’s On the Good, it yields easily Plato’s thesis which Aristotle criticizes iaPhys. 206b27-33. It is reasonable to assume that in his critique of Plato Aristotle has in mind the conceivability argument because, by means of the modal restriction on potential infinity by inverse addition, he avoids inferring the existence of actual infinity fi*om an unending potential operation in Plato’s fashion. ^*®This is the so-called "objects of thought" argument according to which the existence of forms explain the possibility of thought; for an examination of both Platonic and Aristotelian evidence on this argument see Fine (1993) 120ff. ^”There is an obvious affinity between Plato’s argument and Cantor’s "domain principle" according to which each potential infinity presupposes a corresponding actual infinity; see Cantor (1886a) 9 and Hallett (1984) 24flE. 82 In view of Simplicius’ quote of Poiphyrius, which is also related to Aristotle’s On the Good, the fact that Aristotle includes a critique of Plato’s arithmetic infinity in both Phys. 206b3-33 and 207a33-207b20 suggests that his discussion of the complementary infinities by division and inverse addition is intended as a critique of Plato’s infinity principle, "the great and small". This conclusion is further supported by the dialectic nature of Aristotle’s discussion. If this is so, then the cosmological reductio ad absurdum that figures in both passages must also be directed against Plato.“* Aristotle does imply a link between Plato’s arithmetic infinity and actual physical infinity outside the universe: although the Platonic origin of the conceivability argument clearly suggests itself, in aporia (5) this argument justifies not only the actual infinity of numbers (and magnitudes) but also actual physical infinity beyond the cosmos. According to Aristotle’s own testimony, however, unlike the " ^ e formulation of the cosmological reductio ad absurdum also suggests that the argument is designed to show a problem with a thesis of Plato’s. Aristotle says that the inverse addition caimot even potentially exceed all assigned magnitudes not unless there is actual physical infinity beyond the cosmos but unless there is such physical infinityaccidentalfy (kutu avp^e^i)KÔç, Phys. 206b20-24). But, since mPhys. 206b24-33 this absurd cosmological consequence is coupled with a problem in Plato’s double infinity, "the great and small", the specification "acddentally" must carry particular weight. InPhys. 203a4-6 Aristotle remarks that Plato and the Pythagoreans posited infinity not as avp^e^riKÔç but as ovaia: since, however, in Phys. 204a20-34 he accuses the Pythagoreans of cutting accross these two mutually exclusive characterizations, it is very probable that the Kara avp^e^TiKÔç in Phys. 206b20-24 suggests the same charge against Plato, all the more since in Phys. 206b24-33 Aristotle attacks the consistency of Plato’s conception of arithmetical infinity. The purpose of the following discussion is to show that for Aristotle Plato is open to the same charge as the Pythagoreans; see also below n.l23. 83 Pythagoreans who posited infinily both in physical bodies and beyond the cosmos, Plato posited neither any physical body nor the forms beyond the cosmos, because forms are not located anywhere, but took infinity, i.e, the "great and small", as a principle of both physical bodies and forms (Phys. 203a4-10). This incongruity is symptomatic of a broader problem concerning the presence of Plato in a book where Aristotle passingly raises the question of actual infinity in mathematicals and intelligibles only to remark that it is an issue outside the scope of Physics F which is restricted to whether there are actually infinite physical bodies(Phys. 204a34-b4). On the face of it, the accoimt of Plato’s concept of infinity in Aristotle’s historical introduction to his discussion seems certainly misplaced: although Plato appears in the company of the Pythagoreans and the pre-Socratic ^vmoKoyoi (Phys. 202b30ff.), nowhere does Aristotle ascribe to him physical actual infinity. Instead, as seen above, he criticizes Plato’s arithmetical infinity (Phys. 206b27-33) and denies the Platonic view that the divisions of a magnitude form an actually infinite number (Phys. 207b 10-15), a notion refuted in Phys. 204b7-10. Thus Aristotle is interested in Plato’s concept of infinity qua arithmetical infinity despite the fact that inPhys. 203a4-10 Platonic infinity is only a principle of physical things and forms. The problem, however, is easily resolved if for Aristotle Plato’s "great and small", infinity as a principle of both numbers and forms, commits Plato to actual infinity in physical things as well. This would explain not only the formulation of aporia (5) in view of the evidence in Physics F but also the presence of the cosmological reductio ad absurdum inbothPhy^. 206b3-33 and 207a33-b20: in setting 84 out to debunk Plato’s infinity principle Aristotle focuses on its implication of actual physical infinity, the main target ofPhysics P, which he has already refuted inPhys. 204a34-205a28. Here one should take into accoimt that Aristotle clearly conceives of the Timaean receptacle as parallel to the "great and small" of Plato’s "unwritten doctrines": Plato in the Timaeus claims that matter and space are identical viktip Koti ttiv t u v t o eipoci] because he considers the "receptacle" and space one and the same thing [ t o yap iiÆToàai'STiKop Kcà t ^ p xûpap êi»Kcà ravrôp]. Although he characterizes the receptacle differently there [in theTimaeus] and in the so-called "unwritten doctrines", nevertheless he declared place and space [top t&itop kuI t^p xt^poep] the same thing (Phys. 209bll-16).“’ What Aristotle considers a different characterization of the Timaean receptacle in Plato’s "unwritten doctrines" is the "great and small" as is clear firom Phys. 209b33- 210a2: Incidentally, Plato should explain why forms and numbers are not in a place, if place is the receptacle (pedeKUKop), be it either "the great and small" or the matter, as he wrote in the Timaeus. Since Aristotle construes the Timaean receptacle as place or matter,"" the ^^^Testimonia Platonica 54A in Gaiser (1963). "“Against the identification of the receptacle with matter see Taylor (1928) on Tim. 52b4 (cf. Ross (1936) 566; for an approach more favorable to Aristotle’s view see Claghom (1954) 5ff. If Plato himself did not understand the receptacle as a material substratum (which is not improbable since in Phys. 209bll-16 the Timaean receptacle exemplifies the view that takes t o i t o ç as Siâarqpa t o v peyéOovç), it is 85 presumed equivalence between the receptacle and the "great and small" must underlie his account of Platonic infinity in Phys. 203a4-10 according to which infinity is both in physical bodies and forms,But, on Aristotle’s unified reading of the Timaeus and the "unwritten doctrines", Plato’s denial that forms are spatially located is not home out Plato fails to draw carefully a crucial distinction: the difference between the receptacle and the "great and small" is a moot point. As a result, in Phys. 209b33-210a2 Aristotle almost imputes to Plato the spatial location of not only forms but numbers as well since the "great and small" is a principle of both. The problem Aristotle sees in Plato’s Pnnzipienlehre eq>lains why Plato figures among the Pythagoreans and the pre-Socratic concerned, the upshot of Aristotle’s critique mPhys. 209b33-210a2 is that Plato fails to give an account of how actual infinity in intelligibles is not tantamount to actual physical infinity in the fashion of the Pythagoreans.'^ By Aristotle’s lights the possible that Aristotle was led to this interpretation by Plato’s metaphorical language: twice Plato calls the receptacle /«Jti/p (7zm. 50d3, 51a4-5) so that Aristotle could press his point that Plato understands the receptacle as the material principle by analogy to the biological reproductive process. "'According to Ross (1936) 566 the parallelism between the Timaean receptacle and the "great and small" indicates that in the "unwritten doctrines" Plato extended to forms Timaeus’ analysis of sensibles as consisting in a formal and a material principle. Such an analysis is hinted at in Met. 987bl8-25 and 988a7-15. '"The same conclusion can also be derived firom Aristotle’s critique of Plato in Met. 989b29-990a32 where, after criticizing the Itythagorean identification of physical things with numbers, remarks that Plato too identified physical things with numbers and their causes but, unlike the Pythagoreans, held intelligible numbers as the causes of physical numbers (?!), i.e. physical things. This passage clearly supports the above 86 actually infinite, ontologically independent number refuted in Phys. 204b7-10 eo ipso implies an actual infinite spatial location and, therefore, an actually infinite body (cf. Phys. 205a7-28 and b24-206a8). But, if Plato’s conceivability argument turns out supporting actual physical infinity as well, Aristotle is all but justified in deriving such an infinity from the absence of the modal restriction on potential infinity by inverse addition which, like Plato’s conceivability argument, would thus result in the actual infinity of numbers. interpretation and shows that Aristotle is ready to push the alleged vagueness in Plato’s Pnnzipienlehre too far: he explicitly accuses Plato of a version of Pythagoreanism and even takes "Platonic" physical numbers as physical things, an interpretation that obviously ascribes to Plato actual physical infinity. A trace of this Aristotelian interpretation is most probably to be found in Theophrastus’ Met. 6al5- bl7, where he notes that Plato "derived" place, void (?) and infinity from the indefinite two, i.e. "the great and small". Infinity here must mean "physical infinity" since it appears along with place (it cannot refer to the infinity principle in intelligibles since it is derived fi'om this principle, i.e. the indefinite two). PLEASE NOTE Page(s) not included with original material and unavailable from author or university. Filmed as received. 87 UMI CHAPTER V ACTUAL INFINITY AND GREEK MATHEMATICS The above discussion has shown that, contrary to the received doctrine, the cosmological reductio ad absurdum iaPhys. 20653-33 and 207a33-b20 does not intend to banish arbitrarily large potential extensions from geometry: since it stems diedecticcdfy from what Aristotle regards as a problem in Plato’s account of infinity, it cannot contravene Aristotle’s explicit acknowledgment of such extensions elsewhere. InPhysics P, moreover, his interest in mathematical infinity is certainly accidental. The primary focus is indeed on whether there exists actual physical infinity, as he explicitly says inPhys. 204a34-b4: mathematical infinity comes in only because it coincides with physical infinity for the Pythagoreans and, more importanly, for Plato too, according to Aristotle at least. But, although his Auselrumdersetzung with Plato is understandably colored by Platos’s later ontology, the particular way in which Aristotle refutes the notion of an actually infinite Platonic number is not ontologically loaded: It is also impossible that a number is ontologically independent (Kexupiv/téi'oç) and [actually] infinite. Because a number is what can be numbered or what has number. Thus, if what can be numbered is possible to be numbered, it is also possible to traverse the [actual] 88 89 infinite (Phys. 204b7-10).‘® ‘^Plato’s realist construal of numbers as a completed, actually infinite structure is manifest in Phd. 104a7-b4 where, contrasting odd and even numbers, he refers to the latter as the "entire other series of numbers" (àraç b Irepoç ah anxoç tov àpidpov); cf. also the exercise in division at Pol. 262d6-263al. The infinity of numbers is also implied in Rep. 525a4-10. There Plato treats the objects of arithmetic as exemplifying the fundamental problem which the division method is put forth to address in Phil. 18a6-b2: "we see the same thing as one and infinite in multitude (àveipa to irkrjdoç). - Therefore, I said, if this holds for the one, [a] it holds for the totality of numbers too? - It does [b]". Plato of course is rather reticent about the exact meaning of both the "one" and the "infinite multitude" as applied to numbers. Since in Kg?. S26al-7 he introduces the notion of undifferentiated unit, it is clear firom both Phil. 56c9-e3 and Aristotle’s testimony in Metaphysics M and N that there is an infinite stock of such units, each being "one", to bear out claim [a], [b], however, does not follow necessarily firom [a], because, leaving aside the infinite multitude for the time being, "one" obviously cannot mean "unit" in the case of numbers. Plato’s conclusion shows a logical fallacy in his philosophy of arithmetic which Aristotle points out and criticizes in Met. 1084b2-32, a failure to disthiguish between two senses of "one", i.e. unit, out of which a number is made, and the formal unity of the number as a set of units (cf. esp. Met. 1084b21-22). Reading this conflation in Rep. S2Sa4-10 both explains the problematic jump firom [a] to [b] and provides the sense in which numbers can be conceived as "ones" within Platonic arithmetic. Plato must be committed to an infinite multitude of such formal unit(ie)s but, since Aristotle makes clear that form numbers are countable, albeit differently firom mathematical numbers (Met. 1082b34-36), this infinite multitude must correspond to an infinite number. Two Platonic passages bear out this conclusion. In the so-called "generation of numbers" atPaim. 143clff. Plato claims that if there is "one", there is necessarily number and, therefore, many, indeed an infinite multitude of, beings: for number becomes infinite because of multitude and by partaking of being {Parm. 144a4-7). Backed by strong realist assumptions the argument presupposes a pre-Euclideah equivalent to El. Def. 7.1 & 2 and considers the natural numbers as an infinite multitude. That Plato does conceive of this infinite multitude as a numbered multitude in the sense of El. Def. 7.2 is showed by a detail in the discussion of not-being in the Sophist. From the two premisses that no being be related to not-being (Soph. 237c7-8) and that all numbers are beings 90 Building on the characterization of number in Phys. 223a24-25, which reflects the Greek concept of number as "numbered set",‘“ this argument shows that Aristotle’s rejection of actual infinity in numbers is not motivated by spurious, as it were, cosmological or ontological considerations. But falling back on his realist ontolojgr of numbers Plato could have easily dismissed Aristotle’s point as Russell’s "mere medical impossibility" to traverse the actual infinite. Aristotle himself must have realized the weakness of his argument because, when he refutes once more the notion of an ontologically independent, infinite number mM et. 1083b36-1084a7, the thrust of his argument is much more forceful: since all numbers are either even or odd, i.e. of the form 2n or 2n+l respectively, an actually infinite untraversable number can be neither, so that the notion of such a number turns out arithmetically incoherent since Plato himself admits that odd and even are fundamental {Soph. 238al0) the Eleatic Stranger concludes that not-being cannot be correlated with a numbered multitude {t\^6oç àpiOpov, Soph. 238b2-3). It is thus to be expected that not-being does presuppose a numbered multitude, if not-being is to be taken as being, the final solution to the problem of not-being: indeed, the Eleatic Stranger’s assertion, just before the declaration of the victory over Parmenides, that not-being is a form counted off waong the many beings {hv&piBpov tG)v voKkwp bvrap eîôoç ep. Soph. 258c2-3) certainly harks back to the correlation between beings and numbers in Soph. 238al0. But the Stranger has already posited that for each F not- F is infinite in multitude {airapop TkrjOei, Soph. 256e5-6) and number {ampapra t o p àpiOpôp, Soph. 257a5-6). This must be the correlate numbered multitude, all the more since it appears immediately after the requisite characterization of not-being as being first obtains {Soph. 256dll-e3). 124See especially Klein (1968) 46ff. 91 arithmetical properties (Rep. 510c2-d3).‘“ This is again a purely mathematical argument with no ontology involved and its force is manifest from the fact that only Cantor's theory of transfînite numbers eventually subverted it.‘“ But, as long as one stays within Greek arithmetic, Aristotle’s rejection of an actually infinite number in DC 271b33-272a3 and Phys. 207bl0-15 is formally secure without any thorny philosophical cormnitment to the ontology of numbers.*” '^Collaterally Aristotle also argues that the Platonic generation of numbers can produce only odd or even numbers so that admitting an actually infinite number compromises the internal coherence of Plato’s account of arithmetic. Aristotle need not give a reason why the actually infinite number cannot be either odd or even, as Armas (1976) 178-179 requires, because neither an odd nor an even number can be actually infinite (thus it would be absurd to identify an actually infinite number with an odd or even number, as she suggests in objecting to Aristotle’s argument). Annas, moreover, is wrong to claim that Aristotle’s point is a matter of temperament. The philosophical inability to make sense of the notion of an actually infinite number does merely lead to marvel at the ineâability of the infinite, as she comments, but is mathematically useless. Cantor’s achievement lies exactly in his insistance on making rigorous arithmetical sense of the notion of actually infinite numbers, not in awe-struck wondering at a magnum mysterium. '“Ultimately Aristotle’s argument rests on the absence of a tertium quid that could accomodate an actually infinite number. Cantor (1883) 178 argued that, since a natural number cannot be both even and odd, this does not imply that an infinite number cannot be both even and odd: the smallest infinite ordinal w is both even and odd since it equals 1+w and 2+w. '”Given our ignorance about the discussions that certainly went on in the Academy, we need not assume that Plato, or a Platonist mathematician like Eudoxus (see below ch. 6), should have been impressed by Aristotle’s argument in Met. 1083b26-1084a7. In the fourteenth century, for example, thinkers like Robert Grosseteste, William of Auvergne and Henry of Harclay, explained the inequalify of lines on account of the unequal actually infinite totalities of points they consist in (see Maier (1966)). Bearing a notable, albeit intuitive, similarity to Cantor’s 92 k IVN %\\\ \l \\ \ \ \ \ \ \ \ N \ % N \ % \ \ \ % \ \ \ \ V V N l \ \ \ » \ N N 1 V \ \ » » > \ 1 ^ \ \ V V \ N l V > N % 1 \ N T Fig.5 Aristotle’s argument against an actually infinite cosmos (DC 272b25-28) transfînite numbers, such an approach might be thought to avoid Aristotle’s objection -there is no problem with an actually infinite number of the form 2n if n is another actually infîmite number (cf. Grosseteste (1963) 91-94). Had Plato held this view, he could have denied that there is onfy one actually infînite number as Aristotle assumes. But Aristotle could have responded that this view depends on the first actually infînite number which still turns out arithmetically incoherent. He could also have asked about the cardinality of all numbers, both fînite and actually infinite. In this case Plato might have fallen back on scientific optimism, arguing that future developments in mathematics could formally accomodate the recalcitrant numbers, exactly as Eudoxus had developed a proportion theory for both commensurable and incommensurable magnitudes. Nevertheless, it is instructive to note that in 1693 Newton still adhered to the above fourteenth century view (see Turnbull (1961) 239), although critics already in the fourteenth century had undermined it by exhibiting a biunique mapping of the points on a square’s diagonal with those on the side of the square (see Murdoch (1962) 25). 93 In view of DC 271b33-272a3(5), the rejection of actual infioity in numbers would immediately extend to geometrical magnitudes too, although the Aristotelian corpus contains no explicit argument to that effect. But another proof of the fînitude of the universe in DC 272b25-28 implies that Aristotle would have equally strong reasons to offer against actually infinite geometrical magnitudes. As in DC 271b26- 272a7, the purpose of this proof is to show that an actually infinite body cannot rotate. Assuming an actually infinite straight line E and a point F not on E (Fig.5) Aristotle argues that, if A is a point in the other side of E, the straight line FA is actually infinite and rotates so as to describe a circle with center F, then FA cannot cross to the other side of E, since both E and FA are actually infinite; as a result, FA cannot obviously describe a circle. This reasoning contravenes Euclid’s third postulate, "[let it be postulated] to describe a circle with any center and any distance", although Aristotle employs it fireely at Meteor. 3.5 (376b9-10). There can be no doubt, however, that in DC 272b25-28 the postulate fails because FA is assumed to be actually infinite. No matter whether the postulate appeared as such in the pre- Euclidean Elements of Aristotle’s time or Aristotle refers to it as an unstated geometrical assumption, he seems to question the postulate’s allowing for indefinitely large circles, which implies, in Heath’s words, "the fundamental assumption of the infinity of space":*** for neither Heath nor Euclid himself nor, one can safely infer, Euclid’s predecessors did elaborate on the meaning of "any radius" which can easily, as Aristotle assumes, be an actually infinite straight line FA. ‘“ Heath (1956) 1, 200. 94 The general structure of Aristotle’s argument furnishes another indication that this cosmological proof implies a critique of the geometrical assumption in question. As Aristotle assumes a point A on the other side of the actually infinite straight line E and argues that a circle with center F and radius FA cannot be drawn, similarly m El. 1.12"^ Euclid assumes by virtue of the third postulate that, if AB is an infinite straight line, F a point not on it and A a point on the other side of AB, there can be drawn a circle with center F and radius FA. It cannot be accidental that the proof in DC 272b25-28 has exactly the same premisses asEl. 1.12, except for Aristotle’s assumption that FA is actually infinite too, but the contrary conclusion: Aristotle must have grafted his cosmological argument on El. 1.12, a problem which according to Proclus was first investigated by the fifth century astronomer Oenopides for astronomical purposes,"" and, prompted by the unclear formulation of what later became Euclid’s third postulate, added the extra assumption in order to obtain an absurdity of the desired type. It is very plausible that El. 1.12 drew Aristotle’s attention not only because of its astronomical origin but also because it is one of the propositions m the Elements that evoke the notion of infinity, although it has not been e)q}licitly characterized either in a definition or an axiom like Pappus’: since in the Elements of Aristotle’s day the precise characterization of infinity was most probably a moot point as in Euclid’s Elements, if actually infinite straight lines like *^To a given infinite straight line to draw a perpendicular firom a given point which is not on it". Pr. EL Com. 283.7-10 (Friedlein). 95 AB are to be admitted in geometry, there is nothing that prevents FA from being actually infinite as well, especially given the vagueness of the expression "any distance" in the third postulate. It is easy to see that the thrust of Aristotle’s argument lies in the tacit assumption which is crucial to the construction in El. 1.12. By evoking the third postulate Euclid notoriously assumes that the circle with radius FA and the infinite straight line AB must necessarily meet in two points, an assumption which presupposes a continuily principle:"' "if a point moves in a figure which is divided into two parts and if it belongs at the begirming of the motion to one part and at the end of the motion to the other, it must during the motion arrive at the boundary between the two parts.""* But in Aristotle’s argument the point A cannot belong to the other side of E and, consequently, it cannot arrive at the boundary, i.e. the straight line E. For Euclid the continuity principle must have been intuitively implicit in the construction of the circle allowed by the third postulate: as the point A moves towards AB along with the rotating line FA, FZ; (Zj’s being the intersections of FA and AB as A moves towards AB) is always less than FA but there is eventually a point Zg so that FZ^ = FA, since FA is assumed finite and, moreover, Zg is on AB which isex hypothesi infinite; after reaching Zg, A passes to the other side of AB. Aristotle, however, argues that, since both FA and E are actually infinite, the point A caimot move to the other part of E as FA rotates: since FA is assumed to be "'See Heath (1956) 1, 272. '**Killing’s formulation quoted by Heath (1956) 1, 235. 96 actually infinite in the sense of DC 281a33-34/301bl4-16, there cannot be a FZ^ = FA but all FZn are less than FA.‘” There is enough evidence to conclude that the scope of Aristotle’s argument in. DC 272b25-28 goes well beyond the finitude of the cosmos, suggesting that actual infinite magnitudes are unacceptable in Greek geometry where the notion of space is left to intuition: admitting actually infinite straight lines in EL 1.12 results in the paradoxical conclusion that the primitive construction of drawing a circle is not forthcoming. Euclid of course does not need an actually infinite AB. Assuming that AB is infinite collaterally guarantees the intersection of the circle and the straight line AB‘^ but one needs only to assume that AB is potentially infinite in Aristotle’s '^^Milhaud’s objections to Aristotle’s argument are unfounded: "N’ est-il pas permis de dire que ces démonstrations ne sont pas le fait d’un géomètre, habitué aux rotations de lignes, aux variations d’ angles, indépendantes de la longueur de 1’ élément qui tourne? Soit w, dirait-il, 1’ angle de CAE avec une direction fixe CX, perpendiculaire à BB’, par exemple; co variant de zéro à 360 degrés, le rayon fait un tour complet" (hfilhaud (1903) 382). Milhaud’s objection, however, assumes that the angles in the given configuration are Archimedean but, since the formulation of the Archimedean axiom for straight lines and angles are equivalent (see above, n.65), this is clearly not the case. *^In his comments on El. 1.12 Proclus interprets the assumption that AB be infinite as a means of making sure that the point F, which is not on AB, does not lie on the same straight line as AB; the infinitude of AB rules out the availability of space where such a problematic point could have been located (In Pr. EL Com. 284 (Friedlein)). The problem brought up by Proclus could be avoided if Euclid postulated that F and B e.g. be not coUinear when AB is produced but this stipulation clearly could not have secured the intersection of AB and the circle with radius FA. Since Euclid still feels the need to specify that F is not on AB, it is preferable to assume that the infinitude of AB guarantees the intersection of the straight line and the circle to be drawn. 97 sense: if TA > AB, by the Archimedean axiom in Phys. 266b2-3 there is an AB; > FA as is required for the straight line and the circle to intersect. Not only does Euclid apply the Archimedean axiom in El. 5.8 as an unstated assumption but it is also indicative that the intersection of the circle and the straight line AB can be proved using the notion of greatest lower bound, i.e. the conditional completeness of real numbers which is equivalent to the Archimedean property.*^ Thus, as is the case with the two arguments against the notion of an actually infinite number in Phys. 204b7-10 and Me/. 1083b36-1084a7, Aristotle would have valid reasons to argue that actual infinity cannot be accomodated within Greek geometry, irrespective, it should be stressed once more, of any cosmological or ontological considerations. Aristotle, therefore, could, and most probably did, object to actual infinity in mathematics not on ontological grounds or as a metaphysical article of faith but by questioning the coherence of the notion of an actually infinite multitude or magnitude within Greek mathematics. In this light Aristotle himself would have favored the addition of (a version of) Pappus’ infinity axiom to the Euclidean axioms, an addition not at all reminiscent of e.g. Berkeley’s attempt to reformulate geometry along empiricist lines: not only would this addition have spelled out the Archimedean axiom, which, as shown above (ch. 2), turns out indispensible to Euclidean geometry, it would have also delineated formally the notion of infinity which in Euclid’s Elements is left undefined thus causing a problem in cormection with El. 1.12. This is undeniably a minor problem that does not afiect the substance of Euclid’s Martin (1982) 230-233 and 236. 98 geometry and likewise an Aristotelian influence on Euclid would have only added a minor formal touch to the Elements. But, if this conclusion is true, it is a far cry from Knorfs view who, as already seen above, has claimed that Aristotle’s theory of the infinite shows remarkable insensitivity to the issues that might have occupied the geometers of his generation. If Euclid and his predecessors knew of his theory, they chose wisely to disregard it.'“ As far as the exhaustion method is concerned, I have criticized this conclusion in ch. 1 and, in view of Aristotle’s attitude towards arbitrary extensions and especially his objections to actual mathematical infinity, Knorr’s thesis, which relies on an uncritical acceptance of Hintikka’s views, is indeed unjustified. Nevertheless, the second horn of this thesis, without its negative coimotations of course, seems to be correct: as already remarked, Euclid does not show any interest in the exclusive characterization of mathematical infinity as potential that would betray Aristotelian influence. Indeed, to better appreciate the originality and the historical importance of Aristotle’s attitude to mathematical infinity one need also view his rejection of actual infinity vis-à-vis his contemporary mathematicians’ attitude to infinity. Knorr made the contrast to further denigrate the mathematical significance of Aristotle’s potential infinity: In fact, it appears to me more plausible to view Aristotle’s discussions in the Physics as an effort to save the concept of the infinite, in the face of a movement among the geometers “«Knorr (1982) 122. 99 of his day to give up that concept.’®’ For Knorr this thesis suffices to explain the "geometers’ abandonment of the infinite per SB when Aristotle himself neither advises nor follows this course."”® This conclusion is striking because, as shown above in ch. 1, the exhaustion method, which developed in Aristotle’s day, is a prime example of potential infinity, as Aristotle himself recognizes in Phys. 207b27-34. Aristotle’s contemporary mathematicians, therefore, cannot be said to have abandoned potential infinity. The question now is whether there really existed "a movement among the geometers of Aristotle’s day to give up" the concept of actucd infinity, the very course Arsitotle’s himself did advise and follow. It seems that, if one is to gauge fi*om Euclid’s Elements, Aristotle’s contemporary mathematics did not relinquish much. At Euclid’s time the notion of infinity per se was not yet jettisoned, as is shown by El. 1.12, 1.22, the definition of parallels and the parallel postulate. Now, had Euclid abandoned actual infinity, mathematics would have forgone exactly as much as Aristotle. But Euclid’s lack of concern about a characterization of infinity as potential, e.g. by means of an axiom like Pappus’, suggests otherwise. A notorious case in point is El. 3.16 according to ”’Knorr (1982) 121. ”®Knorr (1982) 121. Knorr is not right in claiming that "save for the context of parallel lines, "infinite" has been eliminated firom the geometers’ vocabulary altogether" (121). Apart firom the definition of, and propositions about, parallel fines, the term "infinite" appears in El. 1.12, as said above, and 1.22 whereas the concept of infinity plays a crucial role in both El. 9.20 and 12.14, as will be shown below. 100 which the cornicular angle, formed by a circle and its tangent, and the angle of the semicircle are respectively less and greater than any acute rectilinear angle."' By Aristotle’s lights such a characterization introduces actual infinity in geometry. Proclus brings forth the problem in a poignant way by remarking that, if the cornicular angle is a magnitude and if homogenous magnitudes have a ratio to each other qua being able to exceed each other by multiplication (El. 5 Def. 4), then the cornicular angle cannot have a ratio to an acute rectilinear angle: it is not an Archimedean magnitude because no multiple of it can ever exceed any acute rectilinear angle.*'*" But, since there is strong evidence that Euclid did take angles as magnitudes,*'** he then includes in the Elements without qualms a blatant *^®The nature of the cornicular angle was a subject of controversy firom the thirteenth to the seventeenth century: for a brief overview see Heath (1956) 2,39-43. ***//! Pr. E l Com. 121.24-122.7 (Friedlein). The context of Proclus’ remarks is the much discussed issue of the Aristotelian category to which angles belong (quality, quantity or relation); for a survey of the various proposals see Heath (1956) 1,177- 179. *'**See Heath (1956) 1, 178. It is of course possible to deny that the cornicular angle is a magnitude and thus circumvent the problem. In this vein J. Wallis, Newton’s contemporary, denied that the cornicular angle is a quantity (see Heath (1956) 2,42) but, if this is so, the formulation oiE l. 3.16 is meaningless -there is no point in comparing a magnitude, a rectilinear angle, with something which is not a magnitude, a cornicular angle. Proclus made a desperate attempt to save the phenomena arguing that angles belong to the Aristotelian categories of quantity and quality so that they can be inhomogenous despite the fact that they are magnitudes (In Pr. El Com. 124.2-13 (Friedlein)). There is of course no evidence that Euclid considered cornicular and acute rectilinear angles inhomogenous, apossiblity that can be plausibly ruled out because nowhere Eudlid makes comparisons of inhomogenous magnitudes like the one iuE/. 3.16. But, even if this comparison amounts implicitly 101 counterexample to the assumption central to the exhaustion method. The problem becomes more exacerbated when one considers that, had Euclid had any concern over actually infinite magnitudes, it would have been very easy to omit firom El. 3.16 the reference to the problematic angle which does not play any role in the Elements. His decision to include it, on whatever grounds,certainly betrays a lack of foundational concern about the notion of infinity. The same situation is observed in the case of infinite multitudes which appear in El. 10 Def. 3 and 10.115, the first positing an infinity of straight lines commensurable and incommensurable with a given straight line, the second showing that from a given medial straight line there arises an infinity of irrational straight lines. One can easily infer that Euclid understands infinity in its potential sense since in. El. 10.115 he shows that, ifK^'*p is a medial line and a a rational line, then is a new irrational line.‘“ Nevertheless, an actually infinite multitude does appear explicitly in El. 9.20: oi xpcJTOt àpiBudi irXeiouç eial vavroç rod TporeOévToç vXrjdovç rpûiTUiv àpiOpSsv. As already the anonymous scholiast remarked, the theorem amounts to the infinity of prime numbers but its formulation clearly implies the actually infinite, completed multitude of these numbers and it is noteworthy that the to a characterization of inhomogeneity, Euclid still admits actual infinity - inhomogenous magnitudes are actually infinite with respect to each other. ‘^^Cf. the comments in Heath (1956) 2, 4 and 39. In view of Aristotle’s proof of El. 1.5 in AnPr. 41b 13-22 it seems that "mixed angles" might have played some role in pre-Euclidean geometry; cf. Mueller (1981) 187. % . Heath (1956) 3, 255. PLEASE NOTE Page(s) not included with original material and unavailable from author or university. Filmed as received. 102-125 UMI 126 application of this argument in DC 299all-b7 to show that, on a certain theory, physical bodies cannot have weight unless a point is divided (DC 299b4-7). Since Aristotle starts with the assumption that physical bodies have weight qua being divisible (DC 299a22-23), he carries on the same argument as in his objection to Eudoxus’ theoiy in Me?. 1076b4-ll but, in line with the arguments in Physics Z, the distinctive thesis he leads to this absurdity is the composition of physical bodies out of points, i.e. planes consisting in lines which are composed of points (DC 299a25- 30). That this thesis yields the absurdity inMet. 1076b4-ll as well is the only interpretation of this passage along purely Aristotelian lines. As already Alexander of Aphrodisias had realized, Aristotle here draws on the Platonist nature of Eudoxus’ theory. In his argument against Plato’s own approach, immediately after the critique of Eudoxus’ version, Aristotle takes for granted that a Platonist mathematical solid contains planes "put together" (Met. 1076bl6-24). Soon thereafter, arguing that mathematicals cannot have any claim to substancehood, he supports his objection on the usual grounds that bodies are not made out of planes, lines and points (Met. 1077a31-36). Although in DC 299al-b7 this is explicitly an indivisibilist thesis, reading it into Aristotle’s refutation of Eudoxus as the evidence suggests does not mean that Eudoxus was an indivisibilist or that Aristotle wilfully implied so -like Aristotle, Eudoxus must have been aware of the geometrical nonsense inherent in in Phys. 241al5-23. 127 indivisibilism.*®* In GC 316b25-27 Aristotle clearly evokes the compositio ex punctis as only one of the absurd consequences he sees in a familiar assumption; a body’s being everywhere divisible at the same time has been actualized yielding an "infinitely ^®^See the mathematical arguments against indivisibles in On Indivisible Lines and Aristotle’s comment in DC 271b9-ll which is nicely borne out by Berkeley’s attitude to geometry in his Philosophical Commentaries’, see Jesseph (1993) 57ff. The most impressive mathematical arguments against indivisibles arise frominconunensurability and this is most probably what Aristotle has in mind in DC 271b9-ll; cf. Newton (1934) 39: "It may also be objected that, if the ultimate ratios of evanescent quantities are given, their ultimate magnitudes will also be given: and so all quantities will consist of indivisibles, which is contrary to what Euclid has demonstrated concerning incommensurables in the tenth Book of his Elements”. According to the above interpretation Aristotle must have construed Eudoxus’ doctrine as indivisibilism, although, like Newton’s, it had nothing to do with the mathematically problematic indivisibles. An interesting question is whether Aristotle did the same with what he calls inMet. 992al7-24 Plato’s doctrine of "indivisible lines". This is an attractive possibility because Aristotle makes clear that Plato substituted the "indivisible lines" for the traditional geometrical points. A good reason for which Plato would like to get rid of the points is to avoid the compositio ex punctis charge that Aristotle levelled against the realist view of the divisions of a line: Plato would respond that there is no compositio ex punctis involved here but rathercompositio ex lineis -this is suggested by Aristotle’s remarks that Plato insisted on such lines as "principles of lines". Now it is hard to believe that the author of the Theaetetus would characterize the lines in question as "indivisible". It is probable, however, that he conceived a line as made up not of "indivisible lines" but something analogous to Newton’s "evanescent divisible quantities" (Newton (1934) 38), i.e. quantities without "any determinate magnitude...but such as are conceived to be always diminished without end" (Newton (1934) 39). If Plato did entertain such a concept (which is not impossible given that Newton himself built on intuition rather than mathematics), then his line elements could easily be conceived as indivisibles, especially by a critic like Aristotle. Unfortunately Aristotle does not offer enough information on this particular piece of Platonic doctrine but it seems implausible that Plato espoused indivisiblism. 128 infinite" number of divisions/^ although nobody could complete such a supertask (GC 316a20-23). Regarding the infinite divisions of a solid as completed independently firom operational capabilities would certainly enter Eudoxus’ or any brand of Platonism on the ground floor, all the more since, as seen above, Aristotle explicitly objects to the construal of a magnitude’s infinite divisions as a completed, ontologically independent number in the Platonist fashion. There is adequate evidence to show that Plato conceived of numbers as a completed infinity and there is no reason why Eudoxus would not have adhered to such an important ingredient of mathematical Platonism, ff one wants to understand Aristotle’s argument in Met. 1076b4-ll consistently with parallel Aristotelian passages but without foisting on Eudoxus the ungeometrical doctrine of indivisibles, this must be the property of Eudoxean mathematicals Aristotle reduces to absurdity, aided, on the one hand, by his own conception of divisions as points, lines and planes and, on the other, by what he perceives in Met. 1076bl6-24 as the realist commitments of a Platonist. 184eiç,! iivpia pvpiaKiç (GC 316a21-22). CHAPTER Vn ON TEffi ORIGIN OF THE EXHAUSTION METHOD One need not, of course, agree with Aristotle.**® Nevertheless, his critique allows a glimpse of Eudoxus’ realist philosophical commitments which make it implausible that the mathematician would have reworked Hippocrates’ technique in the wake of Aristotle’s rejection of actual infinity. As already argued,**® Eudoxus’, **®It is doubtful whether Eudoxus would be forced to admit that the actual infinity of a magnitude’s divisions necessarily leads to the absurd conclusion that a point (or a line or a plane) is divisible. He could easily point out that the anthyphaeresis of two incommensurable magnitudes lends itself to a Platonic construal, i.e. it can be considered as an actually infinite process without any mathematical absurdity (for an example of an anthyphaeretic expansion see Fowler (1990) 37-38). Also, one is by no means bound to accept Aristotle’s account in GC 316b 19-27. Eudoxus could have very well argued that Aristotle’s own concept of divisibility underlies his account of the Platonic construal of divisibility leading it to absurdity. A Platonist does not need the distinction between being potentially divisible and actually undivided in order to account for the divisibility of magnitudes: a magnitude’s divisions exist actually in toto and dividing a magnitude simply operates on these independently existing divisions. If one wants to retain the Aristotelian construal of points and not do away with them as perhaps Plato did (see above n.l83) to avoid the compositio ex punctis the Platonist has only to assume that between any two divisions or points there be found another one. On this assumption the actual infinity of these divisions or points does not imply that the magnitude is composed out of points. One can of course only speculate about Eudoxus’ answer to Aristotle’s possible question whether the number of these divisions is odd or even; cf. above n.l27. **®Cf. above n.127. 129 130 or Plato’s for that mather, mathematical realism need not have been disturbed by Aristotle’s cogent arguments against actually infinite totalities, a conclusion strongly supported by the completed infinity of prime numbers in El. 9.20. Consequently, contrary to Knorr, Eudoxus had all the reasons not to object to actual infinity, as is indeed shown by the Euclidean evidence discussed above, unless, of course, Eudoxus was at some point disgusted by infinities, like Maclaurin.'^^ It will be shown below that Eudoxus could have devised the exhaustion method for reasons totally different &om a concern over actual infinity. But, if one reads Phys. 207b27-34 consistently with the availlable evidence, by bringing up exhaustion proofs as a counterexample i87iiwhen I was very young I was an admirer too of infinities; and it was Fontenelle’s piece that gave me a disgust of them or at least confirmed it, together with the reading of some of the Antients more carefully than I had done in my younger years" (in Tweedie (1922) 74). It is not necessary that Eudoxus was disgusted by infinities and reexamined Hippocrates’ method because it could easily lend itself to an indivisibilist construal so that the circle or cone ultimately coincides with a polygon or a pyramid (cf. above rh27). Eudoxus could have argued that, in the proof of El. 12.2 e.g., firom the fact that all the parts of the circle’s surface can be shown to be in constant ratio, it does not follow that the circumference of this figure consists in ultimate straight lines -the actual infinity of such two-dimensional parts does not imply that it ends with a one-dimensional straight line which takes the place of an ultimate arc. There is an interesting historical parallel to such an attitude. Newton (1934) 31 gave an intuitive characterization of the notion of limit arguing that, if in two curvilinear figures equal multitudes of rectilinear figures are inscribed, whose sums are in a given ratio (via El. 12.5), then, if the size of these rectilinear figures diminishes and their multitude increases ad infinitum approaching the curvilinear figures, the latter are in the ratio of the sums of the rectilinear figures. Newton here certainly captures the fundamental idea behind Knorr’s reconstruction of the technique that was supplanted by the exhaustion method; he denies, though, that the curvilinear figure ultimately consists in indivisible lines (cf. also below n.l98 and 199). 131 to aporia (2), Aristotle cannot plausibly imply that Eudoxus developed the exhaustion method as a reaction to Hippocrates’ appeal to actual inSnity. He bolsters his insistence on potential infinity contending, in view of aporia (2 ), that actual infinity plays no role in exhaustion proofs which rely solely on potential infinity; but he also adds the further conclusion that, if mathematicians do not use actual infinity in proofs, they do not need it, exactly because actual infinity enters even El. 12.4 but without ■ contributing anything to the proof; as shown above, this situation is characteristic of actual infinity in El. 9.20 and 3.16 as well. This argument complements nicely the justification of potential infinity in Met. 1048b 14-17 on account of its contribution to mathematical proofs and, in addition to Aristotle’s arguments against actual mathematical infinity, constitutes a respectable philosophical position which is consistent with mathematical practice but leaves well behind his contemporary mathematicians’ unquestioned acceptance of actual infinity, amply testified by Euclid’s Elements and Eudoxus’ realism. To a large extent, the philosophical importance of Aristotle’s theory of infinity lies exactly in his independence from the mathematics of his day and this study has, I hope, shown that, contrary to the impression one gets from the scholarship on the issue, Aristotle’s independence is neither ungainful nor detrimental: not only does insistence on potential infinity not threaten Euclidean geometry, either per se or on Aristotle’s premisses; but his arguments against actual infinity show, correctly, that infinity in this sense does not mesh well with Greek mathematics. It is not surprising that Aristotle’s contemporary mathematics did not come 132 to grips with the problem of infinity. Before Eudoxus’ development of the exhaustion method actual infinity could have appeared in the proo 6 of only a handful of theorems, namely Ef. 12.2,5,10,11 and 12. As shown above, El. 12.5,10,11 and 12 can be proven along the lines of Hippocrates’ proof otE l. 12.2 without violating any "standards of rigor", in view of the naive inductive step in El. 9.20 and, especially, 12.4. These Euclidean counterexamples to Knorr’s hypothesis show that the exhaustion method could not have been developed as a rigorization of an unrigorous technique because there were no pressing mathematical problems to which rigorization would have responded. Questions about infinite summations gave the impetus for the rigorization of the calculus because the analogy between infinite and finite sums resulted in unacceptable conclusions.*^^ Nothing of that sort, however, can turn up in the tiny set of proofs that would have required an infinite summation before the development of the exhaustion method. As far as infinity is concerned, the mathematicians of Aristotle’s day could have adhered to the pre-theoretic realism that characterizes working mathematicians*^^ and was philosophically articulated by Eudoxus.*^ All in all, viewing Eudoxus’ exhaustion method as a rigorization of *«*a. Kitcher (1981) 490. * % Maddy (1992) 1-2. *®®Cf. Grabiner (1981) 22 on the unimportance of rigor in eighteenth century analysis: "There was no "scandal" demanding immediate attention: there were no contradictions serious enough to halt the progress of mathematics. Even in the light of modem knowledge, these mathematicians made surprisingly few errors. In part this was because the infinite series they treated were usually power series with bounded coefficients, which behave very much as the analogy with polynomials would 133 an older technique projects the historical realities of eighteenth and early nineteenth centuries on the fourth century: but, as the evidence firom Euclid’s Elements unambiguously suggests, Aristotle’s contemporary mathematicians accepted, perhaps naively, actual infinity because they did not have topical concerns to reject it, whereas Aristotle did have topical grounds, albeit strictly philosophical, to argue against actual mathematical infinity. If a later era saw in the exhaustion method tliepar excellence rigorous handling of mathematical infinity, this does not imply that Eudoxus developed the method because of a concern over mathematical infinity. Unless of course one is ready to believe that Hippocrates, and/or Democritus, formulated El. 12,2. and 10 respectively ex adyto tamquam cordis responsa dedere, Knorr is right in hypothesizing that the exhaustion method supplanted an older technique. But this does not self-evidently imply that Eudoxus found fault with the appeal to actual infinity in the older proof-technique. In his account of the origins of the exhaustion method Knorr paid attention to only one aspect of the method, the limiting argument, which, in its obvious affinity with the modem definition of limit, justifies the historical relevance of the exhaustion method to discussions of mathematical infinity.'” One, however, must also account for the double reductio ad absurdum, the characteristic technique of exhaustion proofs, which in itself has lead one to ejqpect, even in the absence of a general theory of convergence. Also, the functions that these mathematicians studied often arose firom physical models and thus were relatively well behaved. Experience must have quickly shown that certain types of arguments led one astray and therefore simply should not be used". '” Cf. above n.l9; in what follows the outline of exhaustion proofs given in this note should be kept in mind. 134 nothing to do with infinity. Indeed, although it is impossible to have an exhaustion proof without the double reductio ad absurdum, in Book 12 of Euclid’s E/emen/s there is an exhaustion proof without a limiting argument, solely consisting in the double reductio ad absurdum. This proof, moreover, stands out among the other exhaustion proofs as the only one whose demonstrandum can be obtained without the infinitary technique that must have led Democritus and Hippocrates to the proof of El. 12.10 and 12.2 respectively. What is more important, both the demonstrandum and the particular technique of this proof, the double reductio ad absurdum, can be shown to follow from a geometrical construction. Eudoxus, therefore, could have obtained the formalism of the exhaustion method, and, as will turn out, substitute it for the older infinitary proofe, irrespective of any worries about mathematical infinity. In the proof oiE l. 12.18 x and y are two spheres that must be shown to have to each other the triplicate ratio of their diameters X, Y and the proof consists in the double reductio ad absurdum one expects in an exhaustion proof. One assumes that (x, y) 9 ^ (X^, Y^) and that there is a fourth proportional z < y, a sphere within y and with the same center as y such that (x, z) = (X \ Y*). After y*, a polyhedron inscribed in y such that it does not touch the inner sphere z (i.e. y* > z), is obtained, another polyhedron x* is inscribed within the sphere x and it is (x*, y*) = (X \ Y*) via the porism of El. 12.17. Therefore, it is (x*, y*) = (x, z) and, since x* < x, then y* < z; but y* is greater than z by construction. The assumption z > y is then reduced to z < y, as in the typical exhaustion proofs (except, of course, El. 12.10). To obtain, however, the required y* > z one does not need to invoke the 135 bisection principle as in exhaustion proofs. Since z is a sphere within y and with the same center, one simply constructs a polyhedron between y and z so that it does not touch z: this polyhedron is the required y*, since obviously y* > y - z by construction. To construct such a polyhedron is the problem in EL 12.17 and it presupposes the construction in EL 12.16: given two circles with the same center, to inscribe in the larger one a polygon with an even number of equal sides so that it does not touch the smaller circle. The bisection principle is invoked in EL 12.16 and, therefore, it is because of its deductive dependence on EL 12.16 that EL 12.18 has been categorized as an exhaustion proof. But, even if one starts the proof of EL 12.18 from the construction in EL 16, the application of the bisection priciple does not resemble exhaustion proofs at all: in EL 12.16 successive bisection of an arc only guarantees the existence of an arc less than a given one, a step required to ensure that the polygon to be constructed does not touch the inner circle, not that, given a y„ and a yj such that y, >1 / 2 (y - yo) and y, = yo + yi, there is a y„ < y - z, which is the y* required for the reductio ad absurdum. What, therefore, is absent from the proof of EL 12.18 is exactly the limiting argument, the step in an exhaustion proof which is associated with the rigorous handling of mathematical infinity. This absence can be explained by the given configuration, i.e. because z is taken as a sphere within y and with the same center so that EL 12.17 and its porism guarantee a y* > z such that (x*, y*) = (5^, Y^). According to Mueller this step allows to avoid the limiting argument and "actually 136 simplifies the reasoning firom the standpoint of Euclidean geometry."'^ On this account Eudoxus could have chosen between the typical exhaustion proof, complete with a limiting argument, and the more simplified proof we have in the Elements. The absence of the limiting argument firom the proof of El. 12.18 is thus explained away as a choice of deductive convenience when Eudoxus was already in possession of the exhaustion method. Nevertheless, one can plausibly argue that the absence of the limiting argument marks El. 12.18 and its proof as the context within which the double reductio ad absurdum technique of exhaustion proofs arose firom Eudoxus’ geometrical studies, irrespective of problems concerning umrigorous infinite processes. Assuming that problem solving was the primary focus of geometrical research,^®® it is easy to see howEl. 12.18 could have been obtained as a deductive offshoot of the complicated construction ini?/. 12.17. After the successful completion of this construction, which presupposes the auxiliary construction in El. 12.16, it would have been trivial for Eudoxus to obtain the porism of El. 12.17. Since the polyhedron is constructed m.El. 12.17 as a sum of pyramids whose sides are radii of the sphere, it is easy to conclude firom the porism of El. 12.8^®^ and firom El. 5.12 that, if a similar polyhedron is inscribed in another sphere, the two polyhedrons are to each other in the ratio of the spheres’ diameters. The interesting point now is the *®2MueUer (1981) 246. ‘®®The importance of problem solving in Greek geometry has been argued persuasively by Knorr (1993) 350-351. ‘®^"Similar pyramids which have polygonal bases are to one another in the triplicate ratio of their corresponding sides". 137 ensuing configuration; two nested spheres, a polyhedron inscribed in the outer sphere and another polyhedron, similar to the first and inscribed in another sphere. This is exactly the configuration required for the Euclidean "exhaustion" proof oiE l. 12.18 and Eudoxus could have obviously arrived at this configuration without knowledge of the exhaustion method. If, moreover, x* and y* are the two similar polyhedrons, X, y the two spheres and y, z the two nested spheres, Eudoxus would have hardly failed to notice the blatant proportion-theoretic absurdity implicit in the configuration. It cannot be (x, z) = (x*, y*), since this implies x > x* and z > y* whereas by construction y* > z: alternatively, since x > x* and y* > z, it cannot be (x, z) = (x*, y*). In the Euclidean proof of El. 12.18 this is the first horn of the double reductio ad absurdum of the exhaustion method, a step which Eudoxus could have obtained by means of a simple inspection of the above configuration, without any foreknowledge of the exhaustion method and, most importantly, without any need to circumvent an unrigorous infinitary argument. The discovery of the demonstrandum oiEl. 12.18 would have also followed suit: assuming that (x, z) < (x, y) firom the diagrams,*®* Eudoxus already had half the proof that (x, y) = (x*, y*), since (x, z) ?£ (x*, y*) by construction. It remained only to show that (x, z) ^ (x*. *®*This might seem an arbitrary assumption but it could have been borne out by the relative sizes of the spheres in the diagrams Eudoxus was working with. It is undoubtedly a heuristic assumption which would have been no more necessary in the formal presentation of the reductio ad absurdum after the result had been obtained. Notably the well-defined ordering or ratios as equal, greater or less is a prominent characteristic of the Eudoxean proportion theory as reconstructed by Knorr (1978) (see esp. 219). 138 y*), when z > y, an assumption which Eudoxus could have quickly reduced to the previous one. If (x, z) = (x*, y*), then (z, x) = (y*, x*) and, therefore, for y there is a fourth proportional sphere w such that (z, x) = (y, w) with x > Consequently, (y, w) = (y*, x*) and, since x > w, the iititial hypothesis reduces to the proportion-theoretic absurdity observed in the configuration: one need only visualize y*, X* as polyhedrons inscribed in spheres y, x and w as a smaller sphere within x so that X* > Cl), i.e. the proportion (y, On this account the careful inspection of the configuration that resulted after Eudoxus obtained the porism of El. 12.17 led naturally not only to the formulation of the demonstrandum in El. 12.18 but also to particular technique for its proof: a double reductio as absurdum by means of a fourth proportional, i.e. the technique that characterizes exhaustion proofs. Nowhere is a desire involved to avoid unwanted infinitary arguments: the only assumption one need make is that Eudoxus, or any Greek geometer for that matter, would have studied exhaustively the particular ‘®®In the proof of El. 12.18 Euclid argues that the existence of o) < x such that (z, x) = (y. Cl)) has already been demonstrated, an obvious reference to the lemma after El. 12.2. But, as Heath (1956) 3, 375 remarks, the genuiness of this lemma is suspicious. One does not necessarily need the lemma to secure the existence of w < X. Even if this lemma is not dismissed as a post-Euclidean deductive £riU, Eudoxus would not have required it to carry out the second part of the double reductio ad absurdum at the heuristic stage of his work. 139 geometric configuration he was working on.*’’ The claim, moreover, that, first, Eudoxus arrived ztE l. 12.18 heuristicaily byimagining the spheres as infinitely sided- polyhedrons*’® and, second, that the study of the geometric configuration rigorized *”Needless to say, in Greek "diagram" (ôiâypafjina) means "geometrical proof. As is clear firom Met. 1051a21-29, Aristotle is fully aware of the fact that observing the properties of the diagram is a crucial tool of mathematical discovery. *’®Ini)C 307al6-17 Aristotle seems to attribute to Democritus the conception of a sphere’s surface as composed out of planes: "According to Democritus, the sphere too cuts, as a kind of angle, since it is mobile". Does this statement imply that Democritus thought of the surface of a sphere as angular and, consequently, that the sphere consists in pyramids? In view of the evidence this interpretation does not appear very cogent. In DC 306b29£f. Aristotle discusses two atomist conceptions of fire, one according to which this element is composed of pyramids and another according to which fire consists in spheres. The latter, as is obvious from DA 403a, was put forth by Democritus; the former can only belong to Plato, although Aristotle does not say so in DC 306b29ff. According to DC 307al-3, these atomists conceived of the atoms of fibre as spheres and pyramids respectively because they associated burning and heating with angular shape and, moreover, "the entire sphere is an angle" whereas the pyramid "has the acutest angles". Aristotle’s paradoxical characterization of sphere is certainly relevant to his remark in DC 307al6-17. But in Greek geometry there is only one sense in which the sphere can be characterized as an angle: any of the sphere’s great circles forms with its tangent a hornlike angle (EL 3.16). Since Democritus had written a work On the Contact o f a Circle or Sphere, which most probably dealt with the problematic nature of this angle (see Heath (1956) 2, 40), it is far more plausible to assume that Aristotle’s comment in DC 307al6-17 refers to hornlike angles, not to a view of a sphere’ surface as consisting in infinitesimal planes. Having associated burning and heating with angular shape, as Aristotle reports, Democritus all but naturally conceived of fibre’s atoms as spheres, since a sphere touches a surface in a hornlike angle, the smallest acute angle, as one still reads in Euclid’s EL 3.16, and, therefore, the sharpest possible. 140 the result of his intuitive heuristics begs the question.^” Intuitive heuristics is traditionally viewed as a quick trick, a sleight-of-hand for obtaining results with ^^^Knorr (1982) 140-141 seems to attribute a sort of indivisibilism to Eudoxus but the evidence in support of such a thesis is scant. Knorr’s only argument is that the wording of the limiting argument in EL 12.5 is "different from the wordings used elsewhere in the book" (141) (cf. Knorr (1978) 236): but from the substitution of "to divide" for "to cut" and similar variations it certainly does not follow that "one might entertain the view that Eudoxus had indeed used a method of parallel sections but that his followers chose to modify his proof (!) (Knorr (1982) 141). Were Knorr right, Eudoxus would not have seen that El. 12.5 can be proven by an exhaustion type limiting argument, like e.g. El. 12.2, which is very hard to believe. All in all, there is no reason to assume that Eudoxus would have had to resort to indivisiblism, even if he had obtained EL 12.5, 11 or 12 by means of Hippocrates’ technique, as reconstructed by Knorr (cf. above n.170), before he hit upon the exhaustion method. Generally, there is no cogent evidence to attribute indivisibilist proof-techniques even to Democritus. From Archimedes’ own indivisibilist technique in his Method and Plutarch’s famous report of Democritus’ puzzle about a cone cut by a plane parallel to its base (De Com. Not. adv. St. 1079 E ( = Democritus DK B 155) it has been inferred that Democritus himself, anticipating "Cavalieri’s principle", applied an infinitesimalist technique in his proof oîE l. 12.7; see Heath (1922-1923) and (1956) 3, 368; cf. also Luria (1933) 138ff., Man (1954) 19ff. and van der Waerden (1961) 137-138. But Plutarch’s report might veiy well refer to a mere logical puzzle by Democritus with no solution implied and irrelevant to the atomist’s mathematical researches (cf. Mendell (1985) 339-340); other evidence for Democritus’ indivisibilism is similarly flimsy (cf. above n.l98). Since Democritus was a mathematician, if one is to judge from the titles of his works (cf. Heath (1981) 1, 178ff.), it is not improbable that he had already obtained the Euclidean proof of EL 12.7 which can easily provide a proof of EL 12.10 by means of an infinite summation (see above n.l70). When Archimedes credits Eudoxus with the proof of EL 12.7, he most probably implies that Eudoxus provided the exhaustion proof oiE l. 12.5 on which the proof otE l. 12.5 depends (Democritus could have obtained the proof oiE l. 12.5 too by means of an infinite summation which does not involve infinitesimals; cf. above n.169). 141 expediency in contrast to the painstaking and long-winded rigorous proof.^ But such a distinction obviously does not apply to the very simple situation at hand. After Eudoxus arrived at the porism of El. 12.17, it was the available geometric configuration itself that rapidly led him to mathematical discovery. The single pair of polyhedrons already inscribed in the spheres immediately suggests the strategy to show that the polyhedrons are in the ratio of the spheres, i.e. a double reductio ad absurdum: as seen above, the first horn of the reductio for z < y is already implicit in the figures and, after it has been obtained, it can only lead one to test the hypothesis z > y. To conceive of this proof in seventeenth century terms as a complicated rigorous substitute for a quick sleight-of-hand is plainly an unwarranted anachronism.™* Having obtained the double reductio ad absurdum in the manner I have suggested, a structural similarity between the proof oiE l. 12.18 and those oîE l. 1 2 .2 , 5,11 and 12 would have drawn Eudoxus’ attention: not only has the demonstrandum the same form, (x, y) = (a, /3), but in the proofs of El. 12.2, 5, 11 and 12^“ there ™This is clear in Newton’s defense of his method of "first and last ratios of quantities" against the ready charge of indivisibilism (Newton (1934) 38). ™*Mueller (1981) 246 has remarked that the distinction between the logical rigor of the exhaustion method and the cumbersomeness of its geometric technique is a product of a later era and does not apply to the situation in Euclid’s Elements. ^“Since El. 12.2 and 10 were known before Eudoxus, it is not inconceivable that all theorems proven by means of the exhaustion method in Euclid’s Elements had been obtained before Eudoxus. Archimedes’ reports in the introduction to his On the Sphere and Cylinder^ Quadrature of the Parabola and Method allow to attribute to Eudoxus the exhaustion proofs of certain theorems, namely El. 12.2,10 and 18, not 142 is also an infinite sequence of diminishing similar x*, y* within x, y so that, for each X*, y*, (x*, y*) = (a, jS), exactly as ia.El. 12.18 there is a pair of similar entities x*, y* within x, y so that (x*, y*) = (a, jS). This stuctural similarity would have prompted Eudoxus to prove El. 12.2, 5, 11 and 12 by means of the double reductio ad absurdum: given the infinity of (x*, y*) = (a, j 8 ) in the pre-Eudoxean proof it is an easy matter to invoke the bisection principle, already employed in El. 12.16, to argue that there is a y* > z.*® El. 12.10 also lends itself to a proof along basically the same lines. On this account Eudoxus did refashion Hippocrates’ older technique but not because he perceived foundational problems with the application of El. 5.12 to an actually infinite set of terms, as Knorr has it: he was rather motivated by the wish the formulation of these theorems’ demonstranda. Although the reconstruction of the origins of the exhaustion method proposed in this paper assumes that Eudoxus both formulated and proved at least E l 12.18, there is no reason why pre-Eudoxean mathematicians, like Democritus or Hippocrates, could not have obtained E l.\2 5, 11 and 12 as well, since the method employed by Hippocrates in his proof oiE l. 12.2 according to Knorr can also furnish the proofs of the other theorems. ^“Knorr (1978) 205 has argued that the bisection principle was first formulated and applied in the context of exhaustion proofs and was then imported to Eudoxus’ proportion theory he has reconstructed. The fact, however, that in exhaustion proofs Euclid argues that x > a and y > J8 from (x, y) = (a , j3) by first alternating the proportion is explained by Knorr as a trace of the original Eudoxean proportion theory (Knorr (1978) 197). Thus, on Knorr’s account, the exhaustion method presupposes Eudoxus’ proportion theory which, however, cannot be developed without the bisection principle (see Knorr (1978) 193): how, then, could this principle have been transplanted from the exhaustion method to Eudoxus’ proportion theory? If Knorr’s reconstruction of the Eudoxean proportion theory is correct, it is more plausible to assume that the bisection principle first appeared in Eudoxus’ proportion theory. 143 to homogenize the technique in the proofs of a group of theorems that shared structural similarities. Contrary to Knorr, the bisection principle does not remove "the need to appeal to naive notions of limits and infinite processes",^ like the one Eudoxus and Euclid admit in El. 12.4: as in El. 12.16, the bisection principle serves to justify an existential assumption, that there is ay* > z, necessary to carry out the double reductio ad absurdum^ If a later era saw in the exhaustion method the par excellence rigorous means of handling the formally recalcitrant notion of infinity, this does not mean that the method itself was originally conceived as such. It is of course impossible to prove this hypothesis with historical evidence but 204Knorr (1982) 127. ^®*The proportion-theoretic application of the bisection principle (see above n.203) agrees with its use in El. 12.16. In both cases the principle guarantees not convergence but only the existence of a certain magnitude required for the particular proof or construction -it does exactly the same in exhaustion proofs as weU after one has shown how to constmct a convergent sequence in constant ratio. It is noteworthy, moreover, that the application of the bisection principle in exhaustion proo 6 does not appeal to the potential infinity of divisions, unlike Aristotle’s formulation of the principle in Phys. 266b2-4. One might of course argue that there is no difference between mention and use, or rule and application. Nevertheless, since there is no evidence that Eudoxus and his fellow mathematicians were concerned with the notion of infinity, it is preferable to assume that the application of the bisection principle is an intuitive way to obtain a magnitude less than another one, i.e. to justify an existential assumption: by applying the principle Eudoxus did what every sausage-seller in the Athenian agora would have assumed as matter-of- factly true, even if he bolstered intuition by his Platonist construal of division. Plato in his "divided line example" and Antiphon in his quadrature of the circle must have built on the same intuition and, in view of Plato’s example, it is obvious that Aristotle’s theory of potential infinity was bound to have come about, even if Eudoxus had never applied the bisection principle. 144 in view of the above discussion it seems cogent to single out the proof of El. 12.18 as the context where the exhaustion method originated. Although it employs the formalism of the exhaustion method, the proof of El. 12.18 is marked off from those of El. 12.2, 5,10,11 and 12 not only by the absence of a limiting argument but also by its being the only Euclidean exhaustion proof which, as shown above, can be obtained without a technique that foreshadows Newton’s "method of first and last ratios". But, since pre-Eudoxean mathematicians must have employed such a method and, moreover, Euclid, and, therefore, Eudoxus too, had no problem with its crucial step, the proof of El. 12.18 stands out as the context where the formalism of the exhaustion method arose in order to supplant the older method solely for concerns of formal presentation. The obvious advantage of this hypothesis over Knorr’s lies in that it resolves the problem of Euclid’s lapse from "Eudoxean" deductive rigor in El. 12.4 and 9.20: one need not impute a formal blunder to Euclid simply because the exhaustion method did not arise as an attempt to banish uniigorous appeals to actually infinite processes. On this account the common, and justified, perception of the exhaustion method as a rigorous means of handling mathematical infinity does not reflect the origins of the method. This perception can be plausibly said to originate with Aristotle’s remark in /% j. 207b27-34, although of course Aristotle does not bring in the notion of rigor. To fully appreciate the historical import of this passage, the main objective of this monograph, one must take into account that Aristotle’s contemporary mathematicians had not abandoned actual infinity: as the adverb pw 145 in Phys. 207b30 suggests, the development of the exhaustion method in Aristotle’s time, irrespective of any concern over actual infinity, was a happy coincidence that allowed him to astutely deny any justification of actual infinity on account of mathematical practice. To an extent, the force of Aristotle’s counterargument must also lie in Eudoxus’ Platonist construal of infinity as actual. It is appropriate to close with some general comments on the relationship betwen Greek philosophy and mathematics since it is in this context that Knorr appealed to the traditional view of Aristotle’s theory of infinity, following in the footsteps of Milhaud according to whom Aristotle’s theory "n’ est le fait d’un esprit profondément pénétré de la pensée géométrique du V® et du IV® siècle."^“ As is very often the case, it is easier to pose the question of a philosophical influence on the development of Greek mathematics than answer it, simply because the required evidence is not there. Knorr has undeniably done much to show the untenability of certain older views, like the foundational crisis in Greek mathematics, precipitated by the discovery of incommensurability and finally resolved by Eudoxus,^"^ or the crucial influence of Eleaticism on the development of Greek mathematics.^"^ Unfortunately, in his 1982 article Knorr chose to deny any possible philosophical ^“ Milhaud (1903) 392. ^®^See Knorr (1975) ch.9. One still reads in 1991, however, that Eudoxus’ exhaustion method is "the classical resolution to the paradoxes of dichotomy and of the existence of incommensurables (Borowski & Borwein (1991) 201; as the crossreference to "dichotomy" shows, the authors understand the exhaustion method as a solution to Zeno’s paradoxes); cf. also Stillwell (1989) 37 and Moore (1995) 100. 208i See Knorr (1981). 146 influence on the mathematical understanding of infinity by appealing to the urunathematical nature of the primary potential source of such influence, Aristole’s theory of potential infinity. As shown above, this is a serious distortion of the evidence, despite the fact that Aristotle did not indeed influence his contemporary mathematicians’ attitude to infinity. The really important task is not to determine the debt, if any, of Greek mathematics to philosophy but rather to properly appreciate the views of philosophers, especially those of Aristotle, that pertain to mathematics. 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