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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 it two 2's other than the 2-itself; if not, the
2-itself will be a part of 4 and one other 2 will be added. And
similarly 2 will consist of the 1-itself and another 1; but if this is
so, the other element cannot be an indefinite 2; for it generates
one unit, not, as the indefinite 2 does, a definite 2.
Again, besides the 3-itself and the 2-itself how can there be
other 3's and 2's? And how do they consist of prior and posterior
units? All this is absurd and fictitious, and there cannot be a
first 2 and then a 3-itself. Yet there must, if the 1 and the
indefinite dyad are to be the elements. But if the results are
impossible, it is also impossible that these are the generating
If the units, then, are differentiated, each from each, these
results and others similar to these follow of necessity. But (3) if
those in different numbers are differentiated, but those in the same
number are alone undifferentiated from one another, even so the
difficulties that follow are no less. E.g. in the 10-itself their
are ten units, and the 10 is composed both of them and of two 5's. But
since the 10-itself is not any chance number nor composed of any
chance 5's--or, for that matter, units--the units in this 10 must
differ. For if they do not differ, neither will the 5's of which the
10 consists differ; but since these differ, the units also will
differ. But if they differ, will there be no other 5's in the 10 but
only these two, or will there be others? If there are not, this is
paradoxical; and if there are, what sort of 10 will consist of them?
For there is no other in the 10 but the 10 itself. But it is
actually necessary on their view that the 4 should not consist of
any chance 2's; for the indefinite as they say, received the
definite 2 and made two 2's; for its nature was to double what it
Again, as to the 2 being an entity apart from its two units, and
the 3 an entity apart from its three units, how is this possible?
Either by one's sharing in the other, as 'pale man' is different
from 'pale' and 'man' (for it shares in these), or when one is a
differentia of the other, as 'man' is different from 'animal' and
Again, some things are one by contact, some by intermixture,
some by position; none of which can belong to the units of which the 2
or the 3 consists; but as two men are not a unity apart from both,
so must it be with the units. And their being indivisible will make no
difference to them; for points too are indivisible, but yet a pair
of them is nothing apart from the two.
But this consequence also we must not forget, that it follows that
there are prior and posterior 2 and similarly with the other
numbers. For let the 2's in the 4 be simultaneous; yet these are prior
to those in the 8 and as the 2 generated them, they generated the
4's in the 8-itself. Therefore if the first 2 is an Idea, these 2's

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