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I am interested primarily in physics, and I am generally self-taught in mathematics. However, this implies an inaptitude for rigorous proof. While I am confident that I can grasp the concepts and results well enough, I loath the idea of sitting through just one book on a particular subject, and study it cover to cover. - I do not want to inherit the author's idiosyncrasies, is what I'm saying.

Had I lived in other times, I would not have had a choice, but given the information torrent of our time, I want to shape my education on a mathematical subject through exploration of different means. That being said, having a textbook as a primary exercise and linearity provider (what concept proceeds the other), and multiple others (including the internet in general) to shape my knowledge of the subject, and study results from the definitions I have learned.

There have been many times where I could not see the solution to a simple, nevertheless, problem mainly because I relied on one and only textbook, which did not mention a certain lemma, or some other helpful intuitive result. I wish to abolish that through a more unorthodox approach to learning.

What would your advice be?

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Read all of this users questions. –  Git Gud Mar 17 '13 at 21:37
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Possible duplicate of Examining every mathematical result in purely formal, ZFC language. –  Git Gud Mar 17 '13 at 21:38
    
@Git: That was deleted -- unfortunately, because I think some good answers could be written (and I was writing one when it disappeared under me). True enough the OP didn't quite know what he was asking for there, but there's no same in that -- it's a chance of being told. User67170, would you be opposed to undeleting it? –  Henning Makholm Mar 17 '13 at 21:45
    
@HenningMakholm The deleted question, this one and the one on this link were similar. It's unfortunate that the OP decided to delete the question, because it had some comments that could have been helpful (if not for him, at least to some other people). –  Git Gud Mar 17 '13 at 21:48
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It's not clear how you can avoid that in any subject. If you study English history, you can't get a book that covers all aspects. These are not idiosyncrasies, they are necessary choices that any writer has to apply to material. You can't avoid it as a student because you can't avoid it as a writer. Even in physics, you are stuck with the material that the writer chooses to cover. –  Thomas Andrews Mar 17 '13 at 21:51
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There is no law stating that once you start reading a math book you have to read it cover to cover. Read the parts that interest you.

If I understand what you hope to do you will not inherit some author's idiosyncrasies you will create your own. Perhaps this will be a good thing. In any event, if you get stuck because you don't know some lemma you could post a questions something like: I am trying to prove X. This is how far I have gotten. Is there a lemma or technique that would help me finish proving X?

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First off, there is no royal road to learning mathematics, as Euclid said to Ptolemy. But you should also be aware that very few people read math books linearly, from cover to cover. Much more often people might go straight to page 90 to look up a result they are interested in, and back-fill as necessary. Also: consult many different books to see different approaches.

But this is not to imply that you do not have to submit to a discipline if you really want to learn mathematics. What you might try to do is write mathematics down for yourself somewhere, according to your lights, but do it honestly (with precise definitions and statements, and careful, convincing proofs). You will probably get stuck, at which point you can see what others have done before you to surmount the difficulty. Also, solve plenty of exercises!

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