Learning the Mathematical Process

Watching Mathematicians at Work (AI generated)

The Smithsonian Natural History Museum has a FossiLab where visitors can peek through windows watching scientists prepare fossils for conservation. Maybe we should have a similar exhibit at math museums or universities. How else can we learn what mathematicians do? 

In 2025, artificial intelligence has achieved gold medal status at the International Mathematical Olympiad but so far has only contributed modestly in finding new theorems. Of course, finding and proving new theorems requires a different set of skills than competition problems but it goes further than that.

The Internet has considerable text and video on how to solve math competition problems that machine learning systems can train on. On the other hand, mathematical research papers usually have little more than theorems and proofs. Maybe some intuition. Rarely do papers go into the thinking process and the false steps that one takes until one finds the proof. For some problems I've spent weeks proving a theorem but only the last day's work gets written up.

Now I doubt many mathematicians would give up their privacy and time to train AI systems to take over their jobs, but just suppose we wanted to do so. We could equip every mathematician with a camera recording every mathematical conversation and everything they write, especially the ideas that don't pan out. We can transcribe it all and feed it into an ML system. But it probably won't be enough.

Trouble is most mathematical breakthroughs just happen inside of people's heads. If you ask a mathematician how they came up with the clever idea that led to a major new result, they can rarely truly explain the process behind it. Not unlike neural nets. 

If machines can't learn to prove theorems by watching mathematicians, perhaps the route mathematicians take: A grad school slog towards PhD research and learning from endless failure.