(greek)

sacs YADA KAW a. 21. Hideya Yamakawa O ZHNOQN ITY@ATOPIZE! 226 — wa Pythagorean philosophy / ed. by Konstantinos |. Boudouris. Athens : International Center for Greek Philosophy and Culture, 1992. 257 p. ill. index). (Studies in Greek philosophy ; 7). » papers read at the third international conference on Greek philosophy, Samos, August 1991.

O ZHNQN IIYOATOPIZEI II. The orientation for the paradox Achilles Suppose that the swift runner Achilles coul d overtake a tard y tortoise at a spot n, and now on his back triumphantly says as follows: Achilles: «I conquered you. Now, you shou ld admit your loss». Tortoise: «I cannot bélieve my state of affai rs. M ake clear your way of running, so I may concede my loss». Achilles: «Well, listen to my saying and satisfy yourself of your situation. My way of running was such a skill ful one as follows; HIDEYA YAMAKAWA In the first step, I covered a distance O ZHNQN IIYOATOPIZEI distance greater than the half of that which was left; thus ah I. Preface i Hermann ee i attac ks on plurality» aper «Zeno of Elea’s Zeno’s fragment B3 as on view his ed wee llows:! u | or that f i «One glance at the text is sufficient to make us realize substitute ‘limited’ and ‘unlimited’ (scil. in number) we could almost | ted’ and ‘inexhaustible’». ‘ dike rene to Fränkel, the subjects of Zeno’s eve e er pen tomy and Ac hilles reported in Le Aristotl j all those of Dicho — l problems of ‘exhau Z, chap. 9, were no other than the l agree I I ly, | Basical de. e. ustibil ibilityity ’ of the magnitud ‘inexha , ue on I n, opinio his o aten and here in my paper, developing ti = discuss the paradoxes Dichotomy and Achilles in connec | f incommensurability. question a into calls es Achill ur Zeno in his a. en l incommensurability (or irrationality) of the unequa by n two runners, and moreover, it seems to be very no Mi De t wan ms sù ues there after the model of some mathematical er jn ee think of way orean which originated in the earlier Pythag DE ae ess the following, firstly, I would like to support reconstruction of Achilles and, secondly, I would like to | ichotomy. tion of the paradox Dic inati xamina i atical 7 tn liberal use of the Pythagorean way of mathem Cason on xes parado thinking, Zeno, in my view, formulates his ma neen Zeno’s way of argumentation in paradoxes makes | ass presupposition of some Pythagorean thought patterns. I «Zyvov mv0ayogilel». greater than half of the distance which is showed as a length betw een the Starting point and your final spot here; and in the seco nd step I covered a repeating the process continually, now, I am on your back comfortably. Did you understand?». Tortoise: «Oh, I see. Your way of running was really based on the Axiom of Archimedes. | Achilles: «Not ‘Archimedes’, but Elements X. prop. 1.2 You are not enlightened concerning the history of math ematics». Tortoise: «Don’t mind such a trifling matte r. Now, I recommend you to listen to Zeno’s argument. He says: «To 6oadvtatov oùdérote xatahnponoet ar Oéov Und tod Taxiorov Eungoodev yao avayxaiov EAdeiv To Ötöxov bOEv bounoe tO pedyov, bor dei ti nQoëgeuv avayxaiov To 6oadvtegov. (Even the slowest runner, when he is going, can never be overtaken by even the faste st, for, inevitably, the pursuer must first reach the point from which the pursued started, which means that the slower runn er will always keep ahead”) [Arist. Phys. Z 9, 239b 14-18] If your way of running were Zeno’s, I could never be overtaken by you, since he says that the two distances covered by you and I were incommensurable». | II. Zeno’s Subsoil, Surface soil and Enem ies Before attempting to interpret Achilles in line with Fränkel’s thesis, I would like to identify Zeno’s stand point, his weapons and his enemies. The spanish philosopher Jose Ortega y Gass et made an apt remark concerning the matter. I epitomize the esse nce of his argument? .

A thinker’s thought has always its subsoil, surface soil enemies. The subsoil and as an aged strata of the collective thinking ignored by and as a birthplace of the thinker’s thought is ordinally t and an produc him. On the contrary, the surface soil is a recent ally established fact of which the thinker is aware and gives princip the thinker his approval. Taking his position on the surface soil , about thespeak not does he gives birth to his original thought. But, surface soil, and in opposition to which one can O ZHNQN IIYOATOPIZEI IV. Reconstruction of the Achilles paradox : turn towards the Achilles. I remember here a brilliant achievement of Rademacher and Töplitz'? They used the Proposition X.2 in order to reconstruct an archaic method = D formulate his own original doctrine. Now, let us identify Zeno’s subsoil, surface soil present here the following as a working hypothesis: 229 and enemies. I I. Subsoil: *. The tradition from Ionian philosophy to Pyrhagorean philosophy si II . Surface soil: . (1) Parmenidean philosophy as a legacy to be succeeded’ he was not (2) Weapons utilized to attack his enemies but of which always an inventor. 1) Reductio ad absurdum E B A [fig. 1] as a way of argumentation’. 2) The method of successive subtraction dvôvpaigeois or avtavaigeois which originated in the earliest stage of the history of nts 3) The archaic number theory as seen in Eleme VII and IX, the Pythagoreans’ activity ?. H among all (1) The theory of even and odd® , and gnomon 9, (2) The way of construction of figured number using which does vividly show that the side and the diagonal of the unit square are linearly incommensurable. They utilized the above diagram: | In the above diagram, the figure CDFE is repeated in imi figure FEIH. This figure is a kind of gnomon which er infinitum" . the theory of even 5) Rather hot news about the proof which utilize al of a square diagon and and odd in order to show that the side are linearly incommensurable!!. Now, allow me here to use a slightly different construction than Rademacher & Töplitz. Suppose that when Achilles traverses the distance AB, the tortoise too traverses the distance BC and takes his position at C at the same time. Now, let the ratio of AB to ZC be equal to the ratio of the length of a side to the length of the diagonal u. en we can reconstruct Achilles after the model of III. Enemies But according It is very difficult to identify Zeno’s real enemies. diagonal [d]. Mark off the length AG [= AB = n der on Fl the perpendicular GJ from G to the side BE; then EG = GI = BI = 4) The theorem of Pythagoras !° to Plato’s remark in his Parmenides, Zeno’s enemies were none other than the pluralists who ridiculed Parmenides’ argument and were the empiricists who were confident of the reality of the sensible who defended their position on the xaliytoomoc xéhevôos. world”? . They Given a square ABEF let AB be a side [= a d-a [let it be a;]. Hence EI = BE - BI = a-a [let it be a, ]. | Now BI and EI respectively form the side BC [ = a,] and diagonal BH (= d;) of a new square BCHI. The procedure can be repeated for the smaller square CDJK too, thus a smaller side a and diagonal d, will appear for the square CDJK again. Eisntiusing

in this way, we get smaller and smaller squares and the process of successive subtraction will never terminate. O ZHNQN IIYOATOPIZEI 231 Thus, a is subtracted from d, the difference a; is confirmed; subtracted from a, the difference d, is confirmed; a, is a, is subtracted from d;, the difference az is confirmed, and so on. This is no other than the process of successive subtraction between two unequal magnitudes, i.e. the distances covered by Achilles and the Tortoise . Therefore, we may safely conclude that our swift Achilles runs in line with the Pythagorean method of dv0vpatoeous. V. (fig. 2] In the context of Achilles, this implies the following: when Achilles reaches the point B (the distance covered by Achilles = u), the tortoise reaches the point C at the same time (the distance covered by Tortoise = d). And when Achilles reaches the point C (the distance covered by Achilles = a ), the tortoise reaches to point D (the distance covered by Tortoise = dj), and so on. Thus each distance covered by the Tortoise and Achilles is schematized as The relevance to the theory of even and odd Now we can turn our attention to another aspect of the Achilles. In the above process, if we should give certain numerical expressions to the respective lengths of the sides and the diagonals, we necessarily become aware of a serious contradiction. By the way, it goes without saying that such a way of representing length as a certain amount of numbers is a typical feature of the earliest Pythagorean way of thinking. Let a square ABCD be given, let AC be the diagonal (=d). The length of the side AB is a, and a=a]+d;. Mark off G and H respectively on the sides BC and CD so that the lengths of BG and DH are d; s. and the lengths of CG and CH are a, s. Mark off E and F on the sides AD and AB respectively; so that the straight lines EG and FH are produced. Let Al and EF be the diagonals on the square AEIF. A follows: E J RATIO OF DISTANCES The Ist step . o D 7 N, SUCCESSIVE SUBTRACTION PROCESS : d a K d The 2nd step a : di a = d-a dj The 3rd step do : d A = didq = a - 0 = A - QQ The 4th step a3 : d3 az = d,- a da = The n th step Ont © Ane An = dh- On daar = On - Intl O2- 03 ° di mt © È

O ZHNQN ITYOATOPIZEI 233 are prime one another. Given the square ABCD, let the length of a side of the square be Then, from the fact AC? =2BC?, we can conlcude that AC? (=d?) is a and let AC (=d) be the diagonal. Let BD be another diagonal of the square. Suppose that the side and the diagonal are prime one another. The line BD bisects the line AC at E. Draw the mark off F and and perpendiculars to the BC and CD respectively G respectively. F and G bisect the sides BC and CD respectively. EFCG is a new and smaller square. Let EC be a diagonal of the square EFCG. Let FG be another diagonal of the square. The line FG bisects the line EC at H. Draw the perpendiculars to the CF and CG respectively and mark off / and CG bisect the sides CF and J and J respectively. J respectively. HICJ is a new and smaller square. The process can be repeated successively as though the gnomon for the diminishing squares were continually applied, so that the Suppose that the lengths of AC and BC even, thus d is even and that BC (=a) is odd. Now, from the figure, the length of AC (=d) is 2-; ua On the one hand, in the square CHIG, CP = 2CG?, thus dj? = 2a/’. Therefore dj? is even ‘thus d; is even. On the other hand, from our hypothesis, the length of BC (=d, +a) must be odd, thus d; + a; is odd, while d; is even ; therefore a; must be odd. Let GH be joined; then the diagonal GH bisects at K another diagonal CI. Thus the lengths of ZK and CK are tespeanvely 1/2 d;. But, since CG (=a 7) =2CK?, it must be concluded that a is even, so that a; is even. The same ais odd and even at the same time. Itis impossible. Similar recurrent construction can be continued infinetely in the diminishing squares, and at each step we can verify that any side or diagonal is expressible as even and odd at the same time. points E, H, K, are marked on the diagonal AC, and in parallel with this process the sequence of the diminishing distances DEE VI. Dichotomy The fact that Zeno’s paradoxes are uıumuara of the Pythagorean way of mathematizing is verified more clearly by the examination of Zeno’s first paradox on motion i.e. the Dichotomy. A D 1/2d, 1/4d, 1/8d, ... 1/2" d is formed. Now, it is very easy to see the following: (1) AC? = 2BC? so, AC? is even, thus, therefore, (2) BC?= 2CE? so, BC? is even, thus, therefore, (3) CE?= 2CF? so, CE? is even, thus, therefore, (4) CF?= 2CH? so, CF’ is even thus, therefore, ee E G A B F fig. 5] J 7 L C AC is even, BC is odd. BC is even, CE is odd. CE is even, CF is odd. CF is even, CH is odd. This process can be repeated ad infinitum. In each step we discern an impossible fact that the same length can be expressible as even and odd at the same time. In the above reconstruction of Zeno’s paradox Achilles and Dichotomy, the archaic Pythagorean method was used. So, it is not too much to say that Zeno here in the paradoxes of Achilles and attacks motion after the fashion of the earlier Dichotomy Pythagorean method. Thus, we may safely conclude that Znvoy xv- Bayooiteı.

O ZHNQN IIYOATOPIZEI 235 there. But even Here and There are very close together, nothing can prevent us from making the Something here and the Something there small enough to allow for VII. Epilogue third thing to be squeezed in between them. The operation can be repeated The Hungarian scholar, Arpad Szabó, has earnestly tried to explain «why a predominantly empirical mathematical tradition suddenly and for no apparent reason became anti-empirical and antivisual» in the history of Greek mathematics, and gave his answer as follows: «I believe that the influence of Eleatic philosophy was responsible for the rejection of empiricism and visual evidence in Greek mathematics, as well as for the introduction of indirect proof. The construction of Greek mathematics as a deductive system was a result of this same influence and that, had it not been for the philosophy of Parmenides and Zeno, it would not have been possible to build up so ingenious a system as Euclid’s Elements »'°, This is a very attractive statement; and this is also very similar to Burkert’s and Philip's judgements on the development of Greek mathematics. But, I think that his doctrine has some major defects in his following theses;! (1) The method of indirect proof is a monopoly of the Eleatics. (2) The Eleatics could hardly have adopted the method of indirect proof from mathematics to philosophy. Concerning Szabö’s thesis (1), the philological studies of today can cite some significant refutals from the Pre-parmenideans. Concerning Szabó’s thesis (2), we can neither affirm nor deny it immediately because of the almost entire deficiency of the early fifth century’s mathematical materials. But if we may detect some underlying mathematical structures or mathematical way of thinking in Eleatic philosophy, we can indirectly reject Szabó’s thesis (2) too. My task in this paper was precisely this. indefinitely without reaching a limit. The premise of plurality and divisibility does not admit the assumption of an ultimate indivisible unit, and strict logic does not allow a gradual transition from the very small to the extended. For it is the nerve of many of Zeno’s arguments that lack of magnitude must be radically distinguished from any magnitude, however small... . The idea of decreasing quantitities, ... recurs in Zeno’s dichotomy, Achilles, and frag. B1. The closest parallel, however, is to be found in Plato’s Parmenides in a dialogue, which follows Zeno’s book closely in theme and in treatement». 2. Elements X. prop. 1 says: «Two unequal magnitudes being set out, if from the greater there be subtracted a magnitude greater than its half, and from that which is left a magnitude greater than its half, and if this process be repeated continually, there will be left some magnitude which will be less than the lesser magnitude set out.» See T. L. Heath, The Thirteen Books of Euclid’s Elements, Vol. III pp. 14-16. 3. See José Ortega y Gasset, Origen y epílogo de la filosofia in Obras completas de José Ortega y Gasset, Revista de Occidente 2 ed., 1965. 4. See my book The Origin of Philosophy and Science, Sekai-Shisô-sha, 1987. pp. 16-106; especially see, chapter 6 «Parmenides’ challenge», pp. 76-106. 5. Fragment 8, 1-6 says: «uôvos è Erı pd00g Ôdoïo Asineraı, dc or: tavty © ri onuat Eaoı zohhà wah’, dc dyévnrov gov xai dvóhÂedgóv éotiv, 00Aov povoyevés te wat Argeuts 70° atédeotov, oùdé nor úv odd’ Eorau, Enrei viv éoriv duod nav, Ev, ovvegéc: tiva yao yévvav dibnoea adtod;». It is a noticeable fact that among many signs there are signs «ovlov uovoyevés » and «árgeués ». Being is entire-unique and unmoved, since it is now all together, NOTES one, indivisible. «Év », it seems, gives a firm basis for the entirety of Being (0d4ov uovoyevés ) and «ovveyéc » for the unmovingness of Being, since the not-one being could mean the plurality divided into many parts (# entire-unique) and since in turn the divisible thing implies that it cannot be now all together and so implies 1. Hermann Fränkel in his «Zeno of Elea’s attacks on plurality» in Allen and Furley, Studies in Presocratic Philosophy, pp. 104-105 interprets B3 as follows: «In order to explain the second part of the fragment, there seems to be left only the alternalive of operating with things, or parts, of indefinite magnitude. If we assume plurality, i.e. divisibility, of any unit, some part of it is here and some other part is also its generation (or origin). Thus we may identify the problem of refutability of plurality and indivisibility of Being as a legitimate legacy for Zeno which was handed down from Parmenides. 6. R. E. Allen in his book Plato’s Parmenides . Translation and Analysis, Basil

Blackwell 1983, has put it: «As a pattern of argument, reductio ad absurdum did not originate with Zeno. It is used by Parmenides, and it is a characteristic and pervasive method of Greek mathematics, one which may well antedate not only Zeno but Parmenides» (p. 70); and further commenting on this topic he says:«Heath suggests that the substance of Euclid, Elements VII-IX, which relies on indirect proof in many places goes back to the Pythagoreans and that there is clear indication that number theory had been reduced to elements by the time of Archytas (c. 430365 B.C.). TBEE ii, 294-95» (p. 303). According to K. R. Popper’s «Back to the Presocratics» we can discern a typical type of argumentation by reductio ad absurdum in «Anaximander’s theory of the suspension». Popper asserts as follows: «How did Anaximander arrive at this remarkable theory? Certainly not by observation but by reasoning. His theory is an attempt to solve one of the problems to which his teacher and kinsmen Thales, the founder of the Milesian or Ionian School, had offered a solution before him. I therefore conjecture that Anaximander arrived at his theory by criticizing Thales’ theory. This conjecture can be supported, I believe, by a consideration of the structure of Anaximander’s theory. Anaximander is likely to have argued against Thales’ theory (according to which the earth was floating on water) on the following lines. Thales’ theory is a specimen of a type of theory which if consistently developed would lead to an infinite regress. If we explain the stable position of the earth by the assumption that it is supported by water —that it is floating on the ocean (Okeanos )— should we not have explained the stable position of the ocean by an analogous hypothesis. But this would mean looking for a support for the ocean, and then for a support for this support. This method of explanation is unsatisfactory: first, because we solve our problem by creating an exactly analogous one; and also for the less formal and more intuitive reason that in any such system of supports or props failure to secure any one of the lower props must lead to the collapse of the whole edifice» (Conjecures and Refutations, The Growth of Scientific Knowledge, p.139). Anaximander’s way of refutation of Thales’ theory was a type of reductio ad absurdum which discloses an absurdity as an infinite regress. We should pay attention to a significant fact that the tradition of Ionian philosophy originally had its anti-empirical and anti-visual aspect as a notable feature. The most secure method in order to refute a hypothesis is to locate on the contradictory consequences from the hypothesis. Anaximander realized well this matter. The argumentation by reductio ad absurdum is never the monopoly of Eleatic philosophers. For example, we are justified in detecting a type of reductio ad absurdum in Xenophanes’ attack on the anthropomorphic conception of the gods. Especially, in fragments B14 or B15 we may point out a thought pattern as reductio ad absurdum. But, here I would like to reconstruct Xenophanes’ indirect proof from the doxography. Arist. Rhetorica., 1399b put it: 2 2 m 2 1 «otov Eevopdvns éleyev 6tt Ouoiws àoe6odoiv oie yevéoOar pdoxoviec 7 = 4 Ed eu è O ZHNON IIYOATOPIZEI 237 tovs Oeoùg toig dnobaveiv Aéyovoiv. aupotéows yao ovp6aiver un eivar Tovc Oeovs mote». (For example, Xenophanes used to say that «those who assert that the gods are born are as impious as those who say that they die; for in both cases it follows that the gods at some time fail to exist.) The argument can be reconstructed as a type of reductio ad absurdum follows: as . Thesis: To assert that the gods had birth is impious. 1) Suppose that the assertion «the gods had birth» is not impious. 2) But, whoever asserts that «the gods die» is impious, because of the proposition «the gods die» implies a time when the gods are annihilated. 3) In the same way, the assertion «the gods had birth» implies a time when the gods do not exist. 4)Whoever asserts that there is a time when the gods do not exist is impious. Therefore, to assert that the gods had birth is impious. Xenophanes’ argumentation thus reconstructed not only possesses a perfect bonestructure of the indirect proof, but also in its entirety corresponds to Parmenides’ negative argumentation of the generation and the extinction of Being in the fragment 8. 7. Euclid, Elements X. prop. 2: «If, when the less of two unequal magnitudes is continually subtracted in turn from the greater, that which is left never measures the one before it, the magnitudes will be incommensurable (doduuerga )». Concerning the fact that the theory of successive subtraction originated in the Pythagorean study of musical proportion see a fine description by A. Szab6, The Beginnings of Greek Mathematics, Akademia Kiad, Budapest 1978, especially pp. 199-203. 8. O. Becker in his «Die Lehre vom Geraden und Ungeraden im neunten Buch der eukl. Elemente», Quell. u. Studien zur Gesch. d. Math. Abt. B., Bd. 3, 1936, pp. 533-553 has decidedly verified that the theory of even and odd originated in Pythagorean wy popogia. 9. See W. K. C. Guthrie, A History of Greek Philosophy Vol. 1, pp. 239-251. See also Theo Gerard Sinnige, Matter and Infinity in the Presocratic Schools and Plato, pp. 73-76. 10. See B. L. Van der Waerden, Science Awakening Part I, Chapter 3. It is a firmly verified fact that the theorem of Pythagoras was known by Babylonians even more than 1000 years before Pythagoras’ acme. 11. We need not suppose that a proof of the lineal incommensurability of the side and the diagonal of a unit square was given later than Zeno’s or Parmenides’

time. R. E. Allen says as follows: «We know too little about the history of 239 O ZHNQN IIYOATOPIZEI 16. Op. cit., p. 218. mathematics in the fifth century to speak with confidence, but there is no compelling reason to assign a date late in that century to the discovery of incommensurability, and quite a good reason of supposing that it was known by about 465, and perhaps earlier. For if the Parmenides is to be dated around 450, and if Zeno at that date was nearing forty, and if Zeno wrote his book in his twenties, then 465 would seem to be an approximate terminus ad quem for the discovery of incommensurability, if Zeno’s work betrays familiarity with it.Now, Zeno’s Fragment 3, to take but one example, argues that things that are infinite, since there are always other things between things that are, and again others between them, and this implies a relatively precise concept of the infinite divisibility of magnitudes, and of the property of density, as we should call it, which attaches to such divisibility. But infite 17. Op. cit, p. 219. Szabó says: «The earliest application of indirect proof which occures in the I have encountered in my study of Greek language and culture nowaday that min didactic poem of Parmenides. On the other hand, it is agreed without exception earliest known mathematical applications of this technique are all gy suggests the IE of a later date. This simple observation about chronolo d sane nend in develope conjecture. Unless the method of indirect proof was d exclusively among mathematics and in Eleatic philosophy, it must have originate ics since it on the Eleatics. They could hardly have adopted it from mathemat Contra Szabö’s date.» late ively comparat a not appear to have been used there until theses, see my note (4). divisibility is an intelligible, not a perceptual, property, and it is difficult to see how Zeno could have known of it unless he had a mathematical proof of it; that HIDEYA YAMAKAWA proof must surely have been the proof of incommensurability, which implies of any PROFESSOR OF PHILOSOPHY ST. ANDREWS UNIVERSITY OSAKA magnitude that it has no least measure. It may well be, indeed, that a proof of incommensurability along the line of Euclid X. 117 may have given not only Zeno but Parmenides their confidence in reductio ad absurdum, their faith in reason, and their equally strong distrust of the senses and common opinion. If this were so, then Parmenides must have known the proof when he wrote ‘the Way of Truth, putting the terminus ad quem of the discovery perhaps as early as 485». R. E. Allen, Plato’s Parmenides, Translation and Analysis, Basil Blackwell, 1983, pp. 210-211. 12. I agree basically with A. H. Coxon’s assertion: «Zeno and Melissus are concerned to defend Parmenides’ non-physical monism against all such analyses and accordingly use the term ‘many’ to denote not simply any plurality but the plurality of physical substances». (The Fragment of Parmenides, A critical text with introduction, translation, the ancient testimona and a commentary, Van Gorcun, 1986, Appendix I Zeno’s argument about magnitude p. 255). 13. H. Rademacher and O. Töplitz, Von Zahlen und Figuren, Berlin 1930, p. 258 used Elements X. prop. 2 to show that the side and diagonal of a square are linearly incommensurable. My figure is different from theirs, but the idea is basically the same. 14. Utilizing Elements IX prop. 23 Th. G. Sinninge has vividly shown that a recurrent figure illustrates the absurdity of the supposition. It must be conceded that my reconstruction of Dichotomy owes its idea to his excellent suggestion. 15. A. Szab6, The Beginnings of Greek Mathematics, p. 217.