There are many here who both believe that soon AI will be superhuman at maths, solving all Millenium problems and beyond, and yet also reject any claims that maths is 'dead', and believe that mathematics will thrive.
I'm curious, for those people, what is maths in that future?
First of all, let me state that I'm not sure that I am one of those people who believe that mathematics will thrive in the near or far future, but this has been on my mind for some time and I figured I would get out my own thoughts on the issue.
To begin, we should first distinguish between mathematics as a topic of study, and mathematics as a cultural artifact produced by humans. In the first case, I think it is quite obvious that as long as AI does not immediately kill everyone and start pursuing some non-aligned non-mathematical goal, then mathematics-as-topic-of-study will flourish as the AI begins to both prove every single theorem we can put into the machine and simultaneously invent new mathematical objects and theories. Considering that mathematics is infinite and there are always new things to discover, AI will never run out of interesting things to discover and we can be happy knowing those things.
But I don't think that is what the question above is getting at - really what we care about is the human institution of mathematics. That is, we care about human mathematicians continuing to be able to get value out of math. How do human mathematicians get value out of math? Well I'm not sure but I could try defining some reasons.
- They do it for a paycheck, for rent, to get food in their bellies.
- They value mathematical truth in-and-of itself, they just want to know what is true and to understand it themselves.
- They value a sort of status in the mathematical community, they want to be well-known, they like clout.
- They value a sense of hard work, they like putting effort into difficult challenges.
- They want to solve real problems in the real world, and they just use mathematics to do it.
Ok, that isn't even close to all possible reasons, but we can start tackling them one at a time.
1: This is a serious problem, not just for mathematicians, but for all white-collar work and eventually all human beings who work. We will need a massive political solution to redistribute wealth from winners to losers to prevent people from starving.
2: If what they value is just the truth values, then the above scenario where AI is proving everything is a paradise. But if the value is understanding, then there are at least two problems. The first is that AI might make "ugly" proofs which we are incapable of following. I expect that to be a minor problem because AI for math systems will eventually figure out really robust ways of simplifying proofs. The second issue is that humans may find it easier to cheat than to actually obtain true understanding, like riding a helicopter to the top of a mountain instead of climbing it. I think people with this problem will benefit from AI eventually, as we will be able to use AI to better align humans with their own interests. This can be simple akrasia-reducing medications like Retatrutide, or more exotic things like (consensually!) cyborgifying humans with machines.
3: I think that this will have to turn into a sort of status between human beings who deliberately choose to not use AI to solve their mathematical problems. To be honest, I'm not too concerned about trying to protect motivations that are strictly based on status. It seems anti-social to care specifically about your "ranking" in comparison to other human beings.
4: In addition to the answer for 2, I think that AI for maths will introduce many new hard problems. Maybe the future is the AI giving humans challenges that it can solve, but we can't, and we solve those challenges for the same reason that we run marathons despite cars being faster.
5: If what you care about is solving real problems in the real world, then AI for math is a godsend. It's not just that you can suddenly tackle so many more challenges, but it's the ironic way in which knowing mathematics becomes (at least before ASI) an incredibly valuable skill. Consider this hypothetical: One day, a superintelligent alien crash lands on Earth. She knows everything and is a near-perfect reasoner but she has one fatal flaw: she only speaks Esperanto and refuses to speak in any other language. What would the value of knowing Esperanto be in this situation? Well, considering that the alien knows how to solve effectively every problem on earth, she knows how to design fusion reactors and superconductors and cures for every possible disease, knowing Esperanto immediately becomes the most valuable skill on the entire planet. I think human mathematicians will soon find themselves in a situation like this, but instead of an alien it will be a narrow mathematical superintelligence. The demand for human mathematicians will shoot up until the Hayekian problem of local knowledge can be solved, and the AI can be equipped with both the sensors and understanding of the rest of the world.
I don't know that that will be the case - it could be that we get narrow mathematical superintelligence and then 12 picoseconds later we have full superintelligence. I don't know how it generalizes. If the current verifier gap continues, then it seems at least plausible we could end up in that situation for some time.
To conclude, I've proposed 5 possible values for human mathematicians, and I've found that at least 4 out of the 5 values are either maintained or enhanced significantly. That seems like "thriving" for human mathematicians. I definitely didn't cover all possible values in this response though, and maybe some critical value will not be kept in this future world.