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

MORE ABOUT LIMITS 291
Do not fail to observe how the tremendous generalization, so copiously enriching our human world of mathematical ideas, was brought about: we were operating in a
certain domain,-the domain of ratios; we were there
employing the notion of limit; using the notion, we found
ourselves looking for a limit where we were suddenly and
painfully astonished to find there was none; we were
baffled, we wondered, felt a need,-the need of a deeper
view, of a larger vision, of a more embracing conception
to extricate us; and we found it-how? By means of the
limit idea; that which got us into the difficulty got us
out of it and, in doing so, gave us a larger world.
Did the limit concept compel the generalization? No;
such generalization is never compelled,-it is suggested,
recommended, stimulated, even urged,-but not compelled as a conclusion from premises,-generalization always involves an act of will,-a choice between a smaller,
meaner view and a larger, nobler one; and in the present
instance it was, you see, the notion of limit that gave
man's will the necessary suggestion, incitement and
guidance.
There is another aspect of the matter which you as
philosophers must on no account fail to notice very
carefully, for it is a phenomenon of all genuine generalization. It is this: the world of the real numbers, though
itself a strictly actual world once it is found, yet is, for
any possible point of view in the domain of ratios, a
strictly ideal world-ideal in the just sense that, though
it is suggested by phenomena in the domain of ratios, it is
itself wholly outside thereof and can in no wise be attained
by pursuing sequences, however endless, within the
domain; a generally neglected fact of the utmost importance, pot Qnly inr diwçussing the spiritual bearings of