Mathematics and Physics: Tension with Interaction

It is pretty common for us to hear about interactions between mathematics and physics. If one search for such phrase on the web, one will find many articles that delve into such a topic. Perhaps it is not that surprising the close link between physics and mathematics as they co-evolve earlier under the name of natural philosophy (see Evelyn Lamb’s Symmetry article “The Coevolution of Physics and Math“). The term natural philosophy has gone out of fashion due to the divergent paths made by mathematics, physics and philosophy itself (see problem of calls for return to natural philosophy by Arran Gare in his article, “Natural Philosophy and the Sciences: Challenging Science’s Tunnel Vision“). Thus having physics and mathematics emerge as separate domains but yet interacting in some nominal fashion, may not be surprising.

So I find it refreshing to find an article about tension between mathematics and physics instead i.e. Miklos Redei, “On the Tension Between Physics and Mathematics“, J. Gen. Phil. Sci. 51 (2020) 411-425. Redei is a Professor of Philosopher of Science at LSE and has written numerous articles on philosophy and foundations of physics. On my wishlist is his book on “The Principle of the Common Cause“, co-authored with G. Hofer-Szabo & L.E. Szabo. Why are tensions more interesting? They tend to drive new ideas in both fields of physics and mathematics. Redei classified the tensions into two classes:

• Tension of Type I: Unavailability of ready-made mathematical concepts needed for physics;
• Tension of Type II: Availability of mathematical concepts but dismissive of rigour for further development pf physics.

In the former, it is clear that there is a need for new mathematical ideas and the historical examples given are calculus for Newtonian mechanics and spectral theory of self-adjoint operators for quantum mechanics. The other two examples are perhaps the interesting ones as they are yet unsolved, namely, ergodic theory for statistical mechanics and operator-valued tempered distributions for quantum field theory. The case of quantum field theory is not surprising and there are many attempts of putting quantum field theory on rigorous basis by axiomatizing or through constructive quantum field theory. A relative recent review on CQFT can be found in Summers’ “A Perspective on Constructive Quantum Field Theory” (arXiv: 1203.3991). For ergodic theory, this is lesser known to me and Redei mentions the reference of Szasz’s “Boltzmann’s Ergodic Hypothesis, A Conjecture for Centuries” (book is open access).

For Type II Tension, my first impression is that it is not totally unrelated to Type I, with quantum field theory come to mind. For instance, functional integrals are often used in field theories without rigour (see here). Redei mentions the Jaffe and Quinn article “Theoretical Mathematics: Towards a Cultural Synthesis of Mathematics and Theoretical Physics” which sparks debates. See responses here. How tension II can lead to progress is not as clear as tension I but mathematical rigour can always uncover new problems that may lead to new progress. An example that Redei mentioned is von Neumann’s own dismissive behaviour over the Hilbert space formulation of quantum mechanics in favour of an operator algebraic approach. Redei wrote an article on this: Why John von Neumann Did Not Like the Hilbert Space Formalism of Quantum Mechanics (and What He Liked Instead). I’m recalling that Bob Coecke always made reference to this von Neumann’s reaction in favour for a more intuitive categorical/pictorial approach. Another reference that was also mentioned is that of Valente’s “John von Neumann’s Mathematical “Utopia” in Quantum Theory“. So will need to read these papers to understand von Neumann’s standpoint.