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

290 MATIEMATICAL PHILOSOPHY
upper limit of T, is the ratio -, but we must not confound
this rational 2 (which is a segment of ratios) with the
cardinal 2 nor with the integer 2, nor with the ratio }
(commonly denoted by 2) nor with any other number that
blundering custom may yet denote by the same symbol;
the irrational number or segment represented by V is
denoted by I/2; and so on analogously for analogous
a a
cases; to each ratio a will correspond a rational number,
b b
so that, for example, to the ratio - will correspond the
conceptually distinct rational number 2; to the irrational
numbers, however, no ratios will thus correspond; the
rationals and the irrationals taken together we will call
real numbers; these may be arranged in the order of
increasing magnitude by a sequence S', and, if we then
let F's range be, not the class of ratios less than the ratio 2,
but the class of rationals less than the rational 2, F will
indeed have 2-rational 2-for upper limit; and so at
length the mystery is dispelled-what fooled us before
was our confounding the familiar class of ratios with the
then unknown, yet vaguely felt, class of rationals, corresponding to but logically distinct from the ratios.
Well, what is it that has happened here in our racial
history? I hope you see that what has happened is this:
we have made a leap, an immense forward leap, in the
course of mathematical evolution; we have made a great
number-generalization; we have, that is, extended our old
familiar well-established concept and name of numbers
so as to make it include and cover two immense new
varieties, namely, the rationals,-which are as multitudinous as the infinite host of our old traditional ratios,and the irrationals,-which may be shown to be infinitely
more in multitude than all the old numbers taken together.