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

318 MATHEMATICAL PHILOSOPHY
garded by mathematicians of the conservative and reactionary type with a good deal of suspicion as being, if not
crazy, at least a bit queer, over-romantic, and unsound,
it is now constantly employed as a great convenience by
mathematicians everywhere and even by physicists (say
in the kinetic theory of gases) quite without apology.
The literature of the subject is large and growing. In
Sommerville's Bibliography of Non-Euclidian Geometry,
Including the Theory of Parallels, the Foundations of
Geometry, and Space of n Dimensions (19 1) there are
listed 1832 references on n dimensions.
The concept has not, indeed, so great scientific and
spiritual dignity as some others,-as that of function, for
example, or relation or transformation or group or invariance or infinity or limit,-yet it is a very grave notion,
and it has, moreover, a certain double distinction: it is,
I mean, one of the few among the important concepts in
modern mathematics that philosophers have seriously
grappled with and one of the still fewer that have piqued
the curiosity of the educated public. The results of such
popular curiosity are themselves a little curious. Not
long ago, for example, I heard and read an address on
hyperspace which a professional astronomer had ventured
to make before an audience of university students. It
was not a happy performance; not only did the speaker
confound the idea of n-dimensional space with that of
non-Euclidean space, but he made it pathetically evident
that he had grasped neither the one idea nor the other,
nor did anyone in the evidently interested audience appear to observe the fact. I have cited this instance because, if a reputable astronomer can err so egregiously
in a matter that is not remote from the field of his special
studies, we ought not perhaps to be astonished at the