Written: Sept. 10, 1999, [= Elul 29, 5759]
So-called `respectable' (and would-be respectable, like the Proc. of the AMS) math journals only accept complete proofs, and the PAMS even explicitly states that: `Inconclusive attempts are not acceptable'.
Mathematics was, and largely still is, proof-oriented. In other words, the primary object is proof, and a mathematical fact, even if it is more certain than E=mc^2, that has not been completely and rigorously proved, is given the derogatory name: `conjecture'.
Proof-Oriented math is analogous to procedure-oriented programming. Computer scientists realized that a procedure is just one out of a plethora of objects, and hence C++ and other object-oriented programming languages are now thriving.
The computer revolution has spurred a new kind of math: `quasi-empirical' and experimental (whose central object is the mathematical FACT). Also the amazing symbiosis and dramatic success of theoretical physics gave rise to a whole culture of `non-rigorous proofs'. Borwein et. al even proposed a philosophy and methodology of experimental math, analogous to experimental science. Mathematical purists like Jaffe and Quinn warned for a strict Apartheid between `rigorous' and `theoretical' (i.e. non-rigorous). Gregory Chaitin showed that all provable results are ipso-facto trivial, and most mathematical facts are either undecidable, or unprovable in real time. Hence Chaitin endorses empirical math.
Things don't have to be so polarized, and we should get rid of our true-or-false, black-or-white, `almost does not count', Boolean mentality. In 1993, I proposed semi-rigorous math, but even this is too Boolean. A mathematical fact is semi-rigorous if there exists a computer program that can prove (or disprove) it in a finite time, and that has been verified in many special random cases. But this is almost as good as rigorous.
I would like to legitimize a new kind of proof: `The Incomplete Proof'.
The reason that the output of mathematicians is so meager is that we only publish that tiny part of our work that ended up in complete success. The rest goes to the recycling bin. (And nowadays is just rm-ed) . The old joke of the difference between a mathematician and a sociologist being that the former only needs paper, pencil, and a waste-paper basket, while the latter does not have a need for the waste-paper basket, comes to mind. Also recall Johnnie von Neumann's maid's astonishment at the working habits of her master, who `seems like a reasonable man, but he scribbles all day on pieces of paper, and then dumps them all in the garbage'.
What a pity! If von Neumann's maid would have salvaged all that material, I am sure that we would have found lots of treasures there. Marvin Knopp told me that the late Emil Grosswald had a file-cabinet labelled `failed attempts', that unfortunately is lost for ever. I am sure that many great (and lesser) minds had brilliant ideas on how to prove the Riemann Hypothesis, but just because they could not complete it themselves, they never published them, and we are doomed to never see them. Perhaps all they needed was a little help from Shalosh?, or from Andrew Wiles? Why did Paul Cohen stop publishing at the age of 30? my guess is that he was trying, and probably still is, to prove RH. I would love to be able to see his `failed attempts'. I am sure that Louis de Branges's many `wrong' proofs of RH and other conjectures are as chuck-full of brilliant ideas as is his proof of Bieberbach. But I will never get to see most of them. (Luckily, de Branges managed to publish some of his incomplete proofs, but I do not have the patience or background to go through them.)
So here is my revolutionary proposal. Publish all your (good) thoughts and ideas, regardless of whether they are finished or not. Of course, if you only need one more lemma to complete the proof of RH, you are welcome to wait a few months until you succeed, but if you still can't prove it, in, say, two years, PLEASE, publish it anyway. I promise you that I will still give you most of the credit for proving RH, and phrase my abstract as follows: `We prove a technical lemma needed in Joe Doe's brilliant proof of the Riemann Hypothesis', rather than: `We prove the RH, by proving a statement that was shown by Joe Doe to be equivalent to it'.
But what will the ref and editor say? Luckily, the world-wide-web has freed us from the tyranny of these narrow-minded and nasty creatures. Publish everything in your website. To make it easier for others to fill-in your gaps, make your proposed proof structured like in my seminal (and luckily for me, complete) Proof of the Alternating Sign Matrix Conjecture . Just indicate which vertices in the proof-tree are already proved and which ones still need a proof. This was how my proof was checked by many people, but it also could have been constructed that way, by equally many people or computers. The proof-designer would get most of the credit, before and after the last nail, and the individual provers of the various (sub)^i-lemmas would get credit too, and be immortalized. BUT, This Is Not a Pinjata! The prover of the last unproved (sub)^i-lemma should not get any more credit than the other provers, unless his or her part were particularly difficult.
Also don't be bashful about web-publishing your software development efforts. The Maple packages that are posted in my website are but a fraction of my efforts. I have many more, geared to prove RH, P!=NP, gamma irrational, and other still-in-progress projects. I should make them generally available, right now! (and not wait until finishing them up). So my (Jewish) New Year resolution is to follow my own advice.