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Pages of Metaphysics

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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.

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
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

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