Best book to re-learn my advanced math I’m not a complete beginner. I’ve got a PhD in Mathematics Education which gave me a pretty strong Master’s degree level of pure math knowledge.
That was 30 years ago, and I’ve forgotten tons. Seeing Michael Spivak’s Calculus book mentioned all over, I thought I’d give it a try. The problems seem from pretty simple to impossible.
I’d like to get my analysis chops back in order. I have my copy of Taylor & Mann for Advanced Calculus, Churchill Downs for Complex Analysis, and a few others I’ve collected here and there.
So I’d like your informed opinion: should I try to regain my math chops using Spivak, or use a more traditional Analysis text—maybe not as traditional as Taylor and Mann, but maybe Understanding Analysis by Abbott.
 A: Spivak is good, also so are Tao's books.  I mostly recommend reading a bunch of books and using the ideas that make the most sense to you.  I usually try to have a primary book that I like the most, and a few others that I read a little more quickly just to see how other authors handle the same topic.
A: One reason people use Spivak's book for self-study is that it has a very extensive solution manual containing the answers to the exercises, so if you are studying by yourself you can look up all the answers. Now there is stackexchange so the solutions manual is less relevant, although some people might be embarrassed to admit when they got stuck on question 1.1 or something. Spivak also grades the difficulty of the problems with stars, which are quite accurate about the relative amount of time the problem will take.
It was published in 1966 so it's already "traditional". Also Spivak was the course book for first-year analysis at my university, so I really don't think it's some kind of unconventionial book (or perhaps I went to a weirdo university). Anyway I just went and looked at the university's page and they are still using Spivak. Spivak was ahead of his time because he also invented "gender-neutral pronouns".
If you do decide to go with Spivak it would be smart to start from the beginning and do the "easy" problems and then work through to the end. People who start in the middle of the book end up like those weightlifters who only work on one part of the body, so they have giant biceps and tiny legs or something. Then they can do all kinds of complicated-looking proofs but they cannot do the simple parts of problems.
Spivak also does not cover set theory. Sometimes you need to understand at least the basics of that.
Another opinion: When I mentioned that Spivak was the course book at my university, a person who attended another place told me that he thought that "Spivak has no soul". I have absolutely no idea what he meant by that comment.
