How do mathematicians come up with new proofs?

[docnet]

TL;DR Summary

How are research-level proofs written?

I watched an interview of Yitang Zhang and he said "the way to prove a finite limit of bounded gaps between primes came to him during $30$ minutes in an afternoon", and he worked alone and did not collaborate with others during his research time.

After looking up the proof, I am in disbelief he worked alone. What baffles is how one person could write $50$ pages of what feels like an enormously complicated and difficult mathematical maze to end up with the final result "so and so is the lower limit of so and so". I can't believe that so much work is done just to prove the final result because so many independent steps are taken, that don't seem to be obviously connected to the final result at all. But every step is nit-picky, deliberate, and brings the logic one step closer to the desired result. Are there mathematicians who could even read the entire proof and understand everything in it?

My main question is, do research-level proofs in mathematics such as "bounded gaps between primes", or "Harnack's Inequality for the Ricci Flow" or "Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere." really come as a result of a one-person's genius, like everyone makes them out to be?

 

[mfb]

The key ideas of a proof can be summarized much shorter than the fully worked-out proof. You can give a mathematician working in that field a one-page summary (or maybe even shorter) and they'll be able to reproduce the proof based on that summary.

At the time of the "30 minutes in an afternoon", Zhang had already spent quite some time working on that problem and related problems. He had some preliminary results, he was obviously aware of the older results by his colleagues, had discussed the problem with them many times and so on. At the end of these 30 minutes he didn't have the 50 pages of proof written down, but he had an idea how to combine all these things, plus a few things he expected to be able to proof later, in a way that would lead to a proof.

 

[lavinia]

If one thinks hard and continuously about something for a long time, it seems that the mind works on its own unconsciously. Ideas just pop up seemingly out of nowhere. One "sees" the relationship. However working out the details may take a huge amount of work.

 

[weirdoguy]

Ideas just pop up seemingly out of nowhere.

I'm not a "working scientist" example, but I had numerous situations, where the solution to a problem simply came to me during a... dream.

 

[berkeman]

during a... dream.

In a jacuzzi for me...

(but as pointed out before, I'd put in hundreds of hours on that problem before the final aha moment)