Speedrunning Mathematics?

Around this time last year, I would have scoffed at the prospect of any AI model solving major math problems, let alone a gargantuan problem like the Navier-Stokes problem. But here we are.

I would like to share my personal thoughts about what this means to me, as someone who is actively applying to PhD programs this cycle and as someone who is trying to knock on the doors of the world of academic mathematics.

At first, whenever my friends sent me news about some AI breakthrough for some mathematical problems, I actively shunned them. To me, it didn’t matter. Basically, it was all noise to me. Yes, some AI model made progress on some kind of combinatorics problem that just required a lot of enumeration or something like that? Ok, cool. If this sounds a bit too naive, then you would be right. I wanted to protect myself from the existential dread.

Back in May, it was announced that ChatGPT disproved a longstanding conjecture surrounding Erdos’s planar unit-distance problem. That is around the time when I simply couldn’t continue to brush these off. That was when I realized maybe these AI models will be able to solve some kind of a big mathematical problem one day in the near future. I guess I just didn’t know that that day would come within the next 4 months.

Over the summer, I was working on my own mathematical problem on monomial ideals and characteristic independence of Betti numbers at California State University, Chico. It was a great experience and I had a lot of fun researching. Every now and then though, during the summer, some similar news would break out where some AI model solved big mathematical problems, including more Erdos problems. None of us quite knew what to make of it, so we just kind of just shrugged it off and went on about our own business of researching mathematics.

Now, it has come to a point where OpenAI has announced what it claims is a solution to the Navier–Stokes Millennium Prize Problem. The frightening part for me is not whether this particular proposed solution ultimately survives mathematical scrutiny, but rather the short amount of time it took going from an elementary model that routinely struggled with undergrad level mathematics back in 2022 to genuinely tackling one of the most important mathematical questions only 4 years after its introduction. For a human, to even consider touching the problems of this magnitude, you would have to first go through 12 years of schooling, go through 4 more years of undergrad, then go through 5-6 years of PhD, and maybe a postdoc or two (or three), for anywhere between 3 and 6 years, and then finally you become a professor and can maybe start to think about these kinds of problems. Obviously, there is nothing stopping PhD students (or even undergrads) from thinking about how to solve the millennium problems, but the sheer magnitude of the problem would preclude the vast majority of aspiring mathematicians from even considering the problem.

My friend messaged me “RIP math” after sending me the news about OpenAI’s Navier-Stokes announcement. I replied, “RIP math done by humans*”. It was partly a joke of course, but it captured something real about how I feel about this matter. I do think we live in one of the most exciting moments as mathematicians. These models are capable of performing some of the most extraordinarily difficult mathematical tasks only within a matter of hours, whereas a human might take years to try to compute them or think deeply enough about them to get close to solving them. In a pure mathematical sense, these models are enriching our understanding of the universe, and inevitably, I do think that is genuinely a good thing for us. After all, we are doing mathematics because we are genuinely curious about these problems, right?

There is another aspect to this too, of course. Part of the unease comes from realizing that there may soon be important intellectual tasks at which machines outperform even the smartest or strongest humans. I’ve read a lot of people going crazy on subreddits such as r/math claiming that math will go down as something like a ‘hobby’ area instead of being an actually active research area led by humans, much like how something like go used to be deemed as strictly a human endeavor, only for Lee Sedol, one of the best go professionals in the world, to be defeated by AlphaGo in 2016. While I think this is a bit of an overreaction, I can’t confidently say that there is no future at all where that could come to fruition.

Mathematics is especially exposed to these advances because so much of its raw material, like definitions, conjectures, proofs, examples, can be represented symbolically and manipulated, completely divorced from the physical world, unlike experimental sciences where a lot of empirical evidence and input could be necessary. In other words, mathematical research does not require a laboratory. But mathematics is much more than just the production of formal proofs. The process of mathematical research involves choosing worthwhile questions, carefully constructing definitions and examples, deciding which ideas are worth pursuing, and developing the taste to know what kinds of questions can be interesting in the first place. What would the role of human mathematicians be, however, if producing proofs were no longer the bottleneck in mathematical progress? There is a world in which these qualities prove essential to pushing forward the frontiers of human-led mathematics, and there is also a world in which they are deemed largely inessential to mathematical progress.

I admit that I’ve also used AI in my research. In my own research on characteristic independence of Betti numbers of monomial ideals in five variables, I used AI to help generate code for computational verification of the finite squarefree cases underlying the argument. The models wrote the code in under 5 minutes and I was able to verify quickly. It was genuinely helpful and it saved a lot of time. Although I know basic Python, producing the verification code myself would have taken considerably longer, so AI genuinely did improve the quality of my research, and I acknowledged that in the acknowledgement section of my paper.

I guess my feelings about these advancements are multifaceted and maybe that’s why I’m having a hard time coming up with a coherent opinion about this. I want to wholeheartedly support the advancement of mathematics as a discipline, even if it’s done by these AI models, but I have a hard time doing so. There is something inherently beautiful about struggling with mathematical questions for years on end to finally culminate with a beautiful proof. I watched a video where Andrew Wiles talks about his struggles for years but that he enjoyed every minute of it, however hard it had been. I, much like other mathematicians, find beauty in having the privilege to struggle with problems to come up with solutions that have never been proposed before.

I have only just begun to experience that myself, and I hope there’s much more of it ahead of me. However, now there is a real chance that those moments of joy will be stripped away by all these AI models’ advancements. I no longer feel comfortable assuming that the problems we regard as untouchable today will remain beyond AI systems for decades. Who knows what the next big breakthrough made by these AI models will be?

Maybe this exposes my selfishness. If my goal as a mathematician is simply to know more mathematics, then I should, in theory, welcome a machine that can discover in hours what would take me years. But I do not only want the theorem at the end. I want to be one of the people who gets to struggle toward it. Perhaps what frightens me is precisely that it will advance so quickly that there will be less room for humans to experience discovery for themselves.

In a world where patience, persistence, and sustained attention are increasingly becoming traits of a bygone age, mathematics always struck me as one of the few places where these qualities are still honored. AI seems to be disrupting this gradually, and if this trend continues, I wonder if we will all become just ‘speedrunners’ in the future.

Speedrun Riemann Hypothesis any%?