This paper is a discussion, written as a result of a request of Professor Gonseth, of certain points concerning the philosophy of mathematics. It is a revision of my previous discourse, on this subject, which I now regard as inadequate. The argument is based directly on my contact with mathematics without benefit of any technical acquaintance with philosophy. I have not attempted to confine myself with what is novel; but the paper is intended to be self-contained.

The principal thesis is that mathematics may be conceived as an objective science which is independent of any except the most rudimentary philosophical assumptions. It is a body of propositions dealing with a certain subject matter; and these propositions are true insofar as they correspond with the facts. The position taken is a species of formalism, which may be called empirical formalism.

The problem of mathematical truth

There are three principal types of opinion as to the subject matter of mathematics, viz. realism, idealism, and formalism. We shall consider here the realist and intuitionist views, leaving formalism for the next section.

According to realism, mathematical propositions express the most general properties of our physical environment. Although this is the primitive view of mathematics, yet, on account of the essential role played by infinity in mathematics, it is untenable to-day.

On the idealistic view mathematics deals with the properties of mental objects of some sort. There are various varieties of this view according to the nature of these mental objects.