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

VARIABLES AND LIMITS 235
the terms variable and constant? The great majority of
professional mathematicians take the meanings of these
terms mainly for granted; they do so because they are so
occupied in teaching mathematics or in extending its
superstructure as to have but little or no interest in
"the nice sharp quillets of the law " as revealed in its
logical foundations; in the foregoing lectures I have followed the actual practice of mathematicians in respect to
the two terms in question; I have, that is, freely employed
the terms, not indeed quite unconsciously, but without
apology and without explanation, believing that such use
would lead to no appreciable misunderstanding. Now,
however, before attempting to give a formal definition of
the great term " limit," it will be worth while, I believe,
to glance at its shadowy background-to reflect a little on
the meanings of the yet greater notions upon which the
concept of a limit depends.
Meaning of the Terms Fariable and Constant.-What is
the mathematical meaning of the term variable? It is
natural to answer, and to answer very confidently, that a
variable is something that varies or changes, like the
position of an object in motion, the time of day, the
length of a burning cigar. We are going to see that this
answer, though perfectly natural, is entirely wrong. In a
previous lecture I drew your attention to the fact that
mathematicians habitually employ highly figurative
speech; especially that they are constantly employing
dynamic terms in describing static facts. In particular
I pointed out that it is a common practice of mathematicians to use the dynamic term transformation,-suggesting change, variation, transmutation,-to denote what is
in fact a static thing, namely, a relation-something that
is unchanging, eternal. There is thus a striking incon