Philosophy of Physics Seminar: Rami Jreige (Bristol): Why is Physics unrigorous?

Mathematicians often claim that physicists are unrigorous in their mathematical work. Simplistically, mathematical rigour is defined as the logical derivation of results from axioms. Given this definition, physics clearly does not follow these methodologies and therefore appears unrigorous. Yet this definition of rigour, or any alternative that has been proposed, does not withstand scrutiny. This raises the question: why, then, is physics considered unrigorous by mathematicians? This presentation examines Witten’s 1989 proof of the invariance of the Jones polynomial in order to explore what, in physics, counts as rigorous and what does not.
Although the influence of physics on mathematics has recently received attention, Witten’s proof, recognized with the Fields Medal in 1990, is one of the few modern cases, if not the only one, in which a mathematical problem was proved using physical reasoning. Yet despite being treated as rigorous (as evidenced by the award), the proof nonetheless provoked controversy within mathematical circles. This is significant because it illuminates what aspects of physics are considered mathematically rigorous, and by extension, what aspects are not.
Though a more rigorous proof was later presented, this episode exposed a genuine divide among mathematicians regarding rigour. What some regarded as rigorous, others refused to accept. Witten’s use of a generally covariant quantum field manifold to prove the invariance of the Jones polynomial on an invariant topological manifold became a focal point of contention. This underscored the differing ways in which objects are used in mathematics and physics: in mathematics, objects tend to be implicitly defined and immutable, whereas in physics they are often explicitly constructed, with new properties emerging depending on the model. This difference in object-use was widely discussed in relation to the notion of rigour in mathematics and its contrast with physical reasoning.
This presentation will investigate which aspects of Witten’s argument were deemed rigorous (and therefore were not in contention) and which were not. In this context, if a precise definition of rigour proves elusive, we can at least offer a clearer picture of what rigour is not. In this way, we are a step closer in understanding what role rigour could play in physics.