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

214 MATHEMATICAL PHILOSOPHY
specified point invariant is such a sub-group; the set
leaving two given points unchanged is another. How is
the latter related to the sub-group leaving only one of the
two points invariant? Is there a displacement leaving
three non-collinear points invariant? Do the displacements leaving a line unchanged constitute a group? Such
questions are but samples of many that you will find it
profitable to ask and to try to answer.
For the sake of emphasis, permit me to repeat the two
big questions: (I) Given a group of transformations,
what things are unchanged by them? (2) Given something-an object or property or relation, no matter what
-that is to remain invariant, what are the groups of
transformations, and especially the largest group, that
leave the thing unaltered? You may wish to say: I
grant that the questions are interesting, and I do not
deny that they are big-big in the sense of giving rise to
innumerable problems and big in the sense that many of
the problems are difficult; but I do not see that they are
big with importance. Why should I bother with them?
In reply I shall not undertake to say why you should
bother with them; it is sufficient to remind you that as
human beings you cannot help it and you do not desire to
do so. In the preceding lecture, we saw that the sovereign
impulse of Man is to find the answer to the question:
what abides? We saw that Thought,-taken in the widest
sense to embrace art, philosophy, religion, science, taken
in their widest sense,-is the quest of invariance in a
fluctuant world. We saw that the craving and search for
things eternal is the central binding thread of human
history. We saw that the passion for abiding reality is
itself the supreme invariant in the life of reason. And
we saw that the bearings of the mathematical theory