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[1086a] [1] Further, the fact that the leading authorities1 disagree about numbers indicates that it is the misrepresentation of the facts themselves that produces this confusion in their views.Those2 who recognize only the objects of mathematics as existing besides sensible things, abandoned Ideal number and posited mathematical number because they perceived the difficulty and artificiality of the Ideal theory. Others,3 wishing to maintain both Forms and numbers, but not seeing how, if one posits these4 as first principles, mathematical number can exist besides Ideal number, identified Ideal with mathematical number,—but only in theory, since actually mathematical number is done away with, because the hypotheses which they state are peculiar to them and not mathematical.5 And he6 who first assumed that there are Ideas, and that the Ideas are numbers, and that the objects of mathematics exist, naturally separated them. Thus it happens that all are right in some respect, but not altogether right; even they themselves admit as much by not agreeing but contradicting each other. The reason of this is that their assumptions and first principles are wrong;and it is difficult to propound a correct theory from faulty premisses: as Epicharmus says, "no sooner is it said than it is seen to be wrong."7

We have now examined and analyzed the questions concerning numbers to a sufficient extent; for although one who is already convinced might be still more convinced by a fuller treatment, [20] he who is not convinced would be brought no nearer to conviction.As for the first principles and causes and elements, the views expressed by those who discuss only sensible substance either have been described in the Physics8 or have no place in our present inquiry; but the views of those who assert that there are other substances besides sensible ones call for investigation next after those which we have just discussed.

Since, then, some thinkers hold that the Ideas and numbers are such substances, and that their elements are the elements and principles of reality, we must inquire what it is that they hold, and in what sense they hold it.

Those9 who posit only numbers, and mathematical numbers at that, may be considered later10; but as for those who speak of the Ideas, we can observe at the same time their way of thinking and the difficulties which befall them. For they not only treat the Ideas as universal substances, but also as separable and particular.(That this is impossible has been already shown11 by a consideration of the difficulties involved.) The reason why those who hold substances to be universal combined these two views was that they did not identify substances with sensible things.

1 Alexander preferred the reading πρώτους, interpreting it in this sense; and I do not see why he should not be followed. Ross objects that πρῶτος is used in the chronological sense in 16., but this is really no argument. For a much more serious (although different) inconsistency in the use of terms cf. Aristot. Met. 12.3.1.

2 Speusippus and his followers.

3 Xenocrates and his followers.

4 Unity and the indeterminate dyad; for the difficulty see Aristot. Met. 13.7.3, 4.

5 Cf. Aristot. Met. 13.6.10.

6 Plato.

7 Epicharmus, Fr. 14, Diels.

8 Aristot. Physics 1.4-6.

9 The Pythagoreans and Speusippus.

10 Aristot. Met. 14.2.21, Aristot. Met. 14.3.2-8, 15, 16.

11 Aristot. Met. 3.6.7-9.

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