Mathematical philosophy, a study of fate and freedom; lectures for educated laymen, by Cassius J. Keyser.

314 MATHEMATICAL PHILOSOPHY
him by means of such a series as, ~, i,..., which
his own words indeed suggest, he would prbbably have
said in effect: "Composed of man-made symbols like
words, your series is not and never can be endless; to
speak of the sum of a non-existing endless series is meaningless; moreover, even if we supposed the series to be
endless, to be summable and to have I for its sum, this
would be neither finite nor infinite, for it would not be
a magnitude, inasmuch as the summands are themselves
not magnitudes but are merely empty abstract symbols;
if I be said to be a magnitude, in the sense of representing
a magnitude, then indeed, if magnitude I be composed
of two equal magnitudes, I grant that ~ will be a magnitude in the same sense (of representing one); if all the
symbols be magnitudes in that same sense, the summation
of the series of abstract symbols may be said to be the
summation of an endless (infinite) series of magnitudes;
but otherwise, not; and now what I have contended in
my poem is that, if your magnitude I be finite, not more
than a finite number of the symbols in the series can be
magnitudes, and this contention, denying the endless divisibility of finite magnitude,-especially denying that an
atom has an infinitude of parts,-is based on physical
considerations-on grounds other than that advanced in
the passage you have quoted from my argument." If
Lucretius thus replied to you, what suitable rejoinder, if
any, could you make?
I shall not attempt to recount here, much less to estimate, those other grounds. It must, however, be said, in
passing, that one of them is, in point of kind, almost perfectly represented by the following words of Clerk Maxwell (Theory of Heat, p. 285):