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

350 MATHEMATICAL PHILOSOPHY
non-Euclidean geometry, and what, if I discussed the
subject at all, it would be best to say. Well, the course
of my meditation respecting the first question ran as follows: the birth of non-Euclidean geometry was and is one
of the most momentous events in the history of thought;
no other has served to reveal in so clear a light the nature
of logical Fate and the nature, scope and limitation of
intellectual Freedom; no other has so well disclosed the
distinction,-which is radical and cannot be obliterated,
-between the world of conception and the world of per'ception,-the world of pure thought and the world of
sensuous experience; no other has so clearly defined the
great problem of ascertaining how the two worlds are
interrelated; the matter in question,-the advent and
nature of non-Euclidean geometry,-is one of the few
great mathematical matters that professional philosophers
have seriously sought to \understand and it is one that
has at the same time persistently haunted the imagination
of the educated portion of the non-mathematical public;
many, very many, have been the attempts to explain the
subject in the daily press, in spoken lectures, in magazines
and in books; nevertheless, outside of mathematical
circles, understanding of the matter, it must be owned, is
meagre; for making the matter clear to the man-in-thestreet adequate means has not yet been found. What, I
asked myself, is to be done? Must we in this case relinquish the hope of successful popularization? And I
answered, no, we must keep on trying; for I vividly recalled Gergonne's noble dream and the world's great
hope-Democracy; so my mind was set swarming with
the considerations adduced in the prelude; and that is
why I have presented it here.
Do I flatter myself with the belief that in this lecture