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

POSTULATES 41
changeably according to the taste of the author. It has
not always been so; the term axiom, for example, was
long used to denote "self-evident proposition," which is a
kind of proposition that modern mathematicians have not
been able to discover. But I shall not detain you with
an historical account of the terms, interesting and instructive as their history is. It gives me pleasure to say,
however, that, if you feel drawn thereto, as I hope you
do, you will find much more than an ample clue to it in
the introduction to Dr. T. L. Heath's superb edition of
Euclid's Elements where these terms and kindred matters
are set in the bright light of critical commentary from
the days of Plato down to the present time. In passing,
let me add, by way of indicating an opportunity, that
this work of Dr. Heath, like other works of his, attains
a high degree of excellence in a type of activity in which
our American mathematical scholarship has been singularly lacking; not because American mathematicians have
lacked facilities or ability, for these they have not lacked,
but because the universities in which they have received
their training and have done their work have not yet acquired the requisite atmosphere and spirit.
A postulate is one thing; a system of postulates is
another. In defining the former, I have by no means
defined the latter. It is not easy to do so with logical
precision: it is, I mean, not easy to give an abstract
definition of the generic concept denoted by the term,
postulate system; and I shall not attempt it at this point,
for it presupposes study of the concept as actually revealed in mathematics and so has its proper place at the
end of the study. Here, at the beginning, we must be
content with definition by example, with what Professor
Enriques, in his Problems of Science, has called concrete