Context: I intend to study mathematics and do research as a career. I am studying analysis and abstract algebra now and I shall begin my undergraduate studies shortly.

Should I opt for physics as minor for better mathematical intuition?

I have been told my friend that physics may lend one intuition into a few mathematical structures though I am not sure. I can only think of differential equations as an example. In particular, please tell me if physics can serve as a source of motivation and if it is crucial enough.

  • If you don't want to study physics, I don't see why you should. If you don't find it interesting, you probably won't remember enough for it to be useful later anyway. If you do find physics interesting, go ahead and study it. It can be useful for motivation, intuition, applications, in a number of fields of mathematics. – Peter Shor Jul 2 '13 at 3:37
  • You didn't tell us your level of study (late/upper- undergrad?) or anything about your institution - or exactly what you mean. If you're already interested in maths, then you don't need motivation; it's true that physics has often been a motivation for maths, or at least that they have developed together (Calc and Newtonian mech, for example). I think the answer depends a lot on how much you like physics and how you like your school's physics dept (I like physics, but hated the undergrad teaching I saw). +1 to StasK, especially that CS might be a good choice. – hunter2 Jul 2 '13 at 8:56
  • It is not only differential equations! Whole linear algebra (and a lot of functional analysis), a lot of group theory and thinks related to symmetry, countless things related to 'approximations', many aspect of meta-approach where you rather need to tinker with assumptions to get the result, than just set assumptions and only see what follows... But I guess you get intuition only if you put heart in it, at least a bit. – Piotr Migdal Jul 2 '13 at 9:35
  • @hunter2 I shall begin my undergraduate study shortly. – user7586 Jul 8 '13 at 13:14
  • Hmm. Then, unfortunately, your question doesn't belong here (tricky, I know - I just realized this when I was told a few days ago). Otherwise/more prosaically ... There's good answers here. / Keep an open mind, try and sit in on a variety of lectures (CS, Phys, etc.) in your first term to see what you like. / Note that 'topic' and 'department' aren't the same; you might like to read some books on topics, rather than getting the minor. / Give some good hard thought to the prospect of a career in academia/the math department. – hunter2 Jul 8 '13 at 13:34

While essentially seconding StasK's points, I'd be inclined to make a stronger claim about the utility of looking at not-so-elementary physics. That is, in addition to all the "mechanics" applications of the 18th and 19th centuries, and Maxwell's late-19th century electromagnetism (which provided a huge impetus to ideas about vectors!), many aspects of quantum theory focused attention on differential equations which have proven to be important examples of mathematical phenomena, in addition to applications to physics. This important-example-intensely-studied phenomenon continued with Bargmann's and Wigner's studies on representation-theory of specific Lie groups, which provided the backdrop for Harish-Chandra's vast program. This specificity was in marked contrast to the "generalism" that mathematicians of the time were embracing, e.g., Weil, Godement.

I think it continues to be the case that physical considerations suggest very-specific examples meriting intense study... which provide test cases for "purely" mathematical ideas.

Witten's (and others') relatively recent "physics" programmes have had a large impact on algebraic geometry (moduli problems, mirror symmetry).

Although I'm also fond of the crypto application/motivations of algorithmic number theory, the breadth and depth seems not as great as the math-phys connection, although of course the elapsed time is much less than for math-phys.

(The optimization and math econ, and comp sci and category theory applications/connections are less familiar to me.)

Still, I must confess that I dropped an undergrad physics minor while studying mathematics, because it seemed dreary to me at the time. Partly this was due to my inability to see the physics ideas underlying the tricks to evaluate integrals, but perhaps partly due to the accidentally-dumbed-down viewpoint promulgated in the physics courses ... presumably aiming at "accessibility".

And, yes, there is a similar common risk/disappointment in mathematics courses that give up ideas for the sake of "tractability". The risk is that it gives the wrong impression.

(Yet, yes, sometimes I've been told that what seems to me insanely fussy detail-mongering is the very essence of mathematics, and that perhaps I insufficiently appreciate "proof".)

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There's a range of applications for mathematics around many disciplines. In the XX century, physics was definitely the biggest consumer of mathematics, from the photoelectric effect and the black body spectrum in the early century through nuclear bombs and space exploration of the second half. The XXI century looks to shape around biology and life sciences.

  1. Physics leans heavily on PDE, real and complex analysis. Some areas may require abstract algebra, but they could turn out to be somewhat exotic (quantum Hall effect and other solid state physics stuff), and you would need to study physics for about 5 years to get to understand what it is if you are starting from ground zero. (School physics IS ground zero, in my books.) Until you know what Green's function is, there may be little point approaching physics for you.
  2. Economics leans heavily on real analysis and optimization. There's some use of abstract algebra, although again to get to the areas where it is really needed (welfare economics, may be some very abstract macro), you need to get very deep into grad school in economics.
  3. There's quantitative biology, in which separate fields may require way separate math tools: ecology uses some PDEs, while protein structure is computation that could be using abstract algebra, too, to describe the spatial structures (where it overlaps somewhat with material science).
  4. Computer science is another big obvious consumer of mathematics, and abstract algebra is very immediately used in various codes. If you want to have an immediate gratification from having learned simple groups, you can go ahead and figure out how the PGP algorithm works.
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  • Another comment about computer science is that studying (and programming) algorithms can give you a very good feel for growth models of different functions, which shows up all over the place in the math world. – Charles Staats Jul 2 '13 at 14:25
  • @CharlesStaats, I am not quite sure what you mean here. Can you elaborate? – StasK Jul 5 '13 at 20:57
  • I mean, roughly, that you develop a good sense of which functions dominate which others as x goes to infinity. This is often handy for making estimates (in the technical sense). – Charles Staats Jul 6 '13 at 5:20
  • This may come from some serious mathematical physics... as well as series expansions, basis function representations and other approximations. But, again, that's five years from ground zero. I would not expect even "calculus-based" physics, which would be the first class offered, by default, to user3498582397523754, even mention these in passing. – StasK Jul 6 '13 at 13:11

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