—the reasons being (a) that two solids
cannot occupy the same space, and (b) that on this same theory all
other potentialities and characteristics would exist in sensible
things, and none of them would exist separately. This, then, has been
already stated;but in
addition to this it is clearly impossible on this theory for any body
to be divided. For it must be divided in a plane, and the plane in a
line, and the line at a point; and therefore if the point is
indivisible, so is the line, and so on.For what difference does it make whether
entities of this kind are sensible objects, or while not being the
objects themselves, are yet present in them? the consequence will be
the same, for either they must be divided when the sensible objects
are divided, or else not even the sensible objects can be
divided.Nor again can entities of
this kind exist separately.For if besides sensible solids there are to be other solids which
are separate from them and prior to sensible solids, clearly besides
sensible planes there must be other separate planes, and so too with
points and lines; for the same argument applies. And if these exist,
again besides the planes, lines and points of the mathematical solid,
there must be others which are separate;for the incomposite is prior to the composite,
and if prior to sensible bodies there are other non-sensible
bodies,then by the
same argument the planes which exist independently must be prior to
those which are present in the immovable solids. Therefore there will
be planes and lines distinct from those which coexist with the
separately-existent solids; for the latter coexist with the
mathematical solids, but the former are prior to the mathematical
solids.Again, in
these planes there will be lines, and by the same argument there must
be other lines prior to these; and prior to the points which are in
the prior lines there must be other points, although there will be no
other points prior to these.Now the accumulation becomes absurd; because whereas we get only one
class of solids besides sensible solids, we get three classes of
planes besides sensible planes—those which exist separately
from sensible planes, those which exist in the mathematical solids,
and those which exist separately from those in the mathematical
solids—four classes of lines, and five of points;with which of these, then, will
the mathematical sciences deal? Not, surely, with the planes, lines
and points in the immovable solid; for knowledge is always concerned
with that which is prior. And the same argument applies to numbers;
for there will be other units besides each class of points, and
besides each class of existing things, first the sensible and then the
intelligible; so that there will be an infinite number of kinds of
mathematical numbers. Again, there are the problems which we
enumerated in our discussion of difficultiesAristot. Met.
3.2.23-27.: how can they be solved?