What now?

In recent months, there have been a number of non-trivial mathematical results proven in part, or in whole, by AI systems.

About a month ago, I started writing a blog post (the one you are currently reading) outlining some thoughts and concerns about the future direction of mathematical research. Due to other commitments (primarily the completion of my master's thesis), I left it temporarily incomplete, with intentions to finish it once I arrived home and recovered from the jet lag.

Then, two days ago, I attended an "AI in Mathematics" panel at UNSW, followed by discussions with my former colleagues there about what mathematics research should look like in the age of AI. I can honestly say I found the whole experience rather frustrating. There seems to still be a significant number of people in our mathematical community who do not fully grasp the threat that AI poses to our discipline, intellectual satisfaction, and ultimately our livelihood.

Then, yesterday morning, I woke up to the news that OpenAI claim to have a solution to the Navier-Stokes existence and smoothness problem, accompanied by a 166-page write-up of the solution and a Lean certificate.

The way this happened is somewhat unusual. OpenAI had not been working on the Navier-Stokes problem for a significant amount of time. Rather, rumours were growing that Anthropic had solved a major open problem, and OpenAI, presumably not wanting their competitor to be the first to publish such an achievement, directed a significant effort towards a proof of the Navier-Stokes problem using private models. These rumours were in regard to Levent Alpöge and Tristan Buckmaster, the former of whom is an employee of Anthropic, who were working on related problems they had hoped would extend to Navier-Stokes, and doing so in an LLM-assisted way. In a matter of days, OpenAI had a solution. This, of course, raises a big question: did OpenAI's model train on data given by Tristan in his LLM-assisted approaches to the problem? I recommend reading Tristan's account of events, which shows OpenAI's unwillingness to answer this question clearly, and that had Levent not been an employee at Anthropic, they would have been willing to collaborate rather than compete. Instead, we have an instance of a tech company hearing about two individuals making serious progress on an open problem and putting a huge amount of computing power and energy towards scooping them, purely because of the risk it would otherwise pose to their financial position. I feel the need to state the obvious here: mathematics research cannot have healthy collaboration in an environment in which the major players are tech companies trying to compete for higher valuations for when they ultimately go public.

With some remarks on these recent events out of the way, I continue with concerns about the position of AI in mathematics research more generally.

Why Us?

In popular culture, the development of solutions to difficult mathematics problems is often considered the pinnacle of human intellectual achievement. As Gauss put it, "mathematics is the queen of the sciences." Make no mistake, mathematics has become a target for OpenAI and Anthropic as a propaganda tool: a way to showcase the capabilities of their models in a domain which is notoriously difficult for the average person but unusually well suited to the problem.

Mathematics is a verifiable domain which is still intensely creative, which makes it unusually well suited to AI. A proof can be efficiently verified to be correct when sufficiently well specified, yet the creative side makes it infeasible to use a conventional computer program to generate correct proofs efficiently.

These AI developments have led to the mathematical research community being one of the first wrestling with big questions about the future of their research. One manifestation of this is the Leiden Declaration on Artificial Intelligence and Mathematics, which sets out advice and guidelines that hope to preserve what we value in the subject in the age of AI.

While I admire the effort, I am less than optimistic about the future of mathematics. I can't help but feel a profound sense of hopelessness about the future of the subject that I love.

Why study mathematics?

Last year I visited the van Gogh Museum in Amsterdam, in which hangs one of van Gogh's beautiful paintings of sunflowers. No attempt to explain the beauty and magnificence of that painting will suffice with the actual experience of being there and seeing it in person.

Mathematics is not about obtaining the next proof for the assurance that the statement is true; it is about understanding the process, and the sense of discovery one feels along the way. One's ability to understand the mathematics of others hinges on their own ability to solve non-trivial problems. If this were not so, then we would not instruct students to work on mathematics problems for which known solutions already exist.

I like to think we value mathematics in the same way we value art. Even though I can't draw a straight line from the problems I am currently working on to struggles of the everyday person, neither can the singer or the painter, but I hope we as a society still see value in their endeavours.

Andrew Wiles' solution to Fermat's Last Theorem is not going to cure cancer or solve climate change, yet we see his work as an unmatched accomplishment by the human mind. If we could obtain solutions to problems of this difficulty with minimal human effort through the use of AI systems, how is this any different from flipping through a Sudoku book and just running each puzzle through an online solver, or looking up the ending of a movie online while watching it in the cinema?

What AI is doing is crushing the life and soul of doing mathematics, and leads me wondering what the point of doing mathematics would be in a world where AI forces us to maximise the rate at which we obtain new proofs over working through hard problems for pure intellectual satisfaction.

Think about what the future of mathematics research looks like for the everyday working mathematician, who is pushing our understanding of mathematics forward with every paper they write, but unlikely to solve a problem close to the scale of the Riemann Hypothesis. Their research output is either going to be no longer competitive in the academic job market, or they will have to willingly destroy the intellectually satisfying aspects of their work.

Taking it to the extreme, if all mathematics could be done by AI, and humans need not be involved, then what is the purpose to studying mathematics at all? If we humans just receive proofs from the "great AI" and work to understand and check them (and probably check them less effectively than another AI system could), then not only is there no longer going to be a way to make a career out of mathematics, all we have achieved for ourselves is destroying all of the rewarding parts of the job.

AI is already destroying the discourse around mathematics. There is something deeply satisfying about meeting with other mathematicians, discussing what you have been working on: the struggles and the achievements. The eroding of productive discourse about the technical aspects of the field is well underway in the software industry. Conversations about interesting optimisation techniques, unique algorithms, and the design of systems have been drowned out by AI enthusiasts gawking at what they have "achieved" with little to no intellectual engagement in the problem at all.

This is not too dissimilar to what is happening in the art world. It is discouraging (to put it lightly) to spend hours practising your art form and developing a style, only to have an AI system reproduce it in a matter of seconds.

This leads me to the primary concern that I have about the future of AI in mathematics: using AI to a significant degree in mathematics research eliminates the fun of the job; not using AI, in a world where it continues to push the boundaries of our mathematical knowledge, will render one unemployed. In the end, it will likely render us all unemployed, or worse, unemployable.

Part of me wonders if the mathematical community has had it too good for too long. Have we all been able to sell this lie to people who fund research at universities that our research is important for society, when in reality, it is just an excuse to do something fun and intellectually rewarding? What is the point to studying mathematics? Is it really just a bullshit job that does nothing for the world in the end?

"AI is just a tool"

I have had many discussions with people over the past few weeks about this very topic, and one argument that keeps coming up is that "AI is just a tool, and that tool needs human guidance and intuition to be used effectively."

To me this portrays an extreme level of short-sightedness. I invite the reader to consider the prompts used in the AI-generated counterexample to the Dinitz-Garg-Goemans conjecture.

[User]: Construct a counterexample to general (non-planar) case of Dinitz Garg Goemans conjecture. You should do a breakthrough and find a structured counterexample.
[ChatGPT]: ...
[User]: please continue research and find a complete unconditional counterexample
[ChatGPT]: ...
[User]: Continue the search. Have a clear strategy obtained from deeper understanding of the problem structure.
[ChatGPT]: ...
[User]: it's enough of partial results. let's finish with a complete unconditional counterexample
[ChatGPT]: (solution)

This, dear reader, is the level of technical knowledge and insight now required to make progress in mathematics.

I have heard arguments like "but AI is limited to specific kinds of problems and proofs." I am also very well aware that the recent collection of problems solved by OpenAI likely represents a small subset of the problems they attempted to autonomously solve, and thus likely represents a sample which are, for one reason or another, unusually well suited to current AI systems.

However, this narrative, as I see it, appears to be an attempt to hold on to a small remaining amount of hope for the value of human insight into mathematics. I am not saying AI can currently replace the job of a mathematician, or even that it can do a significant part of their job. What I am pointing out is that AI has gone from being unable to consistently do single-digit arithmetic to making breakthroughs on well-established problems in less than four years. If you can't see where this is headed, I don't know what to tell you... And maybe I am wrong, but are you willing to bet your career on it? Are you willing to bet the future of one of the oldest intellectual disciplines on it?

People have told me that we still need humans to ask good questions and guide the AI systems in the right direction. Why does anyone think this is the exclusive ability that humans have that cannot be replicated by AI? Am I supposed to believe an AI system can solve a Millennium Problem but can't ask good questions to another AI?

For the last few years there has been a constant moving of the goalposts from people who wish to assure themselves that AI is not a threat to their career or lifestyle. I've heard a lot of "oh yes, it is impressive, but it will never be able to do X." Then when AI does do X, we get "well yes, but it will never be able to do Y."

AI systems are becoming better at mathematics faster than any human on the face of the earth. When will people stop thinking about what the world will look like in 3 months and worry about whether or not their job will exist in 10 years?

This kind of self-assurance is not limited to the everyday mathematician or non-mathematician. In some comments about the disproof of the Unit Distance Conjecture, Tim Gowers writes the following.

... [I] thought that the model had proved an upper bound of n^(1+o(1)). ... I spent the evening adjusting my world view: if AI could come up with a proof like that, then maybe it would be all over for mathematicians very soon. The next morning I and the other authors of this paper received an email about the result, and only then did I understand that it had disproved the conjecture rather than proving it, which came as a big relief. It is interesting to reflect on why it should have been a relief ... The short answer is that, without knowing anything about the solution, I could more easily imagine a model coming up with a counterexample while still lacking some essential mathematical capabilities than imagine it coming up with a proof.

The issue I have with this assessment is that producing a counterexample to the Unit Distance Conjecture would be a career-making proof for any mathematician. We should not feel relieved about what AI cannot yet do, and instead be concerned about what it can. I can only assume that if an AI system solved Goldbach's conjecture tomorrow, the next day mathematicians will be saying, "well this is impressive, but it won't determine if P = NP" (this sentence was one that I wrote before OpenAI's work on the Navier-Stokes problem, so I suppose we will see if that is sufficient to squash this kind of argument; I suspect it finally might be).

I study mathematics because I enjoy working on difficult problems. People have told me "we still need mathematicians to read, understand, and check the output of AI systems." I'm sorry, but if the job of a mathematician becomes prompting AI systems and checking its output, I do not want to be a mathematician.

I honestly think if I were in my final year of high school right now, I would consider becoming a plumber or an electrician. If the intellectually rewarding jobs disappear, I may as well earn good money before the trades become oversaturated too.

Where does that leave us?

For most of my life I had hoped to make mathematics a career. Now I fear that I am watching the death of the subject in real time and am powerless to fight against it. In all this, we have to ask ourselves: what have we achieved? How has any of this made our world a better place to live?

What happens if we live in a world where the entire mathematics industry is held hostage by two tech companies that are destroying the planet, stealing intellectual property, and farming our personal data? What about where future students become incapable of doing mathematics at the level we can now because they were robbed of the opportunity to genuinely struggle with difficult problems?

I don't have an answer to this. I don't know what I am going to do. I am looking into my future with more uncertainty than I ever have in my life. On some days, it brings me to the point of tears; on others, I wake up with an intense feeling of anxiety. For someone like me, who feels like his purpose in life is so deeply tied to the value he can provide in his work, these latest developments leave me feeling hopeless, directionless, and depressed.

If you, dear reader, are feeling down about how AI is shaping our world, I am sorry. I hope that this blog post at least helps you feel heard and understood. If you want to engage with me on this topic, I invite you to send me an email.

If you want to hear the thoughts of others who share similar concerns, I encourage you to read (any or all of) the following: