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. 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. So that the subject of our discussion will be not
whether they exist but how they exist.
2
That it is impossible for mathematical objects to exist in
sensible things, and at the same time that the doctrine in question is
an artificial one, has been said already in our discussion of
difficulties we have pointed out that it is impossible for two
solids to be in the same place, and also that according to the same
argument the other powers and characteristics also should exist in
sensible things and none of them separately. This we have said
already. But, further, it is obvious that on this theory it is
impossible for any body whatever to be divided; for it would have to
be divided at a plane, and the plane at a line, and the line at a
point, so that if the point cannot be divided, neither can the line,
and if the line cannot, neither can the plane nor the solid
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