The Complete Aristotle (eng.)

perchance he had not been
indivisible, can belong to him even apart from these attributes.
Thus, then, geometers speak correctly; they talk about existing
things, and their subjects do exist; for being has two forms-it
exists not only in complete reality but also materially.
    Now since the good and the beautiful are different (for the
former always implies conduct as its subject, while the beautiful
is found also in motionless things), those who assert that the
mathematical sciences say nothing of the beautiful or the good are
in error. For these sciences say and prove a great deal about them;
if they do not expressly mention them, but prove attributes which
are their results or their definitions, it is not true to say that
they tell us nothing about them. The chief forms of beauty are
order and symmetry and definiteness, which the mathematical
sciences demonstrate in a special degree. And since these (e.g.
order and definiteness) are obviously causes of many things,
evidently these sciences must treat this sort of causative
principle also (i.e. the beautiful) as in some sense a cause. But
we shall speak more plainly elsewhere about these matters.
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4
    So much then for the objects of mathematics; we have said that
they exist and in what sense they exist, and in what sense they are
prior and in what sense not prior. Now, regarding the Ideas, we
must first examine the ideal theory itself, not connecting it in
any way with the nature of numbers, but treating it in the form in
which it was originally understood by those who first maintained
the existence of the Ideas. The supporters of the ideal theory were
led to it because on the question about the truth of things they
accepted the Heraclitean sayings which describe all sensible things
as ever passing away, so that if knowledge or thought is to have an
object, there must be some other and permanent entities, apart from
those which are sensible; for there could be no knowledge of things
which were in a state of flux. But when Socrates was occupying
himself with the excellences of character, and in connexion with
them became the first to raise the problem of universal definition
(for of the physicists Democritus only touched on the subject to a
small extent, and defined, after a fashion, the hot and the cold;
while the Pythagoreans had before this treated of a few things,
whose definitions-e.g. those of opportunity, justice, or
marriage-they connected with numbers; but it was natural that
Socrates should be seeking the essence, for he was seeking to
syllogize, and ‘what a thing is’ is the starting-point of
syllogisms; for there was as yet none of the dialectical power
which enables people even without knowledge of the essence to
speculate about contraries and inquire whether the same science
deals with contraries; for two things may be fairly ascribed to
Socrates-inductive arguments and universal definition, both of
which are concerned with the starting-point of science):-but
Socrates did not make the universals or the definitions exist
apart: they, however, gave them separate existence, and this was
the kind of thing they called Ideas. Therefore it followed for
them, almost by the same argument, that there must be Ideas of all
things that are spoken of universally, and it was almost as if a
man wished to count certain things, and while they were few thought
he would not be able to count them, but made more of them and then
counted them; for the Forms are, one may say, more numerous than
the particular sensible things, yet it was in seeking the causes of
these that they proceeded from them to the Forms. For to each thing
there answers an entity which has the same name and exists apart
from the substances, and so also in the case of all other groups
there is a one over many, whether these be of this world or
eternal.
    Again, of the ways in which it is proved that the Forms exist,
none is convincing; for from some no inference necessarily follows,
and from some arise Forms even of things of which they think there
are no Forms. For according to the arguments from the sciences
there will be Forms of all things of which there are sciences, and
according to the argument of the ‘one over many’ there will be
Forms even of negations, and according to the argument that thought
has an object when the individual object has perished, there will
be Forms of perishable things; for we have an image of these.
Again, of the most