In the spring of 1915 the author was invited to deliver a course of lectures before a club of graduate students of the University of Illinois on the subject The Philosophy of Mathematics. This club consisted of students who had had an ordinary college course in mathematics, for the most part. This fact tended to narrow the field to be covered, inasmuch as the more difficult questions of mathematical philosophy had to be omitted. It tended to widen the field in the way of making it intelligible to all students of fair mathematical knowledge, which could be accomplished best by consider ing mathematics constantly in its historical development. This class of readers is the one directly addressed in the lectures. The large class of secondary and collegiate teachers of mathematics is also addressed to a great extent.