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

THE PSYCHOLOGY OF MATHEMATICS 415
mean, of discovering the relations between the data of
sense and the conceptual objects of science,-the problem,
in other words, of ascertaining whether, how, and to
what extent such conceptual and hypothetical objects
(points, instants, space, time, atoms, electrons, ether,
etc.) can be replaced by objects actually constructed out
of sense-given data and having the properties demanded
by science. If such constructions can be made, science
will be able to dispense with many hypotheses or many
"as ifs." That problem, it is evident, is a truly great
one.
In this lecture (as also in preceding ones)' I have
repeatedly emphasized the importance of a certain psychological distinction which I have called the distinction
between imagination and conception. It is today well
recognized by all mathematicians. They are accustomed
to designating it,-not quite happily, I believe,-as the
distinction between "intuition" and "analysis." It is the
distinction between the power of the mind to picture
and its power to think. We have seen that failure to
make it has often retarded the progress of mathematics
directly and that of kindred sciences indirectly. It is
absolutely essential to the philosophy of science; without
it the history of thought cannot be understood. I am
here reminding you of the matter because the considerations with which I intend to close will incidentally shed
new light upon it.
I wish to call your attention to certain contrasting
psychological phenomena that seem not to have found
recognition in the literature of psychology. I shall present them without attempting to explain them. What I
wish to point out is that, in relation to space, conception
or thought is symmetric in its representations and that