I just graduated with my bachelors in mathematics last year, so I have little experience in writing huge, very involved proofs. The longest proof I've ever written was about 10 pages, but it wasn't even 10 pages "long" in the way I'm talking about, it just had many cases.

When I have a general idea of a proof, I sometimes prove the theorem in one attempt. Rarely, I completely have a rock-solid proof in my mind before writing anything. Other times, I simply deduce "what I can" and hope that it leads me to my result, or that it at least sheds light on additional facts I need to investigate in order to prove the theorem.

In the last case, sometimes all of my deduction was quite useless in showing the result, and so I start over and take the proof in a different direction, or research theorems that may help me.

Then, I hear about these incredibly long proofs done by world-class mathematicians, and sometimes, I even hear of extremely long proofs (thousands of pages) that were contributed to by many mathematicians working together.

My question is: how are these long proofs "planned?" Is a general outline formulated after much discussion? Surely it is not feasible to get 200 pages in and realize your proof is not leading to the result. How do you embark on a 1000-page journey for proving some theorem? I suspect there are meetings, preliminary work, plans, and things like that.

Is there a general technique for planning these long proofs that I should familiarize myself with as I continue to move forward?

Thank you

  • $\begingroup$ Related: math.stackexchange.com/questions/406108/… $\endgroup$
    – Asaf Karagila
    Jan 3 '15 at 16:07
  • $\begingroup$ Doop, surely most "long proofs" are really just "short proofs" with all the lemmas "spilled" into the main proof. Do you have any particular examples in mind of long proofs that aren't decomposable into lemmas? $\endgroup$ Jan 5 '15 at 6:42
  • $\begingroup$ Is every proof not decomposable into lemmas? You could plausibly call every conclusion throughout the proof a lemma. I don't see how decomposition into lemmas is meaningful. $\endgroup$ Jan 6 '15 at 1:56
  • 1
    $\begingroup$ See here and here. $\endgroup$ Jan 6 '15 at 23:29
  • $\begingroup$ @AndresCaicedo, very nice :) $\endgroup$ Jan 8 '15 at 0:54

I think that many mathematicians (including me) create long proofs similarly to you, so this is a general way. Therefore I cannot add much to my answer to the linked question. Sometimes long proof are guided by a priori intuition, as Poincaré wrote, sometimes they are a posteriori glued together, for instance, as a classification of finite simple groups. Sometimes in long proof we just follow standard algorithm, for instance, while we solve PDE. One more approach I used a couple of years ago, while solving a next open problem I had no clear intuition, but I wrote a complex speculations in order to see the ideas more clearly, and, finally, I got it.

But sometimes the work done will be idle, and I don’t know a perfect way to avoid that. One of insurances can be a priori estimation: when you prove the lemmas which you write, will you be able to prove the theorem? As George Polya recommended to start a proof form the end, although this is a “symmetric” situation.


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