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[1076a] [8]

We have already explained what the substance of sensible things is, dealing in our treatise on physics1 with the material substrate, and subsequently with substance as actuality.2 Now since we are inquiring whether there is or is not some immutable and eternal substance besides sensible substances, and if there is, what it is, we must first examine the statements of other thinkers, so that if they have been mistaken in any respect, we may not be liable to the same mistakes; and if there is any view which is common to them and us, we may not feel any private self-irritation on this score. For we must be content if we state some points better than they have done, and others no worse.

There are two views on this subject. Some say that mathematical objects, i.e. numbers and lines, are substances; and others again that the Ideas are substances.Now since some3 recognize these as two classes— [20] the Ideas and the mathematical numbers—and others4 regard both as having one nature, and yet others5 hold that only the mathematical substances are substances, we must first consider the mathematical objects, without imputing to them any other characteristic—e.g. by asking whether they are really Ideas or not, or whether they are principles and substances of existing things or not—and merely inquire whether as mathematical objects they exist or not, and if they do, in what sense; then after this we must separately consider the Ideas themselves, simply and in so far as the accepted procedure requires; for most of the arguments have been made familiar already by the criticisms of other thinkers.And further, the greater part of our discussion must bear directly upon this second question—viz. when we are considering whether the substances and first principles of existing things are numbers and Ideas; for after we have dealt with the Ideas there remains this third question.

Now if the objects of mathematics exist, they must be either in sensible things, as some hold; or separate from them (there are some also who hold this view); or if they are neither the one nor the other, either they do not exist at all, or they exist in some other way. Thus the point which we shall have to discuss is concerned not with their existence, but with the mode of their existence.

That the objects of mathematics cannot be in sensible things, and that moreover the theory that they are is a fabrication, has been observed already in our discussion of difficulties6

1 The reference is presumably to Aristot. Physics 1.

2 In Books 7-9.

3 This was the orthodox Platonist view; cf. Aristot. Met. 1.6.4.

4 Xenocrates and his followers.

5 The Pythagoreans and Speusippus.

6 Cf. Aristot. Met. 3.2.23-30.

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