The Complete Aristotle (eng.)

impossible that substance and that whose
substance it is should exist apart; how, therefore, could the
Ideas, being the substances of things, exist apart?
    In the Phaedo the case is stated in this way-that the Forms are
causes both of being and of becoming. Yet though the Forms exist,
still things do not come into being, unless there is something to
originate movement; and many other things come into being (e.g. a
house or a ring) of which they say there are no Forms. Clearly
therefore even the things of which they say there are Ideas can
both be and come into being owing to such causes as produce the
things just mentioned, and not owing to the Forms. But regarding
the Ideas it is possible, both in this way and by more abstract and
accurate arguments, to collect many objections like those we have
considered.
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6
    Since we have discussed these points, it is well to consider
again the results regarding numbers which confront those who say
that numbers are separable substances and first causes of things.
If number is an entity and its substance is nothing other than just
number, as some say, it follows that either (1) there is a first in
it and a second, each being different in species,-and either (a)
this is true of the units without exception, and any unit is
inassociable with any unit, or (b) they are all without exception
successive, and any of them are associable with any, as they say is
the case with mathematical number; for in mathematical number no
one unit is in any way different from another. Or (c) some units
must be associable and some not; e.g. suppose that 2 is first after
1, and then comes 3 and then the rest of the number series, and the
units in each number are associable, e.g. those in the first 2 are
associable with one another, and those in the first 3 with one
another, and so with the other numbers; but the units in the
‘2-itself’ are inassociable with those in the ‘3-itself’; and
similarly in the case of the other successive numbers. And so while
mathematical number is counted thus-after 1, 2 (which consists of
another 1 besides the former 1), and 3 which consists of another 1
besides these two), and the other numbers similarly, ideal number
is counted thus-after 1, a distinct 2 which does not include the
first 1, and a 3 which does not include the 2 and the rest of the
number series similarly. Or (2) one kind of number must be like the
first that was named, one like that which the mathematicians speak
of, and that which we have named last must be a third kind.
    Again, these kinds of numbers must either be separable from
things, or not separable but in objects of perception (not however
in the way which we first considered, in the sense that objects of
perception consists of numbers which are present in them)-either
one kind and not another, or all of them.
    These are of necessity the only ways in which the numbers can
exist. And of those who say that the 1 is the beginning and
substance and element of all things, and that number is formed from
the 1 and something else, almost every one has described number in
one of these ways; only no one has said all the units are
inassociable. And this has happened reasonably enough; for there
can be no way besides those mentioned. Some say both kinds of
number exist, that which has a before and after being identical
with the Ideas, and mathematical number being different from the
Ideas and from sensible things, and both being separable from
sensible things; and others say mathematical number alone exists,
as the first of realities, separate from sensible things. And the
Pythagoreans, also, believe in one kind of number-the mathematical;
only they say it is not separate but sensible substances are formed
out of it. For they construct the whole universe out of
numbers-only not numbers consisting of abstract units; they suppose
the units to have spatial magnitude. But how the first 1 was
constructed so as to have magnitude, they seem unable to say.
    Another thinker says the first kind of number, that of the
Forms, alone exists, and some say mathematical number is identical
with this.
    The case of lines, planes, and solids is similar. For some think
that those which are the objects of mathematics are different from
those which come after the Ideas; and of those who express
themselves otherwise some speak of the objects of mathematics and
in a mathematical way-viz. those who do not make the Ideas numbers
nor