Even if ‘unit’ means not a quantity but what is indivisible in quantity, it refers to quantity and thus confirms Aristotle’s point.
10QO* 1-2. woddds . .. evavtudcers. Alexander (812. 19) mentions two. (1) If substance is quantity, then since quantity is an accident, substance will be an accident. (2) Substance as substance will be a substratum ; as quantity it will be zz a substratum.
CRITICISM OF THE THEORY OF NUMBERS (ch. 2. 10908 2—6. 1093? 29).
(A) Zhe theory that mathematical numbers exist separately (ch. 2. 1090 2—3. 10gI® 12),
1090* 2, (1) How are we to be convinced that the numbers exist? For the believers in Zdeas they act as causes of existing things, since each number is an Idea.
7. But why should we believe one who sees the difficulties about the Ideas but posits mathematical number; what causal value has this? It is not asserted to be the cause of anything, but a self-subsistent entity, nor does it turn out to be the cause of anything ; the theorems of arithmetic are true of sensible things, and do not imply self- subsistent mathematical number.
16. (z) Those who hold that there are Ideas and that these are numbers (Plato) fail to show why there must be Ideas, and therefore why self-subsistent number must exist.
20. (6) The Pythagoreans, because they saw many attributes of numbers belonging to sensible bodies, thought things must be numbers —not separately existing numbers but numbers of which things were made.
25. (c) Those who believe in mathematical number only (Speusip- pus) cannot say this; they only said that the objects of the sciences could not be sensible things. We maintain that they are. If mathe- matical objects existed apart, their attributes would not be found in bodies.
30. The Pythagoreans are free from objection on this score, but in constructing bodies out of numbers seem to be speaking of other bodies than those we perceive
35. while those who treat number as self-subsistent are open to the objection made above (1. 29).
b5. (d) Some treat the limits, point, line, and plane, as separate entities. But (i) at this rate the limit of a walk should be a substance, and (ii) at all events the limits do not exist apart from the things of which they are limits.
1g. (2) One might point out that the prior genera contribute nothing to the later; (a) if number did not exist, spatial magnitudes could still exist for those who believe in mathematical objects only (Speusippus) ; and if these did not exist, the soul and sensible things could still exist. But nature is not episodic like a bad tragedy.
20. (4) The believers in. Ideas (Xenocrates) escape this objection, for they construct magnitudes out of matter and number, lines out of the number two, planes out of three, solids out of four, But (i) are these magnitudes Ideas, or what are they? They contribute nothing to sensible things, any more than the mathematical objects referred to above (I. 15).
27. (ii) No mathematical proposition is true of them, unless one starts a new set of assumptions.
32. (c) The first thinkers who believed in both ideal and mathe- matical number (Plato) cannot tell us how the latter exists. They make it intermediate between ideal and sensible number. If it is derived from the great and small, it is the same as ideal number; if not, the elements are getting rather numerous. If the formal principle in both kinds of number is a One, how does the One take these two forms, if at the same time number cannot on Platonic principles be derived from anything but the One and the indefinite dyad?
091° 5. The theory is evidently like Simonides’ ‘long story’, which slaves. spin when they have nothing sound to say. The great and small seem to complain of their ill-treatment; for they cannot generate any number but two and its powers.
1090* 2—109QI® 12 is a discussion of the doctrine of separately exist- ing numbers, covering much the same ground as that covered in M. 2, 3, and it seems impossible to detect the distinction Bz. draws between the two passages: ‘lIllic ipsa rei natura disputandi legem et ordinem praescribit, hic vero eorum philosophorum, qui res mathematicas per se esse statuerunt, sententias singulas respicit et refutat.’ M and N cannot have been meant to form parts of a single treatise; they are independent essays.
4. etotv. Alexander 812. 30 interprets this as eiot xwpiorod, and this is confirmed by Il. 1-13.
480 Commentary
TH pev yap iBdas riepdvy. This applies to Plato (cf. M. 1076" 19n.). 16 € «7h. (1. 7) applies to Speusippus (cf, M. 10764 20-21 n.), Aristotle returns to these two views respectively in ll. 16-20 and in ll. 25-30, and to both alike in 35-» 5, Thirdly, he discusses the views of the Pythagoreans in ll, 20-25, 30-35,
11. o60evdg depends on atriov, which can be supplied from atris |. 13.
15. ka0darep ééxOn, M. 3, especially 1077? 17-22.
16-20 refer to Plato, 20-25 to the Pythagoreans, 25-g0 to Speusippus, 30-35 to the Pythagoreans, 35- § to Plato and Speusip- pus alike, The views of Plato, who believed in both mathematical and ideal number, apart from sensible things, of Speusippus, who believed in mathematical number apart from sensible things, and of the Pythagoreans, who believed in mathematical number existing in sensible things, are to some extent played off against one another. E.g. in» 2 in attacking the view of Plato and Speusippus Aristotle says the Pythagoreans (6 évayriovjevos Adyos) can make out as good a case for the opposite view.
16-19. The best interpretation of this very difficult sentence seems to be got by reading, as Bessarion perhaps did, 74 before xard in|, 17, omitting rd in 1, 18, and reading ¢éorw inl. 19. ‘As for those who assert that the Ideas exist, and that they are numbers, by their assumption —in virtue of the method of setting out each term apart from its in- stances—of the unity of each general term they try at least to give some account of why they believe number to exist.’ I take the subject of Zarw to be number, which is the subject of the whole discussion (cf. ll. gf, 10, 13, 20). It is impossible to say what Alexander read, except that he does not seem to have had 76, Other attempts to deal with the sentence are (1) that of Winckelmann, who keeps the manuscript reading and translates ‘those who posit the Ideas ... try at least to say how and why it is possible, according to the doctrine which separates each kind of thing from its many particulars, to assume each to be a unity’. ‘The objections to this are that (a) 76 is unexplained, and (4) the order in which the words are taken is intolerably unnatural, (2) Bz. suggests card 10 exOcow. » « Aap Bdvew. This leaves the difficult rd, and it neglects a passage which in some respects illustrates the present passage, Z 1031” 21 Kora, THY Exdeow avayKn év tu elvan dppo. (3) Maier proposes TO Kare, tiv exGeow . .. KapBdvew & 1 exaorov. But if, as he says, the subject of éorw (sic) is & 1 exaorov, then ro... AapBdvew is left without a construction, while ifr)... AapBdvew & 1 Exacrov is the subject, the order is highly unnatural. (4) Bullinger proposes AapPdvovres & for AapPdvew 7d év, which gives much the same sense as the reading we have adopted but is somewhat less probable as an emendation, (5) Prof. Joachim proposes 7 xara. rhv éxOeow exaorov mapa ro. TOG AapBdvew, &v te Exacrov weipOvrat KTA.
The reading adopted in |. 18 as being the better attested, mws for mas kai, does not affect the main difficulties of the passage.
17. For the meaning of é«Oeows cf. A. 992100, Z. 1031” 24,
M. 1086" 10, Ps.-Alexander here describes the process very much as Alexander describes it in A. 992” 10. The procedure according to him is as follows: You adduce particular ale@yrd, e.g. Plato, Socrates, Alcibiades, Dion. You then argue ‘ Man is either the same as Socrates or different. If it were the same, it would not be also present in Plato, Therefore it is different from Socrates, and similarly from Plato and from all the other individuals. There is, therefore, one thing apart from the many men, and this is man himself. Similarly with horses, oxen, &c.’
This account of the Platonic éx@ecrs is probably correct.
25-26, rots Se... dprOudv, i.e. Speusippus, cl. M. r076® 20-21 n, These thinkers cannot justify their belief in the substantial existence of numbers by saying that sensible things are composed of them; their own language precludes this (cf. 1. rr), They therefore only said that the objects of the sciences could not be sensible things (adray |. 27 refers to rd aloOyrd ocadara |. 22, dppovia, obpavds, and woAAd dda Il. 24, 25), and must therefore be immaterial but substantial numbers,
28, etiropev, M, 3.
85-37. Ste... Wuxyy must be taken to give the reason for drodap- Bavover, not for yopirrdv rowwdvres ; otherwise elvar . .. yeopirrd elvac is otiose,
37. catver thy woxyy, ‘fawn on, flatter, the soul’, Cf, dadpa yodv dm dpparov | cave we Soph. O. C. 310.
xwptord, The subject of efvar should be rdv dpBudy (ch lL 35 of 8 xopiordy wowdvres). But Aristotle has, not unnaturally, passed in thought to a vague subject such as radra; it is not necessary to adopt Bz.'s conjecture ywopirrdy.
be rot... eat ‘both ...and’, 6 evavriovuevos Adyos Means the Pythagorean argument (*® 20-25).
&pre HropyOy, ® 29.
5. eiot S¢ twes. The persons meant are probably Pythagoreans. They seem to be distinguished from Plato (Z 1o28" rp, 19), and from the Platonists (B. 1002* 8, rr).
Ir, od phy AdAAd answers irregularly to odre 1, 8,
12, lot, se. rd Eoyara ovoriat.
15-16. rd pydev .. . Uerepov. Speusippus’ doctrine is similarly characterized in Z, 1028 ar (where he is mentioned by name) and in A. 1075" 37. rots Td padyparixd pdvov evar daudvors (1. 7) also shows that Speusippus is meant; cf, M. t076% a0-ar n,
1g. eweiodiidys, used in the same connexion in A, 1076" rt.
20-32. A comparison with De An, 404» 16-24 has led some interpreters to suppose that Plato is referred to, but the phrase xuwety rd pabyparixiKat roredy Sias reds SdEas (1, 28) at once suggests Xenocrates (cf. M. 1080» 28, 1086" ro), and this is confirmed by Alexander's inter- pretation of mpooyAtydpevoe fl gt). From the thinkers here referred to Aristotle distinguishes in 1, 32 of mpdros, i.e, Plato (of mporot is replaced by éxeivov in rogt® §).
QI, Todro pev expedyet, ‘this objection escapes’, i. e, fails to hit them,
2673-2 Tt
482 Commentary
There seems to be no quite similar use of éxpevyew in Aristotle, but cf. €ouxey ovtwot ¥s TKOTOU[LEVOLS diahevyety 1093 9.
26-27. od0ev ydp ...cupBdddeTar, 1. e. Recnocaiee assumption of paOnparixa which were Ideas does as little to. explain the sensible world as Speusippus’ assumption of pabywarixa which were not Ideas.
It appears from this passage that as Xenocrates identified ideal with mathematical numbers, he identified ideal with mathematical figures. By idcat d0€au is meant the belief in indivisible magnitudes.
30. paxpotrovety, cf. pyxivey M. 10836 and paxpds dAdyos IOgI® 7.
cuveipew, cf. 1093” 27, De Div. 464” 4, G. A. 71644. It is short for rovs Adyous ovveipew, for which cf. #. NV. 11.47% 21.
BI. TpocyAixspevor Tats Sears TA pOlnpatiKd. mpooyALyouar OCCUTS with a different construction in A. 98626 «i ti mov diéAeure, mpooeyAi- Xovro Tod ovveipomevnv Tacay advrois civar THY Tparyparelav. Alexander interprets (816. 37) ctv ydov7n mpooriOeacr Ta palnpariKd Tals ideas Kal ideas aita mowtov. yAcxouat occurs with the accusative (Pl. Aipparch. 226 ©, and in some manuscripts of Hipp. Z/. ix. 364 Littré), and it seems quite possible for the two elements in zpoo-yAiyoua to govern tais ideas and ra paPnpwatixa respectively, Cf. Hdt. iii, 21 yy addqnv mpookTacGar TH EwuTOv.
37. The manuscript reading, é& &Aou 8€ twos puKpod Kal peyddou: Td yap jeyéOn movet, may be dealt with in various ways. (1) We may omit the colon and the ydép (the reading implied in Bessarion, and possibly in Al. 817. 7 ; but cf. 814. 14). (2) We may read rivos for twos, and translate ‘from what other small and great can he construct mathematical numbers? He already constructs spatial magnitudes out of one other’. But the supplying of ‘out of one other’ is difficult. (3) Christ’s weydAov ov is open to the same objection.
IOQI* 2-3. kal ci... ev. Aristotle here passes to the formal cause and says ‘if some One is the formal principle of each of the two kinds of number, unity will be something common to these two Ones’.