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some sense a cause. But we shall speak more plainly elsewhere about
these matters.
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
accurate arguments, some lead to Ideas of relations, of which they say
there is no independent class, and others introduce the 'third man'.
And in general the arguments for the Forms destroy things for
whose existence the believers in Forms are more zealous than for the
existence of the Ideas; for it follows that not the dyad but number is
first, and that prior to number is the relative, and that this is
prior to the absolute-besides all the other points on which certain
people, by following out the opinions held about the Forms, came
into conflict with the principles of the theory.
Again, according to the assumption on the belief in the Ideas
rests, there will be Forms not only of substances but also of many
other things; for the concept is single not only in the case of
substances, but also in that of non-substances, and there are sciences
of other things than substance; and a thousand other such difficulties
confront them. But according to the necessities of the case and the
opinions about the Forms, if they can be shared in there must be Ideas
of substances only. For they are not shared in incidentally, but
each Form must be shared in as something not predicated of a
subject. (By 'being shared in incidentally' I mean that if a thing
shares in 'double itself', it shares also in 'eternal', but
incidentally; for 'the double' happens to be eternal.) Therefore the
Forms will be substance. But the same names indicate substance in this
and in the ideal world (or what will be the meaning of saying that
there is something apart from the particulars-the one over many?). And
if the Ideas and the things that share in them have the same form,
there will be something common: for why should '2' be one and the same
in the perishable 2's, or in the 2's which are many but eternal, and
not the same in the '2 itself' as in the individual 2? But if they
have not the same form, they will have only the name in common, and it
is as if one were to call both Callias and a piece of wood a 'man',
without observing any community between them.
But if we are to suppose that in other respects the common
definitions apply to the Forms, e.g. that 'plane figure' and the other
parts of the definition apply to the circle itself, but 'what really
is' has to be added, we must inquire whether this is not absolutely
meaningless. For to what is this to be added? To 'centre' or to
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