Aristotle, Metaphysics

It is clear, then, both that the contrary theory¹ can make out a case for the contrary view, and that those who hold this theory must find a solution for the difficulty which was recently raised²—why it is that while numbers are in no way present in sensible things, their attributes are present in sensible things.

There are some³ who think that, because the point is the limit and extreme of the line, and the line of the plane, and the plane of the solid, there must be entities of this kind.We must, then, examine this argument also, and see whether it is not exceptionally weak. For (1.) extremes are not substances; rather all such things are merely limits. Even walking, and motion in general, has some limit; so on the view which we are criticizing this will be an individual thing, and a kind of substance. But this is absurd. And moreover (2.) even if they are substances, they will all be substances of particular sensible things, since it was to these that the argument applied. Why, then, should they be separable?

Again, we may, if we are not unduly acquiescent, further object with regard to all number and mathematical objects that they contribute nothing to each other, the prior to the posterior. For if number does not exist, none the less spatial magnitudes will exist for those who maintain that only the objects of mathematics exist; and if the latter do not exist, the soul and sensible bodies will exist.⁴ But it does not appear, to judge from the observed facts, that the natural system lacks cohesion, [20] like a poorly constructed drama. Those⁵ who posit the Ideas escape this difficulty, because they construct spatial magnitudes out of matter and a number—2 in the case of lines, and 3, presumably, in that of planes, and 4 in that of solids; or out of other numbers, for it makes no difference.But are we to regard these magnitudes as Ideas, or what is their mode of existence? and what contribution do they make to reality? They contribute nothing; just as the objects of mathematics contribute nothing. Moreover, no mathematical theorem applies to them, unless one chooses to interfere with the principles of mathematics and invent peculiar theories⁶ of one's own. But it is not difficult to take any chance hypotheses and enlarge upon them and draw out a long string of conclusions.

These thinkers, then, are quite wrong in thus striving to connect the objects of mathematics with the Ideas. But those who first recognized two kinds of number, the Ideal and the mathematical as well, neither have explained nor can explain in any way how mathematical number will exist and of what it will be composed; for they make it intermediate between Ideal and sensible number.For if it is composed of the Great and Small, it will be the same as the former, i.e. Ideal, number. But of what other Great and Small can it be composed? for Plato makes spatial magnitudes out of a Great and Small.⁷