ARISTOTLE ON THE INFINITE
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does include one and the same magnitude, whatever it is, you
will come to the end of the finite magnitude, for every finite
magnitude is exhausted by continually taking from it any
definite fraction whatever. In no other sense does the infinite
exist but only in the sense just mentioned, that is, potentially
and by way of diminution. 1 And in this sense you may have
potentially infinite addition, the process being, as we say, in
a manner the same as with division ad infinitum ; for in the
case of addition you will always be able to find something-
outside the total for the time being, but the total will never
exceed every definite (or assigned) magnitude in the way that,
in the direction of division, the result will pass every definite
magnitude, that is, by becoming smaller than it. The infinite
therefore cannot exist, even potentially, in the sense of exceed
ing every finite magnitude as the result of successive addition.
It follows that the correct view of the infinite is the opposite
of that commonly held; it is not that which has nothing
outside it, but that which always has something outside it.“
Aristotle is aware that it is essentially of physical magnitudes
that he is speaking: it is, he says, perhaps a more general
inquiry that would be necessary to determine whether the
infinite is possible in mathematics and in the domain of
thought and of things which have no magnitude. 3
‘ But ’, he says, ‘ my argument does not anyhow rob
mathematicians of their study, although it denies the existence
of the infinite in the sense of actual existence as something
increased to such an extent that it cannot be gone through
(dSie^LTTjToi')', for, as it is, they do not even need the infinite
or use it, but only require that the finite (straight line) shall
be as long as they please. . . . Hence it will make no difference
to them for the purpose of demonstration/ 4
The above disquisition about the infinite should, I think,
be interesting to mathematicians for the distinct expression
of Aristotle’s view that the existence of an infinite series the
terms of which are magnitudes is impossible unless it is
convergent and (with reference to Riemann’s developments)
that it does not matter to geometry if the straight line is not
: nfinite in length provided that it is as long as we please.
1 Phys. iii. 6. 206 a 15~b 13.
3 lb. iii. 5. 204 a 34.
2 lb. iii. 6. 206 b 16-207 a 1.
4 lb. iii. 7. 207 b 27.