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Will_Sawin comments on Many Weak Arguments vs. One Relatively Strong Argument - Less Wrong
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Do you have/are you planning an argument demonstrating in detail use of MWA in physics?
I don't know physics, but I think that my post on Euler and the Basel Problem gives a taste of it.
I found this unhelpful because I do math and frequently do non-rigorous math reasoning, which seems to me to have much more of a ORSA argument flavor. Or maybe like two arguments - "here is a beautiful picture, it is correct in one trivial case, therefore it is correct everywhere". My previous understanding was that physicists non-rigorous math reasoning was much like mathematician's non-rigorous math reasoning, and that my non-rigorous math reasoning is typical. So to accept this claim I have to change some belief.
Do you think that the specific example of Euler and the Basel Problem doesn't count as an example of the use of MWAs? If so, I don't necessarily disagree, but I think it's closer to MWAs than most mathematical work is, and may be representative of the sort of reasoning that physicists use.
There might just be a terminological distinction here. When I think of the reasoning used by mathematicians/physicists, I think of the reasoning used to guess what is true - in particular to produce a theory with >50% confidence. I don't think as much of the reasoning used to get you from >50% to >99%, because this is relatively superfluous for a mathematician's utility function - at best, it doubles your efficiency in proving theorems. Whereas you are concerned more with getting >99%.
This is sort of a stupid point but Euler's argument does not have very many parts, and the parts themselves are relatively strong. Note that if you take away the first, conceptual point, the argument is not very convincing at all - although this depends on how much calculation of how many even zeta values Euler does. It's still a pretty far cry from the arguments frequently used in the human world.
Finally, while I can see why Euler's reasoning may be representative of the sort of reasoning that physicists use, I would like to see more evidence that it is representative. If all you have is the advice of this chauffer, that's perfectly alright and I will go do something else.
I don't have much more evidence, but I think that it's significant that:
Physicists developed quantum field theory in the 1950's, and that it still hasn't been made mathematically rigorous, despite the fact that, e.g., Richard Borcherds appears to have spent 15 years (!!) trying.
The mathematicians who I know who have studied quantum field theory have indicated that they don't understand how physicists came up with the methods that they did.
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.
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.
As a tangent, I think it's relatively clear both how physicists tend to think differently from mathematicians, and how they came up with path-integration-like techniques in QFT. In both math and physics, researchers will come up with an idea based on intuition, and then verify the idea appropriately. In math the correct notion of verification is proof; in physics it's experimentation (with proof an acceptable second). This method of verification has a cognitive feedback loop to how the researcher's intuition works. In particular physicists have intuition that's based on physical intuition and (generally) a thoroughly imprecise understanding of math, so that from this perspective, using integral-like techniques without any established mathematical underpinnings is intuitively completely plausible. Mathematicians would shirk away from this almost immediately as their intuition would hit the brick wall of "no theoretical foundation".
If you're really curious, you can talk with my chauffer (who has deep knowledge on this point).