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are not. For there is nothing contrary to that which is primary; for
all contraries have matter, and things that have matter exist only
potentially; and the ignorance which is contrary to any knowledge
leads to an object contrary to the object of the knowledge; but what
is primary has no contrary.
Again, if besides sensible things no others exist, there will be
no first principle, no order, no becoming, no heavenly bodies, but
each principle will have a principle before it, as in the accounts
of the theologians and all the natural philosophers. But if the
Forms or the numbers are to exist, they will be causes of nothing;
or if not that, at least not of movement. Further, how is extension,
i.e. a continuum, to be produced out of unextended parts? For number
will not, either as mover or as form, produce a continuum. But again
there cannot be any contrary that is also essentially a productive
or moving principle; for it would be possible for it not to be. Or
at least its action would be posterior to its potency. The world,
then, would not be eternal. But it is; one of these premisses, then,
must be denied. And we have said how this must be done. Further, in
virtue of what the numbers, or the soul and the body, or in general
the form and the thing, are one-of this no one tells us anything;
nor can any one tell, unless he says, as we do, that the mover makes
them one. And those who say mathematical number is first and go on
to generate one kind of substance after another and give different
principles for each, make the substance of the universe a mere
series of episodes (for one substance has no influence on another by
its existence or nonexistence), and they give us many governing
principles; but the world refuses to be governed badly.

'The rule of many is not good; one ruler let there be.'


WE have stated what is the substance of sensible things, dealing
in the treatise on physics with matter, and later with the substance
which has actual existence. Now since our inquiry is whether there
is or is not besides the sensible substances any which is immovable
and eternal, and, if there is, what it is, we must first consider what
is said by others, so that, if there is anything which they say
wrongly, we may not be liable to the same objections, while, if
there is any opinion common to them and us, we shall have no private
grievance against ourselves on that account; for one must be content
to state some points better than one's predecessors, and others no
Two opinions are held on this subject; it is said that the objects
of mathematics-i.e. numbers and lines and the like-are substances, and
again that the Ideas are substances. And (1) since some recognize
these as two different classes-the Ideas and the mathematical numbers,
and (2) some recognize both as having one nature, while (3) some
others say that the mathematical substances are the only substances,
we must consider first the objects of mathematics, not qualifying them
by any other characteristic-not asking, for instance, whether they are
in fact Ideas or not, or whether they are the principles and
substances of existing things or not, but only whether as objects of
mathematics they exist or not, and if they exist, how they exist. Then
after this we must separately consider the Ideas themselves in a
general way, and only as far as the accepted mode of treatment
demands; for most of the points have been repeatedly made even by
the discussions outside our school, and, further, the greater part
of our account must finish by throwing light on that inquiry, viz.
when we examine whether the substances and the principles of
existing things are numbers and Ideas; for after the discussion of the
Ideas this remans as a third inquiry.
If the objects of mathematics exist, they must exist either in

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