It is clear, then, both that the
contrary theory^{1} 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 raised^{2}—why
it is that while numbers are in no way present in sensible things,
their attributes are present in sensible things.

There are some^{3} 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.
Those^{5} 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 theories^{6} 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}