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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.
Part 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,
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METAPHYSICS 165
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 should be mathematical number, is impossible.
For it is not true to speak of indivisible spatial magnitudes; and
however much there might be magnitudes of this sort, units at least
have not magnitude; and how can a magnitude be composed of indivisibles?
But arithmetical number, at least, consists of units, while these
thinkers identify number with real things; at any rate they apply
their propositions to bodies as if they consisted of those numbers.
"If, then, it is necessary, if number is a self-subsistent real thing, [ Pobierz całość w formacie PDF ]