{\huge How To Check A Proof?}

What to do about claims of hard theorems? \vspace{.5in}

Shinichi Mochizuki has claimed the famous ABC conjecture since 2012. It is still unclear whether or not the claimed proof is correct. We covered it then and have mentioned it a few times since, but have not delved in to check it. Anyway its probably way above our ability to understand in some finite time.

Today I want to talk about how to check proofs like that of the ABC conjecture.

The issue is simple: Someone writes up a paper that ``proves'' that X is true, where X is some hard open problem. How do we check that X is proved?

The proof in question is almost always long and complex. So the checking is not a simple matter. In some cases the proof might even use nonstandard methods and make it even harder to understand. That is exactly the case with Mochizuki's proof---see here for some comments.

Possible Approaches

Let's further assume that the claimed proof resolves X which is the P vs. NP problem. What should we do? There are some possible answers:

• Ignore: I have many colleagues who will not even open the paper to glance at it. Ken and I get a fair number of these, but I do at least open the file and take a quick look. I will send a message to the author---it usually is a single author---about some issue if I see one right away.

• Show Me The Beef: I firmly believe that a proof of an open problem should have at least one simple to state new trick or insight that we all missed. I would suggest that the author must be able to articulate this new idea: if they cannot then we can safely refuse to read it. I have worked some on the famous Jacobian Problem. At one time an author claimed they had a proof and it was just ``a careful induction.'' No. I never looked at it because of the lack of ``beef,'' and in a few weeks the proof fell apart.

• Money: Several people have suggested---perhaps not seriously---that any one claiming a proof must be ready to post a ``bond'' of some money. If someone finds an error they get the bond money. If no one does or even better if the proof is correct, then the money can be donated to one of our conferences.

• Hire: I have seen this idea just recently. The author posts a request for someone to work on their paper as a type of consultant. They are paid a fair hourly rate and help find the error.

• Timeout: An author who posts a false proof gets a timeout. They are not allowed to post another paper or submit a paper on X for some fixed time period. Some of the top journals like the Journal of the ACM already have a long timeout in place. The rationale behind this is very often when an error is found in such a paper the author quickly ``fixes'' the issue and re-claims the result. In Stanislaw Ulam's wonderful book Adventures of a Mathematician he talks about false proofs: Here ``he'' refers to an amateur who often joined Ulam at his habitual coffeehouse: Every once in a while he would get up and join our table to gossip or kibitz $latex {\dots}&fg=000000$ Then he would add, ``The bigger my proof, the smaller the hole. The longer and larger the proof, the smaller the hole.''

• Knock Heads Together: Oxford University hosted in December 2015 a workshop to examine Mochizuki's claimed proof, including contact by Skype with Mochizuki himself. A report by Brian Conrad on the MathBabe blog makes for engaging reading---we could quote extensively from its concluding section 6. This shorter news report cited feelings of greater understanding and promise but lack of definite progress on verifying the proof, noting:

  $latex {\dots}&fg=000000$[N]o one wants to be the guy that spent years working to understand a proof, only to find that it was not really a proof after all.

• Share The Credit: Building on the last point, perhaps proper credit can be given to someone who does spent a great deal of time working on trying to understand a long proof. If they find an unfixable error, then maybe they can publish that as a paper---especially if the error is nontrivial and not just a simple one. If they show that the proof is indeed correct, could they be rewarded with some type of co-authorship? Maybe a new type of authorship:

  P Does Not Equal NP: A Proof Via Non-Linear Fourier Methods
  Alice Azure with Bob Blue

  Here the ``with'' signals that Alice is the main author and Bob was simply a helper. Recall a maxim sometimes credited to President Harry Truman: ``It is amazing what you can accomplish if you do not care who gets the credit.''

Open Problems

What do you think about ways to check proofs? Any better ideas?