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

THE PSYCHOLOGY OF MATHEMATICS 409
it was born or well-nigh born, as we have seen, to the
genius of Epicurus and had the fortune to inspire the
great poem of Lucretius. With this exception, it had
no career in science and none in philosophy-it was
sterile. Why? The same old trouble-a shallow psychology. For, as you know, an infinite class must have
a part containing as many things as the whole class contains. But who ever saw such a class or ever imagined
one? "Nonsense " exclaimed psychology, and the great
conception,-so important for science, for philosophy
'and for rational theology,-slumbered for twenty centuries.
Passing to another field, we find that the development
or generalization of the number concept was greatly
hampered by the same cause. The descriptive terms,"surd" (which means absurd), "irrational," "imaginary,"
and "impossible,"-which were applied to large classes
of numbers that had been literally forced upon the attention of mathematicians by familiar operations, sufficiently
tell the tale. Mathematically those numbers, as we now
know, were quite as genuine, quite as legitimate, as the
ordinary integers and fractions. Why, then, were they
called "surd," "irrational," "imaginary," and "impossible"? Because they encountered a psychology that did
not understand the nature,-the mental nature, —of
mathematical generalization: a psychology which held
that the new "numbers," in order to be legitimate, must
conform to the familiar laws of the old ones and must,
moreover, like the old ones, admit of interpretation or
application in the so-called "actual" world of sense-perception.
The history of many another mathematical development bears similar witness. But we need not pursue the