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

LECTURE Xn
The Group Concept
THE NOTION SIMPLY EXEMPLIFIED IN MANY FIELDS-IS
"c MIND " A GROUP? —GROUPS AS INSTRUMENTS FOR
DELIMITING DOCTRINES-CONNECTION OF GROUP WITH
TRANSFORMATION AND INVARIANCE-THE IDEA FORESHADOWED IN THE AGES OF SPECULATION-THE
PHILOSOPHY OF THE COSMIC YEAR-THE IDEA OF
PROGRESS.
You will recall that near the close of the introductory
lecture I gave a partial list of those mathematical terms
which may be rightly regarded as denoting the pillarconcepts of the science. Among these are function, relation, transformation, invariance and group. In Lecture X
we saw that the first three denote three aspects of one and
the same thing seen from different points of view, and
this thing,-whether we call it function or relation or
transformation,-is sovereign-the central support not
only of mathematics but of the entire edifice of science
taken in the widest sense. In the closest logical connection therewith stand the two great concepts of invariance
and group; so that I can hardly overemphasize the importance of your learning to associate the three notions as
intimately in your thought as they are associated in fact:
Transformation - Invariance - Group. Of transformation I endeavored to give some account in Lecture X and
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