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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.

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I'm just finishing Szpiro's The Kepler Conjecture , which "climaxes" in the Hales-Ferguson proof concerning sphere packing in three dimensions. All told, the proof (which was carried out with extensive computer assistance, since it was treated as an optimization problem) ran for some hundreds of pages. I think Wiles' proof of Fermat's "Last" Theorem is on the order of 250 pages. Paper piles up -- as the development of an argument unfolds, and effort is made to be clear about communicating the reasoning, the discussion can sometimes become rather extensive. (continued) –  RecklessReckoner May 29 '13 at 21:18
    
Hales didn't start out thinking a proof was going to run all that long. It ended up spanning around six years and several published (and unpublished) papers. –  RecklessReckoner May 29 '13 at 21:19

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There are two cases:

  1. 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.

  2. 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.

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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
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Yes, I keep bumping up against that and want to read more about the history of that work. The "length" of the proof can also depend on how one wants to count subsidiary contributions. Wiles' proof is already quite long because he developed new material along the way. But if one were to count the relevant works of the preceding (forty? sixty?) years that fed into it, the length would be a substantial multiple greater. –  RecklessReckoner May 29 '13 at 21:26

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.

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