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Are there any resources that show how are the various proofs of important theorems in mathematics are invented? I don't understand how can anyone come up with this method for proof for example. I want to know some of the process of thinking. I can understand the proof, fill the missing steps and prove easier theorems by myself but there are proofs that although I understand the logic behind them, I don't know how they are discovered.

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    We don't "invent" proofs, we find them. And to be honest, I think the sad answer to your question is just that with experience and imagination, one thinks of new ideas to build proofs of new theorems. Going through history and looking at how mathematicians found inspiration for their proofs can be a pleasure though.2012-12-08
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    Most of us feel that way; we just have different thresholds above which we can just shake our heads. (And looking back, we don’t always even know how we came up with some of our own proofs.)2012-12-08
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    It varies wildly by subject, and who discovered them. For example, Gauss was famous for proving wildly hard theorems, but giving as few details as possible. In any case, the truth is most proofs when they are first discovered probably caused someone much more pain, and were not as straight forward than is apparent when you see it in an article or textbook.2012-12-08
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    Experience. ${}{}$2012-12-08
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    One of the places where scientists frequently get inspired is the bathroom ;-)2012-12-08

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