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

284 MATHEMATICAL PHILOSOPHY
ing it is very easy if we take "scientific " in its stricter
and narrower sense. We may go at once to the heart of
the matter by reflecting a little upon the most rigorously
scientific of scientific subjects and procedures-the Differential and Integral Calculus-and upon its ramifications
and its applications. Some of you have had a beginner's
course in the calculus; others of you, not; I am not
going to offer here an introduction to it but will merely
state succinctly, by way of reminder or of information, a
few such facts respecting it as will make indubitable the
great scientific importance of the term in question. One
of the facts is that the Calculus is primarily and mainly
concerned with what mathematicians call continuous
functions (or variables), and that both functional continuity and functional discontinuity, with which latter
the calculus is also concerned, are not only defined by
means of limits, but are indeed not otherwise definable.
Another of the salient facts is that among the host of
ideas met with in the Calculus three ideas are supremenamely, those denoted by the terms Derivative, Antiderivative (or Indefinite Integral) and Definite Integraland the three essentially involve a limit conception, the
first and third of them directly, the second one indirectly.
It follows, as you see, that in all the multifarious ramifications and applications of the Calculus, whether in differential equations or function theories or geometry or mechanics or astronomy or physics or chemistry or other fields
into which the calculus has found or is inevitably finding
its way, some variety of limit conception is continually
playing an indispensable scientific rôle. Indeed it is only
by prolonged meditation upon the matter that one can
even fairly begin to realize how very deeply the progress
of science and therewith of civilization depends upon