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the 1 is an element and principle of things suppose numbers to consist
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METAPHYSICS 162
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.
Part 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
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METAPHYSICS 163
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 [ Pobierz całość w formacie PDF ]