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

THE PSYCHOLOGY OF MATHEMATICS 413
ness to light,-from doubt to conviction. Obviously sucli
a proposition is not mathematical; it is psychologicalit states a fact respecting the nature of a normal human
mind. Such interpretations of mathematical literature
are psychologically very illuminating; the possibility of
making them is so evident, once it is pointed out, that I
should have refrained from mentioning it except for the
fact of its being commonly overlooked and neglected.
For another example, consider the phenomenon of
generalization,-the process by which the human mind
from time to time enlarges the empire of its rational
activity. What is generalization as a process of mind,
as a mental event? What are the mental phenomena
involved? How? In. what relations? I am not going
to attempt to answer here. I wish merely to propose the
problem to students of psychology. Generalization occurs
in all fields of thought but in mathematics it may be seen
in its nakedness. There, then, is the best place to study
it as a phenomenon of mind. Take, for example, the
striking succession of generalizations by which the domain
of the number concept, which once contained nothing but
our familiar integers, has been gradually extended to
embrace positives and negatives, rationals and irrationals,
reals and imaginaries, cardinals and ordinals, including
the transfinite numbers of Georg Cantor; or take the
no less striking series of generalizations by which the
conception of geometry has been enlarged. As specimens
of generalization, those alluded to are probably the best
to be found in the history of thought. I venture to commend them as such to students of mind. Some of you
are psychologists. If you will study the great process
of generalization by help of the specimens mentioned,
then the rest of us will go to you confidently,-as laymen