Logic & Philosophy of Science Colloquium

No other mathematician has shaped post-war mathematical physics to a larger extent than John von Neumann. In this talk I intend to show that his methodological remarks of the 1940s and 1950s also open up an interesting philosophical perspective on the interaction between mathematics and physics which can be conceived as a pragmatist re-interpretation of Hilbert's program of the axiomatization of the empirical sciences. On this basis, axiomatization can prove fertile both in physically well-entrenched theories, such as quantum mechanics, and in cases where the basic concepts of the science are not yet fully clarified and empirical evidence is still poor, such as game theory. The main reason is that mathematization and axiomatization permit great flexibility and opportunism in concept formation and allow a continuous application of pragmatic criteria of success from the sciences to mathematics proper. In this way even some motives from the foundational debates in mathematics appear in a pragmatist mode without any absolutist ambitions. To make my point concerning the timeliness of von Neumann's opportunistic axiomatics more precise, I shall first investigate which stand he could take in the debates on the character string theorists" "theoretical mathematics" stirred up by Arthur Jaffe and Frank Quinn in 1993/4. Secondly, I ask how von Neumann's opportunistic axiomatics is related to the distinction between mathematical optimism and mathematical opportunism recently drawn by Mark Wilson.