Logic & Philosophy of Science

During the 20th century natural science, physics foremost, has undergone a constant increase of mathematization. This did not only yield practical or calculatory advantages; often genuinely mathematical reasoning has contributed to mathematical progress while theoretical physicists discovered new structures that were of great importance to pure mathematicians. In this seminar an attempt is made to characterize this field of interaction, be it labeled as mathematical physics, applied mathematics or -- so proposed only recently -- as theoretical mathematics. Does this field follow a specific methodology? Can its objects or results be always unequivocally attributed to either mathematics or physics or do they have their own mode of existence -- at least in a pragmatic sense? Do such ontological claims presuppose or indicate a finality or completeness of the physical theory? What is the role of rigor in justifying claims in this field?

The first part of the seminar addresses these questions from a historical perspective. After the invention of differential calculus, the first mathematical principle containing a general framework of the various domains of physics was the Principle of Least Action. It was also a major cornerstone for Hilbert's program of axiomatization of science that was launched with his 1900 problem list and was based on his reinterpretation of the classical view of more geometrico. Hilbert applied the axiomatic method both to theories he had considered as universal and to merely phenomenological theories. John von Neumann not only fulfilled some of the promises of Hilbert's program in concreto, but he also gave it a strongly pragmatic twist. Is this effectiveness of mathematics in physics unreasonable, as his close friend Eugene Wigner held? Rather independently of such ontological questions, the striking (mathematical) successes of string theorists' mathematics has stirred up a debate as to whether this interaction could be dangerous for the identity of mathematics proper.

The second part of the seminar will be devoted to a discussion of concrete examples, such as Hilbert's axiomatization of general relativity, the intentions of von Neumann's hidden variable proof, and the axiomatic approach to quantum field theory. According to the interests of the participants the second half seminar might take a different route as well.

Auditors are welcome. For obtaining a credit the presentation of a written paper is required.

Preliminary reading list for Part I (copies and translations will be supplied, numbers do not necessarily reflect sessions)