Aristotle's Classification of Number in "Metaphysics" M 6, 1080a15-37 TARÃ#N, LEONARDO Greek, Roman and Byzantine Studies; Jan 1, 1978; 19, 1; ProQuest pg. 83 Aristotle's Classification of Number in Metaphysics M 6, l080a15-37 Leonardo Taran AT the beginning of Metaphysics M 6, Aristotle decides to examine the views of those who think that numbers are separate substances and the causes of existing things. l The rest of the chapter falls into two parts: a theoretical account of the different kinds of numbers that can be conceived (I080a15-bll), followed by a historical survey of the views of Aristotle's predecessors concerning the nature of number (1080bll-33), which ends with the contention that all the views outlined are impossible (I080b33-36). The classification Aristotle puts forward in I080a15-37 betrays misunderstanding of the concept of number and also of Plato's ideal numbers or ideas of numbers. Aristotle refers to this doctrine as that of acvILf3I\TJTOt aptOJLot, that is, incomparable or, even better, inasso­ ciable numbers.2 These numbers, however, are not congeries of units, as Aristotle thinks they are, but merely the hypostatization of the universals which constitute the series of natural numbers.3 These points must be made at the outset in order to clarify that it is only because he considers number to be a congeries of abstract monads that Aristotle offers the following theoretical, a priori classification of number according to the nature of the units (1080a15-37): 1 This is the third question announced in the first chapter of M, cf 1076a29-32. The thinkers referred to are the Pythagoreans and the Platonisrs, cf W. D. Ross, Aristotle's Metaphysics II (repr. with corrections, Oxford 1953) 426-27. 8 As L. Robin says (La theorie platonicienne des idees et des nombres d' apres Aristote [Paris 1908, repro Hildesheim 1963] 272 n.1), in associable numbers (i.e. numbers which cannot be added, subtracted, multiplied or divided) is a more appropriate translation of acvp.fJA1JTOI apl8p.ot. Following the usage of most English scholars, however, I refer to them as 'in­ comparable numbers'. a Cf J. Cook Wilson, CR 18 (1904) 247-60; Ross, op.cit. (supra n.l) 427; H. F. Cherniss, Aristotle's Criticism of Plato and the Academy I (Baltimore 1944) 513-24 and The Riddle of the Early Academy (Berkeley 1945) 33-37. In connection with Aristotle's misunderstanding of Plato's concept of number as such, there is no need to distinguish between the ideal num­ bers of Plato's 'earlier theory' and the idea-numbers of the 'later theory' Aristotle ascribes to him. Cf Wilson, op.cit. esp. 249-51 and 253-55; Cherniss, Aristotle's Criticism 513-16. 83 84 ARISTOTLE'S CLASSIFICATION OF NUMBER 15 • I ~." .,.. ° ' .I.. I " -"'A'" avaYIe'T'J 0, H7T€P €C'TW 0 apt fLOC ",VCtC 'TtC leat fL'T'J all. ., " ,- r " _'_\ \ \ _, " ~ J.. I 'TtC €C'TtV av'TOV 'T'J ovCta a/\I\a 'TOV'T av'TO, WC7T€P ",aCL 'TLV€C, " 1 '\ .... I ,.... \~'" fI 'T'J'Tot € vat 'TO fL€V 7TPW'TOV 'Tt av'TOV 'TO 0 €XOfL€VOV, €'T€POV OV 'TcfJ €iS€t EleaC'TOV-lea, 'TOiho ~ '7T' 'TWV fLovaSwv €v8vc ~ I \" " QA f .... \ f ..... V7TapXH leaL €C'TW acvfLt' 'T'J'TOC 07TOLaovv fLovac 07TOtll-0VV 20 I~" '0' '.I.. l:.~ ~ , t:lA ,. ~ fLovaot, 'T'J €V vc €",€s'T'JC 7TacaL leat CVfLt' 'T'J'TaL o7TotaLovv • ~"A' ,. \ 8 \. 8 I O7TOLatCOVV, OLOV €yovcw HVat 'TOV fLa 'T'JfLa'TtIeOV apt fLoV ('v yap 'TcfJ fLa8'T'JfLa'T£lecfJ OUSEV S£a~€pE£ ovSEfLla fLoVaC €'T€pa • I ) .. , \ t:lA ' , ~, I ( " ,,, €'T€pac· 'T'J 'Tac fL€V CVfLt' 'T'J'Tac 'Tac O€ fL'T'J OLOV H €C'Tt fL€'TfX 'T6 ~V 7TPW'T'T'J Tj Svcfc, €7T€t'Ta Tj 'TpLaC lea, OV'TW STJ <> 25 ", , '0 I ,\ ~ , t:lA \ • , • I '0 ~ al\/\oc apL fLOC, HCt O€ CVfLt' 'T'J'Tat aL €V €leac'TctJ apt fLctJ fLovaS€C, orov al 'v 'Tfj SvaSt 'Tfj 7TPW77I aV'Taic, leal. al 'v Tfj 'TptaS£ 'Tfj 7TPW77I aV'Taic, leal. OVTW STJ '7T1. TWV aAAwv apLOfLwv' ai S' 'v Tfj SvcfS£ aVTfj 7Tp6C Tac 'v Tfj 'TptcfSt • ~ , I t:lA • I ~ , \" ~ ", , ~ aV77I acvfLt' "I'TOt, OfLotwC O€ leaL €7Tt 'TWV al\/\wv 'TWV 30 '~€~fjc aptOfLwv' SL6 lea, <> fLEV fLaO'T'JfLa'TtIe6C aptOfL€'iTat fL€Ta 'T6 ~v Suo, 7Tp6C 'TcfJ €fL7TPOCO€V lv, a'\'\o EV, leal. Ta Tpla 7Tp6C Toic SVct TOUTO£C aAAO EV, lea, <> AOt7T6C SE Wcau'TWC OO'TOC SE fL€'Ta T6 ~V Suo ET€pa av€v TOU €VOC TOU 7TPW'TOV, lea, Tj 'Tptac av€v 'Tfjc SvaSoc, <>fLOLWC SE leal. <> 35 a'\'\oc aptOfL6c)' ~ 'T6V fLEV €lvat 'TWV aptOfLWv oloc 0 7TPW­ TOC 'A€X8'T'J, T6V S' olov ol fLCx8'T'JfLaT£1e01. A€YOVCt, TplTOV SE TOV p"I0€VTa TEAEvTa'iov. This is Ross's text. His explanation of lines 17-23 must be accepted, for it is clear from the context that, though syntactically ~ Tae p.~v leTA. in line 23 is coordinate with ijTOt Elva£ leTA. in line 17, in sense it is coor­ dinate with ~ '1T' TWV p.ova8wv leTA. in line 18 and with 7j ev8ve JtPegfje le'TA. in line 20.4 But Ross goes astray in the classification of number he infers from this whole passage. , The syntax could be normalized by emending TaC p.~v cvp.fJ>.r/,rac TaC 8~ p.~ (line 23) to al I-'~V cVl-'fJ>'TJTal al 8~ p.~; but, in view of the absence of any variant, it seems preferable to keep the reading of the MSS. as lectio difficilior. Be that as it may, there can be no question that the hypothesis of line 23 is in sense coordinate with those of lines 18 and 20, since it is clear that the kind of number described in lines 23-30 and 33-35 is such that TO P.EV 7TpWTOV n aVTOV TO 8' €xoP.£vov, ET£POV ov TqJ £i8n EKacTOV. This consideration suffices to refute the interpretation of A. Schwegler (Die Metaphysik des Aristoteles IV [Ttibingen 1848] 311-12), who takes the hypothesis of line 23 as coordinate with that of line 17. (Schwegler's inter­ pretation of 1080a35-37, which he adopts from the ps.-Alexander, simply cannot be got out from the text. Cf. Ross, op.cit. (supra n.l) 426-27.) For similar reasons, I cannot accept the suggestion of J. Annas (Aristotle's Metaphysics, Books M and N [Oxford 1976] 163-64) to excise Ti in line 18. For, if we do excise it, the Ti of line 23 would introduce a different possibility from that introduced by TiTOL in line 17. This would be awkward, however, since the LEONARDO TARAN 85 According to 1080a35-37 (but taking into account also what is said in 1080a17-35) there can be three kinds of numbers: (a) incomparable numbers with units all incomparable, (b) mathematical number with units all comparable, and (c) incomparable numbers with the units of each number comparable with each other but incomparable with those of other numbers. The problem is that after introducing incomparable numbers in lOS0a17-1S Aristotle goes into the nature of the units themselves, so that in 1080a15-35 he appears to be offering the following classification: (a) incomparable numbers (i) with the units all incomparable, (ii) with the units all comparable, and (iii) with the units of each number comparable with each other but incomparable with those of other numbers. But Ross is mistaken, I think, in inferring from 1080a15-37 the following classification: on the one hand, the belief in either Ca, i) or Ca, ii) or (a, iii), and, on the other hand, the belief in all three kinds of numbers. This interpreta­ tion causes him to contend that Aristotle has omitted the belief in three different combinations of numbers and that he has confused incomparable numbers the units of which are all comparable (a, ii) with mathematical number (b) the units of which must necessarily be all comparable.5 It would be more than remarkable, however, if Aristotle were guilty of the confusion Ross ascribes to him, since in the very next chapter of Metaphysics M he states that if all the units are comparable and undifferentiated there is only mathematical number,6 whereas if all the units are incomparable this number cannot be mathematical number.7 And it is implicit even in lOS0a15-35 that numbers such as (a, ii) cannot be incomparable, since all the monads are said to be CVJLfJA'Tj'Tat. We must notice, moreover, that in 1080a21 Aristotle says number described in lines 23-30 and 33-35 would then have to be different from that described in lines 17-20 (and not merely from that of lines IS-20, as it is if we keep the text of the MSS.); so that, apart from saying that the two numbers differ in the nature of their respective units, the emended text would seem to imply that the number of lines 23-30 and 33-35 is not 'TO I-'-£V TTpW'TOV 'TI alhov 'TO 8' 'X0l-'-€VOV, €TEPOV ov 70 €i8H €KaC'TOV (lines 17-1S), which it is. Surely Aristotle would have repeated this phrase in lines 23ffhad he not written rhe if of line 18. The mere facr rhar from line 18 onward Aristotle discusses rhe monads is an indication that the three kinds of numbers described in lines lS-35 are divisions of the class established in lines 17-1S. (It should be added that Annas adopts Ross's interpretation of lOSOa35-37, against which cf my remarks in the text infra.) 6 Cf Ross, op.cit. (supra n.1) 426. His note on 10soblO-11 is also wrong. S Cf Metaph. 10Sla5-7. 7 Cf. Meuzph. 1081a17-21. 86 ARISTOTLE'S CLASSIFICA nON OF NUMBER olov '\Eyovnv €lvcn TOV /La8"1/LaTtKOV apt8/L6v (i.e. the very number which is not and cannot be incomparable) and that (a, ii) could not be such h '\ ..... I ,..... \ ~~ , I rI ~..... J1~ ~ t at TO /LEV TTPWTOV Tt aVTov TO ° EXO/L€VOV, €T€POV OV TCfJ €w€t EKaCTov (lOS0a17-1S).8 That is to say that (a, ii) cannot have a serial order of numerical elements-the very essence of incomparable numbers. In lOS0a3Q-33 Aristotle himself tacitly denies this property to mathe­ matical number when he says that one such number includes another. This is itself a consequence of the statement that in mathematical number all the units are comparable and undifferentiated,9 and we are therefore entitled to infer that the same thing would be true of numbers such as (a, ii). Now it is improbable that Aristotle, having mentioned in 10SOalS- 20 the incomparable monads, went then into a digression concerning the nature of the monads as such, in which the question of the different kinds of numbers was lost sight of; for, though such an interpretation would make sense in itself, Aristotle could hardly have disregarded the fact that lOS0a2Q-21 is still affected by Ka~ ToiiTo KT'\. in 10S0alS.10 One must then agree with Ross's view that in lOS0a20-21 Aristotle does mention incomparable numbers with the units all comparable, but his contention that Aristotle has confused this number with mathematical number must be rejected. The words olov ,\EYOVCW €lvat TOV /La8"1/LaTtKOV apt8/L6v do not support Ross's interpretation, since this sentence in all probability refers merely to the units' being all comparable, not to incomparable numbers as such. Why then does Aristotle mention incomparable numbers with the units all comparable, a notion which is self-contradictory as he him­ self implies? If the text is basically sound, as it seems to be, I submit that he does so for the following reasons. In view of the purely 8 It is remarkable that G. Reale (Aristotele. La Metafisica. traduzione. introduzione e commento II [Napoli 1968] 369 n.7). having seen this last point. nevertheless accepts Ross's interpretation. , Cf Metaph. 1080a22-23. 10 This is fatal to the interpretation of Robin. op.cit. (supra n.2) 272-73 n.258. who tacitly denies that in 1080a20-21 Aristotle refers to incomparable numbers with the units all comparable. He reads into 1080a17-23ff the following classification: (1°) Incomparable numbers (1080aI7-18). (A) with units all incomparable (1080aI8-20); (2°) mathematical number with the units all comparable (1080a20-23); (1 ° B) incomparable numbers with the units of each number comparable with each other but incomparable with those of other nUlTIbers (1080a23ff). (Nor is it possible, I think. to consider lines 20-23 as purely paren­ thetical.) LEONARDO TARAN 87 theoretical nature of his classification in 1080a15-37 and of his refuta­ tion of the separate existence of numbers in 1080b37ff, Aristotle felt he had to mention all the possible views of numbers as separate sub­ stances and as causes of existing things.u So far as we know, no Platonist did believe in incomparable numbers with the units all comparable; but neither did anyone-according to Aristotle himself (lOS0bS-9)-ever posit incomparable numbers with the units all incomparable. Aristotle's classification in lOSOa15-37 is merely for the purpose of a dialectical attack against the diverse Platonistic doctrines of number; it enables him to argue that if numbers actually exist apart from the sensibles, they must belong to one or another of the three categories of incomparable numbers he has set up, all of which he believes to be impossible. Thus in losob37ff he tries to prove that separately existing numbers would have to be constituted by incomparable monads such as (a, i) or (a, iii) and that neither can be the case. If the units are all comparable, however, such a number can only be mathematical number, and mathematical number cannot be incomparable;12 therefore, it cannot have separate existence eitherP Now there is some evidence that in rejecting (a, ii) Aristotle wished to indicate what to him was the absurd implication of Speusippus' doctrine and perhaps also to forestall a modified version of Xenoc­ rates' idea-numbers. Speusippus posited the separate existence of mathematical number, and Aristotle-who also identifies number with mathematical number-attacks him because of his attempt to 'separate' such a number. Thus in 1083aZo-3514 Aristotle argues that if only mathematical number exists, it cannot exist apart from the sensibles; for, if it did, not only would there have to be a first 'one' (as Speusippus is said to have believed), but there would also have to be a first 'two' and a first 'three', etc. (as, according to Aristotle, Speusippus did not believe); in this case, however, Plato's view of number would be the correct one, since these numbers would have 11 cf Metaph. 1080b4-11. 12 cf Metaph. lOSla5-1Z, especially a5-7 (El p.~v ovv 7Taca, cvp.f3A'T}Tal Kal &8uxtjJopo, at P.OVa,8EC, 0 p.a8'T}p.aTLKOC ylyvETa, apL8p.oc Kal ErC p.ovoc), which is an inference from 1080aZZ-Z3 and 30-33. 18 Cf Metaph. lOS3aZ0-35 and my comments on this passage in the text infra. 14 Cf also Metaph. lOS1a5-12. That l083aZ0-35 refers to Speusippus is shown by com­ parison of l080bl4-16 with lozSbzl-Z4, l075b37-1076a3, and 1090b13-zo. 88 ARISTOTLE'S CLASSIFICATION OF NUMBER to be incomparable numbers. In short, Speusippus' view that num­ bers with the units all comparable have separate existence would amount to the absurd notion of incomparable numbers with the units all comparable; for according to Aristotle, unless such numbers are incomparable, they cannot have separate existence. Similarly, the absurdity of a number such as (a, ii) may have its use against an attempt to defend a modified version of Xenocrates' view of number. By identifying the ideas with mathematical numbers Xenocrates was most probably trying to offer a compromise between Plato's ideas and Speusippus' mathematical numbers. In fact, how­ ever, Xenocrates believed in numbers of the class (a, iii);l0 hence Aristotle's contention that this view destroys mathematical number.16 Since Xenocrates nevertheless called his ideal numbers mathematical, Aristotle's mention of the possibility of incomparable numbers (something that the Xenocratean idea-numbers would have to be) with the units all comparable forestalls any attempt to defend Xenocrates on the ground that his idea-numbers are really mathe­ matical numbers with the units all comparable. Thus in lOSla5-7 Aristotle maintains that if all the units are com­ parable and undifferentiated we get only one kind of number­ mathematical-and the ideas cannot be the numbers;17 conversely, in lOS3a17-19 he insists that if the ideas are numbers the units cannot all be comparable. We must still determine the meaning of lOSOa35-37 and its relation to lOS0a17-35. The view described in lOSOa35-37 cannot be "one which believes in the existence of three complete number series of different kinds," as ROSS18 and others believe it is. For it is ostensible from what follows in the rest of chapter 6 of M that, though theo­ retically (a), (b) and (c) are three possible views of number according 16 On Xenocrates' identification of the ideas with mathematical numbers cf Metaph. 10sob22-23 and 28-30 (with Ross's notes on lOSob22-29). 102Sb24-27 (with Ross's notes on 102Sb24 and 26-27), 1069a35 (with Ross's note on 1069a34-36). 1076a20-21 (with Ross's note ad loc.); Ross. op.cit. (supra n.1) Ilxxiv-Ixxv. On Xenocrates' belief in incomparable numbers of the class (a, iii) if. Metaph. 10sob22-23 and 2S-30, where n.b. ovO' 01To,acovv I'0vaoac ovaoa Elva,. 18 Cf Metaph. 10S3bl-8 and IOS6a5-11. where n.b. TOV athov Elo"1TlKOV Kall'a8"11J.aTlKOV ~1Tol7JCav &'p,8I'ov TijJ ;\Oycp, ~1TEi:pycp yE &'v6P"1Ta, 0 l'aO"1l'aTlKOc, KT.l 17 Aristotle's inference (lOSlal2if) that if the ideas are not numbers they cannot exist at all, apart from being unjustified in itself, is vitiated by his misconception about the true nature of number. Cf the references in n.3 supra and the corresponding remarks in the text. 18 Cf Ross, op.cit. (supra n.1) II 427, and similarly p.426. LEONARDO T ARAN 89 to the nature of the respective units, in fact no one has ever held (a),19 some have held (b),20 someone (C),21 others (b) and (C),22 and still others have identified (b) and (C).23 And so it would have been point­ less and inconsistent to have said in lines 35-37 what Ross thinks Aristotle did say, since, apart from having omitted three different combinations of numbers, Aristotle later states that no one ever posited (a) and never mentions anyone who posited (a), (b) and (c). In short, Ross's interpretation destroys the rationale of Aristotle's classification both in itself and in the light of what follows in the rest of Metaphysics M .. There is an alternative interpretation of these lines, however; and that is to take 7"61' fLJv . . " 7"6V 8' .. " 7"pt7"OV 8' as introducing three different conceptions of number. But these numbers have already been mentioned in 1080aI7-35, so that the if in 1080a35 can hardly introduce the second part of the classification which begins in line 17. This if must be corrective; it introduces a summary but more correct account than that given in 1080a17-35.24 There is a break in the sentence which begins in line 17, and the anacoluthon leaves the if7"Ot there without its complement. In other words, in 1080a35-37 Aristotle comes back to his original purpose of stating how many kinds of numbers can be conceived by those who believe that numbers are separate substances and the causes of existing things. He begins once more with the kind of number described in 1080a17-20, and it is noteworthy that lines 35-37 are still dependent on aV&YK'Y] 8' K7"A. in 1080a15-16. But now that the three kinds of monads have been described, there remains to distinguish three possible views of num­ ber according to the nature of the component units: (a) incomparable 18 cf Metaph. lOSObs-9 and lOS 1 a35-36. 20 These are Speusippus and the Pythagoreans. Cf 10S0bl4-21 with Ross's note on 10sob14 and n.14 supra. 21 This is the anonymous Platonist of Metaph. 10S0b21-22 (cf. Ross's note on line 21), and n.b. that according to Aristotle no one believed in incomparable numbers with the units all incomparable, cf n.19 supra. 22 This is the view Aristotle ascribes to Plato. Cf Metaph. 10sobll-14 with 9S7bl4-1S. 28 The view implicitly ascribed to Xenocrates, cf n.15 supra. 24 Robin, op.dt. (supra n.2), also interpreted lines 35-37 as a resume of lines 17-35; but, because of his interpretation of lines 20-23 (cf n.10 supra), failed to see that the if of line 35 introduces a corrective summary of the previous classification. On corrective if at the beginning of clauses (Kuhner-Gerth, Griechische Grammatik II p.297 #3) even when no question precedes, cf Arist. Top. 159all, Eth.Nic. lloob7; H. Bonitz, Index Aristotelicus 313a17-26, esp. 25-26. 90 ARISTOTLE'S CLASSIFICATION OF NUMBER numbers with units all incomparable, (b) mathematical number with units all comparable, (c) incomparable numbers with the units of each number comparable with each other but incomparable with those of other numbers. Given that l080aZo-Z3 and l080a30-33 con­ tain an implicit refutation of the possibility of numbers such as (a, ii),25 it suffices for Aristotle here to mention as the second kind of number mathematical number, according to him the only kind of number that is possible if all the units are comparable and undifferentiated. After chapter 6 Aristotle rejects (a) and (c) and in the case of (b) tries to show that mathematical number, precisely because its units are comparable and undifferentiated, cannot exist apart from the sensibles. To summarize the results of the preceding discussion: Aristotle begins in 1080a17 as if his classification of numbers as separate sub­ stances and as causes of existing things were going to be: on the one hand either (a, i) or (a, ii) or (a, iii), and, on the other hand, (b). But, because in maintaining (a, ii) he mentions mathematical number as an example of number with the units all comparable and undifferen­ tiated and because after mentioning (a, iii) he goes into a rather lengthy digression to explain the difference between mathematical number and incomparable numbers such as (a, iii), he probably felt that it would be anticlimactic to mention (b) as the second and final part of his classification. Hence the break in the construction which begins with 7JTOL KTA. in 1080a17 and the corrective and summary classifica­ tion of 1080a35-37. The most serious difficulty in 1080a15-37 is one of contorted syntax, not of conceptual confusion as Ross thinks. COLUMBIA UNIVERSITY January, 1978 U Cf, with the corresponding remarks in the text. n.12 supra.