Classic Mathematics Books for Lifelong Learners

Depending on your level of math ability you can often learn a lot by reading textbooks and proofs very carefully and closely (my opinion)

Sorta, kinda supposed to learn to "do proofs" by reading books that are done with theorems and proofs and have exercises that ask for proofs.

Commonly the first start on proofs, in either the books or the exercises, is high school plane geometry. The proofs there are commonly in a rigid format which is traditional and maybe good for a start. Later, the good proofs are still fully precise but without the rigid format.

Then high school second year algebra may have some proofs; trigonometry pretty much does; solid geometry does; analytic geometry mostly does. Then first calculus commonly goes a little light on the proofs.

After I went through all that, I took a course in college abstract algebra, and (1) it was all proofs and, best of all, (2) the prof read some of the homework and gave little notes on how to do proofs. E.g., he said to use "since" instead of "because". That was about all I needed in one on one tutoring in how to do proofs.

Do calculus with all the proofs later in pure treatments of advanced calculus, e.g., W. Rudin, Principles of Mathematical Analysis.

As I read proofs by some good authors, especially, Nering, Halmos, Rudin, Fleming, Spivak, Coddington, Tukey, Neveu, but more, I refined how I wrote proofs.

One step I took was, in a proof, when use one of the clearly stated assumptions in the statement of the theorem, mention that are using the assumption, that is, so that can be more sure are actually using the assumption, and where and for what, for yourself and any readers. If don't use the assumption, then either have (A) a more general result and maybe something exciting or (B) an error, and in practice more likely (B)!

Later in a course in measure theory, a prof corrected some of my homework: In measure theory, are working with infinity so much that have to be careful not to, say, subtract one possibly infinite quantity from another one; that is, have to be a little careful slightly to refine some algebraic manipulation skills learned before. It's all quite doable, but if get that far in math then should learn this little point of being careful.

When you have the actual ideas necessary for a proof, the above is about all you need -- or so it seems to me, and I have an applied math Ph.D. and have published papers with theorems and proofs.

For thinking of the ideas needed for an especially challenging textbook exercise or, and mostly for, research, i.e., a new theorem and its proof, commonly my approach is (1) have good enough understanding of the likely main prerequisite material, techniques, and tools, (2) think intuitively, e.g., build little intuitive models, (3) with the little models, or actual reasoning if can, do some fast thought experiments, working quickly, with a lot of intuition and not very precisely, something like "If A is true, then it looks like B is also true, but, gee, B sounds like asking too much, so maybe A isn't true.". I.e., trying to prove things that likely aren't true is at best a long shot!

Can continue by looking at what does/does not hold in simple cases; if can find a counterexample in a simple case, then can give up on a generalization being true!

If the result does hold in a simple case and have a proof it does, then can look at the proof and start to guess where the simple proof would fail in the more general case.

Uh, just because one proof of a simple case doesn't generalize does NOT mean that the more general case doesn't hold! Be careful about that! There start to back off a little, ascend to maybe 2000 feet up, and start to look at the larger picture and what the important parts, assumptions, techniques, are there. Then with this larger view, get some hints for a proof for that more general situation.

If something works in some simple cases, then start to guess, test, understand where generalizations might or do fail.

Use more such simple, intuitive, meta stuff, maybe from what I outlined here or what you can dream up and try.

From such guessing around, may start to have some idea what is true and false and why. Then can set out to do a real proof.

If are still stuck, then try to back off and guess, maybe, what assumption actually holds that so far I have not noticed and not exploited.

Or, a polished, correct proof is very precise; coming up with that proof might have used a lot of intuition, guessing, etc.

For winning the Abel Prize, f'get about my advice!

It depends on your definition of learning. If by learning you mean remembering how one author defined and proved various things then sure. If by learning you mean gaining the ability to do what the author did except for new problems, then at some point you will have to grapple with the fact that coming up with new ideas is a very different mental exercise from recalling things you've read.

A supporting factor in the necessity of exercises for learning to actually do the math is that authors often hide important technical tricks in the exercises.

Of course, a sufficiently intelligent person could learn anything by reading tea leaves and then deriving on their own whatever it was they wanted to learn, so in a sense you can learn math by doing anything.

Proofs from the Book looks superb but does anyone know of another introduction to proofs or compendium of simple proofs suitable for young mathematicians?

I would recommend "Fermat's Enigma" by Simon Singh. It's is book that goes into the heart of how mathematics is sometimes elusive to the most sincere pursuit and at the same time yields itself to a creative mind that chases it with passion.