Das Uberraschende: Wittgenstein on the Surprising in Mathematics

This essay argues that the concepts of the surprising, the interesting, and the change of aspect of things are centrally important, both for Wittgenstein and for the philosophy of mathematics. They need not be psychologized or reduced to other kinds of talk about language without loss. In this vein I discuss Adam Smith on wonder, Wittgenstein's remarks about Godel's theorem in correspondence with Schlick and his remarks on Hardy, and several examples from the history of mathematics that fit nicely with Wittgenstein's ideas, including the theory of Latin Squares (involved in Sudoku, in Euler's famed problem concerning the thirty-six officers) which received surprising rearticulation in the course of the development of modern algebra and in contemporary discrete mathematics. These examples evince a perfectly good sense in which the surprising may be accommodated within our discussions of mathematics without forcing us to adopt either Platonism or eliminative anti-psychologism about the phenomena at issue. The conceptual reframings involved here confirm the usefulness of Wittgenstein's approach to the investigation of surprise in mathematics, which resists construing the notion as everywhere indicating the discovery of new objects or facts.