Book 14
8
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.That the criticism is directed against Speusippus is clear from Aristot. Met. 7.2.4. Cf. Aristot. Met. 12.10.14.
9
But it does not appear, to judge from the observed facts, that the natural system lacks cohesion, like a poorly constructed drama. ThoseXenocrates (that the reference is not to Plato is clear from sect. 11). 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.
10
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 theoriese.g. that of indivisible lines. 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.
11
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.
