The Turning Point in Wittgenstein’s Philosophy of Mathematics: Another Turn

According to Mark Steiner, Wittgenstein’s intense work in the philosophy of mathematics during the early 1930s brought about a distinct turning point in his philosophy. The crux of this transition, Steiner contends, is that Wittgenstein came to see mathematical truths as originating in empirical regularities that in the course of time have been hardened into rules. This interpretation, which construes Wittgenstein’s later philosophy of mathematics as more realist than his earlier philosophy, challenges another influential interpretation which reads Wittgenstein as moving in the opposite direction, from a more realist toward a less realist position. Both of these readings, I argue here, tend to overlook the crucial role of conventionalism in shaping Wittgenstein’s later philosophy. I show that during the transition period, Wittgenstein was first strongly attracted to conventionalism, but then, upon discovery of the rule following paradox, came to realize its weaknesses. Did this realization mark the end of the liaison with conventionalism and a wholehearted acceptance of the empirical regularities account? Illustrating Wittgenstein’s ongoing ambivalence toward conventionalism and pointing to his novel understanding of this position, I answer this question in the negative.