The Complete Aristotle (eng.)

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 other senses as well, and the other objects of sense, will
exist apart; for why should one set of them do so and another not?
And if this is so, there will also be animals existing apart, since
there will be senses.
    Again, there are certain mathematical theorems that are
universal, extending beyond these substances. Here then we shall
have another intermediate substance separate both from the Ideas
and from the intermediates,-a substance which is neither number nor
points nor spatial magnitude nor time. And if this is impossible,
plainly it is also impossible that the former entities should exist
separate from sensible things.
    And, in general, conclusion contrary alike to the truth and to
the usual views follow, if one is to suppose the objects of
mathematics to exist thus as separate entities. For because they
exist thus they must be prior to sensible spatial magnitudes, but
in truth they must be posterior; for the incomplete spatial
magnitude is in the order of generation prior, but in the order of
substance posterior, as the lifeless is to the living.
    Again, by virtue of what, and when, will mathematical magnitudes
be one? For things in our perceptible world are one in virtue of
soul, or of a part of soul, or of something else that is reasonable
enough; when these are not present, the thing is a plurality, and
splits up into parts. But in the case of the subjects of
mathematics, which are divisible and are quantities, what is the
cause of their being one and holding together?
    Again, the modes of generation of the objects of mathematics
show that we are right. For the dimension first generated is
length, then comes breadth, lastly depth, and the process is
complete. If, then, that which is posterior in the order of
generation is prior in the order of substantiality, the solid will
be prior to the plane and the line. And in this way also it is both
more complete and more whole, because it can become animate. How,
on the other hand, could a line or a plane be animate? The
supposition passes the power of our senses.
    Again, the solid is a sort of substance; for it already has in a
sense completeness. But how can lines be substances? Neither as a
form or shape, as the soul perhaps is, nor as matter, like the
solid; for we have no experience of anything that can be put
together out of lines or planes or points, while if these had been
a sort of material substance, we should have observed things which
could be put together out of them.
    Grant, then, that they are prior in definition. Still not all
things that are prior in definition are also prior in
substantiality. For those things are prior in substantiality which
when separated from other things surpass them in the power of
independent existence, but things are prior in definition to those
whose definitions are compounded out of their definitions; and
these two properties are not coextensive. For if attributes do not
exist apart from the substances (e.g. a ‘mobile’ or a pale’), pale
is prior to the pale man in definition, but not in substantiality.
For it cannot exist separately, but is always along with the
concrete thing; and by the concrete thing I mean the pale man.
Therefore it is