Nowadays I see a distinct “line” dividing people working in Mathematics and the Physical Sciences. Why?

Question

The direction in which leading research is heading in these subjects (Math, Physics) is very much different and don't seem to be in tandem. Is this something that developed in more recent times? This is strictly my belief. This query is not meant to target any particular field, but to clarify this stereotype for me.

Historically we have seen many mathematicians contribute to physics and vice versa. For example Newton, Gauss, Einstein, Hilbert, Poincaré, etc. But now this communication between mathematicians and physicists seems to have reduced.

I just want to know if this is true or a misconception. Provide some resources if possible with your opinion. Also, if true, how does history suggest that we work towards their reunification?

Answer 1

You do not say what field of mathematics you are working in, and perhaps there are signs of separation there. Overall however, lively interaction between mathematics and physics is alive and well. John Baez has a blog This Week's Finds in Mathematical Physics, that is full of contemporary examples of it, so does Terence Tao. Nature, a leading journal in empirical sciences, just published a mathematical paper Undecidebility of the Spectral Gap related to one of the millenium problems, which in turn came from quantum field theory.

One example is the Fields medal awarded to Witten for his insights into the mathematical structure of gauge theories and low-dimensional topology that were obtained by combining advanced mathematics with heuristic physical reasoning. Converse example is the mirror symmetry conjecture in algebraic geometry that came from string theory physics and led to foundational work by some of our leading mathematical lights like Givental, Kontsevich, Fukaya, etc., which is ongoing. The Gromov-Witten theory, a major mathematical advance in algebraic and symplectic geometry, grew out of this circle of ideas. Okounkov received Fields medal in 2006 for his mathematical work in it partially done in collaboration with leading physicists like Vafa. More recently, large N duality and holography conjectures received the same treatment.

This is not restricted to frontier quantum physics. General relativity continues to present deep and intricate problems in (pseudo-)Riemannian geometry, and cosmologists like Hawking are adept at proving mathematical theorems about pseudo-Riemannian manifolds, see e.g. work on the chronology protection conjecture. Physical insights by Mandelbrot and Feigenbaum in 1970s led to the modern theory of fractals and chaotic dynamical systems. Perelman's work on the Ricci flow and the geometrization conjecture was as driven by physical inspiration as was Riemann's, one of the central notions in his proof is a new kind of entropy in a statistical canonical ensemble.

Perhaps the best evidence for the link is the 1990s controversy, where some mathematicians became concerned that the link between mathematics and physics became so strong that it was undermining the integrity of mathematics by blurring the lines between mathematical and physical "levels of rigor", and threatening to make mathematical results unreliable. The controversy was sparked by Jaffe and Quinn, who sugested explicit separation of "theoretical mathematics" with lowered standars of rigor, from the body of mathematics proper. Take a look at responses from who's who in modern mathematical physics: Atiyah, Glimm, Maclane, Mandelbrot, Thom, Uhlenbeck, Witten, etc.

Answer 2

You can only answer your question by a bibliometric analysis over the far and recent past. Luckily, there are nowadays quite a lot of tools to evaluate the data. But this is a research question and some has to do this evaluation or you have to look on google scholar, searching in bibliometrics or philosophy of science journals if someone maybe already has done such analysis.

To add to Conifolds answer, there are also physicists with the opinion that there is too much focus on mathematics in cosmology and particle physics nowadays. The recent book Lost in Math by Sabine Hossenfelder shares very interesting interviews with world-leading theoretical physicists.

From a more philosophical/meta-physical point of view, if mathematics is the language of nature or just the best mankind/human brain is able to develop is a neverending discussion. But I find it quite interesting to think about if the progress in math/physics is limited by each other! But I would be very much surprised, if the collaboration between both branches has shrinked in the recent 2 decades since internet boosted scientific publications and collaboration. If you could prove this by bibliometric analysis, this would trigger major questions, as it would seem very counter-intuitive.

Answer 3

Mathematics and physics have a long standing relationship. In some periods there is more interaction than at others. For example, there was a large amount of work in the 70s when physicists and mathematicians realised that some of the tools and theories they had developed had much in common with each other, and would serve to Illuminate.

Although I’m not a big fan of string theory, by all accounts it has helped invigorate many areas of mathematics.