[1082b]
[1]
e.g., one might say that animals are
composed of animals, if there are Ideas of animals. In
general, to regard units as different in any way whatsoever is absurd
and fictitious (by "fictitious" I mean "dragged in to support a
hypothesis"). For we can see that one unit differs from another
neither in quantity nor in quality; and a number must be either equal
or unequal—this applies to all numbers, but especially to
numbers consisting of abstract units.Thus if a number is neither more nor less, it
is equal; and things which are equal and entirely without difference
we assume, in the sphere of number, to be identical. Otherwise even
the 2's in the Ideal 10 will be different, although they are equal;
for if anyone maintains that they are not different, what reason will
he be able to allege? Again, if every unit plus
another unit makes 2, a unit from the Ideal 2 plus one from the Ideal
3 will make 2—a 2 composed of different units1; will this be prior or posterior to 3?
It rather seems that it must be prior, because one of the units is
contemporaneous with 3, and the other with 2.2 We
assume that in general 1 and 1, whether the things are equal or
unequal, make 2; e.g. good and bad, or man and horse; but the
supporters of this theory say that not even two units make
2.If the number of the Ideal 3 is
not greater than that of the Ideal 2,
[20]
it is strange; and if it is greater, then clearly
there is a number in it equal to the 2, so that this number is not
different from the Ideal 2.But this is impossible, if there is a first and second number.3 Nor will the Ideas be numbers. For on this
particular point they are right who claim that the units must be
different if there are to be Ideas, as has been already stated.4 For the
form is unique; but if the units are undifferentiated, the 2's and 3's
will be undifferentiated.Hence they have to say that when we count like this, l, 2, we do not
add to the already existing number; for if we do, (a) number will not
be generated from the indeterminate dyad, and (b) a number cannot be
an Idea; because one Idea will pre-exist in another, and all the Forms
will be parts of one Form.5 Thus in relation to their hypothesis they are
right, but absolutely they are wrong, for their view is very
destructive, inasmuch as they will say that this point presents a
difficulty: whether, when we count and say "1, 2, 3," we count by
addition or by enumerating distinct portions.6 But we do both;
and therefore it is ridiculous to refer this point to so great a
difference in essence.
1 Which conflicts with the view under discussion.
2 The implication seems to be, as Ross says, that the Platonists will refuse to admit that there is a number between 2 and 3.
3 i.e., if numbers are specifically different. Cf. Aristot. Met. 13.6.1.
4 sect. 2-4 above.
5 i.e., the biggest number.
6 This is Apelt's interpretation of κατὰ μερίδας. For this sense of the word he quotes Plut. Mor. 644c. The meaning then is: If you count by addition, you regard number as exhibited only in concrete instances; if you treat each number as a "distinct portion" (i.e. generated separately), you admit another kind of number besides the mathematical. Aristotle says that number can be regarded in both ways.
Aristotle. Aristotle in 23 Volumes, Vols.17, 18, translated by Hugh Tredennick. Cambridge, MA, Harvard University Press; London, William Heinemann Ltd. 1933, 1989.
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