# Rigorous Mathematics

## Main Question or Discussion Point

I've taken a liking to studying mathematics, though I'm a physics major I've always tried to learn things as rigorous as possible whether it's mathematics or physics. Now, I haven't quite gotten to the level where I just breeze through proofs or at least when I study the theorems it still takes a considerable amount of time to understand it.

This takes so much time that I'm confused in what I should study first. When I study physics and I'm given a derivation/proof where I lack the mathematical knowledge or the book kind of skipped the mathematics behind it, I am left confused and that is quite irritating.

That's it, I want to learn the rigorous way, well I think this can be done with proper time management and work but I still want to ask what kind of advices can you guys give me with regards to the kind of approach I could use.

chiro
Hey mathsciguy.

The first question I have for you is what kinds of things are you proving?

If its a physics proof then this may probably be a lot different to a 'math proof'.

Besides the above though, the important things for proofs will be to consider the assumptions that are set, the goal of the proof and anything that aids in the transformation from initial statement(s) to final statement(s).

Now sometimes the transformations may just some algebraic tricks like substitution of other identities as well as some approximation tricks like through taylor series expansions, or even by setting terms to a constant or to an approximate term (like taylor series but in other ways).

The particular type of approximation may get advanced in that specific bounds based on theoretical results could be used which are substituted. For example if you had a factorial in your formula then you might use Stirlings approximation which puts in a form of the factorial that is based on well known transcendental functions and others that have a well known taylor series expansion.

I don't think you will have to do the kinds of proofs you do in mathematics like say proving forms of convergence for some class of functions or proving existence theorems.

It doesn't mean that your proofs are 'easier', just that they have a different focus to the kinds of things that mathematicians focus on.

So yeah in conjunction with the above, it also needs to be asked about what specific subjects you are talking about. If you give us some of this information I gaurantee the other posters here will give more specific advice that will be more of a benefit as opposed to broader advice which may not have the same effect.

I think this is a dilemma many mathematically-minded physicists have. There really are two kinds of physicists: those who don't really care about math (they just use it fairly loosely, don't care about the proofs, and memorize mathematical facts rather than deriving them), and those who really do want to understand the math deeply.

Studying physics quickly leads you into all kinds of advanced math--even sophomores in undergrad physics will learn to use spherical harmonics, fourier transforms, legendre transformations, gaussian/fourier integrals, etc. These are all advanced topics that you'd be lucky to cover rigorously in four years as an undergrad math major.

One of my favorite examples of this kind of thing is when a physics teacher writes something like this on the board:
$$\int_{-\infty}^{\infty}\sum_{n=1}^\infty F_n(x)dx$$
Almost invariably they'll quickly swap the summation symbol and integral symbol, hoping that nobody in the class notices... Rarely they'll mutter something like "And in case you're wondering, this series is uniformly convergent." Uniform convergence is a big deal to math majors, and typically they don't cover it until a senior-level analysis course (they'll usually dwell on uniform convergence for about a week in an analysis class).

Another example is Lie groups. Physicists use Lie groups all the time, but even after two senior-level undergrad courses in linear algebra and abstract algebra, you'd be lucky to even touch on Lie groups (they typically spend the entire semester on finite groups and the trivial infinite groups like Z, R, C, etc.).

One last example: the "dirac delta function". Mathematicians cringe at that phrase, since it's really a distribution rather than a function. The fact that physicists use it ALL the time and call it by the erroneous name just shows that physicists are more concerned with quick and dirty tools rather than a mathematically correct understanding.

The point I'm trying to make is that to study mathematics rigorously, you take a LOT longer to get to the advanced topics. You could spend years trying to rigorously understand fourier transforms, so it's not recommended you try to learn all the math rigorously before using it in physics.

But knowing the math definitely helps you with physics. I definitely don't want to give the impression that it's useless to learn the math rigorously. Learning contour integration, PDEs, and linear (matrix) algebra the correct mathematical way will really help any physicist.

The fact is that if you're a typical physics student, there will always be more math to learn, and the more physics you study, the more math you'll feel you need to understand. The only way you could ever avoid being in that situation is by devoting years and years to math (like getting a Ph.D. in math) and only THEN turning to physics--e.g. David Hilbert.

@Chiro: I've studied a few proofs myself, and yes I, know how different mathematical proofs are from physics derivations/proofs. The latter tends to rely greatly on physical analysis and assumptions.

To answer the last question, I'm actually still a 1st year taking up mostly intro courses and calculus. Though I'm still in my 1st year I want to understand both physics and math as deep as I could. It's just so irritating to leave myself with hanging question especially in physics when books tend to skip certain parts in derivations. It's so irritating I find myself researching most of the time about how certain steps are done until I move on.

@Jolb: Yes this might be a dilemma, but I'd rather think this is something of benefit for me too in the long run.

chiro
To answer the last question, I'm actually still a 1st year taking up mostly intro courses and calculus. Though I'm still in my 1st year I want to understand both physics and math as deep as I could. It's just so irritating to leave myself with hanging question especially in physics when books tend to skip certain parts in derivations. It's so irritating I find myself researching most of the time about how certain steps are done until I move on.
You have the right attitude in that if you continue in this kind of thing, you'll find that being able to work with incomplete information will set you up to be able to do this in a higher capacity, so hang in there.

I can understand how it is difficult especially when you are in a situation where you have time constraints like having to get homework, assignments and specific learning with specific objectives done in a very short time window.

But yeah remember that being able to deal with incomplete, missing, and even sometimes partly wrong information is a very valuable life skill that will be immensely helpful for all kinds of situations in the future.

The only way you could ever avoid being in that situation is by devoting years and years to math (like getting a Ph.D. in math) and only THEN turning to physics--e.g. David Hilbert.
And because you have spent so much time learning Mathematics, some 'quick and dirty' physicist, like Einstein, walks away with the main prize...

@Chiro: I've studied a few proofs myself, and yes I, know how different mathematical proofs are from physics derivations/proofs. The latter tends to rely greatly on physical analysis and assumptions.

To answer the last question, I'm actually still a 1st year taking up mostly intro courses and calculus. Though I'm still in my 1st year I want to understand both physics and math as deep as I could. It's just so irritating to leave myself with hanging question especially in physics when books tend to skip certain parts in derivations. It's so irritating I find myself researching most of the time about how certain steps are done until I move on.

@Jolb: Yes this might be a dilemma, but I'd rather think this is something of benefit for me too in the long run.
I want to recommend you some books or sites you can look at. But could you first mention some of the things you want to see rigorous proofs of? Just give some examples.

In any case, it would be good to study some real analysis. The calculus book by Spivak is a good start, or "understanding analysis" by Abbott should also be a good first book.

And because you have spent so much time learning Mathematics, some 'quick and dirty' physicist, like Einstein, walks away with the main prize...
But Einstein didn't get the main prize because of his 'quick and dirty' mathematics, which actually took him 10 years I believe to fully construct GR, he got the 'prize' because of his ideas. There is a difference.

But Einstein didn't get the main prize because of his 'quick and dirty' mathematics, which actually took him 10 years I believe to fully construct GR, he got the 'prize' because of his ideas. There is a difference.
Case in point: Hilbert sat in on one of Einstein's lectures and gleaned Einstein's ideas, then went home and derived the Einstein Field Equations even before Einstein did! (Thankfully Hilbert was gracious and gave Einstein the credit.) Hilbert didn't struggle with the math for 10 years like Einstein did, probably because he took the approach of learning ALL the math and THEN moving to physics. The physical ideas were what Hilbert needed help with, and that is why we credit Einstein.

Stephen Tashi
I want to learn the rigorous way,
The "rigorous way" doesn't define a unique way. There are levels of rigor. For example, many math professors in pure math departments don't wish to bother with the extremes rigor possible in discussing set theory.

Although the idealistic picture of pure mathematics is that it is developed in a perfectly rigorous and logical fashion, step by step, each theorem building on the things that were proven before, I've never met anyone who had this development laid out completely in his mind. People know bits and pieces of it.

In learning math the rigorous way, the first thing you'll have to do is develop your own tastes about what rigor is. Most people find it helpful to do a brief study of symbolic logic and set theory. This is the deep end of the rigor pool. However, I think it is a mistake to plan to work you way through a big book on set theory and logic in order to "learn the basics". I advise you look at a small book. I also advise you that it's usually impossible to learn math in a step by step logical fashion, throughly mastering each idea before proceeding. Obviously if you don't have some grasp of the earlier material, you will be lost later. But if you have extreme tastes in rigor, you won't cover much material by learning each topic to perfection before going on.

But could you first mention some of the things you want to see rigorous proofs of? Just give some examples.
Okay, I only have a few that's spontaneously at the tip of my head now. In mathematics it's pretty much proofs on limit theorems in epsilon-delta notation, the techniques in differentiation and integration (I haven't studied them yet but if I had the time, I would). In physics, some parts on the speed of angular precession, the kepler's laws (I haven't studied this extensively too, but I think I saw a part where it just had some kind of leap in the mathematics, or somewhere they assumed I have the knowledge some advanced mathematics). I'm sorry that's all I can say right now, but I hope you get the idea.

The "rigorous way" doesn't define a unique way. There are levels of rigor.
Sorry, that must have came off as a bit arrogant and too idealistic. But yeah, I do want to learn with as much rigor or as axiomatic as I can, I am emphasizing those parts since I know it is again too ideal of me, or even pretentious to say that I want to learn with rigor.