Terence Tao: uncertainty

Transcript context

…One big question I have is how plausible is it that if we just keep training AIs—they get better and better at solving problems in Lean—that they will continue to solve more and more impressive problems, and then we will be surprised at how little insight we got from some Lean solution to proving the Riemann hypothesis or something. Or do you think it is a necessary condition of solving the Riemann hypothesis, even by an AI that is doing it entirely in Lean, that the constructions and definitions created in the Lean program have to advance our understanding of mathematics? Or could it just be assembly code gobbledygook? We don’t know. Some problems have been basically solved by pure brute force. The four color theorem is a famous example. We have still not found a conceptually elegant proof of this theorem, and maybe we never will. Some problems may only be solvable by splitting into an enormous number of cases and doing brute force, uninsightful computer analysis on each case. Part of the reason we prize problems like the Riemann hypothesis is that we’re pretty sure a new type of mathematics has to be created, or a new connection between two previously unconnected areas of mathematics has to be discovered to make this work. We don’t even know what the shape of the solution is, but it doesn’t feel like a problem that will be solved just by exhaustively checking cases. Or it could be false actually. Okay, there is an unlikely scenario that the hypothesis is false, and you can just compute a zero off the line, and a massive computer calculation verifies it. That would be very disappointing. I do feel that fully autonomous, one-shot approaches are not the right approach for these problems. You’ll get a lot more mileage out of the interplay of humans collaborating with these tools. I can see one of these problems being solved by smart humans assisted by extremely powerful AI tools. But the exact dynamic may be very different from what we envision right now. It could be a collaboration of a type that just doesn’t exist yet. There may be a way to generate a million variants of the Riemann zeta function and do AI-assisted data analysis to discover some pattern connecting them that we didn’t know about before. This lets you transform the problem into a different area of mathematics. There could be all kinds of scenarios. Suppose the AI figures it out, and latent in the Lean is some brand-new construction which, if we realized its significance, we would be able to apply in all these different situations. How would we even recognize it? Again, a very naive question, but if you come up with the equivalent of Descartes’ idea that you can have a coordinate system unifying algebra and geometry, in Lean code it would just look like R→R, and it wouldn’t look that significant. I’m sure there are other constructions which have this kind of property.…

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