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

254 MATHEMATICAL PHILOSOPHY
once that there are two respects in which they agreethey involve the notion of variable and its range (every
limit is a limit of a variable) and all but one of them
involve the notion of difference. The notion of variable
we have discussed at unusual length and it is, I trust, now
fairly clear. It will be useful, I believe, to say a preliminary word regarding the notion of difference. It is
not my aim to define the general notion; my aim is merely
to enliven a little our consciousness of it. In the background of our human thinking, however refined, however
precise the ideas we are explicitly handling, there lurk
other ideas-shadowy, nebulous, vague-which we have
not defined, which we may not have attempted to define,
which we may not even be conscious of; yet these background ideas give our so-called precise ones all the meaning
the latter have. In the present discussion, the idea of
difference is a background idea.
Any given variable, as we have seen, has a range-a
certain class of things, or objects, called the terms of the
class and commonly spoken of as the variable's " values."
A class being given, each of its terms is comparable with
each in one or more respects: that is to say, each of them
differs from each in one or more respects; the respects,
and hence the differences, may be very definite or fairly
definite or very vague; the differences may be differences
in respect of position or of magnitude (size) or of number
or color or shape or weight or of importance or of dignity
or of beauty or of sensibility and so on; we may, therefore,
speak of kinds of difference as distinguished from amounts
of a given kind. It is essential to note the obvious fact
that, if each term of a class differs from each of its fellow
terms in respect to some specific kind k of difference, it
may happen that the terms in the class differ from some