Two opinions are held on this subject; it is said that the objects
of mathematics-i.e. numbers and lines and the like-are substances, and
again that the Ideas are substances. And (1) since some recognize
these as two different classes-the Ideas and the mathematical numbers,
and (2) some recognize both as having one nature, while (3) some
others say that the mathematical substances are the only substances,
we must consider first the objects of mathematics, not qualifying them
by any other characteristic-not asking, for instance, whether they are
in fact Ideas or not, or whether they are the principles and
substances of existing things or not, but only whether as objects of
mathematics they exist or not, and if they exist, how they exist. Then
after this we must separately consider the Ideas themselves in a
general way, and only as far as the accepted mode of treatment
demands; for most of the points have been repeatedly made even by
the discussions outside our school, and, further, the greater part
of our account must finish by throwing light on that inquiry, viz.
when we examine whether the substances and the principles of
existing things are numbers and Ideas; for after the discussion of the
Ideas this remans as a third inquiry.
If the objects of mathematics exist, they must exist either in
sensible objects, as some say, or separate from sensible objects
(and this also is said by some); or if they exist in neither of
these ways, either they do not exist, or they exist only in some
special sense.
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