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

86 MATHEMATICAL PHILOSOPHY
losophy with its towering heroes, its men of "summitminds": with Plato, for example, who knew perfectly the
mathematics of his time, whose sense and revelation of
its spiritual significance has never been surpassed, and
whose influence in his own and all succeeding ages has
given his name a permanent place in mathematical history;
and with Aristotle, whose discussions of such fundamental
questions as the nature of mathematical definition, hypothesis, axiom, postulate, and subject matter, are of high
value even today and whose great contributions to logic
must now be regarded, in the light of modern symbolic
logic, as being, though he did not know it, genuine contributions to mathematics; and with Descartes, discoverer
of important mathematical propositions, and chief inventor of analytical geometry,-second in scientific power to
only one among mathematical methods; and with Leibniz,
father of modern symbolic logic and co-inveritor with
Newton of the infinitesimal calculus, "the most powerful
instrument of thought yet devised by the wit of man";l
and with Spinoza to whose lot it fell to try the great experiment,-inevitable in the history of thought,-of clothing ethical theory,-highest of human interests,-with
the strength and beauty of mathematical rigor and form,
and, in trying it, to exemplify in a singularly noble way,
the fact that illustrious failures fall to the lot of none
but illustrious men; and with other great philosophic
personalities, if I did not fear to weary you in naming
them, who by their mathematical competence worthily
represent the heroic tradition.
In closing this initial lecture, I desire to indicate in a
general way the sort of topics with which the following
lectures will deal. The endless number of the ideas, or
1See the preface of Professor W. B. Smith's Infinitesimal Calculus.