[laichzeit0]

Step 1: Read Lockhart's Lament: https://www.maa.org/external_archive/devlin/LockhartsLament....

Step 2: Download the Book of Proof: http://www.people.vcu.edu/~rhammack/BookOfProof/ You read through it and do all the odd numbered exercises (the solutions are at the end of the book).

Step 3: Get a book called Real Mathematical Analysis by Charles Pugh and you work through that and attempt as many problems as you can, with a view not to rush through it, but to expand your mind through each problem.

Step 4: Pick any of these books that interest you the most and do the same:

- Calculus by Spivak

- Algebra: Chapter 0 by Paolo Aluffi

- Linear Algebra Done Right by Axler

By then you should have enough mathematical maturity to know what to do next.

[nextos]

I'd also recommend Hubbard & Hubbard for a beautiful and beginner-friendly mix of algebra and analysis.

My preferred starter kit is Rudin plus Halmos or Axler, but treating Rudin as a summary. So a helper would be needed, like Counterexamples in Analysis. This is what Math 55 used to do.

[tgb]

I started with Hubbard & Hubbard. It is truly wonderful (and based on Spivak's other calculus book), but I couldn't imagine learning it without either significant background or a formal class. Even with a 2 semester class devoted to it in college, I ended up not understanding the most advanced concepts (eg. differential forms) until years later when it finally came up again in graduate-level courses. Of course, it was really an excuse to learn to think mathematically, not to learn calculus on manifolds.

[pmiller2]

I don't really recommend Rudin for a true beginner at all (unless, by "as a summary," you mean not really digging into the proofs themselves, in which case any good analysis book will do). Rudin will always try to take the most elegant route to the theorem, regardless if that route goes anywhere near where the rest of the text has been. The result, for me, has been that many of his proofs seem to just meander about for a little while until, at the very end, you arrive at the theorem. It's a bit like driving to work on auto pilot, and just as disconcerting to me.

[nextos]

I should have said as an outline, instead of a summary.

I think the beauty of Rudin is how compact it is. But of course you need an alternative book to be able to digest it.

[j2kun]

Caveat Emptor: Aluffi and Axler are texts meant for vastly different levels of maturity. The former for first-year graduate students, and the latter for first/second-year undergraduate students.

[kinai]

I don't find calculus and lin alg very interesting, school ruined those for me forever I guess. I am mostly interested Logic and Information Theory but I guess the latter doesn't go with a good calculus base.

Thanks for the links, I will read the book of proof and try the exercises, though mostly studied those topics already.

[mietek]

You may be interested in Paul Taylor’s “Practical foundations of mathematics”.