The Complete Aristotle (eng.)

a man and a horse; but those who hold these views say
that not even two units are 2.
    If the number of the 3-itself is not greater than that of the 2,
this is surprising; and if it is greater, clearly there is also a
number in it equal to the 2, so that this is not different from the
2-itself. But this is not possible, if there is a first and a
second number.
    Nor will the Ideas be numbers. For in this particular point they
are right who claim that the units must be different, if there are
to be Ideas; as has been said before. For the Form is unique; but
if the units are not different, the 2’s and the 3’s also will not
be different. This is also the reason why they must say that when
we count thus-’1,2’-we do not proceed by adding to the given
number; for if we do, neither will the numbers be generated from
the indefinite dyad, nor can a number be an Idea; for then one Idea
will be in another, and all Forms will be parts of one Form. And so
with a view to their hypothesis their statements are right, but as
a whole they are wrong; for their view is very destructive, since
they will admit that this question itself affords some
difficulty-whether, when we count and say —1,2,3-we count by
addition or by separate portions. But we do both; and so it is
absurd to reason back from this problem to so great a difference of
essence.
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8
    First of all it is well to determine what is the differentia of
a number-and of a unit, if it has a differentia. Units must differ
either in quantity or in quality; and neither of these seems to be
possible. But number qua number differs in quantity. And if the
units also did differ in quantity, number would differ from number,
though equal in number of units. Again, are the first units greater
or smaller, and do the later ones increase or diminish? All these
are irrational suppositions. But neither can they differ in
quality. For no attribute can attach to them; for even to numbers
quality is said to belong after quantity. Again, quality could not
come to them either from the 1 or the dyad; for the former has no
quality, and the latter gives quantity; for this entity is what
makes things to be many. If the facts are really otherwise, they
should state this quite at the beginning and determine if possible,
regarding the differentia of the unit, why it must exist, and,
failing this, what differentia they mean.
    Evidently then, if the Ideas are numbers, the units cannot all
be associable, nor can they be inassociable in either of the two
ways. But neither is the way in which some others speak about
numbers correct. These are those who do not think there are Ideas,
either without qualification or as identified with certain numbers,
but think the objects of mathematics exist and the numbers are the
first of existing things, and the 1-itself is the starting-point of
them. It is paradoxical that there should be a 1 which is first of
1’s, as they say, but not a 2 which is first of 2’s, nor a 3 of
3’s; for the same reasoning applies to all. If, then, the facts
with regard to number are so, and one supposes mathematical number
alone to exist, the 1 is not the starting-point (for this sort of 1
must differ from the-other units; and if this is so, there must
also be a 2 which is first of 2’s, and similarly with the other
successive numbers). But if the 1 is the starting-point, the truth
about the numbers must rather be what Plato used to say, and there
must be a first 2 and 3 and numbers must not be associable with one
another. But if on the other hand one supposes this, many
impossible results, as we have said, follow. But either this or the
other must be the case, so that if neither is, number cannot exist
separately.
    It is evident, also, from this that the third version is the
worst,-the view ideal and mathematical number is the same. For two
mistakes must then meet in the one opinion. (1) Mathematical number
cannot be of this sort, but the holder of this view has to spin it
out by making suppositions peculiar to himself. And (2) he must
also admit all the consequences that confront those who speak of
number in the sense of ‘Forms’.
    The Pythagorean version in one way affords fewer difficulties
than those before named, but in another way has others peculiar to
itself. For not thinking of number as capable of existing
separately removes many of the impossible consequences; but that
bodies should be composed of numbers, and that this