29. 000 Strovacody povddas Sudda etvar is not strictly relevant here, where spatial magnitudes are being spoken of, but is rather illustrative. Xenocrates speaks of mathematical magnitudes in a non-mathematical way, supposing that the units in one mathematical number (as Plato had supposed that the units in one zdea/ number) are specifically different from those in another, so that a unit of 2 + a unit of 3 would not make 2.
30-33. Aristotle here recurs from 7a pera Tas ideas to numbers, and repeats what he has said in Il. 17—20.
36. paddov 8 tows Odrepa tav érépwy. Aristotle means the view ot Xenocrates, which xe/purra déyerar (1083> 2), combining as it does 800 dpaprias, misdescribing mathematical number and also being open to all the objections against ideal number (1083) 3).
Examination of Plato's view (ch. 7-8. 1083* 20).
1080) 37. (A) We must first examine whether the units are com- parable, and if not, in which of the two senses they are not.
1081* 5. (1) If all are comparable and not different in kind, we get only mathematical number, and the Ideas cannot be the numbers thus produced (for there is but one Idea. of each thing, e.g. of man, while there is an indefinite number of similar numbers, e.g. threes ;
12. but if the Ideas are not numbers, they cannot exist (for the first principles are said to be first principles of number, and the Ideas cannot be classed as either prior or posterior to numbers).
17. (2) If all units are incomparable, (a) the number so produced is not mathematical number (which is composed of undifferentiated units).
21. Nor is it ideal number, for 2 will not be the first product of 1 and the indefinite dyad, and be followed by 3, 4, &c. (the units in 2 not being prior or posterior to one another), since if one unit is to be prior to the other, it will be prior to the 2 which they compose.
29. (4) The units will be prior to the numbers after which they are named, e. g. the third unit (the second in the number 2) will be prior to the number 3.
35. Though no one has supposed the units incomparable in this way, the view agrees sufficiently with the principles of these thinkers, If there is a first unit, there will be priority and posteriority among the units, and similarly among the twos; but though they recognize a first unit and a first two, they do not recognize a second or a third.
bro. (c) If all the units are thus incomparable, there cannot be a ‘two itself’, a ‘three itself’, &c. For whether the units are different or not, the numbers must be generated by successive additions of 1; but if so, they are not generated as these thinkers say they are, from the One and the indefinite dyad.
18. For the number two is a part of the number three, and this of the number four, whereas they generate four from the number two and the indefinite dyad and make it consist of two twos other than the number two;
22. otherwise 4 will consist of the number 2 and another 2, and the number 2 will consist of the One itself and another 1, and if so, the element in 2 other than the One itself cannot be the indefinite dyad, since it produces one unit, not a definite 2.
27. (d) How can there be other twos besides the number 2? How can they be composed of prior and posterior units? These sup- positions are quite fictitious. But if the conclusions are absurd, the first principles must be wrong.
35. (3) If the units in different numbers are different but those in the same number not different, equal difficulties follow.
10821. (a) Since the ideal ten is no ordinary number and the fives in it no ordinary fives, the units in the one five must be different from those in the other; i.e. the theory inconsistently with itself implies that five of the units in ro are different from the other five.
7. If the units in ro differ, there must be other fives in 10 than those we have named, but if so, what sort of tens do they make? These thinkers recognize no other 10 in the number 1o.
11. They must, as we have assumed (I. 3) that they do, suppose the number 4 to be composed of no chance twos, for they say the indefinite dyad received the definite dyad and made two dyads.
15. (2) How can the number 2 be something apart from the two units? Either by participation of one in the other, as ‘white man’, which shares in ‘ white’ and in ‘man’, is apart from them, or by one part being a differentia of the other, as ‘man’ is apart from ‘animal’ and * two-footed ’.
20. (c) The units in 2 or in 3 cannot be one by contact, mixture, or position; there is nothing apart from the units any more than a pair of men is anything apart from the two men. The indivisibility
432 Commentary
of the units makes no difference; points are indivisible, but a pair of points is nothing apart from the single points.
26. (d) The theory implies that there will be prior and posterior twos, threes, &c. The twos in 4 are prior to those in 8, and generated the fours in 8 as 2 generated ¢hem, so that,since the 2 is an Idea, they also are Ideas. :
32. So too the units in 2 generate the units in 4, so that all the units are Ideas and an Idea is composed of Ideas, and therefore that of which the Idea is an Idea is similarly composite, e.g. animals are composed of animals.
br, (ec) To make the units different in any way is absurd and artificial ; unit differs from unit neither in quantity nor in quality, and a number which is neither greater nor less than another must be equal to it and identical with it; if it is not, neither will the twos in 10 be without difference, as the theory supposes them to be.
11. (f) If one unit and another unit always make two, a unit in 2 and a unit in 3 will make a 2. Now (a) this will consist of units differing in kind; () will it be prior or posterior to the number 3? Presumably prior, since one of the units is simultaneous with 3 and the other with 2.
16. We say that one and one (e.g. good and evil) always make two; but ¢hey say that not even one unit and another unit always make a two.
19. (g) The number 3 must surely be greater than the number 2, but if so, it will contain a number equal to and without difference from the number 2; which it cannot, if there is priority and posteriority between any two numbers.
23. (#) On this view the Ideas cannot be numbers. ‘Those who say all units are different are right in supposing this to be implied in there being Ideas; for the Form is unique, but if units are without difference, the twos and the threes will be without difference too.
28. These thinkers are bound to say that in’counting ‘ one, two’ we do not add one to the original one; for then (a) generation would not be from the indefinite dyad, and (8) an Idea would not be pro- duced, since if it were it would contain another Idea, and all the Ideas would ultimately be parts of one Idea.
32. Thus what they say agrees with their hypothesis ; but it destroys many of the truths of mathematics. They will say that there is a difficulty in the question whether we count by successive additions of x or by constructing each number separately. But we do both; it is absurd to suppose a separate kind of number to which the latter process applies.
St. Albert’S College Librar
1083*1. (7) What is the differentia of a number, and of a unit, if a unit has any? Units must differ in respect either of quantity or of quality, but neither is possible. (a) If units differed in quantity, num- bers equal in number of units would differ from each other. Are the first units greater or less than the later? All this is absurd.
8. (8) They cannot differ in quality. They have no qualities, for in numbers quality depends on quantity. They cannot get quality either from the One, which has none, or from the indefinite dyad, which gives quantity.
14. If units differ in some other way, these thinkers ought to have said why this difference must exist, or at least what difference they mean. Thus if the Ideas are numbers, the units cannot be all com- parable, nor incomparable in either of the two ways.
Examination of the views of other Platonists and of the Pythagoreans (ch. 8. 10832 20-» 23),
1083* 20. (Ba) The views of other thinkers are no better, viz. of those (Speusippus) who do not believe in Ideas but in mathematical objects, and make numbers the primary realities, and the One their first principle.
24. For it is paradoxical that there should be a first 1, but not a first 2, 3, &c. If only mathematical number exists, the One is not a first principle (for if it were, it would be different from other ones, and there must then be a two different from other twos); if the One is a first principle, the numbers must be such as Plato supposed them, i.e, incomparable.
35. If both Plato’s doctrine and that of Speusippus lead to impossible results, number cannot exist apart.
by, (Cd) The worst view is that ideal number and mathematical are the same (Xenocrates). This view falsifies the nature of mathe- matical number, and involves further the difficulties incidental to the belief in ideal number.
8. (Bd) The Pythagorean view escapes some difficulties by not making number exist apart, but has peculiar difficulties arising from the supposition that bodies are composed of mathematical numbers.
13. For there are no indivisible magnitudes, or at any rate units have no magnitude, and therefore bodies cannot be composed of them, as the Pythagorean view implies.
19. Thus none of the ways of treating number as self-subsistent is satisfactory; therefore it is not self-subsistent.
2573-2 Ff
434 Commentary
Aristotle first (1080 37) discusses the theory of incomparable num- bers (which, we are told in 1083% 32, was the view of Plato). Then (10832 20) he proceeds to the theory of Speusippus, then (1083 1) to that of Xenocrates, and last (1083 8-19) to that.of the Pythagoreans.
IO81* i. domep Sve(Aopev, 10808 18-20, 23-35.
4. TpOTe = cidyrixG Al. 748. 1. Cf. -1080b22, and aparn dvds 1080* 26 and MN assim, mp@rov phKos, tAaTos, Babos De An. 404» 20, eripdverar tpatar K. 1060) 13.
7. Bz.’s proposal to omit tovs is supported by |. 12, but is not absolutely necessary. As he himself says, we may render the manu- script reading ‘and the Ideas cannot be the numbers thus produced’.
11-12. dot’ oW0ev ... dmovaody, E.g. any of the threes in nine will be the Idea of man as much as any other, and the uniqueness of the Idea will be destroyed.