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

MATHEMATICAL PHILOSOPHY
gruity between what is said, or the manner of saying it,
and what is meant. We are going to see that there is a
like incongruity between the meaning of the mathematical
term " variable " and the manner in which mathematicians
habitually speak of variables. I do not condemn their
manner of speech; I approve of it because it is stimulating
and economical and because it does not, except in certain
very fundamental questions, lead to error; but such
incongruity is a thing which you as philosophers should
carefully notice in the interest of clarity and critical
understanding. Mathematicians do indeed habitually
speak of variables as if the mathematical concept of a
variable were the concept of something whose essential
nature is to suffer change; that is to say, when they use
some symbol, say, x, to denote what they call a variable,
they familiarly speak of the variable x as altering its
value, as increasing or decreasing, as growing large or
growing small, as approaching or not approaching this or
that, and so on; yet, in spite of such a way of speaking,
what they really mean by the term " variable " essentially
involves no idea of change whatever as "change" is
commonly understood. This fact may, I believe, be made
sufficiently evident.
To make it evident let us, in seeking the meaning of the
term 'variable," recur to the idea of propositional
function; for, although some of the things called variables
in the logical theory of propositional functions are not so
called in traditional mathematics, yet whatever is called a
variable in mathematics appears explicitly or implicitly
as a variable in some propositional function; for example,
the variable x in such a propositional function as -x is a
man,-has not yet gained full recognition as a mathematical variable; on the other hand, the mathematical