^{1}disagree about numbers indicates that it is the misrepresentation of the facts themselves that produces this confusion in their views.Those

^{2}who recognize only the objects of mathematics as existing besides sensible things, abandoned Ideal number and posited mathematical number because they perceived the difficulty and artificiality of the Ideal theory. Others,

^{3}wishing to maintain both Forms and numbers, but not seeing how, if one posits these

^{4}as first principles, mathematical number can exist besides Ideal number, identified Ideal with mathematical number,—but only in theory, since actually mathematical number is done away with, because the hypotheses which they state are peculiar to them and not mathematical.

^{5}And he

^{6}who first assumed that there are Ideas, and that the Ideas are numbers, and that the objects of mathematics exist, naturally separated them. Thus it happens that all are right in some respect, but not altogether right; even they themselves admit as much by not agreeing but contradicting each other. The reason of this is that their assumptions and first principles are wrong;and it is difficult to propound a correct theory from faulty premisses: as Epicharmus says, "no sooner is it said than it is seen to be wrong."

^{7}

We have
now examined and analyzed the questions concerning numbers to a
sufficient extent; for although one who is already convinced might be
still more convinced by a fuller treatment,
[20]
he who is not convinced would be
brought no nearer to conviction.As for the first principles and causes and
elements, the views expressed by those who discuss only sensible
substance either have been described in the __Physics__^{8} or have no place in
our present inquiry; but the views of those who assert that there are
other substances besides sensible ones call for investigation next
after those which we have just discussed.

Since, then, some thinkers hold that the Ideas and numbers are such substances, and that their elements are the elements and principles of reality, we must inquire what it is that they hold, and in what sense they hold it.

Those^{9} who posit only numbers,
and mathematical numbers at that, may be considered later^{10}; but as
for those who speak of the Ideas, we can observe at the same time
their way of thinking and the difficulties which befall them. For they
not only treat the Ideas as universal substances, but also as
separable and particular.(That this is impossible has been already shown^{11} by a consideration of the
difficulties involved.) The reason why those who hold substances to be
universal combined these two views was that they did not identify
substances with sensible things.