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

286 MATHEMATICAL PHILOSOPHY
In order to arrive at a fair estimate of the philosophic
significance of limit concepts and limit processes,-in
order, that is, to win a fair sense of their function and
service in the life of Thought taken in all its varieties and
scope,-it is necessary as a preparation to examine the
matter a little further in mathematical light for it is
here and not elsewhere that concepts of limit and limit
processes are seen, and seen at work, in their nakedness
and purity. As beheld in that light, conceptions of limit,
apart from any question regarding their instrumental
value, are objects of no little interest-a fact well worthy
of passing mention, though I do not insist upon it in this
connection. Regarding instrumental value, we have seen
that limit concepts enable us to discriminate and classify
variables with reference to the constitution of their
ranges and to the connections of these with series; we
have seen that limit concepts are essential to the formation
and so to the meanings of innumerable other concepts,
many of them of great import, as that of functional continuity or that of derivative, instanced a moment since;
you know, or (if not) you can quickly learn by glancing
at mathematical literature that limit concepts play an
indispensable, perhaps the chief, rôle in the conduct of
proofs, or demonstrations, in all branches of Analysis and
its applications. I wish now to invite your best attention
to the fact that, over and above the foregoing types of
service, limit concepts render an invaluable service of a
radically distinct kind in connection with that very
familiar yet always strange thing which we are wont to
call " generalization." I mean the kind of generalization
which consists in our somehow contriving so to extend the
meaning of an established concept as to bring within its
enlarged scope,-as under the unity and order of a new