The Complete Aristotle (eng.)

eternal, and, if there is, what it is, we
must first consider what is said by others, so that, if there is
anything which they say wrongly, we may not be liable to the same
objections, while, if there is any opinion common to them and us,
we shall have no private grievance against ourselves on that
account; for one must be content to state some points better than
one’s predecessors, and others no worse.
    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. So that the subject of our discussion will be not
whether they exist but how they exist.
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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.
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