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[1090a] [1] but if they are the same, he who holds this has to face many logical contradictions.

One might fasten also upon the question with respect to numbers, whence we should derive the belief that they exist.For one1 who posits Ideas, numbers supply a kind of cause for existing things; that is if each of the numbers is a kind of Idea, and the Idea is, in some way or other, the cause of existence for other things; for let us grant them this assumption.But as for him2 who does not hold this belief, because he can see the difficulties inherent in the Ideal theory (and so has not this reason for positing numbers), and yet posits mathematical number, what grounds have we for believing his statement that there is a number of this kind, and what good is this number to other things? He who maintains its existence does not claim that it is the cause of anything, but regards it as an independent entity; nor can we observe it to be the cause of anything; for the theorems of the arithmeticians will all apply equally well to sensible things, as we have said.3

Those, then, who posit the Ideas and identify them with numbers, by their assumption (in accordance with their method of abstracting each general term from its several concrete examples) that every general term is a unity, make some attempt to explain why number exists.4 Since, however, their arguments are neither necessarily true nor indeed possible, [20] there is no justification on this ground for maintaining the existence of number.The Pythagoreans, on the other hand, observing that many attributes of numbers apply to sensible bodies, assumed that real things are numbers; not that numbers exist separately, but that real things are composed of numbers.5 But why? Because the attributes of numbers are to be found in a musical scale, in the heavens, and in many other connections.6

As for those who hold that mathematical number alone exists,7 they cannot allege anything of this kind8 consistently with their hypotheses; what they did say was that the sciences could not have sensible things as their objects. But we maintain that they can; as we have said before. And clearly the objects of mathematics do not exist in separation; for if they did their attributes would not be present in corporeal things.Thus in this respect the Pythagoreans are immune from criticism; but in so far as they construct natural bodies, which have lightness and weight, out of numbers which have no weight or lightness, they appear to be treating of another universe and other bodies, not of sensible ones.9 But those who treat number as separable assume that it exists and is separable because the axioms will not apply to sensible objects; whereas the statements of mathematics are true and appeal to the soul.10

1 Plato and his orthodox followers.

2 Speusippus.

3 Aristot. Met. 13.3.1.

4 I have followed Ross's text and interpretation of this sentence. For the meaning cf. Aristot. Met. 14.2.20.

5 See Introduction.

6 Cf. Aristot. Met. 14.6.5.

7 Cf. Aristot. Met. 14.2.21.

8 i.e., that things are composed of numbers.

9 See Introduction.

10 The statements of mathematics appeal so strongly to our intelligence that they must be true; therefore if they are not true of sensible things, there must be some class of objects of which they are true.

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