I have a background in theoretical physics, but no rigorous training in math besides the introductory coursework I took in college a while ago. I would like to become more proficient in math, though -- mostly for fun. I thought I'd start with real analysis because it seems so central. How accessible is the "baby Rudin" book, and would that be a good place to start? How would you recommend going about this? I should say I don't know much about mathematical formalism (mathematical logic, set theory, proofs, and so forth) but I'm very eager to learn. Life is short, and I want to develop this interest!
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$\begingroup$ Good choice. Both Rudins are decent books. Also you may try an algebra book and a geometry book. $\endgroup$– markvsFeb 3, 2022 at 1:55
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$\begingroup$ I think that Ch. Pugh "Real mathematical Analysis" beats Rudin, especially if you want to study on your own. $\endgroup$– SalcioFeb 3, 2022 at 2:06
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3$\begingroup$ What's really important about math is working from the ground up. So I would start with the basics-- logic and set theory. Before you cover those properly learning real analysis would probably be unnecessarily difficult. You don't need to get to a very high level in set theory and logic (unless you find it interesting), just what is covered in an introductory first year course for mathematicians (often called discrete math). $\endgroup$– SnawFeb 3, 2022 at 2:31
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$\begingroup$ @Snaw +1 What do you recommend for that (a specific online set of lectures, book, notes, etc.)? $\endgroup$– user95199Feb 3, 2022 at 4:20
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1$\begingroup$ @user2561523 I like the book "Naive Set Theory" by Halmos, but it covers more than what you need. If you search for discrete math on youtube you'll find a lot of material including real courses from universities such as MIT. You can probably skip stuff related to graph theory which is usually covered in such a course, it likely won't be relevant for your studies for now. Other than this most courses should cover pretty much the same topics -- they start with logics and move on to set theory, make sure they get as far as cardinal numbers and that should be all the background you need. $\endgroup$– SnawFeb 3, 2022 at 8:15
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The process you should take depends on a few things. What's your background with proofs? If you're not very experienced in reading and writing proofs, starting with baby Rudin will be a bit difficult - you might be better served starting with an easier intro analysis text like Abbott or Ross. If you're comfortable with proofs, you could start with Rudin and see how it goes (but expect it to be difficult regardless). You should try to as many exercises as you can, and if at all possible find people to discuss with; analysis is difficult, and it helps to have somebody to run your proofs by and who you can talk with about the material.
With a background in theoretical physics, after learning some real and complex analysis you might be interested in learning more about differential geometry and functional analysis. Functional analysis helps to make rigorous a lot of quantum mechanics, and similarly the language of differential geometry comes up quite often in classical mechanics and relativity. Rudin has a good text on functional analysis as well, and I quite like John Lee's series on manifolds. With a bit of differential geometry behind you, you might also appreciate Spivak's text "Physics for Mathematicians: Mechanics I".
If you're just looking to learn math for its own sake, you need not limit yourself to analysis as a starting point! You can get into abstract algebra, elementary number theory, and logic/set theory without a background in analysis - the only limiting factor, again, is your comfort with proofs. (If you find yourself wanting to learn one of those topics instead I'd be happy to give more book recommendations.)
EDIT: In terms of starting out with proofs, Rosen's book "Discrete Mathematics" will give you the language you need to get started in terms of basic logic and set theory. You could also jump straight into analysis with Abbott - in my opinion it's friendly enough for beginners that you could learn proofs as you go.
For abstract algebra, Dummit & Foote's algebra text is terrific but definitely requires some fluency in reading and writing proofs (and should definitely be supplemented with another algebra text for a first read). After getting down the fundamentals of proof-writing you could start on that book; others to supplement with could be Carter's "Visual Group Theory", Artin, or Fraleigh (my undergrad course in algebra used Fraleigh, but it felt a little too gentle for my taste - but especially for a first introduction to advanced math, it might be worth checking out!).
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$\begingroup$ +1 Thank you! My background with proofs is nearly non-existent, limited mostly to a few exercises in college and not a particular course devoted to it. What would you suggest there in terms of logic/set theory, since that seems perhaps to be a good place to start? And what would you recommend for abstract algebra? I have a decent command of linear algebra from a computational perspective only. $\endgroup$– user95199Feb 3, 2022 at 4:23
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1$\begingroup$ No problem! I've edited the answer to give some references for algebra and logic/set theory. $\endgroup$– csch2Feb 3, 2022 at 16:25