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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.

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. What
difference, then, does it make whether sensible things are such
indivisible entities, or, without being so themselves, have
indivisible entities in them? The result will be the same; if the
sensible entities are divided the others will be divided too, or
else not even the sensible entities can be divided.
But, again, it is not possible that such entities should exist
separately. For if besides the sensible solids there are to be other
solids which are separate from them and prior to the sensible
solids, it is plain that besides the planes also there must be other
and separate planes and points and lines; for consistency requires
this. But if these exist, again besides the planes and lines and
points of the mathematical solid there must be others which are
separate. (For incomposites are prior to compounds; and if there
are, prior to the sensible bodies, bodies which are not sensible, by
the same argument the planes which exist by themselves must be prior
to those which are in the motionless solids. Therefore these will be
planes and lines other than those that exist along with the
mathematical solids to which these thinkers assign separate existence;
for the latter exist along with the mathematical solids, while the
others are prior to the mathematical solids.) Again, therefore,
there will be, belonging to these planes, lines, and prior to them
there will have to be, by the same argument, other lines and points;
and prior to these points in the prior lines there will have to be
other points, though there will be no others prior to these. Now (1)
the accumulation becomes absurd; for we find ourselves with one set of
solids apart from the sensible solids; three sets of planes apart from
the sensible planes-those which exist apart from the sensible
planes, and those in the mathematical solids, and those which exist
apart from those in the mathematical solids; four sets of lines, and
five sets of points. With which of these, then, will the
mathematical sciences deal? Certainly not with the planes and lines
and points in the motionless solid; for science always deals with what
is prior. And (the same account will apply also to numbers; for
there will be a different set of units apart from each set of
points, and also apart from each set of realities, from the objects of
sense and again from those of thought; so that there will be various
classes of mathematical numbers.
Again, how is it possible to solve the questions which we have
already enumerated in our discussion of difficulties? For the
objects of astronomy will exist apart from sensible things just as the
objects of geometry will; but how is it possible that a heaven and its
parts-or anything else which has movement-should exist apart?
Similarly also the objects of optics and of harmonics will exist
apart; for there will be both voice and sight besides the sensible
or individual voices and sights. Therefore it is plain that the

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