I used to pooh-pooh the bias that e.g. physicists had for mathematical elegance, feeling that it’s mainly an arbitrary / aesthetic preference instead of something grounded in physical truth.
Thinking about it more now, to the extent our mathematical structures came to be partly as a reflection of us solving problems in the real world, they likely do encode a frequency prior on the likelihood of a particular type of solution to be correct. That is to say, we built the math that most compactly solves existing problems. The set of existing problems is ceteris paribus a reasonable prior for new problems, hence compact math is more likely to contain the “true” solution for new problems than very elaborate math.
Not an ironclad rule, not always true, but a reasonable prior.
The most impressive case of elegance guiding a physicist to truth is Einstein. I don’t know the history of his thought in detail, but my understanding is that it was most of what he used to figure out special and general relativity. Typically when physicists wax poetic about elegance, they are usually talking about milking symmetries for as much info as you can.
Feynmann once said (possibly in Surely You’re Joking?) that this trick, like others before it, is now something everybody tries—so you shouldn’t expect to find an untouched $100 bill there. Accordingly, my understanding is indeed that physicists being guided by mathematical elegance (especially in regards to symmetries/conservation laws) was really kicked off by Einstein. It plausibly even influenced Noether’s theorem (relating symmetries to conservation laws, one of the most beloved theorems in physics), given how (iirc) she was working with him.
I (naively) have thought symmetries are also about representational invariance. Something about swapping the axis shouldn’t change any measurable outcome, implying certain kinds of symmetry.
It sounds like your naive view was better than mine if you already had that idea, as that’s basically Nate’s point. I see the general ‘definition’ of symmetry in math or physics as “if I change X in Y way, it doesn’t ‘really’ change”. In physics symmetries usually mean there’s a conservation law. Specifically that’s only true for continuous ones, but you can still get important stuff out of the discrete ones like the CPT theorem or fermions vs bosons for exchange symmetry
I used to pooh-pooh the bias that e.g. physicists had for mathematical elegance, feeling that it’s mainly an arbitrary / aesthetic preference instead of something grounded in physical truth.
Thinking about it more now, to the extent our mathematical structures came to be partly as a reflection of us solving problems in the real world, they likely do encode a frequency prior on the likelihood of a particular type of solution to be correct. That is to say, we built the math that most compactly solves existing problems. The set of existing problems is ceteris paribus a reasonable prior for new problems, hence compact math is more likely to contain the “true” solution for new problems than very elaborate math.
Not an ironclad rule, not always true, but a reasonable prior.
Elegance is also related to description length, so really any reasonable prior will have to incorporate it so some degree.
The most impressive case of elegance guiding a physicist to truth is Einstein. I don’t know the history of his thought in detail, but my understanding is that it was most of what he used to figure out special and general relativity. Typically when physicists wax poetic about elegance, they are usually talking about milking symmetries for as much info as you can.
Feynmann once said (possibly in Surely You’re Joking?) that this trick, like others before it, is now something everybody tries—so you shouldn’t expect to find an untouched $100 bill there. Accordingly, my understanding is indeed that physicists being guided by mathematical elegance (especially in regards to symmetries/conservation laws) was really kicked off by Einstein. It plausibly even influenced Noether’s theorem (relating symmetries to conservation laws, one of the most beloved theorems in physics), given how (iirc) she was working with him.
One possible (highly speculative!) reason why elegant symmetries seem to be what the universe runs on is that a symmetry implies that there are more programs (or whatever form the hypotheses should actually come in) of similar complexity that give the same predictions (imagine how easy it is to make a raytracer use a different point as the origin), and so gets more probability under the simplicity prior. You have to assign more complicated stuff lower probability in the limit just to have probabilities that add to at most 1, so this conclusion doesn’t seem sensitive to the precise details (I say this because I find it likely that the ‘grown up’ way to do simplicity priors will look a bit different from the current way, and so don’t want to rely on anything that only holds given specific details like using Turing machines instead of some other computational model).
Neat idea—and that makes sense!
I (naively) have thought symmetries are also about representational invariance. Something about swapping the axis shouldn’t change any measurable outcome, implying certain kinds of symmetry.
It sounds like your naive view was better than mine if you already had that idea, as that’s basically Nate’s point. I see the general ‘definition’ of symmetry in math or physics as “if I change X in Y way, it doesn’t ‘really’ change”. In physics symmetries usually mean there’s a conservation law. Specifically that’s only true for continuous ones, but you can still get important stuff out of the discrete ones like the CPT theorem or fermions vs bosons for exchange symmetry