Doper Quote: "I'm strong in mathematics, but weak in physics." Please explain.

[Mathochist]

@Anne Neville

Mathematics involves two very different skill sets:

1. Arithmetic, algebra, and related stuff that 90+% of people think of when you say “mathematics”.

2. Proofs.

Being good at the first doesn’t mean you’ll be good at the second. A lot of math majors at my school left the major when they came to the first class that really taught proofs- they’d thought #1 was what math was, and they were good at that, but not good at proofs.

Actually, I’d say that #1 is practically incidental. It’s just the only arena that behaves purely rationally enough to really get anywhere with #2.

[CalMeacham]

I’ve known quite a few physicists and engineers who seem to know things intuitively and nabvigate “by the seat of their pants”, rather than by rigorous math. At one company I worked at I developed a reputation for being good at math, because I took the time and effort to prove things that the other scientists and engineers sort of assumed or hand-waved their way past. It’s not that they didn’t know math (I saw ample evidence to the contrary), but in some cases I thjink they felt they couldn’t be bothered, or else this particular application was in a blind spot. It was regarded as siognificant that I worked through these matters.
An awful lot of scientists have a particular blind spot in the area of probability and statristics, which I feel isn’t sufficiently well-taught at the graduate level. When I was in grad school, I had to answer the question "What is the probability that the nearest neighbor of a particle is a distance *r[/r] away, given an average uniform density of these throughout space? I was surprised at the number of incorrect answers I got, including one in a paper I dug out. To their credit, the methods of solution they all gave me gave the correct order of magnitude of the result, but most of them did not give the correrct expression. It was such an interesting and obscure problem that I devoted an appendix of my thesis to it. (There were various correct solutions. Chandrasekar gets top credit with a two-line proof of the correct formula.)

[Chronos]

For example, when I was an undergrad, I had a friend who was studying General Relativity. I had taken a linear algebra course, so I knew what a vector was, but he would mention something called a covector. Yet when I asked him what this was, he could provide nothing more than that it was “something like a vector, but different.” Yet he was working with vector, covectors, and tensors everyday. This became more mysterious when I later learned the very simple definition of “covector” (and, as a matter of fact, covectors are just vectors, at least as mathematicians understand them; what makes some vectors “co-” has to do with their relationships to other vectors). That my friend could reason correctly while doing physics problems involving covectors when he didn’t know this definition almost seems magical to me. In this example, I suppose that there is a lot of learning computations by rote, but it’s hard to believe that that accounts for all of it. I can only conclude that physicists learn to use mathematics correctly in the same way that I have learned to use prepositions correctly in English. I can do it, but I can’t explain how I do it.

This particular example is good for illustrating one of the key differences between mathematicians and physicists. Physicists want to know how to do something, but mathematicians want to understand why that method works. To a physicist doing relativity, a covector is one for which one writes the indices “downstairs”, that is, as subscripts. Using the Einstein Summation Convention, a notation which is very useful for dealing with tensors, that’s all you need to know: Some objects have indices upstairs, some have indices downstairs, and upstairs and downstairs indices interact in particular ways. Almost everything we need to know about tensors is how to work with them, and almost everything we need about how to work with them is encoded in Einstein notation, so that’s what we learn and use. Mathematicians will learn more rigourous definitions, and we physicists generally encounter those definitions once or twice in a class, but we don’t use them, so we don’t remember them

[Mathochist]

@Chronos

This particular example is good for illustrating one of the key differences between mathematicians and physicists. Physicists want to know how to do something, but mathematicians want to understand why that method works. To a physicist doing relativity, a covector is one for which one writes the indices “downstairs”, that is, as subscripts. Using the Einstein Summation Convention, a notation which is very useful for dealing with tensors, that’s all you need to know: Some objects have indices upstairs, some have indices downstairs, and upstairs and downstairs indices interact in particular ways. Almost everything we need to know about tensors is how to work with them, and almost everything we need about how to work with them is encoded in Einstein notation, so that’s what we learn and use. Mathematicians will learn more rigourous definitions, and we physicists generally encounter those definitions once or twice in a class, but we don’t use them, so we don’t remember them

More directly to the point, physicists seem to think that indices actually mean something, while a mathematician knows they’re completely artificial (artifactual?).