Philosophy of Mathematics Seminar


Philosophy of Mathematics seminar


Benjamin Marschall, Trinity College, Cambridge

Carnap, the Conventionalist

What is Carnap’s philosophy of mathematics? Heavyweights like Gödel, Quine, and Putnam read Carnap as a linguistic conventionalist about mathematical truth. Among the new wave Carnap scholars of recent decades, however, the conventionalist interpretation has been unpopular. They have argued that Carnap’s unique metaphilosphy, which is based on the principle of tolerance and the method of explication, excludes a conventionalist reading. Ricketts even thinks that Carnap “gives up philosophy of mathematics” altogether by rejecting the questions about the nature of mathematics thinkers like Frege, Hilbert, and Wittgenstein wanted to address.

I will oppose this trend by arguing that Carnap’s position is most perspicuously described as conventionalism, albeit a non-standard form. I proceed by responding to two arguments purporting to show that Carnap cannot be a conventionalist. The first argument is based on Carnap’s rejection of “language-transcendent” metaphysical facts. The second is based on the observation that, for Carnap, the aim of philosophy is not primarily to describe reality but rather to improve our conceptual apparatus. While these considerations indeed exclude some forms of conventionalism, I will show that they naturally lead to the conclusion that Carnap is what I call a normative commitment conventionalist: He thinks that we should adopt a language with rules that commit its users to accept all true and reject all false mathematical statements. It will prove illuminating to compare this view with the descriptive form of conventionalism Jared Warren recently defended in his Shadows of Syntax.