THE REVIEW OF SYMBOLIC LOGIC Volume 5, Number 3, September 2012
ON ONTOLOGY AND REALISM IN MATHEMATICS HAIM GAIFMAN Philosophy Department, Columbia University
Outline The paper is concerned with the way in which “ontology” and “realism” are to be inter- preted and applied so as to give us a deeper philosophical understanding of mathematical theories and practice. Rather than argue for or against some particular realistic position, I shall be concerned with possible coherent positions, their strengths and weaknesses. I shall also discuss related but different aspects of these problems. The terms in the title are the common thread that connects the various sections. The discussed topics range widely. Certain themes repeat in different sections, yet, for most part the sections (and sometimes the subsections) can be read separately. Section 1 however states the basic position that informs the whole paper and, as such, should be read (it is not too long). The required technical knowledge varies, depending on the matter discussed. Aiming at a broader audience I have tried to keep the technical requirements at a minimum, or to supply a short overview. Material, which is elementary for some readers, may be far from elementary for others (my apologies to both). I also tried to supply information that may be of interest to everyone interested in these subjects. Section 2 illustrates the application of the proposed methodology by considering various positions, some of which represent main schools in the philosophy of mathematics. There are some technical items, but I hope that the gist of the discussion should be obvious from the more elementary examples. Some material from Section 2 is used in Section 6. The more philosophical parts are the discussion of Quine’s bound variable criterion (Section 3), the “ontology-ideology” distinction (Section 4), and the short discussion of some attempts to ground mathematics in “concrete” entities (Section 5). Section 6, “Realism in mathematics,” which is about half of the paper, is divided into several subsections that treat a variety of subjects, from Hilbert’s program and the import of Godel’s¨ incompleteness results to the significance of computer-aided proofs. Section 6.4 discusses certain connections with epistemic issues, which arise through the notion of a “natural mathematical question.” The relation between epistemic and ontological issues is rather involved, and I cannot do it full justice here; the paper is intended to have an epistemic twin under the name “How do we know mathematical truths?” Section 7 contains some final observations that bring together some of the themes addressed in various parts of the work.
Received: July 19, 2011.