One might fasten also upon the question with
respect to numbers, whence we should derive the belief that they
exist.For one^{1} who posits Ideas, numbers supply a kind of
cause for existing things; that is if each of the numbers is a kind of
Idea, and the Idea is, in some way or other, the cause of existence
for other things; for let us grant them this assumption.But as for him^{2} who does not
hold this belief, because he can see the difficulties inherent in the
Ideal theory (and so has not this reason for positing numbers), and
yet posits mathematical number, what grounds have we for believing his
statement that there is a number of this kind, and what good is this
number to other things? He who maintains its existence does not claim
that it is the cause of anything, but regards it as an independent
entity; nor can we observe it to be the cause of anything; for the
theorems of the arithmeticians will all apply equally well to sensible
things, as we have said.^{3}

Those,
then, who posit the Ideas and identify them with numbers, by their
assumption (in accordance with their method of abstracting each
general term from its several concrete examples) that every general
term is a unity, make some attempt to explain why number exists.^{4} Since, however,
their arguments are neither necessarily true nor indeed
possible,
[20]
there is no
justification on this ground for maintaining the existence of
number.The
Pythagoreans, on the other hand, observing that many attributes of
numbers apply to sensible bodies, assumed that real things are
numbers; not that numbers exist separately, but that real things are
composed of numbers.^{5} But why? Because the attributes of numbers
are to be found in a musical scale, in the heavens, and in many other
connections.^{6}

As for those who hold that
mathematical number alone exists,^{7} they cannot allege anything of this kind^{8} consistently with their hypotheses; what they
did say was that the sciences could not have sensible things as their
objects. But we maintain that they can; as we have said before. And
clearly the objects of mathematics do not exist in separation; for if
they did their attributes would not be present in corporeal
things.Thus in this
respect the Pythagoreans are immune from criticism; but in so far as
they construct natural bodies, which have lightness and weight, out of
numbers which have no weight or lightness, they appear to be treating
of another universe and other bodies, not of sensible ones.^{9}
But those who treat number as separable assume that it exists and is
separable because the axioms will not apply to sensible objects;
whereas the statements of mathematics are true and appeal to the
soul.^{10}