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

THE GROUP CONCEPT 215
of invariance upon the universal enterprise of Thought
are the bearings of a prototype and guide. It is evident
that the same is true of the mathematical theory of
groups. Our human question is: what abides? As
students of thought and the history of thought, we have
learned at length that the question can not be answered
fully at once but only step by step in an endless progression. And now what are the steps? You can scarcely
fail to see, if you reflect a little, that each of them,whether taken by art or by science or by philosophy,consists virtually in ascertaining either the invariants
under some group of transformations or else the groups
of transformations that leave some thing or things
unchanged.
Groups as Instruments for Defining, Delimiting, Discriminating and Classifying Doctrines.-The foregoing
question (2) has another aspect, which I believe to be of
profound interest to all students except those, if there be
such, who are insensate to things philosophical. I mean
that, if and whenever you ascertain the group of all the
transformations that leave invariant some specified object
or objects of thought, you thereby define perfectly some
actual (or potential) branch of science-some actual (or
potential) doctrine. I will endeavor to make this fact
evident by a few simple examples, and I will choose them
from the general field of geometry, though, as you will
perceive, such examples might be taken from other fields.
For a first example, consider the above-mentioned
group D of the displacements of our space. I say that
this group defines a geometry of space, which may be
called the geometry of displacements. It defines it by
defining, or delimiting, its subject-matter. What is its
subject-matter? What does the geometry study? The