Question about long proofs? How do people write 50-page long proofs (and longer)? So they have a target in mind, but I can't get my head around them foreseeing that these 50 pages of work will actually lead them to exactly their target.
 A: When I hear extremely long proofs mentioned, the classification of finite simple groups comes to my mind. One often hears that the complete proof is some 15000 pages long! But note that this is not a single contiguous piece of work, but rather some 500 published articles by more than 100 mathematicians over a period of half a century. So this is an excellent example of a proof


*

*that is unbearably long when starting from the basics

*was started with a clear target in mind: to classify all finite simple groups

*was split up into lots and lots of subresults

A: There are two cases:

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*You start by noticing something, then you prove a small claim, and slowly you add more and more claims, until you end up with a big proof of something impressive.


*You set out to prove a certain thing, then you say "Ah, if only X was true", and when you think about it you realize that X is true, and you prove that. After some finitely many iterations you end up with a complete proof spanning over 50 pages or so.
Sometimes you have to develop an entirely new technology in order to prove something. Then explaining it, and proving all its basics can end up in a small book.
Of course, often when writing complicated proofs one may feel the need to add some less trivial introductory parts, which take more pages.
All in all mathematics is not something that you do from one day to the next, but rather a huge project that you never finish in your lifetime. And the proofs just accumulate.
A: I know three ways to build long proofs. 
The most direct is a work of the mathematical intuition. It is considered by Henri Poincaré in the beginning of “Mathematical Creation”.
The another way is to expand, to generalize our knowledge. 
And the third way is used when results obtained by many authors during many years already imply the searched result. Then you can construct a proof of the searched result, looking from the height over the ways by the authors, and creating a systematical way, proof.  
