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How should a physics student study mathematics?

If some-one wants to do research in string theory for example, Would the Nakahara Topology, geometry and physics book and other geometry and topology books geared at physicists be sufficient for that purpose, or should one read abstract math textbooks e.g. Spivak Differential geometry. What about real analysis and functional analysis (not just the introductory functional analysis chapter that's present in quantum mechanics textbooks)?

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ahmed
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2 Answers2

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I think this is an interesting question. The answer depends very sensitively on what kind of physics you want to work on. If you want to do "fundamental physics" in the vein of people like Edward Witten, then the ability to think both like a mathematician and physicist is probably very valuable. On the other hand, if you are interested in other sorts of problems I'm sure it is enough to have enough of a mental picture of what is going on to come up with useful experiments (and thought experiments) without worrying about having perfect mathematical proofs of everything you use.

It is true that mathematicians and physicists are generally interested in different things. (Pure) mathematicians are interested in proving theorems from basic logical starting assumptions, whereas physicists usually aim to perform some sort of numerical computation to compare with the numerical predictions of experiment. The difference in both points of view probably vanishes "as h-bar goes to zero" for some physicists. The point is, mathematics is interested in intuitions that help build coherent mathematical theories that stand up to logical attack. Physicists want intuitions that can be used to build models that give good experimental predictions up to the tolerance of measurement (which is now pretty high). At the quantum scale, I think that intuitions built on common sense and "physical" experience break down and must be replaced by the more spartan mathematical intuition. (Mathematicians are used to taking less for granted...that's really the only difference.)

The best thing I can say is, physicists are probably "birds" in Freeman Dyson's sense. The best thing to do if you're a bird is follow Michael Atiyah's advice and build up a storehouse of fundamental (simplest nontrivial) examples you can use to test theories. Such examples build intuition (physical and otherwise) and that's what you want. Whatever books you read, carry your collection of basic examples around and check your intuition against these. For physics, this is probably just as valuable if not more than proving the theorems.

I, for one, wish that the perceived gap weren't so large between mathematics and physics. The point of mathematics is to refine one's intuition...not to lose it. If you're losing your intuition, you're doing something very wrong.

Jon Bannon
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As someone who intended to study physics, but ended up studying maths; I found maths, dry & discursive and far removed from what my physical intuition found useful. It was impossible to study :-)

When I returned to Physics, I found the arguments sometimes impossible to follow, as I was always looking for the logical motivation. In other words, my physical intuition had evaporated :-(.

Mathematics books, although there exposition may be clearer, to the mathematician; has differing standards, and is trying to accomplish different things.

I suggest you stick to physics books with the appropriate mathematical technology, so that the primary physical intuition you need to develop isn't displaced. But also dip into maths texts to see what else is going on, or get a mathematician to explain, to see what you're missing out on. Expositary papers are useful.

Historically the links between the two subjects are complex, and fascinating; and I can only expect this to continue, despite the occasional falling out (Gruppenpest & Abstract Nonsense).

Mozibur Ullah
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