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Aristotle, Menaechmus, and Circular Proof

Published online by Cambridge University Press:  11 February 2009

Jonathan Barnes
Jonathan Barnes
Affiliation:
Oriel College, Oxford
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Extract

(a) The Regress: Knowledge, we like to suppose, is essentially a rational thing: if I claim to know something, I must be prepared to back up my claim by statingmy reasons for making it;and if my claim is to be upheld, my reasons must begood reasons. Now suppose I know that Q; and let my reasons be conjunctively contained in the proposition that R. Clearly, I must believe that R (for R cannot give my reasons unless it has my assent);equally clearly, I must know that R (for mere opinion is not nutritious enough to sustain the demanding body of knowledge). Thus if I know that Q, I know that R. But if I know that R, then I must have my reasons, R' for holding R; and, by the same argument, I mustknow that R'. And if R', then R”; and so on, ad infinitum

Type
Research Article
Information
The Classical Quarterly , Volume 26 , Issue 2 , December 1976 , pp. 278 - 292
Copyright
Copyright © The Classical Association 1976

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References

1 See Tht. 209 e - 210 b. (The regress has also been connected with the matter of Socrates' dream at 201 d - 202 d: see G.R. Morrow, ‘Plato and the Mathematicians: an Interpretation of Socrates' Dream in the Theaetetus’, Philosophical Review, 79 (1970), 309–33.) For modern discussion see, e.g., Deutscher, M., ‘Regresses, Reasons and Grounds’, Australasian Journal of Philosophy, 51 (1973), 1–16Google Scholar; Armstrong, D.M., Belief, Truth and Knowledge (Cambridge, 1973), pp.150–61CrossRefGoogle Scholar (of the six reactions to the regress which Armstrong lists, Aristotle considers and rejects (1) and (3), and accepts a version of (4)); Nathan, N.M.L., ‘Scepticism and the Regress of Justification’, Proceedings of the Aristotelian Society, 75 (1974/1975) 77–88.Google Scholar

2 In addition to Arist. APo. A 3 and the texts discussed below, see Metaph.A 2; Theophrastus, Met. 9b 21; cf. Sextus, adv. Math. 8. 347.

3 This, and not , is the right reading (cf. 72b 12; 16) - pace Ross, W.D., Aristotle's Prior and Posterior Analytics (Oxford, 1949), p.514 (hereafter APPA).Google Scholar

4 I discuss here only some aspects of Aristotle's first argument; for further notes on APo. A 3, see Barnes, J., Aristotle: Posterior Analytics (Oxford, 1975), pp.106–12 (hereafter APA).Google Scholar

5 The Academic Amphinomus took this line: see the discussion in Proclus, Comm. in Eucl. 202.9–25; cf. Heath, T.L., Euclid's Elements2; (Cambridge, 1926), i. 150.Google Scholar, n.1 (hereafter E). For a clear illustration of Aristotle's view see EE 1222b 31–7 discussed in Heath, T.L., Mathematics in Aristotle (Oxford, 1949), pp.279–80 (hereafter MA).Google Scholar

6 APo. A 1, 71a 17–21, argues that not all the premisses of a demonstration need be known before the inference is drawn. Aristotle's argument is faulty (see Barnes, APA, p.93); but even so he can st put forward the view that I construct for him - for he insists that at least some of the premisses must be known beforehand.

7 A little misplaced ingenuity will in fact revive the theory; and the sophisticated version of section (e) is not yet dead. But the remaining breaths are few and not worth chronicling.

8 Ross, APPA, pp. 513–14, compares Metaph. 1011a 3–13; 1006a 5–9; 1012a 20–1. In these passages, and hence in APo.A 3, he sees an allusion to Antisthenes (following Maier, H., Die Syllogistik des Aristoteles (Tübingen, 1896–1900), vol. ii b, p.15, n.2)Google Scholar. Similarly Cherniss, H.F., Aristotle's Criticism of Plato and the Academy (Baltimore, 1944), pp. 64–6Google Scholar (hereafter ACPA) - though Cherniss rightl adds that ‘anyone who was inclined to seer ticism would be likely to maintain’ the position Aristotle describes. The Metaphysics passages do not seem to me to deal wit the same issue as APo. A 3 (I note in passir that) Waitz, T., Aristotelis Organon (Leipzig1844–1846), ii. 310, cites two of them as parallels to the thesis of the circular reasoners); nor are Maier's reasons for attaching the Metaph. passages to Antisthenes at all strong.Google Scholar

9 Xenocrates at least once indulged in a logical circle (fr. 76 H); and he also tried to demonstrate definitions (APo.B 4, 91a 35 - b 11: see Barnes, APA, p. 200). Cherniss, ACPA, pp.67–8, suggested that ‘it was from the practice, if not from the express doctrine of Xenocrates, that the belief in universal demonstrability took its rise’. (Cf. Cherniss, H.F., ‘Plato as Mathematician’, Review of Metaphysics, 4 (1950/1951), 395–425, at p.417, n.50 (hereafter PM).) Ross, APPA, p.514, follows Cherniss; but the conjecture is evidently frail. I have not come across any other attempt to put a name to .Google Scholar

10 See Barnes, J., ‘Aristotle's Theory of Demonstration’, Phronesis, 14 (1969), 123–52, at pp. 127–32 - further references are given there; see also I. Mueller, ‘Greek Mathematics and Greek Logic’, in Ancient Logic and its Modern Interpretations, ed. J. Corcoran (Dordrecht, 1974).Google Scholar

11 See Robinson, R., ‘Analysis in Greek Geometry’, Mind 45, (1936), 464–73Google Scholar (repr. in his Essays in Greek Philosophy (Oxford, 1969) ), at p.465Google Scholar; Mahoney, M.S., ‘Another Look at Greek Geometrical Analysis’, Archive for History of Exact Sciences 5 (1968/1969), 318–48Google Scholar, at p.326; Hintikka, J. and Remes, U., The Method of Analysis (Dordrecht, 1974), pp.37–8.CrossRefGoogle Scholar

12 The interpretation of this passage is difficult; see Heath, MA, pp. 27–30; id., ‘On an Allusion in Aristotle to a Construction for Parallels’, Abhandlungen zur Geschichte der Mathematik 9 (1899), 155–60; Toth, I., ‘Das Parallelenproblem im Corpus Aristotelicum’, Archive for History of Exact Sciences 3 (1966/1967), 249–422, at pp. 256–74.Google Scholar

13 Pappus' text is printed in Thomas, I. (ed.), Greek Mathematical Works (London, 1939 - Loeb Classical Library); ii. 596–8Google Scholar; see also pseudo-Euclid, in Euclidis Opera, iv.364–6 H. On the modern controversy see esp.: Heath, E 137–42; F.M. Corn-ford, ‘Mathematics and Dialectic in the Republic’, Mind 41 (1932), 37–52 and 173–90; Robinson, op.cit., Gulley, N., ‘Greek Geometrical Analysis’, Phronesis, 3 (1958), 1–14; Mahoney, op. cit.; Hintikk and Remes, op.cit. (see my revīew in Mind 85 (1976)).Google Scholar

14 See Cornford, op. cit., pp. 43–50; contra Robinson, R., Plato's Earlier Dialectic 2 (Oxford, 1953), p.166.Google Scholar

15 See Proclus, Comm. in Eucl. 211. 18–22; Acad.Ind.Herc. 15–7 M; Diogenes Laertius, 3. 24 (= T 17–18b in Gaiser, K., Platons ungeschriebene Lehre (Stuttgart, 1963))Google Scholar. Mugler, C., Platon et la recherche mathiinatique de son e'poche (Strassburg/ Zurich, 1948), ch.5, argued that Plato invented the method of analysis; Chemiss's PM is a long review of Mugler, and it disputes his ch. 5 at pp. 414–9. But I stillcannot see good reason to doubt that Plato applauded the method of analysis and commended it to his Academic geometers.Google Scholar

16 Proclus, Comm. in Eucl. 67.8–12 = Menaechmus, fr. 4 Sch. (see Schmidt, M.C.P., ‘Die Fragmente des Mathematikers Menaechmus’, Philologus, 42 (1884), 72–81)Google Scholar. Here and hereafter I use Morrow's translation of Proclus: Morrow, G.R., Proclus - A Commentary on the First Book of Euclid's Elements (Princeton, 1970)Google Scholar. On Dinostratus see Allman, G.J., Greek Geometry from Thales to Euclid (Dublin, 1889), pp.180–93Google Scholar; Cantor, M., Vorlesungen über Geschichte der Mathematik 2 (Leipzig, 1894), i. 233–4Google Scholar; Loria, G., Le scienze esatte nell' antica Grecia 2 (Milan, 1914), pp. 160–4Google Scholar; Waerden, B.L. van der, Science Awakening (Groningen, 1954), pp.191–3.Google Scholar

17 The texts bearing on this are collected in Gaiser, op.cit., pp.460–74. Cherniss, PM, is still the best survey of the texts bearing on Plato's attitude to contemporary mathematics; see also Heath, T.L., A History of Greek Mathematics (Oxford, 1921), i. 284–315 (hereafter GM).Google Scholar

18 See pseudo-Eratosthenes in Eutocius, Comm. in Arch. 3. 88.5 - 90.29 H (see esHiller, p. E., Eratosthenis Carminum Reliquiae (Leipzig, 1872), pp.122–37). Other texts are referred to by Gaiser, op.cit., p.474.Google Scholar

19 See e.g. Heath, GM i. 244–70; Simon, M., Geschichte der Mathematik im Altertum (Berlin, 1909), pp.192–203Google Scholar; Wolfer, E.P., Eratosthenes von Kyrene als Mathematiker and Philosoph (Groningen, 1954), pp.4–12Google Scholar; R.Böker, ‘Würfelverdopplung’, RE IXA 1 (1961), 1193–1223. Texts in Thomas, op.cit. i. 256–308; a modern presentation in Eves, H., An Introduction to the History of Mathematics 3 (New York, 1969), pp.82–4.Google Scholar

20 See Pseudo-Eratosthenes in Eutocius, Comm. in Arch. 3, 88.17–23 H. The proof is simple: Let x, y, be mean proportionals between a, b; i.e. let a.: x = x:y = y:b. Then ay = x2, bx = y2, xy = ab. Hence x = y2 lb; and ay = y4 /b2. Thus ab2 = y3. Now let a = 2b. Then 2b3 = y3. Thus if b is the side of the given cube, then y is the side of its double.

21 See Eratosthenes in Proclus, Comm. in Eucl. 3.20–3 = Menaechmus, fr. 3 Sch; more fully in Eutocius, Comm. in Arch. 3. 96, 10–27 (see Hiller, op.cit., p.130; cf.Eudoxus, D 24 L). On Menaechmus' discovery see Cantor, op.cit. 231–3; Heath, GM ii. 110–6; Neugebauer, O., ‘The Astronomical Origin of the Theory of Conic Sections’, Proceedings of the American Philosophical Society 92 (1948), 136–8.Google Scholar

22 See Bretschneider, C.A., Die Geometrie und die Geometer vor Euclides(Leipzig, 1870), pp.155–63Google Scholar; Gow, J., History of Greek Mathematics (Cambridge, 1884), pp.185–8Google Scholar; Allman, op.cit. pp.153–79; Cantor, op.cit. 217–18; Heath, T.L., Apollonius of Perga - Treatise on Conic Sections (Cambridge, 1896), pp. xvii-xxxGoogle Scholar; (hereafter ATC); Zeuthen, H.G., Gescbicbte. der Mathematik im Altertum und Mittel-alter (Copenhagen, 1896), pp.191–9Google Scholar; Loria, op.cit., pp.149–57; Heath, GM i. 252–5; Simon, op.cit., pp. 208–10; Kliem, F., ‘Menaichmos’ RE xv. 1, 1931, 700–1Google Scholar; Mugler, op.cit., pp.324–8; Michel, P.H., De Pythagore á Euclide (Paris, 1950), pp.252–3Google Scholar; van der Waerden, op.cit., pp.162–5; Boker, op.cit., cols.1211–13; Lasserre, F., The Birth of Mathematics in the Age of Plato (London, 1964), pp.119–23.Google Scholar

23 See Gow, op.cit., p.186, n.3; Allman, op.cit., p.165; Loris, op.cit., p.153, n.2.

24 And he had an independent interest in philosophy, if he is identical with (Suda, s.v. ). For the identification see Martin, T.H., Theonis Smyrnaei Liber de Astronomia (Paris, 1849), pp.58–60; Allman, op.cit., p.153, n.1; Loria, op.cit., p. 149, n.2; Michel, op.cit., p.252. Against: Bretschneider, op.cit., p.162; Schmidt, op.cit., pp.77–8.Google Scholar

25 For Aristotle's view see Cael. 279b 32 - 280a 10 = Speusippus, fr. 54 a L. See, e.g., Niebel, E., Untersuchungen über die Bedeutung der geometrischen Konstruktion in der Antike, Kantstudien, Ergänzungshefte 76 (Köln, 1959), pp.89–103.Google Scholar

26 See esp. Zeuthen, op.cit., pp.88–91; cf. Becker, O., Grundlagen der Matbematie (Freiburg/Munich, 1964), pp.90–1.Google Scholar

27 Amphinomus is mentioned a fourth time by Proclus as having discussed different types of (Comm. in Eucl. 220.9).

28 On see esp. Burkert, W., ‘CTOIXEION’, Philologus 103 (1959), 167–97 (for Menaechmus see pp.191–6); for Aristotle's analysis of the term see Metaph. A. 3.Google Scholar

29 At 73.3 the text is corrupt, but the sense is not in doubt. (The simplest emend tion is perhaps excision of .)

30 Morrow's insertion.

31 See, e.g., Burkert, op.cit., p.191. Malcolm Brown is inclined to see the bulk of fr. 5 Sch. as a genuine quotation from Menaechmus. I am sceptical: some of the terminology (e.g. ) is late.

32 See Mugler, C., Dictionnaire historique de la terminologie géométriquen des Grecs (Paris, 1958), pp.245–6 (hereafter DG).Google Scholar

33 Cf. Speusippus, fr. 30 L; and see Lang, P., de Speusippi Academici Scriptis (Bonn, 1911), pp.28–9; Lasserre, op.cit., pp.31–2 (though I cannot see that the use of implies that mathematics is reduced to ‘an exercise in logic’).Google Scholar

34 APo. A 24, 85b 38; B 17, 99a 19; see Heath, MA, pp. 62–4. I guess that the theorem was proved by Menaechmus.

35 I accept O. Becker's palmary conjecture of for the manuscript reading . Friedlein prints by mistake for .

36 On Proclus' sources see Tannery, P., La Géométrie grecque (Paris, 1887), pp.71–5Google Scholar Heath, GM i. 35–8; E i. 29–45; Wehrli, F., Eudemos von Rhodos, Die Schule des Aristoteles 8 (Basel, 1955), pp.114–15.Google Scholar

36 The question is complicated. Plutarch, Quaest.conviv. 718 F (= Menaechmus, fr. 2 Sch. = Eudoxus, D 28 L = T 21a in Gaiser, op.cit.) says that Plato reproached Eudoxus, Archytas, and Menaechmus (cf. Plutarch, Vit.Marc. 14.5 = Eudoxus, D 27 1. T 21b in Gaiser, op.cit.). Ps.-Eratosthenes reports of …… (Eutocius, Comm. in Arch. 3. 90.4–11 H = Menaechmus, fr. 10 Sch. = Eudoxus, D 25 L). These passages suggest the following story: Eudoxus and his friends, having provided a geometrical solution to the Delian problem, attempted to apply their solution to the practical construction of a cube. Their attempt was unsuccessful - and it aroused the intellectual disapproval of Plato. (See esp. Heiberg, J.L., ‘Jahresbericht - griechische und romische Mathematik’, Philologus, 43 (1884), 467–522,Google Scholar at p.475; van der Waerden, op.cit., pp.163–5.) Very different accounts have, however, been canvassed (e.g. Schmidt, op.cit., pp.78–80; Allman, op.cit., pp.170–2; Heath, ATC, pp. xxix-xxx; Niebel, op.cit., pp.112–33); and the continuation of Plutarch's text is in any case corrupt. The commonplace that Menaechmus ‘engaged in mechanics’, which I repeat in the text, is thus neither very clear in meaning nor very well supported by the ancient evidence (see Tannery, op.cit., pp.79–80, who is sceptical of the whole story). On the history of mechanics from Plato to Archimedes see esp. Solmsen, F., Die Entwicklung der aristotelischen Logik und Rhetorik (Berlin, 1929), pp.130–5 (hereafter ELR).Google Scholar

38 See Schmidt, op.cit., p.78; Allman, op.cit., p.154; Loria, op.cit., p.149, n.3; Heath, GM i. 252; van der Waerden, op.cit. p.190. Contra: Bretschneider, op.cit., p.163.

39 On Alexander's tutors see the texts assembled in Düring, I., Aristotle in the Ancient Biographical Tradition (Goteborg, 1957), pp.284–99Google Scholar; cf. Jaeger, W., Aristotle 2 (Oxford, 1948), pp.120–3Google Scholar; Chroust, A.H., Aristotle (London, 1973), i. 125–32.Google Scholar

40 In the Menaechman sense an element is ‘like a lemma’: elsewhere (Comm. in Encl. 211.5) Proclus defines ‘lemma’ as ‘a proposition requiring proof’; but in fr. 5, the word has its more general sense of ‘assumption’, (see Mugler, DG, pp. 261–2).

41 AaB represents ‘A belongs to every B’ or ‘Every B is A’ (for this way of writing syllogistic propositions see Patzig, G., Aristotle's Theory of the Syllogism (Dordrecht, 1969), pp.49–50). x, y, z, are variables ranging over the syllogistic relations a, e, i, o. On syllogistic circles see further Barnes, APA, p.108.Google Scholar

42 References in Bonitz, Index Aristotelicus, 414a 6–14; see esJoachim, p. H.H., Aristotle on Coming-to-be and Passing-away (Oxford, 1922), pp.254–66Google Scholar; Solmsen, F., Aristotle's System of the Physical World (Ithaca, 1960), pp.420–39Google Scholar. On the idea of cycles of generation see also Mugler, C., Deux themes de la cosmologie grecque (Paris, 1953).Google Scholar

43 See further Barnes, APA, pp.228–9. I punctuate the lines differently from Ross.

44 If I am right, GC B 10–11 and APr. B 5–7 were written before APo. A 3; but tha is hardly a startling hypothesis. (Solmsen holds, in ELR, p.145, n.2, that APr. B 5–7 recants APo. A 3: that is incredible, particularly in the light of 73a 6–20; and Solmsen only suggests it because it follows from his general (and mistaken) theory that APo. was written before APr.)

45 The final draft of this paper benefited from the critical scrutiny of my colleague, Robert Delahunty; an earlier version was rea( at a seminar organized by Professor David Balme, where it was the object of several helpfully sceptical animadversions. My major debt is to Malcolm Brown, of Brooklyn College: finding by chance that we shared an affection for old Menaechmus and believed him to have been an important influence on Aristotle, we planned to compose a joint paper. As our thoughts progressed, they diverged; and the conclusion of this paper is, alas, only mine. But that conclusion would not have been reached without Malcolm Brown's original enthusiasm and continued encouragement; and I thank him warmly.