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August 15th, 2013, 01:07 PM   #1
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Book explaining proof development

Hi,
do you please know any book that explains how particular proofs of the given theorems were developed (created)?

The process of evolution of the proof of a theorem is often not so straightforward, so there are some ideas at the begining, these ideas are made to be more precise and after some clean-up of the ideas and leaving only the final argumentations the proof is made complete. Often the reader cannot see the auxiliary objects that helped to develop the proof (or these objects are part of the proof but there is no explanation why do they have the presented shape and how was the shape "discovered") and the reader also cannot see the process of development of the proof - but this can be often more interesting than the proof itself.

To give some (trivial) example - let's say that we have "theorem" that the equation x^2 -11x + 30 = 0 has solution. The proof (that is not explanatory but such proofs are common) can be: set x:=5 and it can be easily verified that this solves the equation. The proof is correct, as much compact (short) as possible but it gives no clue how it was developed (discovered) - where the number 5 come from? What I am looking for is explanation how the proof was discovered, ie. in this case using discriminant there is general method for solving quadratic equations (or even derive these method) and showing how can this method be used to find the solution of the equation. (Well, this example is trivial but I hope it will help to show what I mean.)

So that's why I am looking for some book that will explain this process of proof development (creation). Or to take it even further - the process of theorem development (when we even don't know theorem that is to be proved and we are trying to formulate theorems on our own.).

Thank you for any suggestion
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August 15th, 2013, 01:58 PM   #2
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Re: Book explaining proof development

That's a broad question! I think Polya's How to Prove It is the standard reference.
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