Objectivity and Truth in Mathematics: A Sober Non-platonist Perspective

Abstract

The paper is concerned with the sources of objectivity and truth in mathematics. They have been traditionally located in the seemingly platonist nature of mathematical thought, by logicians like Frege, Russell, Gödel, and mainstream mathematicians. But platonism has had its powerful detractors in general philosophy as well as in the tradition of constructive mathematics and proof theory, where the need was felt to redefine the very notion of truth. There are two aspects of this situation, ontological and epistemological. Regarding the former, the paper argues that the debates revolving around platonism in the philosophy of mathematics have suffered from a House of Mirrors effect: both platonism and anti-platonism appear to be still under the same spell of medieval ontological thinking as they merely reflect the opponent’s position by negating it. The general cure is to naturalize ontology by viewing mathematical concepts (like other scientific concepts) as theory-bound abstract cultural artifacts. On the epistemological side, the challenge of mathematical constructivism has been to deny meaning to concepts going beyond what can be justified by constructions. While agreeing with the anti-platonist impetus, I argue here that constructivists tend to subscribe to a verificationist philosophy that fails to do justice to the remarkable robustness of classical mathematical practice. By reflecting on the sources of the latter, we can still explain objectivity and defend a substantive notion of truth in mathematics within the overall framework of scientific realism.