14. This isthe first use in the AZe/aphysics of the phrase adpucros dvas. Doubt has been felt as to whether the phrase refers to the doctrine of Plato, or to that of his followers. The description of the material prin- ciple as the ‘great and small’ is certainly ascribed to Plato (cf. A. 987» 20, 26, 988813, 26). In N. 1088b 28 Aristotle says eiot d€ tes ot dvdda pv dopictov rrovodat TO pera TOD Evds oTOLXElov, TO 8 dvicov SvoXepaivou- ow eddAdyws 8a TA orp Baivovra ddvvata. I.e., the adoption of the indefinite dyad as material principle seems to be described as an amendment of the description, which we can safely ascribe to Plato, of the material principle as the unequal. On the other hand the un- equal and the indefinite dyad are coupled as belonging to the same theory in N. 1088215. So, too, in 10836 23-32, N. rogob 32— 1091%5 the great and small and the indefinite dyad seem to be both referred to Plato. Cf. Theophr. AZe/. 33, Hermodorus ap. Simpl. Phys. 244..30—248. 18, and Al., Simpl., Syr., Asc. passim. The reference of the indefinite dyad to Plato was doubted or denied by ‘Susemihl (Genet, Entwickl. ii. 532 ff.), Zeller in Plat. Stud. (222), Trendelenburg (De Jd. et Num. 48-51), Heinze (Xenocr. 16-15). ‘On the strength of two passages (which are inadequate for the purpose) Heinze maintains that the phrase originated with Xenocrates (Theophr. Met. 11, Plut. De An. Procr. ii. 1, 2, 1012 DE). Zeller later gave up his doubts, and there is no reason to distrust the evidence of Hermo- dorus, Alexander, &c. In N. 1089®% 35, though the expressions ‘ great and small’ and ‘indefinite dyad’ are distinguished, there is nothing to show that they were not used by the same thinkers to designate the same thing. And 1088» 28 does not tell us that ‘the indefinite dyad’ was a later phrase than ‘the unequal’, but merely that some thinkers (Xenocrates is probably included) retained the former, while discarding the latter because it made one of the first principles something merely relative.
Zeller thinks that Plato described only the material principle of zdeal and mathematical numbers as the indefinite dyad. But the silence of MN{as§to the derivation of sensible things from it proves nothing, since these books are not concerned with the derivation of sensible
things. The PArlebus certainly describes the dzeipia as the material cause in all oteéa, without special reference to numbers (ravta ra viv évta év TH wavti 23.C 4). On the whole subject of the indefinite dyad cf. Robin 641-654.
15. at dpxat kat ta otorxeta = 7d ey Kal Svas 7% ddpiotos. The argument seems to be: ‘the only principles put forward by these thinkers are put forward as principles of number. If, then, they are also the principles of the Ideas, which they are clearly meant to be, the Ideas must be (1) identical with numbers, which we have shown they are not, or (2) prior or posterior to, causes or effects of, numbers, which they evidently cannot be, since they are composed of a different kind of units. Therefore the Ideas are left without any dpyai at all’. The order of the words is against Apelt’s proposal to interpret 1. 15 ‘and the principles (sc. of the Ideas) are said to be also the elements of number’,
17. Aristotle passes now to the view according to which even the units in one number are incomparable with one another. Considering that this view had found no supporter (1080 8, 1081 35), the space Aristotle devotes to it (10814 17— 33) is disproportionate.
21-29 is a difficult piece of argument. Alexander supposes I]. 21-23 to mean that if the units are incomparable each with each, the numerical series will be destroyed because the numbers will be all formed simultaneously (749. 19), and this view is adopted by Robin (note 285, iii). But this is contrary to Aristotle’s usage, according to which ‘incomparability’ implies the very opposite of simultaneity ; TO pev TpOTdv TL avrod Td 0 exouevov 1080®17 is synonymous with dovpBAnros ib. 19. Alexander, continuing to misunderstand the passage, thinks that Aristotle should have said (1. 23) 7 yap dua ai... povddes yevvvrat 7) ody dua. This means that Alexander is quite at sea. Bz, perceives the general nature of the argument as it stands in the received text, viz. that Aristotle offers two proofs to show that the supposition of units incomparable each with each is contrary to the Platonic view of ideal number as forming the series 1, 2, 3, 4, &c. (ll. 21-23), one proof being given in Il. 23-25, another in Il. 25-29 ; and that really what Aristotle professes to show (that ovx éorai 7 dvas tpatn) is proved only by the second proof. The argument can be made right by the alteration of one word—by reading ézeé for éreira in |. 25 (éxevra has probably come in through the influence of érera in |, 22). Then the argument runs thus: ‘For the two will on this hypothesis not proceed first from the one and the indefinite dyad, and then the other numbers, as the Platonists say “2, 3, 4””—for they generate the units in the first (i.e. the ideal) two simultaneously—since if the one unit in the two were prior to the other (as it must be, on the hypothesis of incomparability, cf. ro80 17, 19), it would be prior also to the two which is composed of them. Thus the order would be not, as they say, I, 2, 3, 4, but 1, first unit in 2, 2, second unit in 2, first unit in 3, &c. For the combination éret ei cf. 10874 21.
24. 6 mptos eimdy, sc. roy TOV ciddv dpiOuor etvai (cf. 1.21), A com-
Ef 2
436 Commentary
parison with 1086411, N. rogo>32, where Aristotle is referring to a doctrine which marks Plato off from Speusippus and Xenocrates (cf. 1076% 19-21 nn.), shows that here also Plato is meant.
ef dviowy (icac0évrwy yap éyévovto). The unequals are the great and the small (1083 23, N. rog1%24). Plato, according to Aristotle, represented the One as producing the units in 2 by equalizing the great and the small. But Aristotle speaks with some hesitation as to how this was done (1083 23-25). It is noteworthy that the material principle is never spoken of as ra dvuca but always as 76 avicov, and it seems probable that Plato did not think of two things, the great and the small, but of one thing, the great-and-small, i.e. indeterminate quantity, and that he represented this as simply being determined into the successive numbers by the operation of the One or formal principle. Cf. Introduction, Ixi f.
30-31. Tv dAd\wv . . . éxeivo, sc. the first unit in 2.
31-32. tpitov ... ev, sc. the second unit in 2.
32. The principal clause begins irregularly with dore, as often in Aristotle. Cf. 2nd, Ar. 873° 31-44.
33. The reading of AP Al., zAéxovrar, would require a strange per- version of order, the antecedent of dy being then ai povades. The reading of EJ}, Néyovrat, gives an excellent sense and must be adopted. ‘The units will be prior to the numbers after which they are called ; the third unit (i.e. the second unit in 2) will be prior to the number 3, and so on.’
byi-g. tds te yap... mpatov. ‘It is natural that there should be prior and posterior units, if there is also a first unit or first one ’—sc. the ideal one.
6-8 is a parenthetical recurrence to the point made in ® 21-29,
8-10. It is doubtful whether Aristotle’s attack is quite fair. The Platonists spoke of the Ideal One as the first One not in the sense that it was the first member of a series of units, but in the sense that it was the principle of the whole class of units. The word ‘first’ is ill- chosen, since it seems to make the universal a member of the class which it constitutes; but there is not necessarily any serious confusion in the thought.
12-14. ‘Whether the units are without specific difference or not, number must be counted by addition’, i.e. each number must be arrived at by adding 1 to the previous number. This is a successful enough appeal to common sense, but is something of a petrtzo principir. The Platonists simply denied the premise that the numbers were reached by addition, and gave quite a different account of their generation.
17-19. ‘The numbers cannot be generated as they try to generate them, out of the indefinite dyad and the One; for three is generated not from the indefinite dyad and the One but from the number two and a unit.’
21. aX’ does not, as Al. 753. 9 and Bz. suppose, introduce a possible objection to the previous argument. It points out the con- tradiction between the actual facts (Il. 18-20) and the Platonic account
M. 7. 1081430 — 10827 Ii 437
(ll. 21, 22). Jaeger’s addition of «i is ingenious, but not strictly necessary.
22-26. ‘If the two 2’s in 4 are not distinct from the 2 itself, 4 will be composed of the 2 itself and another 2, and similarly 2 of the One itself and another one; so that the element other than the One itself will not be (as they said) the indefinite dyad, since the second element generates one unit (the second unit in the 2), not (as the indefinite dyad does) a definite dyad.’
According to the Platonic account (as represented by Aristotle) the indefinite dyad ‘received the definite dyad and made two — (1082* 13).
27-33. Aristotle passes here from particular arguments to a general protest against the absurdity of the position created by supposing all units, even those in the same number, to be incomparable.
30. dromd. Alexander (754. 12) and Syrianus (131. 1) may have read advvata, or this may be their interpretation of déroma. For drora kat tAacparwdn and for the meaning of wAacparwdy cf. 1082) 2—4.
BI. dvdyxn 8 wth. I.e., if each number is derived not (as common sense says) from the previous number by the addition of 1, but (as the Platonists say) from the One and the indefinite dyad, each number is as it were a special creation, differing in kind from all others, and thus we get an ideal two, an ideal three, &c.
1082? 1. otov ydp, ‘for, for example’.
2-4. What does Aristotle mean by saying that the ‘ 10 itself’ is not any chance number nor composed of any chance 5’s or units? The meaning Seems to be that, the ro itself being an ideal number, the numbers contained in it must be numbers of a special kind, viz. ideal numbers, just as the units in it are supposed by these thinkers to be of the special type described in 1081 35-37. Now two Ideas cannot be specifically the same; therefore the two 5’s in 10 differ specifically ; and therefore the units in them differ specifically; thus five of the units in 10 differ specifically from the other five, which is contrary to the hypothesis we are examining. ‘There seems to be no allusion to the presence of numbers other than 5 in ro (Al. 755. 12), nor to the specific difference between ro and the 5’s in it (Bz.).
7-11. In Il. 8, 9 (after ay), ro it is not clear (as Bz. thinks) that Alexander read écovrar; it seems better to keep the reading of all the good manuscripts, évécovrar, which Alexander may be merely misinter- preting. év 7 dexdév in |. 11 rather confirms évécovra, and quite a good interpretation may be given to the word. If the units in the 10 differ specifically, will there be no other 5’s in the ro than the two already mentioned? (1) We can hardly suppose that there are not. Aristotle does not say why, but the reason obviously is that if you take, say, three units from the first 5 and two units from the second, you will get a new 5 different from the original two. (2) If there are. these other 5’s in the 10, what sort of 10 will they constitute? Apparently another ro in the 1o itself, but the Platonists do not suppose that there is any such thing.
438 Commentary
11-15. Aristotle now proceeds to confirm what he has already used as a premise (I. 3), viz. that the 10 itself is not composed of any chance 5’s, He infers the.mode of composition of the ro from the mode of composition of the 4. This, according to the Platonists, is not produced by successive additions of specifically like units, but by the action of the indefinite dyad, which received the definite dyad or
‘2 itself’ and made two dyads.