These suggest that the physicists who invented this theory reasoned in a very different way from how mathematiicans usually do.
That part seems obvious: physicists treat math as a tool, it does not need to be perfect to get the job done. It can be inconsistent, self-contradictory, use techniques way outside of their original realm of applicability, remove infinities by fiat, anything goes, as long as it works. Of course, physicists do prefer fine tools polished to perfection, and complain when they aren’t, but will use them, anyway. And invent and build new crude ones when there is nothing available off the shelf.
As a mathematician, it’s possible to get an impression of the type “physicists’ reasoning isn’t rigorous, because they don’t use, e.g. epsilon delta proofs of theorems involving limits, and the reasoning is like a sloppier version of mathematical reasoning.”
The real situation is closer to “physicists dream up highly nontrivial things that are true, that virtually no mathematicians would have been able to come up with without knowledge of physics, and that mathematicians don’t understand sufficiently well to be able to prove even after dozens of years of reflection.”
But mathematicians also frequently dream up highly nontrivial things that are true, that mathematicians (and physicists) don’t understand sufficiently well to be able to prove even after dozens of years of reflection. The Riemann hypothesis is almost three times as old as quantum field theory. There are also the Langlands conjectures, Hodge conjecture, etc., etc. So it’s not clear that something fundamentally different is going on here.
I think that the Langlands program is an example: it constitutes a synthesis of many known number theoretic phenomena that collectively hinted at some general structure: they can be thought of “many weak arguments” for the general conjectures
But the work of Langlands, Shimura, Grothendieck and Deligne should be distinguished between the sort of work that most mathematicians do most of the time, which tends to be significantly more skewed toward deduction.
From what I’ve heard, quantum field theory allows one to accurately predict certain physical constants to 8 decimal places, with the reasons why the computations work very unclear. But I know essentially nothing about this. As I said, I can connect you with my friend for details.
No, but my impression is that the physics culture has been more influenced by the MWA style than mathematical culture has. In particular, my impression is that most physicists understand “the big picture” (which has been figured out by using MWAs) whereas in my experience, most mathematicians are pretty focused on individual research problems.
That part seems obvious: physicists treat math as a tool, it does not need to be perfect to get the job done. It can be inconsistent, self-contradictory, use techniques way outside of their original realm of applicability, remove infinities by fiat, anything goes, as long as it works. Of course, physicists do prefer fine tools polished to perfection, and complain when they aren’t, but will use them, anyway. And invent and build new crude ones when there is nothing available off the shelf.
What I was highlighting is the effect size.
As a mathematician, it’s possible to get an impression of the type “physicists’ reasoning isn’t rigorous, because they don’t use, e.g. epsilon delta proofs of theorems involving limits, and the reasoning is like a sloppier version of mathematical reasoning.”
The real situation is closer to “physicists dream up highly nontrivial things that are true, that virtually no mathematicians would have been able to come up with without knowledge of physics, and that mathematicians don’t understand sufficiently well to be able to prove even after dozens of years of reflection.”
But mathematicians also frequently dream up highly nontrivial things that are true, that mathematicians (and physicists) don’t understand sufficiently well to be able to prove even after dozens of years of reflection. The Riemann hypothesis is almost three times as old as quantum field theory. There are also the Langlands conjectures, Hodge conjecture, etc., etc. So it’s not clear that something fundamentally different is going on here.
I agree that the sort of reasoning that physicists use sometimes shows up in math.
I don’t think that the Riemann hypothesis counts as an example: as you know, its truth is suggested by surface heuristic considerations, so there’s a sense in which it’s clear why it should be true.
I think that the Langlands program is an example: it constitutes a synthesis of many known number theoretic phenomena that collectively hinted at some general structure: they can be thought of “many weak arguments” for the general conjectures
But the work of Langlands, Shimura, Grothendieck and Deligne should be distinguished between the sort of work that most mathematicians do most of the time, which tends to be significantly more skewed toward deduction.
From what I’ve heard, quantum field theory allows one to accurately predict certain physical constants to 8 decimal places, with the reasons why the computations work very unclear. But I know essentially nothing about this. As I said, I can connect you with my friend for details.
Most physicists most of the time aren’t Dirac, Pauli, Yang, Mills, Feynmann, Witten, etc.
No, but my impression is that the physics culture has been more influenced by the MWA style than mathematical culture has. In particular, my impression is that most physicists understand “the big picture” (which has been figured out by using MWAs) whereas in my experience, most mathematicians are pretty focused on individual research problems.