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

228 MATHEMATICAL PHILOSOPHY
The fact that the precise formation ofthe mathematical
concept of group is of so recent date is all the more curious
because an idea closely resembling that of group has
haunted the minds of a long line of thinkers and is found
stalking like a ghost in the mist of philosophic speculation
from remote antiquity down even to Herbert Spencer.
1 refer to those worldwide, age-long, philosophic speculations which, because of their peculiar views of the universe, may be fitly called the Philosophy of the Cosmic
Cycle or Cosmic Year. This philosophy, despite the spell
of a certain beauty inherent in it, has lost its vogue,
To-day we are accustomed to thinking of the universe as
undergoing a beginningless and endless evolution in course
of which no aspect or event ever was or ever will be
exactly repeated. In sharpest contrast with that conception, the philosophy of the cosmic cycle regards all
the changes of which the universe is capable as constituting an immense indeed but finite and closed system of
transformations, which follow each other in definite succession, like the spokes of a gigantic revolving wheel,
until all possible changes have occurred in the lapse of a
long but finite period of time-called a cosmic cycle or
cosmic year-whereupon everything is repeated precisely,
and so on and on without end. This philosophy, I have
said, has lost its vogue; but, if the philosophy be true,
it will regain it, for, if true, it belongs to the cosmic
cycle and hence will recur. The history of the philosophy
of the cosmic year is exceedingly interesting and it would,
I believe, be an excellent subject for a doctor's dissertation. The literature is wide-ranging in kind, in place and
in time. Let me cite a little of it as showing how closely
its central idea resembles the mathematical concept of a
cyclic group.