Theories, Stories, Rules-of-Thumb

This mostly summarizes some stuff I posted in a Comments thread at Uncertain Principles.

It comes out of my attempts to learn, this year, a lot of what Hannah Shapero calls "math in bad faith.".

The thing is, I got into this business for explanations. As opposed to calculations. Nothing frustrates me more than formalisms and mathematical tricks that I'm supposed to use "because they work." I mean, it's nice to know that the universe behaves predictably, but I want to know that it also behaves rationally, that there are reasons for particles to take such and such a path (which you can describe exactly by such and such a method.) The natural response to this is, "But we don't know the reasons why the universe obeys Newton's laws and Maxwell's laws and the Schroedinger equation and relativity... Only that it does. You just have to content yourself with proofs that the answers you get are consistent with those rules."

But I just don't trust mathematics that much. Look, I can understand, "a force is exerted upon an electron, and so it moves." Forces making things move are a part of my everyday experience. I know what that means. I know how it works, what it looks like. But I don't understand, "The electron goes over there in order minimize its 'action'". Or "The field looks like this because the potential has to satisfy Green's theorem." Unless the electrons and fields are doing little calculations and adjusting themselves accordingly, this doesn't explain anything.

So I'm still not happy with Lagrangian mechanics, even though I can apply it with reasonable success (assuming I pick the proper coordinate system, which I never do the first time). Why should this apparently meaningless quantity called "action" be minimized? (Actually, there's a paper by a professor at my old university that seems to hint at an answer for this, at least when combined with Feynman's path integral stuff, but I don't understand it.) E&M is even worse. J.D. Jackson, bane of my existence, is constantly appealing to theorems in vector calculus, approximation methods, series expansions, etc. to justify physical laws like "There can be no electric field inside of a conductor." I want to know what physical thing prevents it, J.D. I want you to tell me a little story about electrons moving around.

I used to believe that the people who came up with the theories must have had such stories in mind, and that my professors were being willfully perverse in teaching these theories as though they were the rules to a meaningless mathematical game, but recently, I have learned better. (It's funny when you find out the reason you don't understand something is not because you're dumb, but because nobody really understands. Makes you feel a little better about yourself, but on the other hand, it's so disappointing, because now you may never know... As a TA, I like to make a big deal over this stuff [and I try to cop to the stuff that someone understands, but not me. I wish my professors would.] I also tell them they'd better do a good job testing Newton's laws and the gravitational constant in lab, because without new experimental evidence every day, we'd really have no reason to believe these things were true...)

Anyway, it seems that historically, the equations often come first, and then the stories. There's a lot of theory we wouldn't be able to use if we demanded that they come with little movies we could run in our heads, and I do know that now, even as I still go around demanding that someone tell me a story I understand... It seems that there still isn't a really satisfying explanation of the least-action principle, or anyway, not one that I've heard. Max Planck didn't have a clue why energy should be quantized as he described (and no one does now, either.) The second law of thermodynamics -- " Entropy must increase" -- was discovered before anyone had a real definition for entropy in terms of the distribution of particles (it was incredibly frustrating to ask, in my thermodynamics-without-stat-mech class, "So what is entropy?" and be told "It's what you get when you integrate ds=dq/T." As though that's an answer! But it was the only answer 19th century engineers had, and they designed steam engines anyway...) Mendeleev designed a periodic table and made predictions before anyone knew what atoms were. People knew rates of certain diseases were higher in mosquito-infect areas before anyone came up with the germ theory of disease. Copernicus figured the earth revolved around the sun before Newton explained why, and he didn't even figure it for more than a convenient trick for doing calculations, a mathematical formalism, if you will...

But the thing is, sometimes wrong theories fit the facts as well, like the Bohr atom, or epicyles. So fitting the facts may not be enough after all... I like this line of argument, because it allows me to protest that the storyless theories I don't like to learn aren't real science after all, but are in the same category as epicycles. They can become real, but only if someone gives me a decent interpretation, dammit.

However my friends Andy and Sarah point out that you can actually rule out the Bohr atom, and epicycles, on the grounds that they only fit a special case (the hydrogen atom and our solar system, respectively) and aren't general. If something fits the facts and is general, I'm going to have to call it real science after all.

So, are stories really necessary at all? What do we want science to do? Should I quit, and concentrate on writing novels?