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From time to time, people announce papers claiming to have settled long open problems like $P$ vs. $NP$. There have been many attempts, reading them is time-consuming, and finding bugs in their arguments is not easy, ... . This brings up the following question:

When would you read a paper claiming to have settled a famous long open problem like $P$ vs. $NP$? What are your criteria to consider such an announcement as serious?

EDIT: I am mostly interested in the case that the paper is in your area and is not written by a crank but by a mathematician with previous publications in reputable journals (although not necessary in the same area or a related one).

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closed as off topic by Harry Gindi, Hailong Dao, Noah Snyder, Akhil Mathew, Felipe Voloch Aug 9 '10 at 0:41

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I think that this is a moral duplicate of mathoverflow.net/questions/6912 . At least, I would answer it in exactly the same way. –  Greg Kuperberg Aug 8 '10 at 21:02
    
Thank you Greg. I think that they are quit different, that asks if people really check these claims, this one is asking when they would check them. Your answer there is the kind of answer I am looking for: "Learning from reading it". But I want to hear other factors also. By the way, I would be happy to merge this question with that one and make that a community wiki if it is possible to do so. (There is a new claim of $P \neq NP$, and I have heard that more than one expert in complexity theory consider this one to be serious.) –  Kaveh Aug 8 '10 at 21:16
    
There is also an implicit question: criteria that would make you not read the paper, e.g., if the author is not a mathematician. –  Kaveh Aug 8 '10 at 21:29
    
I am mostly interested in the case that the paper is in your area and is not written by crank. –  Kaveh Aug 8 '10 at 21:30
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There has been blog post on Scott Aaronson's blog a while ago. scottaaronson.com/blog/?p=304 –  wood Aug 8 '10 at 22:45
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3 Answers 3

When would you read a paper claiming to have settled a famous long open problem like $P$ vs. $NP$?

If the paper claimed to resolve $P$ vs $NP$, I'd begin reading it right away. For instance, I'm currently looking at this paper. But that is only because I have a good chance of understanding the work. If the paper claimed to resolve any other Clay Millennium Prize problem, I'd defer to others.

What are your criteria to consider such an announcement as serious?

(a) It's not written in Microsoft Word

(b) The abstract, title, and opening paragraphs do not convey obvious misunderstandings

For $P$ vs $NP$ proofs, criteria (a) and (b) work about 99% of the time. (Seriously.) I'm still checking to see if the above link passes criterion (b). It's a long abstract.

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For the link above the first thing I checked was whether the paper cites Razborov-Rudich. –  Qiaochu Yuan Aug 8 '10 at 22:40
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Yep, that's important to check, for a proof that wants to go through random models of k-SAT... and it passes that check as well. –  Ryan Williams Aug 8 '10 at 22:43
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Dick's post: rjlipton.wordpress.com/2010/08/08/… –  Kaveh Aug 9 '10 at 4:23
    
That paper in pdf format: hpl.hp.com/personal/Vinay_Deolalikar/Papers/pnp12pt.pdf :) –  Kaveh Aug 9 '10 at 7:09
    
ok, highly commend/appreciate your declared flexibility/enthusiasm, now what about a proof outline? =) –  vzn Feb 18 at 22:38
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Published in a respected journal?

Unless the solution claims to use mathematics where I have some particular expertise, that is probably the only place I would read such a paper.

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I think the problem with the first one is that it would be really hard to publish such a result in a reputable journal before people working in that area are convinced that it is correct. –  Kaveh Aug 8 '10 at 21:25
    
On the other hand, I have heard from people that they would not consider a paper using usual well-known stuff as serious, the reason being that many experts in the area familiar with them were unable to solve it, and we also have some negative results that prove that using such and such techniques can not settle the question (based on some widely believed conjectures). –  Kaveh Aug 8 '10 at 21:25
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@Kaveh: when they decide whether to publish a paper in a reputable journal, they don't take a poll of researchers working in the area; they solicit one or two particular reserrachers in the area to act as referees, who then read the paper very carefully. So I suppose "When you have agreed to be a referee" is one sufficient reason to take a paper seriously. –  Pete L. Clark Aug 8 '10 at 21:42
    
@Pete: I think what you say is correct for normal questions. But not for long open questions like Poincaré conjecture or $P$ vs. $NP$. The problem is that almost no expert in that area is ready to read say a hundred pages and spend time to find a bug in it or announce that it is correct. Let's consider Poincaré conjecture. Was Perelman work published in a reputable journal before people in the area were convinced that the solution is correct? –  Kaveh Aug 8 '10 at 22:02
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Kaveh, your assertion that "almost no expert in that area is ready to read say a hundred pages and spend time to find a bug in it or announce that it is correct" is wrong. Firstly, experts have read over long papers claiming proofs of big conjectures, even if the history was somewhat problematic: Bieberbach conjecture, Seifert conjecture, 4d smooth Poincaré conjecture, Fermat's last theorem, the Kepler conjecture, etc. Secondly, it's highly unusual $\textit{in mathematics}$ to have an expert publicly certify that a certain proof is correct. BTW, refereeing doesn't provide such a certification. –  Victor Protsak Aug 8 '10 at 22:38
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just caught this at slashdot...thought I would make it my first post at MO.

http://gregbaker.ca/blog/2010/08/07/p-n-np/

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This came up in the comments to this (closed) question: mathoverflow.net/questions/34953/… –  Yemon Choi Aug 9 '10 at 0:28
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