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.
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