Can we discover new theorems by analyzing their proofs?

[nucl34rgg]

"Necessity is the mother of invention"

Most theorems are invented to be used to prove something so that ultimately, people can use it for something else (maybe more proofs or physics/engineering).

 

[mathwonk]

One of my professors suggested analyzing the proof of a theorem you have learned, and noticing whether it really uses all the hypotheses it has. If not, you can prove a more general theorem by omitting that unnecessary hypothesis. Sometimes your theorem will prove less, but still be new.

Here is an example: George Kempf proved that when one considers the birational map from the symmetric power of a curve to the theta divisor in its jacobian, the tangent map is a birational surjection from the normal bundle of the fiber, to the tangent cone of the theta divisor.

https://www.jstor.org/stable/1970910?origin=crossref&seq=1

My colleague and I noticed that in the case of a Prym variety, where one again has a map from a divisor variety to the theta divisor of the Prym variety, even though it is not birational, the tangent map on the normal bundle is sometimes still a surjection onto the tangent cone of the theta divisor. This allows one to understand the tangent cone of the theta divisor almost as well as in the birational case. The essential similarity was the fact that in both examples, Jacobian and Prym, the fibers of the original parametrizing map were smooth subvarieties, i.e. the derivative of the parametrizing map vanished only along tangents to the fiber. Using that in the case where the Prym divisor variety was also smooth, which always holds for Jacobians, gave the new result, by essentially the same proof as the old one. I.e. the key point in finding a new theorem, was noticing that the key property was not one of the overtly mentioned hypotheses, but one of the facts hidden in the proof. In the words of my professor, "try to find a theorem whose proof proves more than it claims to."

https://www.math.uga.edu/sites/default/files/inline-files/sv2rst.pdf

It is not at all easy to find interesting new theorems however by this or any other strategy. This example is several decades old, and even with the collaboration of brilliant colleagues, I contributed to producing less than one paper a year during my career. Kempf's beautiful theorem itself built on insights he unearthed in a wonderful paper of Andreotti and Mayer, and others shown him by David Mumford.

That history is discussed in the appendix to this paper:
https://www.math.uga.edu/sites/default/files/inline-files/onparam.pdfanother idea that relates to your post #29, is to read the statement of a theorem in your book and then close the book and try to prove it yourself. This forces you to think of the idea for the proof, or at least how to begin. This exercise is usually not completely successful, but even if you only get the very faintest first idea, you already have jumped the hurdle of finding out how to begin. And if you in fact prove it yourself, often by a somewhat different proof, you may have already proved a new result, maybe a slightly stronger version. I read this advice in a famous article of Zariski, a pioneer of american algebraic geometry.