Math Knowledge for Physics: Role of Theory & Experience

[Noxide]

I decided to take a math course based solely on proofs, no such courses are in my undergrad requirements and i thought it would be interesting experience.

It seems like physicists have a working knowledge of mathematics rather than a really deep knowledge of it. I realized this after trying to show all of my work on a simple fraction simplification problem. I didn't know how. I know how to get the right answer, but I didn't actually know how to show the formalisms to my work, and explain the steps/associate them with theorems. This applies to almost all of the math I know. Maybe this is a flaw in my own math abilities, but it seems like a lot of my colleagues also don't quite have a full understanding of the math they are using, even though they are really good at using it. Is this a common occurence? I want to get into theoretical physics, and I'm wondering if I should load up on more theoretical math courses, or if a really good working knowledge is sufficient for getting into the deeper theoretical stuff. I really do understand what I'm doing, I just don't know how to show it/explain it to others which strikes me as a problem in the future.

I know I'm posting this on the math forums and am likely to get a somewhat biased answer, but how much of a role does deep theoretical knowledge of math play in physics?

 

[HallsofIvy]

That depends upon the area of physics. Group Theory, a very abstract field of mathematics, is important in Quantum Physics. Differential Geometry, and, to a lesser extent, I think, Topology, are used in General Relativity.

You might be better served posting this in the Physics section and let the physicists tell you what mathematics they use.

 

[mdnazmulh]

it seems like a lot of my colleagues also don't quite have a full understanding of the math they are using, even though they are really good at using it. Is this a common occurrence?

Oh yeah, it’s very common among engineers and physical scientists.

This applies to almost all of the math I know. Maybe this is a flaw in my own math abilities

No it's not your flaw at all. It's our limitation. Today every subject’s domain is increasing at enormous rate but our life span is not increasing significantly. So naturally we are compelled to focus on a specific area because we don’t have that much time to cover all aspects of knowledge. Our short life time is the major constraint in front of us. If we could live 200/300 years on earth, we may have had solid knowledge of different subjects at the same time.
Math is a tool in engineering and physics, chemistry and other natural sciences. Each time we apply any theorem solving physical problems, our sole focus remains on solving the problem not on proving that theorem we are using to solve. We know the proof for that theorem exists, so we don’t worry to prove it again. We leave this task on the shoulder of pure mathematicians.
After all, Physics whatever it is theoretical or experimental is connected to nature which is a physical science based on experimental observations. Yes I agree sometimes there arise cases in theoretical physics where pure mathematics dominates for a while.
So having interest in pure mathematics ( proof theory, formalism like mathematical logic, axiomatic set theory) is fine when you want to become a theoretical physicist. But you should not overemphasize it in the field of physics such that you deviate from your primary focusing point which is to explain natural phenomena with the help of mathematical formulation.
You have a glaring example in front of you- string theory. It has an elegant mathematical construction purely a theoretical mathematical theory attempting to explain our nature in an unified boundary. But is it a theory of physics or philosophy? For earning a title 'theory' in physics, the theory must be tested. And there's no instrument mankind has made so far that can test string theory. So, string theory is not accepted as a theory of Physics by many physicists. It's mathematically sound and solid but how about its physical interpretation?