Truth in Mathematics: The Question of Pluralism

Abstract

The discovery of non-Euclidean geometries (in the nineteenth century) undermined the claim that Euclidean geometry is the one true geometry and instead led to a plurality of geometries no one of which could be said (without qualification) to be “truer” than the others. In a similar spirit many have claimed that the discovery of independence results for arithmetic and set theory (in the twentieth century) has undermined the claim that there is one true arithmetic or set theory and that instead we are left with a plurality of systems no one of which can be said to be “truer” than the others. In this chapter I will investigate such pluralist conceptions of arithmetic and set theory. I will begin with an examination of what is perhaps the most sophisticated and developed version of the pluralist view to date—namely, that of Carnap in The Logical Syntax of Language—and I will argue that this approach is problematic and that the pluralism involved is too radical. In the remainder of the chapter I will investigate the question of what it would take to establish a more reasonable pluralism. This will involve mapping out some mathematical scenarios (using recent results proved jointly with Hugh Woodin) in which the pluralist could arguably maintain that pluralism has been secured.

“… before us lies the boundless ocean of unlimited possibilities.”

—Carnap, The Logical Syntax of Language

I would like to thank Bill Demopoulos, Iris Einheuser, Matti Eklund, Michael Friedman, Warren Goldfarb, Daniel Isaacson, Øystein Linnebo, John MacFarlane, Alejandro Péres, and Thomas Ricketts for helpful discussion.