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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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    $\begingroup$ Abstract algebra will be useful as well. $\endgroup$
    – icurays1
    Commented Dec 4, 2012 at 16:52
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    $\begingroup$ Possible duplicate: physics.stackexchange.com/q/234/2451 and links therein. $\endgroup$
    – Qmechanic ♦
    Commented Dec 4, 2012 at 16:59
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    $\begingroup$ But learning how to prove theorems is essential to understand the theorem and how one can apply it to various problems $\endgroup$
    – ahmed
    Commented Dec 4, 2012 at 16:59
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    $\begingroup$ I intend to do string theory $\endgroup$
    – ahmed
    Commented Dec 4, 2012 at 17:00
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    $\begingroup$ @ahmed: that's entirely dependent on the theorem. Some proofs are more enlightening than others. $\endgroup$
    – Zo the Relativist
    Commented Dec 4, 2012 at 17:18

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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.

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  • $\begingroup$ Great answer! thanks. I've a question what if I want to apply what I read in math books to interesting problems in physics (e.g While reading math book on algebraic topology where should I go to apply the ideas I learn in condensed matter theory for example? Or apply differential geometry ideas to D-branes ,Ads spacetime etc without necessarily having the required physics background (String theory, QFT)I know QM,electrodynamics,CM and in QFT canonical quantization and S-matrix stuff but nothing in QED $\endgroup$
    – ahmed
    Commented Dec 4, 2012 at 20:05
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    $\begingroup$ Feynman once said that in research you just keep learning more and more about a topic until you end up learning something no one else knows. Focus on questions, not topics. I think it may be a good idea to really think hard about the physical situation and learn how others have modelled it, and come up with a question. The needed mathematics should come from the need to do something very concrete, e.g. solving some system of equations. You don't want to use a machine gun to kill a fly. Introduce math when it is clearly needed. If I'm wrong about this, hopefully someone else will chime in. $\endgroup$
    – Jon Bannon
    Commented Dec 4, 2012 at 20:24
  • $\begingroup$ @ahmed: Remember, Einstein came up with GR thinking about concrete physical problems, he got his friend Grossman to teach him the relevant maths; he also commented that once the mathematicians got hold of his theory, he himself no longer understood it :) $\endgroup$
    – Mozibur Ullah
    Commented Dec 5, 2012 at 2:56
  • $\begingroup$ @Bannon: there isn't just a conceptual/teleological/justification barrier, but also a language barrier; names/notation is used differently. But then the same thing infests maths if you move from one discipline to another. It would be nice if people could agree on names & notation in a gentlemanly fashion. $\endgroup$
    – Mozibur Ullah
    Commented Dec 5, 2012 at 3:03
  • $\begingroup$ @Mozibur: I agree about the language barrier. I think it's there because of a conceptual difference in scope, mostly. It would be interesting to collect the major differences in how mathematicians and physicists think about math, say, in a blog post. Something collaborative may be nice. A succinct expression of the differences may uncover precisely why we cannot talk more. (Absence of "mathematical phenomenologists" is part of why math even has this problem internally...I agree with that, too.) $\endgroup$
    – Jon Bannon
    Commented Dec 5, 2012 at 12:06
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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).

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    $\begingroup$ The general conclusion I came to was that mathematicians stopped working once they showed that a solution existed and was unique, rather than actually working to find the solution. $\endgroup$
    – Zo the Relativist
    Commented Dec 4, 2012 at 17:36
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    $\begingroup$ @Schirmer: Thats a recent innovation :). Its not sheer laziness on their part, there's a point to that attitude... $\endgroup$
    – Mozibur Ullah
    Commented Dec 4, 2012 at 17:46

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