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

410 MATHEMATICAL PHILOSOPHY
matter further. The evidence now before us is sufficient.
It is perfectly clear that in course of the centuries the
progress of mathematics has been much retarded, sometimes arrested for long periods or diverted from its
natural course, by a psychology which, in things mathematical, often did not know a knee from an elbow.
Is mathematics retarded by that cause today? I
believe that mathematical research is not much thus retarded-at all events not directly. No doubt the number
of mathematicians who are also expert psychologists is
very small. But research mathematicians usually, though
not always, understand the psychology of their own science
well enough to recognize a mathematical idea as being
such, wherever and whenever it occurs. If it be a new
one and be found to be interpretable in the world of
perception or in the world of imagination, they are
thereby rejoiced, naturally so; but if it be not thus interpretable, as it may not be, they are not so psychologically unenlightened as to refuse it hospitality on that
account. The history of their subject has taught them
better.
But mathematical research and the dissemination of
mathematical knowledge are very different things. In
respect of the latter, I have no doubt that, if teachers
of mathematics were better trained in psychology and
especially in the psychology of mathematics, their teaching would be far more effective; for questions of logic
would then be seen and set in clearer light, less frequently
confused with psychological considerations, while the
latter, presented as such, would often contribute to the
instruction a light of their own. Will you allow me a
word of personal experience? I count it a great good
personal fortune that as a young man I received mathe