Are the traditional philosophies of mathematics really incompatible?

Abstract

Having been under the impression that most mathematicians do not care about the foundations of their subject, I was amazed by the heat generated by this topic in recent issues of the Mathematical Intelligencer, particularly in the letters to the editor (see, e.g., Paris [25]). The purpose of this article is to marshal a number of facts that support a certain philosophical thesis, which I hope to persuade at least some readers to share. I would like to argue that, contrary to widely held opinion, the traditional philosophies, logicism, formalism, Platonism, and intuitionism, if stated with sufficient moderation, do not really contradict each other, although I still have some reservations about logicism. This idea was first proposed in our book [17] by Phil Scott and me and elaborated for a philosophical audience in collaboration with Jocelyne Couture [5]. The present discussion owes a considerable debt to both co-authors. For background material on the traditional mathematical philosophies, the reader is referred to the standard references Benacerraf and Putnam [1], Hintikka [10], and van Heijenoort [30]. There are a number of problems a philosophy of mathematics should address. Perhaps the most important of these are: How is mathematical knowledge obtained (epistemology), and why can it be applied to nature? However, we shall here confine attention to another problem: What is the nature of mathematical entities (ontology) and of mathematical truth? The best-known mathematical philosophies have given different answers to this ontological question (see [5]), which we shall summarize here in rather abbreviated form.