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

898 MATHEMATICAL PHILOSOPHY
Some Questions.-Hereupon certain questions naturally supervene. One of them is this: How can the
symbols or elements or terms that constitute the mathematical continuum be effectively employed in studying
such a magnitude as pressure, for example, or gravity or
velocity? The answer seems to reside in two considerations: one of them, which I have hitherto mentioned in
these lectures, is the fact that the various kinds of the
magnitude in question are each of them conceived to be
composed of, or decomposable into, parts or elements
matching in a one-to-one way the elements or terms that
constitute the mathematical continuum, and related
among themselves as the terms of the continuum are
related among themselves; the second consideration is
the fact that natural science is concerned, not with the
constituents of a magnitude, but only with the relations
among them. This second consideration, which so easily
escapes attention, is one of those fundamental matters
which Professor Poincaré never wearied of insisting upon.
See his Science and Hypothesis, for example.
Another natural question is this: Does the replacement
of the sensible or of the rational continua by continua
patterned on the model of the Grand Continuum guarantee us against all difficulties resembling the kind represented by the possibility of two lines crossing without
intersecting? The answer is, no. This particular kind
of difficulty has indeed been overcome by means of the
Grand Continuum. But there remain to surprise us other
difficulties of a somewhat similar kind. Let us glance at
one of them. Consider a sensible curve and a sensible
straight line. We can always dispose them so that they
will have a common part without crossing. Let us now
replace them, in thought, by corresponding conceptual