A guide to tell if a proof is a proof

Andrew Wiles is a famous mathematician at Princeton University who solved the long standing open problem that was called “Fermat’s Last Theorem” even before he made it a real theorem. You probably know that this states that

$x^p + y^p + z^p = 0$

for $p>2$ and $x,y,z$ integers implies that $xyz=0$. When I was at Princeton one of my graduate students, Dan Boneh, went to see Wiles about an idea that we had in number theory. After a few minutes Dan was gently–but firmly–asked to leave Wiles’ office. The story of this is another post that I will do on the complexity of factoring of integers.

I raise this great result because the first set of comments on this blog have been about the validity of a claimed proof to a long standing open problem. You may recall that Wiles first proof failed and it took almost another year to repair the problem with his proof. The issue I want to start to talk about is what makes a proof believed? I claim that this is a social process. Indeed I written an entire paper on this decades ago with Rich DeMillo
(see our paper ). I will discuss this in detail again in a later post: I hope I can keep all these promises.

Today I want to say something a bit different. I want to claim that a proof to a long standing open problem must pass a simple test: it must pass the Elevator Pitch Test. That is you must be able to convince an expert that you have a new idea in the time it takes to ride an elevator.
You do not have to convince an expert that your proof is correct. That could take days, weeks, or even months, if the proof is long and complex. No. But you must be able to convince an expert that you have a new idea that gives you the “edge” that no one previously had. The edge that makes it possible for you to have solved the problem. For example, in Wiles’s case the edge was a new approach to Fermat’s problem based on elliptic curves.

I recently looked up the term Elevator Pitch on Wikipedia. This is what they said: “an Elevator Pitch is an overview of an idea for a product, service, or project. The name reflects the fact that an elevator pitch can be delivered in the time span of an elevator ride (for example, thirty seconds and 100-150 words).”

So what does an Elevator Pitch (EP) have to do with a mathematical proof? I believe that solutions to hard open problems must have a short EP that gives the reader some insight into why this solution works when all previous ones have failed. This probably applies to any proof, but it is especially relevant to claimed proofs of difficult open problems.

Let me be clear about what I am not saying. I am not saying that proofs of hard open problems can be summarized into a few words. Of course not. Proofs like Wiles brilliant solution of the famous Fermat Last Theorem are long and difficult. What I am saying is that the author of a proof to a hard open problem must be able to give us an EP. They should be able to, in a few words, explain what the new “trick”, “idea” or “approach” they use that makes their proof work. Statements like:

• It just works.
• It is just a long induction.
• It is too complex to explain.

• are not sufficient. Good examples would be: “I noticed that X is true and so I could prove the result if I could only show Y. For some reason no one before noticed that X was true.” An expert hearing this would say, “Cool, I never noticed that X is true. I can see why this gives you an edge.”