Von Neumann, Empiricism, and the Foundations of Quantum Mechanics1

The issue of the application of mathematics to natural science has received a great deal of consideration in the last few years (see, for instance, Field [1980] and [1989], Hellman [1989], Steiner [1989] and [1998], and Shapiro [1997]). In this paper, I am concerned with two tasks: (1) to provide a formal framework to accommodate one important aspect of this issue − the heuristic use of mathematics in theory construction − and (2) to illustrate how this framework works by focusing on a particular case-study: von Neumann’s use of logic and set theory in his work in the foundations of quantum mechanics. The reason to consider von Neumann’s work derives from the interplay of mathematical, logical and heuristic considerations that he wove together in the process of providing an axiomatization for quantum mechanics and exploring some of its consequences. Moreover, his work provides important insights into the nature of theory construction in physics. As we shall see below, we cannot separate von Neumann’s attitude towards mathematics from his attitude towards logic. He was searching for a unified approach to physics, where logic, mathematics and probability nicely hang together. This approach naturally raises the issue of the cognitive status of logic and mathematics in von Neumann’s view. In addressing this issue, what I want to establish is that there are three major empiricist trends in von Neumann’s work: (i) von Neumann was an empiricist in his view of mathematics (in the sense that mathematical theories are often created from empirical demands); (ii) von Neumann was also an empiricist with regard to logic (in the sense that logic should be inspired and modified by experience; moreover, in his view, there are as many logics as physical phenomena demand). Finally, (iii) in developing his approach to the foundations of quantum mechanics, von Neumann countenanced an empiricist version of the semantic approach (stressing the empirical adequacy of scientific theories, rather