We have already explained what the
substance of sensible things is, dealing in our treatise on
physics^{1} with the material substrate, and subsequently
with substance as actuality.^{2}
Now since we are inquiring whether there is or is not some immutable
and eternal substance besides sensible substances, and if there is,
what it is, we must first examine the statements of other thinkers, so
that if they have been mistaken in any respect, we may not be liable
to the same mistakes; and if there is any view which is common to them
and us, we may not feel any private self-irritation on this score. For
we must be content if we state some points better than they have done,
and others no worse.

There are two views on this subject.
Some say that mathematical objects, i.e. numbers and lines, are
substances; and others again that the Ideas are substances.Now since some^{3}
recognize these as two classes—
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the Ideas and the mathematical
numbers—and others^{4} regard both as having one
nature, and yet others^{5} hold that only the
mathematical substances are substances, we must first consider the
mathematical objects, without imputing to them any other
characteristic—e.g. by asking whether they are really Ideas
or not, or whether they are principles and substances of existing
things or not—and merely inquire whether as mathematical
objects they exist or not, and if they do, in what sense; then after
this we must separately consider the Ideas themselves, simply and in
so far as the accepted procedure requires; for most of the arguments
have been made familiar already by the criticisms of other
thinkers.And
further, the greater part of our discussion must bear directly upon
this second question—viz. when we are considering whether
the substances and first principles of existing things are numbers and
Ideas; for after we have dealt with the Ideas there remains this third
question.

Now if the objects of mathematics exist, they must be either in sensible things, as some hold; or separate from them (there are some also who hold this view); or if they are neither the one nor the other, either they do not exist at all, or they exist in some other way. Thus the point which we shall have to discuss is concerned not with their existence, but with the mode of their existence.

That the objects of
mathematics cannot be in sensible things, and that moreover the theory
that they are is a fabrication, has been observed already in our
discussion of difficulties^{6}