[1086a]
[1]
Further, the fact that the leading
authorities1 disagree about numbers indicates that it is the
misrepresentation of the facts themselves that produces this confusion
in their views.Those2 who recognize only the objects of mathematics as
existing besides sensible things, abandoned Ideal number and posited
mathematical number because they perceived the difficulty and
artificiality of the Ideal theory. Others,3 wishing to
maintain both Forms and numbers, but not seeing how, if one posits
these4 as first principles,
mathematical number can exist besides Ideal number, identified Ideal
with mathematical number,—but only in theory, since actually
mathematical number is done away with, because the hypotheses which
they state are peculiar to them and not mathematical.5 And he6 who first assumed that there are Ideas, and that
the Ideas are numbers, and that the objects of mathematics exist,
naturally separated them. Thus it happens that all are right in some
respect, but not altogether right; even they themselves admit as much
by not agreeing but contradicting each other. The reason of this is
that their assumptions and first principles are wrong;and it is difficult to
propound a correct theory from faulty premisses: as Epicharmus says,
"no sooner is it said than it is seen to be wrong."7We have
now examined and analyzed the questions concerning numbers to a
sufficient extent; for although one who is already convinced might be
still more convinced by a fuller treatment,
[20]
he who is not convinced would be
brought no nearer to conviction.As for the first principles and causes and
elements, the views expressed by those who discuss only sensible
substance either have been described in the Physics8 or have no place in
our present inquiry; but the views of those who assert that there are
other substances besides sensible ones call for investigation next
after those which we have just discussed. Since, then,
some thinkers hold that the Ideas and numbers are such substances, and
that their elements are the elements and principles of reality, we
must inquire what it is that they hold, and in what sense they hold
it. Those9 who posit only numbers,
and mathematical numbers at that, may be considered later10; but as
for those who speak of the Ideas, we can observe at the same time
their way of thinking and the difficulties which befall them. For they
not only treat the Ideas as universal substances, but also as
separable and particular.(That this is impossible has been already shown11 by a consideration of the
difficulties involved.) The reason why those who hold substances to be
universal combined these two views was that they did not identify
substances with sensible things.
1 Alexander preferred the reading πρώτους, interpreting it in this sense; and I do not see why he should not be followed. Ross objects that πρῶτος is used in the chronological sense in 16., but this is really no argument. For a much more serious (although different) inconsistency in the use of terms cf. Aristot. Met. 12.3.1.
2 Speusippus and his followers.
3 Xenocrates and his followers.
4 Unity and the indeterminate dyad; for the difficulty see Aristot. Met. 13.7.3, 4.
5 Cf. Aristot. Met. 13.6.10.
6 Plato.
7 Epicharmus, Fr. 14, Diels.
8 Aristot. Physics 1.4-6.
9 The Pythagoreans and Speusippus.
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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