The Complete Aristotle (eng.)

say that Ideas exist; and others speak of the objects of
mathematics, but not mathematically; for they say that neither is
every spatial magnitude divisible into magnitudes, nor do any two
units taken at random make 2. All who say the 1 is an element and
principle of things suppose numbers to consist of abstract units,
except the Pythagoreans; but they suppose the numbers to have
magnitude, as has been said before. It is clear from this
statement, then, in how many ways numbers may be described, and
that all the ways have been mentioned; and all these views are
impossible, but some perhaps more than others.
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7
    First, then, let us inquire if the units are associable or
inassociable, and if inassociable, in which of the two ways we
distinguished. For it is possible that any unity is inassociable
with any, and it is possible that those in the ‘itself’ are
inassociable with those in the ‘itself’, and, generally, that those
in each ideal number are inassociable with those in other ideal
numbers. Now (1) all units are associable and without difference,
we get mathematical number-only one kind of number, and the Ideas
cannot be the numbers. For what sort of number will man-himself or
animal-itself or any other Form be? There is one Idea of each thing
e.g. one of man-himself and another one of animal-itself; but the
similar and undifferentiated numbers are infinitely many, so that
any particular 3 is no more man-himself than any other 3. But if
the Ideas are not numbers, neither can they exist at all. For from
what principles will the Ideas come? It is number that comes from
the 1 and the indefinite dyad, and the principles or elements are
said to be principles and elements of number, and the Ideas cannot
be ranked as either prior or posterior to the numbers.
    But (2) if the units are inassociable, and inassociable in the
sense that any is inassociable with any other, number of this sort
cannot be mathematical number; for mathematical number consists of
undifferentiated units, and the truths proved of it suit this
character. Nor can it be ideal number. For 2 will not proceed
immediately from 1 and the indefinite dyad, and be followed by the
successive numbers, as they say ‘2,3,4’ for the units in the ideal
are generated at the same time, whether, as the first holder of the
theory said, from unequals (coming into being when these were
equalized) or in some other way-since, if one unit is to be prior
to the other, it will be prior also to 2 the composed of these; for
when there is one thing prior and another posterior, the resultant
of these will be prior to one and posterior to the other. Again,
since the 1-itself is first, and then there is a particular 1 which
is first among the others and next after the 1-itself, and again a
third which is next after the second and next but one after the
first 1,-so the units must be prior to the numbers after which they
are named when we count them; e.g. there will be a third unit in 2
before 3 exists, and a fourth and a fifth in 3 before the numbers 4
and 5 exist.-Now none of these thinkers has said the units are
inassociable in this way, but according to their principles it is
reasonable that they should be so even in this way, though in truth
it is impossible. For it is reasonable both that the units should
have priority and posteriority if there is a first unit or first 1,
and also that the 2’s should if there is a first 2; for after the
first it is reasonable and necessary that there should be a second,
and if a second, a third, and so with the others successively. (And
to say both things at the same time, that a unit is first and
another unit is second after the ideal 1, and that a 2 is first
after it, is impossible.) But they make a first unit or 1, but not
also a second and a third, and a first 2, but not also a second and
a third. Clearly, also, it is not possible, if all the units are
inassociable, that there should be a 2-itself and a 3-itself; and
so with the other numbers. For whether the units are
undifferentiated or different each from each, number must be
counted by addition, e.g. 2 by adding another 1 to the one, 3 by
adding another 1 to the two, and similarly. This being so, numbers
cannot be generated as they generate them, from the 2 and the 1;
for 2 becomes part of 3 and 3 of 4 and the same happens in the case
of the succeeding numbers, but they say 4 came from the first 2 and
the indefinite which makes