14. (2) (a) Some (Speusippus) believe in mathematical number only; (4) the Pythagoreans too believe in mathematical number only, but in the sense that sensible substances are actually composed of extended units—though they cannot tell us how the first extended unit came into being.
21. (C) (a) Another thinker says only ideal number exists, and (6) some (Xenocrates) identify this with mathematical number.
23. There is a similar variety of opinion about lines, planes, and solids. (A) Some distinguish the mathematical lines, &c., and those which come after the Ideas; (#) some say that the mathematical objects exist, and speak mathematically about them (viz. the non- believers in Ideas); (C) others say that the mathematical objects exist, but do not speak mathematically, for they say that not every magnitude is divisible into magnitudes, and not every two units make a two,
go. All who treat the One as a first principle suppose numbers to
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be composed of abstract units, except the Pythagoreans, who conceive of numbers as extended.
33. These are the possible views; all untenable, but perhaps some more so than others,
1080 15-4. The sentence is_irrégular ‘in structure. Aristotle begins (1. 17) by stating what looks as if it were to be the first of a series of alternative hypotheses about the nature of zzmders, but he proceeds to state three possible forms of this one hypothesis, differing in the view they take of the nature of wnz/s (Il. 18, 20, 23), and recurs to numbers only in 1]. 35, where he states as a fresh alternative that there may be three kinds of number having the three kinds of unit respectively ; finally in 1. 37 he classifies the possible views according to a different principle of division. It is noteworthy that he brings forward his classification as one arrived at a priorz, not by enumeration of existing views. This awakens a certain suspicion that he may in his account of the actual views do them some injustice in order to fit them into his ready-made scheme. He says «definitely, however, that each of the views he mentions had supporters, except the view that all units are incomparable (1080) 8, cf. 1081°35). # tas pev KrAr. (I. 23), though grammatically, co-ordinate with rou etvar xrX. (J. 17), is in sense co-ordinate with # él rév povddwy KrX. (1. 18) and with 7 ets epeens xd. (1. 20). The views Aristotle mentions are
(1) the belief in incomparable numbers (1. 17), (2) with units all incomparable (I. 18),
or (4) with units all comparable (1. 20),
or (c) with the units of each number comparable with each other,
but incomparable with those of other numbers (1. 23),
(2) the belief in all three kinds of number, i.e. the kind (1 a), the kind (1 4), and the kind (1 c) (1. 35).
He omits (3) the belief in two kinds of number, (1 a) and (1 4),
’ (1 a) and (1 ¢), or (1 6) and (1¢).
In ll. 21, 36 he confuses (1 4), the belief in incomparable numbers whose units are all comparable, with (4), the belief in comparable numbers (whose units must necessarily be all comparable), for clearly this is what he conceives 6 wabyparixds dpiO.ds to be.
It must not be thought that this passage offers a classification of hypotheses about ideal number in particular, for the classification includes the Pythagoreans (16), who drew no distinction between mathematical and ideal number, and Speusippus ( 14), who believed only in the former. What we have is a classification of all the views which treated numbers as ‘separate substances and first causes of existing things’ (#14). Alexander, failing to notice that the Pytha- goreans enter into the classification, takes it to be a classification of theories of ideal number (e.g. 743. 13), and this leads him to give an absurd interpretation of * 35-37. He takes it to mean that e.g. 3, 4,
5 might be composed of units all of which are incomparable, 7, 8, 9 of units all of which are comparable, 20, 30 of units such that those in 20 are comparable with each other and those in 30 with each other but those in 20 are not comparable with those in 30. It seems clear that the view referred to in # 35-37 is one which believes in the existence of three complete number series of different kinds.
So far as ideal numbers are concerned, the doctrine that they are ‘incomparable’, i.e. incapable of being added or subtracted, multi- plied or divided, is a perfectly sound one which is misunderstood by Aristotle. The ideal numbers are simply the natural numbers, 1, 2, 3, &c., or in other words oneness, twoness, threeness, &c., and these of course cannot be added. You cannot add oneness to one- ness because there is only one oneness; and it is equally certain that you cannot add oneness to twoness. And, further, Aristotle’s notion of numbers as containing units, and the resulting question whether these are comparable or incomparable, is equally mistaken. The number 2 does not contain two numbers 1, for there is only one number 1.
As against the mathematical or ‘intermediate ’ numbers believed in by the Platonists, Aristotle’s objection would have more force. There are no ‘mathematical numbers’ distinct on the one hand from the natural number, and on the other from the particular one-member groups, two-meimber groups, &c., which are the instances of the natural numbers. ‘The weakness of Aristotle’s position is that he believes in mathematical numbers, which do not exist, and does not believe in ideal or universal numbers, which do.
On the conception of dovpBAynto. apiOyot cf. Cook Wilson in Classical Review, xviii. 247-260, especially §§ 2, 3, 5.
1g. doupBAntos. The usage of cvpBardAcv, cvpBAnrtds, dovpBAnros in Aristotle shows that the word must mean ‘incomparable’ ; and things are comparable if and only if they belong to the same kind (Phys. 2488, 24993, Top. 107147, I. 1055%6). Thus dovuBdrnros is practically equivalent to érepov dv ra cide (1. 17), and cupBAyrds can be coupled with adiddopos (108145). Strictly, to say that two things are ovpBAyra is to say that one can be expressed as a fraction of the other, or at least as greater or less than or equal to the other. But in this context cuuBdAyrat seems to mean ‘capable of entering into arithmetical relations with one another—of being added and sub- tracted, multiplied and divided’.
35-37. otos 6 mp@ros ehexOn, cf. ll. 15-203; otoy ot polnpatiKkol héyouat, cf. 20-23 ; tov pnOévta TedeuTatoy, cf. 23-35.
bg, 16 mp@tovy, 10769 38-6 11. Aristotle is now omilting the com- promise of Pythagorean and Platonic views referred to there, and taking account only of the genuine Pythagorean view.
4. Bz. argues that the view that all the numbers are immanent in sensibles has been already mentioned in ], 1, so that 7 wavras eivau is unmeaning. But Alexander evidently had these words (perhaps }) Tavras py etvar as well), though his interpretation of them cannot be
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right; and so have all the good manuscripts. The solution of the difficulty is to treat ot otrws . . . aiofyra as parenthetical. ‘* The numbers must be either transcendent, or immanent .. . either some immanent and not others, or all immanent.’
6. cxeddv is explained by zAqw ra. 1. 8.
4. Gddou twés, ie. the dzeipov in the case of the Pythagoreans, the _‘ great and small’ er indefinite dyad in the case of the Platonists.
IO-Il. o0 yap... cipnpérvous. We shall see how Aristotle can say this if we remember that (a) he confuses (4) above with (1 4), and (8), since the view (1 @) is not held by any one (Il. 8, 9). (3.2) and (34) disappear with it and (3¢) takes the place of (2). There thus remain (A) the belief in (x 4) (Il. 14-21), (2) the belief in (12) (I. 21, 22), (C) the belief in both (Il. 11-14), and (JD) the confusion of the two (ll. 22, 23).
II, ot pév, who believed in both, means Plato and his orthodox followers ; cf. A.g87®14-18. Aristotle thinks of the Platonic 7a peraév as of the type (1 4), though they were more probably of the type (4).
14. ot S€ means Speusippus; cf. 10767 20-21 n.
15. Tov (not 75) mpatoy Tey Svrwv is somewhat strange ; in the light of 1083? 23 (also on Speusippus) 7a 8¢ pabqparixa eivar Kai Tous apil- pods =pérous Tay Gvrev One May conjecture that réy should be omitted.
16. The Pythagoreans, like Speusippus, believed in mathematical number only; but they differed from him, as from all ra in holding numbers to be actually present in ‘things (cf. 237-" 4). Mr. Cornford holds, with much probability, that the Pythagorean doctrine here referred to is not the mystical system of Pythagoras but a scientific system of number-atomism which was developed in the fifth century and was the forerunner of Atomism proper.- He summarizes the main features of this system as follows: ‘(1) there is only one kind of number—namely, mathematical number. (2) This number does not exist separately, but sensible substances are composed of it . . (3) These numbers do not consist of abstract units, but the units are conceived as having spatial magnitude. (4) They are described as “indivisible magnitudes” (1083"13). (5) Things or bodies are identified with numbers composed of the indivisible magnitudes or monads (1083 12 sqq.). (6) The Pythagoreans regarded numbers as generated—the process of generation being, of course, identical with the physical generation of the sensible world (1091* 17 sqq-)’ (CZ. Quart. xvii. 8). Cf. N. 1092» 8-15 n.
19. Atv of povadixéy. The Platonists, like Aristotle, thought of numbers as composed of unextended units; the Pythagoreans thought of them as extended and having extended units. In other words they had not reached the notion of arithmetic as distinguished from geometry. Aristotle uses dpifpos dpiOuqrixos in the same sense as dppos povadixes (108316). Zeller thinks (i.° 483-488) that in stating the Pythagorean units to be extended Aristotle is drawing a mistaken inference from the Pythagorean view that bodies are com- posed of numbers; he admits, however, that the Pythagorean
cosmology implies the treatment of numbers as spatial. But -at the time of the Pythagoreans the notion of non-corporeal reality did not exist, so that they necessarily thought of the units as extended. There is some ground for holding that they called them éyxo: (Burnet, E.G. P.§ 146).
20-21. Stas 5é. . . oixacw. Cf. N. 10912 15, where we learn that the Pythagoreans ‘say that when the One had been put together whether out of planes or out of surface or out of seed or out of they know not what, immediately the nearest part of the infinite began to be drawn and limited by the limit’. The general sense of the present passage is: ‘ The Pythagoreans construct the universe out of numbers having spatial units; but how the first unit was constructed as an extended thing they cannot tell’. ‘ They had not’, as Mr. Cornford observes (CZ. Quart. xvii. 9), ‘reached the position of fully developed atomism, which postulates an indefinite plurality of atoms or monads as an ultimate and eternal fact.’
ai. It is not easy to identify dAdos . . . 71s, who believed in ideal number only. Alexander’s suggestion that it was a Pythagorean can hardly be right, since Aristotle never ascribes the belief in Ideas to Pytha- goreans. It must be a Platonist, but further than this we cannot go. Elsewhere in enumerating the views held Aristotle omits this one (1076? 19-22, 10864 2-13, cf. 1080» 24-30).
Jaeger would remove the difficulty by treating Go: as a variant for eivax and removing eivaz in |. 23 as a later-addition due to the intrusion of €or. But Alexander and Syrianus had our text; and it is not particularly surprising that Aristotle should mention here a view he does not mention elsewhere—a view which is almost certain to have been held by some Platonist.
22. By the vo. who identified ideal and mathematical number Aristotle probably means Xenocrates. Cf. 10762 20-21 n.; Z, 1028> 24 Nn.
24. ot pev answers to of wer in |. 11 and refers to Plato.
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26. of pév answers to ot dé in ]. 14 and means Speusippus and the Pythagoreans.
28. ot 8¢ answers to évoi' dé in 1, 22 and means Xenocrates. (No one is mentioned answering to dAAds ts in 1. 21, and this view is difficult to distinguish from that of Xenocrates.) od réuverOar peyeOos may eis peyeOy (1. 29) is a clear allusion to the doctrine of indivisible lines, of which the main supporter was Xenocrates, though it is also ascribed by Aristotle to Plato (A. 992 20).