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> My own suggested rule of thumb: if the authors cannot convincingly demonstrate that they are able to give a clear, expert-level talk on their results, one that is correct and properly attributed, then the result should not be published. A proof that no human can properly explain should be viewed as incomplete, even if it has been formally verified.
Properly explain is an enormous grey area. Soon, I think, there will be proofs of results that are verified in Lean that are so long that no one will be able to “properly explain”. I don’t think they should be discarded.
Resolution of singularities is a famous theorem of Hironaka. Abhyankar claimed that no one truly understood the proof of the theorem. He said that he and Zariski couldn’t get through the paper with a full understanding. But everyone accepts this theorem as being correct.
Hmm, doesn't it take an expert to explain why those cases are exhaustive, and why the code that checked them is correct?
Tangentially, I'm not a mathematician but I wonder if one "opaque" proof that is too complicated for anyone to understand, but that we know is correct via formal verification, might end up being built on with "transparent" human-understandable proofs. For example, it's my understanding that there are many conjectures that have been proven true conditional on the riemann hypothesis being true. In that case, an opaque proof of the riemann hypothesis would enable those conjectures to be known and built upon
To your first point. There a large number of cases that maps can be reduced to. Very few people have checked these reductions themselves. In 50 years there will be no human alive that will have checked the reductions by hand. Do we then discard the theorem? More importantly, do we trust the people that claim to have checked all the reductions? There are hundreds of cases. I trust a computer verification much more than I’d trust human verification. Humans will likely make mistakes due to the tedium. And some will claim understanding of all cases but be wrong in their understanding in some of the cases.
If Amazon uses AI math to come up with better routing, the cats can benefit from cheaper delivery fees just as much as humans can. No understanding needed.
The human brain is being obsoleted, soon thinking is going to be a recreational activity like weightlifting. If you want to think as a hobby, that's fine, but most people will be free of that toil of unwanted brain labor.
Understanding was critical for the field to progress when only humans were involved but if humans are not needed to make progress, I wonder if we split into two worlds: an AI math-world where amazing new results continue at a rapid pace bottlenecked only by compute/cost and a human math-world where we understand a subset of the AI math-world as a hobby (similar to Stockfish vs human chess).
I am wary of AI in all aspects I am seeing it in but in many ways in mathematics seems to me the least troubling. It will change things in and the field will not be the same. Blacksmithing has not really gone away. You can still work as a farrier, if you like that sort of things. The tools that replaced a man working over a forge with a big hammer are part of a giant industry that is still producing works for the modern world.
[1]: https://bbchallenge.org/8226493
This is also the strategy I use for editing drafts of my books. I bring a printed draft to someplace nice (e.g. coffee shop or park) and read it all carefully, then I transfer the edits back to the .tex sources. I do several passes of this, until I feel the text + explanations are solid.
Reading on screen just isn't the same...
It's the old cliche of "if you only have a hammer every problem looks like a nail". Let's not fall into the trap of thinking that our life needs to be 100% about AI or completely devoid of AI. We can really use this thing to make our lives better.
Instead of wasting time on the question of whether we should use it, let's focus on HOW we'll use it.
And one thing about Tao: it's really refreshing to have an influential genius "around" who isn't a egomaniacal psychopath trying to rule the world through their XYZ corporation but, instead, being a reasonable and well-balanced person. Big fan.