An alternative route to learn mathematics for physics, the "rigorous way"

[jordi]

What I am going to outline should not be seen as a recommendation for most. Probably only a minority would benefit from it.

One possibility to learn the necessary maths to study physics, from a "mathematical physics" point of view is to first study a degree in maths (and maybe also a Master, too).

However, not everybody has the time to do that.

What I am doing now is the following:

First, go to the basics, and study mathematical logic and set theory (including the construction of numbers). I like both books by Goldrei, for that, especially for self-study.

Then, when one understands the completeness theorem, and how the numbers are constructed, one can go to mathematical analysis. I enjoy Apostol's Mathematical Analysis, but for particular-historical reasons. Probably, other books could do the job equally well. In particular, Lebesgue integration is not required. There should be multiple integration (Riemann), though.

The previous steps are intended to give maturity to the student. It is especially important to solve many problems by oneself, and be able to "mentally reproduce" all the theorems and proofs.

So far, everything pretty standard. The key issue of my proposal is to go straight ahead into Szekeres and Hassani books on Mathematical Methods for Physics. Probably Szekeres would suffice, but Hassani is good in some things, too.

I would say that these two books are "rigorous enough", especially if one has accomplished enough mathematical maturity with the Analysis book. Some people say these books are not rigorous enough, but I would disagree. For sure, they have less material in each chapter they cover, than the equivalent "pure math" book, but usually the material there is the one that is going to be used in physics. And, as said, it is rigorous. They key issue of this approach is many proofs of the theorems are not provided (which is an anathema in most math books). But this should be fine for many students, after having accomplished mathematical maturity with analysis. This maturity should lead to search for rigorous theorems, without needing the proof of them, necessarily. And this is exactly where these two books excel. If one needed more material or explicit proofs, sure, go to a math book, but often in physics it is not necessary (at least, up to undergraduate level).

Studying these two books allows to cover a massive amount of math material in a relatively short period of time. Even better, Szekeres deals with classical mechanics, electrodynamics, special relativity, thermodynamics and quantum mechanics, using math vocabulary. As a consequence, a math oriented student can understand better the typical physics books, after studying Szekeres.

Modifications of this recommendation could be: add a calculus book before Apostol (say, the calculus books by Lax), or even substitute completely the analysis book by the calculus book. Also, one could get rid of mathematical logic and set theory, if one is comfortable enough with the explanations on logic and set theory that most calculus / analysis books have, at least summarized in a chapter 0 or in an appendix. Finally, one could use Boas before Szekeres, too, but I do not see this as essential.

 

[fresh_42]

I do not think that starting at such fundamental theorems like the completeness theorem, or even to deal with logic and set theory is good advice for physicists. Those are even for mathematicians negligible. The main difference between mathematics and physics isn't the content, it is the way to approach problems. Physicists need a good foundation in calculus of all kind, differential equations and differential geometry. The rest can be done en passant, although a bit group theory and definitely statistics and stochastics would be an advantage. There is no time for logic. The necessary logic is part of the way things are proven anyway.

The main difference is, and this is exaggerated: physicists use coordinates all the time, and mathematicians hate coordinates.

All calculations in physics take place in a reference frame, Euclidean, Minkowski, or Riemann. But always with coordinates. This means that they have to manage directions all the time, whereas mathematicians usually do not bother any specific bases. This fact requires a different education, or point of view. Hence a master in mathematics, depending on the way it is achieved, can even be a disadvantage.

 

[jordi]

Of course, I do not disagree with you.

But what I have said might be useful to at least two kind of people:

1. Students that find difficult to get motivation to start studying analysis from the axioms of the real numbers. This is extremely unmotivated. One could say that mathematical maturity is getting used to those things. But also one could think that this unmotivated method expels valid people, that could otherwise make good contributions, but that due to their psychological background are unable to learn from unmotivated content. For those people (those asking why why why all the time) a process of mathematical logic -> set theory -> construction of numbers -> analysis could be good, despite the short term burden of having to learn things that, I agree, are not really necessary for physics.

2. People that like to learn, but they are not undergraduate students. I think there are more and more people like that. For those, asking why why why all the time is the motivation that keeps them alive. And for those, learning the curve mathematical logic -> set theory -> construction of numbers -> analysis, in contraposition to starting with the unmotivated axioms of the real numbers, to start doing analysis, is like the difference between death and life.

But of course, if you are able to proceed without all this background, better for you. You can go ahead further with less requirements.

 

[Vanadium 50]

No objection to what you are doing, but I would not characterize it as "learning mathematics for physics".

 

[Fred Wright]

There is no time for logic.

I totally agree with your assessment. Long ago, when I began my University studies with a concentration on physics I told my father that I was interested in learning about complex analysis because I wanted to get a jump on the math required for a physics major. He went to his book shelf, pulled his copy of Whittaker and Watson's "A course of Modern Analysis", blew off the dust, handed it to me and said, "This is what Cambridge expects its teenagers to master." I can only say that it served me well.

 

[jordi]

No objection to what you are doing, but I would not characterize it as "learning mathematics for physics".

I do not disagree with your comment.

Having said that, I repeat the issue of motivation. Maybe some (lucky) students do not need motivation to learn analysis, and they blindly believe the axioms of the real numbers, and everything goes well starting from there.

But some others may be aesthetically baffled by such an approach. And these students might have worse marks than the first ones, not due to the fact they are "worse learners", but just because they need more motivation to learn things.

Eventually, finding new theories requires this kind of motivation. So, could it be that we are selecting out the students that potentially could do better in finding new ideas in the future?

For this reason, this path I suggest could be considered for some (not for everybody).

 

[jordi]

I totally agree with your assessment. Long ago, when I began my University studies with a concentration on physics I told my father that I was interested in learning about complex analysis because I wanted to get a jump on the math required for a physics major. He went to his book shelf, pulled his copy of Whittaker and Watson's "A course of Modern Analysis", blew off the dust, handed it to me and said, "This is what Cambridge expects its teenagers to master." I can only say that it served me well.

This seems like choosing a profession. Nothing bad about this, but in my case, I am not speaking about being successful professionally, but about the pleasure to learn, and the pleasure to be able to answer the why why why questions.