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

INTRODUCTION 15
could feel the weight of such a response, and I did not
make it. It is, you observe, a response in terms of quan.
tity. Quantity is indeed omnipresent in our world; but
so, too, is quality, and of the two things, the latter is perhaps the more universal in its appeal. Algebra is indeed
essential to the theory of quantity and the theory of
quantity is essential to the subjugation of natural resources
to the use of man; of quality, on the other hand, algebra
is not a science but, though it is not a science of quality,
it has a quality, a human quality, to which it owes its high
rank in the spiritual hierarchy of human disciplines. And
so I endeavored, with poor success I fear, to answer the
challenge in terms of quality. I invoked the principle
which in this lecture I have been calling the principle of
humanistic education. I sought, that is, to make it clear
that, in contrast with the practical arts, the science of algebra as a discipline possesses a certain quality by virtue
of which, if the subject be rightly administered, the
student is gradually brought into the presence of one of
those great standards of excellence by which, as we have
seen, distinctively human activity in all its principal types
is to be guided and judged. The standard to which I
refer is, as you have doubtless surmised, the standard of
excellence in the quality of thinking as thinking-the
standard which mathematicians are accustomed to call
Logical Rigor-clarity, that is, precision and coherence.
And now the mention of that great term may serve
to reassure you, should you have begun to suspect that
in the course of this rather long excursion I may have
forgotten the question initiating it. The question is:
How much mathematical training is essential to the appropriate education of men and women as human beings?
I have said that the question admits of a definite answer