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I was reading a book about how mathematics was taught in Cambridge in the 19th century, and it struck me how much physics was included in the syllabus, and it wasn't optional but everyone had to learn it. For example, there were things like planetary theory, hydrostatics, mechanics, astronomy, optics, electricity & magnetism, elasticity. Is there really any point for a student who wants to major in strictly pure math to go through the ordeal (and it was an ordeal back then, especially with those terrifying olden tripos papers) of learning mathematical physics alongside pure math? What about a student now, would it be useful to learn physics given the relations between e.g. quantum physics and the Riemann Hypothesis? If so, what benefits can such a student hope to derive from dabbling in physical topics?

Another question: Is it likely that someone like Andrew Wiles has a thorough grasp of physics, metaphysics, etc., or is it possible to prove great problems in math merely by learning pure mathematics deeply enough.

Thanks

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    $\begingroup$ In my opinion mixedmath gives rather good answers and all in all is a good mod. $\endgroup$
    – Asinomás
    Dec 12, 2014 at 23:53
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    $\begingroup$ You need not learn 'mixed math' to succeed in pure fields, but having a better understanding of the world around you can enhance your creativity. $\endgroup$
    – user142198
    Dec 12, 2014 at 23:54
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    $\begingroup$ Remember, in the early 19th century, there wasn't necessarily as large a distinction between physics and math as there is today. Sure, Cauchy was writing his Cours d'Analyse, but all of the greats in math and physics were making breakthroughs in both fields. We seem to have departed from that somewhat today. $\endgroup$ Dec 12, 2014 at 23:56

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Everything you quoted (planetary theory, hydrostatics, mechanics, astronomy, optics, electricity & magnetism, elasticity) in some form or another are still compulsory subjects in maths departments in universities: for example, most academic universities in Russia (MSU, MPTI, NSU, SPbSU to name the top ones), well-reputed Grandes Ecoles in France (Polytechnique, Mines, ENS Ulm,...), etc.

To give a personal example, back in my university on mech-maths department in addition to covering most major fields in maths and mechanics we still had our fair share of physics, quantum mechanics, and CS.

I agree that the proportion of "maths"/"not strictly maths" changed: the sheer amount of graduate-studies-level knowledge in all those fields has increased very significantly since 19th century, so we simply can't ask out students to learn everything to the state-of-the-art level. However, we can ask them to have at least the general idea of other fields, which is still more or less required in major universities.

As for the potential benefit of having the broad view and knowing something from other fields - the ability to draw parallels, to think out of the box is always a plus. I saw some problems in geometry that had very simple interpretation in mechanics; when talking about Lyapunov functions I saw the eyes lighten up after I told the students that "it is some sort of an energy which decreases over time due to some sort of friction", and vice-versa, for some math students the notion of entropy became clear only when we mentioned the Lyapunov function. This list can go on and go on. The most extravagant case I saw was to apply some form of the Heisenberg uncertainty principle in economics for demand prediction.

The basic physics intuition can help while imposing some additional hypotheses on your initial data for some math problem - if one variable has a sense of density, then we want it to stay non-negative. Many maths problem come from the real world - mechanics, chemistry, biology, physics - and therefore we need to understand what we are looking for, what we can suppose and we can not, etc.

So, I would not say that we departed from the "mixed math": on the contrary, it is still there, up to an evolution of our knowledge. And "mixed math" is quite an asset for our skills, ability to draw conclusions, and for our imagination.

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Only in the twentieth century did a substantial amount of mathematics come into being that was not connected with physics.

  • mathematical logic / proof theory;
  • (some) abstract algebra;
  • computability;
  • type theory;
  • ...

Thus it is now possible to have a mathematical career untouched by physics. There is at least one substantial body of such people. We call them (theoretical) computer scientists.

At the other end of the spectrum, there are many concerned with the analytic and numerical solution of essentially physical problems, not so concerned with shaking the logical tree of theory. We call them applied mathematicians.

May all prosper and do good work.

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