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Kindly comments on The Dilemma: Science or Bayes? - Less Wrong
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You go back and prove it if you can - and are mathematicians special in that regard, save that they deal with concepts more easily proven than most? When scientists in any field can prove something with just logic, they do. Evidence is the tiebreaker for exclusive, equivalently proven theories, and elegance the tiebreaker for exclusive, equivalently evident theories.
And that seems true for all fields labeled either a science or a form of mathematics.
I would say that when they do this, they are doing mathematics instead of science. (By the time that scientists can prove something with logic, they necessarily have a mathematical model of their field.)
If that's the case, and if it is also the case that scientists prefer to use proofs and logic where available (I can admittedly only speak for myself, for whom the claim is true), then I would argue that all scientists are necessarily also mathematicians (that is to say, they practice mathematics).
And, if it is the case that mathematicians can be forced to seek inherently weaker evidence when proofs are not readily available, then I would argue that all mathematicians are necessarily also scientists (they practice science).
At that point, it seems like duplication of work to call what mathematicians and scientists do different things. Rather, they execute the same methodology on usually different subject matter (and, mind you, behave identically when given the same subject matter). You don't have to call that methodology "the scientific method", but what are you gonna call it otherwise?
Fair enough, but I think that the example of mathematics was brought forth because of the thing mathematicians primarily do.
So we could rather say that "the scientific method" refers to the things scientists (and mathematicians) do when there are no proofs, which is to test ideas through experiment, and "the mathematical method" refers to proving things.
Of course, you don't have to draw this distinction if you prefer not to; if you claim that both of these things should be called "the scientific method" then that's also fair. But I'm pretty sure that the "Science vs. Bayes" dilemma refers only to the first thing by "Science", since "Bayes" and "the mathematical method" don't really compete in the same playing field.
I think that's a workable description of the process, but using that you still have the mathematical tendency to appreciate elegance, which, on this process model, doesn't seem like it's in the same place as the "mathematical method" proper - since elegance becomes a concern only after things are proven.
You could argue that elegance is informal, and that this aspect of the "Science V Bayes" argument is all about trying to formalize theory elegance (in which case, it could do so across the entire process), and I think that'd be fair, but it's not like elegance isn't a concept already in science, though one preexisting such that it was simply not made an "official" part of "Science".
So to try to frame this in the context of my original point, those quantum theorists who ignore an argument regarding elegance don't strike me as being scientists limited by the bounds of their field, but scientists being human and ignoring the guidelines of their field when convenient for their biases - it's not like a quantum physicist isn't going to know enough math to understand how arguments regarding elegance work.
Their inherently weaker evidence still isnt empirical evidence. Computation isn;t intrinsically emprical, because a smart enough mathematician could do it in their head...they are just offloading the cognitive burden.
I'd say it's a little more complicated than this. In those fields where solid mathematical models have been developed, there's usually some back-and-forth between experimentalists and theoreticians, and any particular idea is usually considered mildly suspect until both rigorous mathematical models and solid empirical backing exist. New ideas might emerge either from the math or from empirical findings.
These days in physics and chemistry the mathematical models usually seem to emerge first, but that's not true for all fields; in astronomy, say, it's common for observations to go unexplained or to clank along with sketchy models for quite a while before the math properly catches up.