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Cross-posted to Math.SE here.

Physicists are widely respected for using and sometimes even inventing mathematics yet physicists study Physics which is a subject in its own right.

So surely someone studying physics spends less time studying maths than the mathematics student?

If this is the case how does the Physicist achieve the required high level of mathematical maturity?

Is it a case of just knowing what mathematical methods to use and what results to look for or does the physicist study things in pure mathematics like analysis, etc?

I ask this because I am a computer science student who wants to learn mathematics in my spare time, but I think it might be a better idea to learn mathematics like a physicist as a large portion of my time will be taken with my current studies.

Also for clarity I only speak of study up to BSc level.

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please do not cross-post questions across sites. –  EnergyNumbers Sep 8 '13 at 19:16
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So surely someone studying physics spends less time studying maths than the mathematics student?

Of course. There is only finite time to learn, and tradeoffs have to be made.

If this is the case how does the Physicist achieve the required high level of mathematical maturity?

Some do so by taking courses from math departments. This is a healthy thing in general. If all the courses relevant to your bachelor's degree were from a single department, then either that department is remarkably broad in scope or you ended up with a particularly focused (for a negative connotation, read: narrow) education.

Another way to achieve this is by reading more mathematically-inclined books on one's own. There are a number of mathematical physics books that treat physics topics from a "purer," more abstract viewpoint (especially in classical mechanics!). This is really just a specific instance of the more general rule that you only really learn a subject after seeing it from multiple points of view, for example in a heuristic "physics" style and then in a rigorous "math" style.

In any event, I'd say the key is to be curious. When you come across a mathematical statement that seems unjustified, or if it feels like you're not being told all the important parts of the story, you have to probe deeper and seek out the answers.

Is it a case of just knowing what mathematical methods to use and what results to look for...

Certainly once one begins specializing in a particular branch of physics, certain techniques will become more commonplace. In almost all cases, a good researcher is in close contact with others in the field, exchanging ideas. In this way, the mathematical methods that work best in that field are shared among the physicists.

...or does the physicist study things in pure mathematics...

But of course, research is all about doing something new. Going out of your way to learn a new tool could lead to new insights and ways of thinking that were missed by others. Perhaps a physicist is discussing some aspect of research with a mathematician, and the latter points him in the direction of some math topic that addresses similar issues. Of course, making a new connection like that is way beyond what is expected of thinking at the level of a BSc.

...like analysis, etc?

Now if you want to prepare for learning "advanced" math, there is the question of which subjects are important. Here, though, we come to the crux of the issue, which I've been evading all along. Do you want to learn the most "useful" math for a particular application, or are you just interested in math for math's sake.

In the former case, learning from a physics point of view may not be the most conducive to CS. As some people over at Math.SE have already pointed out, CS-math and physics-math tend to have different focuses. Physics leans heavily on analysis and linear algebra, as measures, differentials, functions of and into $\mathbb{R}$, and eigensystems are ubiquitous in pretty much every field of physics. Things like category theory, group theory (understood here to not include representation theory), number theory, and topology (as distinct from geometry) are far less common in physics, but I imagine show up in various parts of computer science. A student training as a mathematician would have a decent exposure to most if not all of the aforementioned subjects.

On the other hand, if you just want to see some cool mathematics, that's fine too.1 The thing is, while some math courses will stand on their own, many of them usually draw on one or more "core" subjects in math, such as group theory and basic abstract algebra, analysis, or topology. For example, the incredible abstractness of category theory may appeal to you (especially if you like the paradigm of functional programming), but diving head-first into an "introduction to category theory for mathematicians" course or text without being familiar with something like group theory (which has many similar ideas but used in a narrower context) could be too much all at once. As a result of prerequisites, this path probably does have more of a time commitment before you start appreciating what you are learning. This is the price for being able to direct your studies any way you choose rather than just following up on leads that arise in your primary studies.


1 Personal note: I studied both physics and math as an undergrad, probably in a 60-40 ratio. Most of the math, however, was just because I liked the subject in and of itself. I studied all sorts of things that have had no direct impact on my (astro)physics career, and probably never will, but I don't in the least regret it. And besides, just learning something about how to think like a mathematician has probably helped me indirectly, as I believe learning a new way of thinking would help any person in any intellectual endeavor.

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Thank you for such a detailed response. I am interested in maths with a 70 - 30 ratio between interest and anything that could be applied to software development. I also find applied math quite interesting. –  user866190 Sep 8 '13 at 19:40
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That is entirely a function of which school you go to. At the University I went to, a science degree in Physics (to the bachelor level) required three years of study in physics. A science degree in mathematics required three years of study in "pure mathematics" and three years of study in "applied mathematics", normally both done together. In either case those requirements were part of the overall requirement to have 8 "units", each comprising a full one year of study in that unit. Most physics majors also did the three years of applied maths, and the first two years of pure maths, for their 8 units. I chose to do all nine of those units, to get to a dual degree in Physics and Mathematics.

Doing physics, without parallel level studies in mathematics, is just silly.

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