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

INTRODUCTION 87
notions, or concepts,-as they are variously called,which enter as components into the stately edifice of
mathematics, though they are all of them, in a sense, indispensable to it, yet differ very widely in respect of their
place and rank, their dignity and structural service. Examination of the gréat edifice makes it evident that some
of them,-a relatively small number of them,-have the
distinction of being related to it as central supporting
pillars. Among the chief of these are the concepts denoted by the terms: Function-Propositional Function
-Implication-Proposition-Class-Relation-Postulate
System-Doctrinal Function-Doctrine-Variable-Limit
-Number-Finitude-Infinity- Transformation-Group
-Invariance. It is with such pillar-concepts,-which
are obviously not coordinate in rank,-that I purpose
to deal, and I shall deal with them primarily as concepts,
explaining them with constant regard to clarity, with a
minimum of technical symbols, and with a view, not
alone to their mathematical meanings, but to their significance and use in outlying fields of thought. But I shall
not endeavor to expound, in the proper sense of the
term, the great technical doctrines that have grown up
about them as subject matter, for such exposition would
demand, as you know, not merely one course, but many
courses, of lectures. You will rightly infer that, though
proof or demonstration may not be entirely absent, it will
not be permitted to detain us too long, much less to
dominate the discussions.
Let me say, finally, that the course is not designed to
be, in the stricter and narrower sense of the term, a course
in the philosophy of mathematics. It aims at being at
once something less and something more: less, in that it
does not endeavor to begin with the most ultimate of