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

50 MATHEMATICAL PHILOSOPHY
sure, as the prosperity of a throat specialist upon the
moisture of the climate, as the attraction of material
particles upon their distance asunder, as prohibitionary
zeal upon intellectual distinction and moral elevation,
as rate of chemical change upon the amount or the mass
of the substance involved, as the turbulence of labor
upon the lust of capital, and so on and on without end.
This familiar notion of mutual dependence and mutual
variation thus exemplified in every turn and feature of
life and the world, is indeed a powerful concept; it is,
in a sense, the sole subject matter of science; its scientific name-function-was first pronounced, it is said,
by Leibniz; in modern mathematical analysis, it has
played a dominant rôle, giving both name and character
to certain great branches, as the theory of functions of
real variables and the theory of functions of complex
variables. Yet, powerful as it is, this Leibnizian conception, as employed in traditional mathematics, is far
inferior in scope to that denoted by propositional function, which indeed embraces the former as a special case.
What, then, are we to understand by this great term?
The answer, describing rather than strictly defining,
is that a propositional function is any statement containing one or more real variables, where, by a real variable,
is meant a name or other symbol whose meaning, or
value as we say, is undetermined in the statement but
to which we can at will assign in any order we please one
or more values, or meanings, now one and now another.
I fear that what I have just said is too general to be quite
intelligible. The idea can be made sufficiently clear,
however, by some simple examples-by concrete definition-provided you will understand that the examples
are to the general concept in question as a burning match