In The Wake Of The Latest Unprecedented AI Proofs, What Now For Mathematics And Mathematicians?

In The Wake Of The Latest Unprecedented AI Proofs, What Now For Mathematics And Mathematicians? from the age-of-wonders-and-terrors dept As arguments rage about the possible or real threats of AI, there is one domain that AI has already turned on its head, and in the space of just the last few weeks: mathematics. The first hint of what was coming arrived in May, when OpenAI announced that one of its models had disproved a famous mathematical conjecture. In August, the company shared a list of ten more AI-generated mathematical results, “each of which resolves or makes substantial progress on a long-standing open problem.” But the real bombshell arrived on 8th September, when it announced that an “internal OpenAI system” had come up with a solution to the Navier–Stokes Millennium Prize Problem: The Millennium Prize Problems represent some of the deepest questions at the frontier of mathematics. The question of whether smooth three-dimensional fluid motion can break down has remained unresolved for roughly 90 years. Although there is no doubt that this represents a major advance in AI mathematics, there is still controversy about who exactly should get credit for the solution of the Navier-Stokes problem. Zvi Mowshowitz has an excellent rundown of what we know and what we don’t know about this saga. But much more important is the effect this achievement has had on the mathematical community. For example, the Caltech Mathathon was already considering “how can we responsibly use AI tools to augment human understanding of mathematics?”: We are assembling 100 teams of mathematicians to answer this question. Our objective is to provide frontier models for the math community to use, as opposed to solely AI corporations. But on 10th September, two days after OpenAI revealed its Navier-Stokes solution, current and former Caltech mathematicians published an “Open Letter about the Mathathon”. In it they warned that: Two prominent AI companies, Anthropic and OpenAI, will supply participants with 2 million dollars in AI credits. This event is likely to have destructive impacts for the mathematical community. They listed a number of concerns, and called on the organizers to suspend this event. In response, the organizers admitted that worries “the event could incentivize rushed, poorly understood mathematics, place verification burdens on the broader community, and amplify unhealthy incentives around AI-generated results” were valid, at least partially. They went on to clarify what they had done to address the concerns. Following this brouhaha, OpenAI dropped its sponsorship of the event. Those doubts about the Mathathon’s encouragement of the use of AI tools in mathematics are part of much wider soul-searching in the mathematics community. Even before OpenAI announced the Navier-Stokes solution, the mathematician Max Weinreich wrote a paper entitled “The crisis of AI-generated mathematics” in which he presented the case for “total opposition to the use of artificial intelligence in mathematics.” He is on the organizing committee of the Association for Human Mathematics, which wants to “organize mathematicians to center human understanding and protect against the threat of artificial intelligence.” Another mathematician, Daniel Litt, wrote of “The End of Mathematics”. Alongside these and many other posts on the topic, thousands of mathematicians around the world have signed declarations and open letters that call for action to address the challenges posed by the use of AI in mathematics research. The Leiden Declaration was published in June of this year, and currently has over 4,000 signatories. A declaration on the “Math and AI” site entitled “A Severe Misalignment of AI in Mathematics” was published on 11th September, but already has around 8,000 endorsers. Even the more narrowly focused Open Letter about the Mathathon has over 2,000 supporters. The “Math and AI” declaration identifies the central problem as follows: We are witnessing a general threat to intellectual work, with misalignment between the outcome of the use of AI and its initial purpose. In many fields and activities, years of training have traditionally served not only to produce a final answer or product, but also to develop understanding and the ability to formulate new questions and ideas. However, building on a vast body of previous human work, AI systems are becoming increasingly capable of producing the results of such work directly, and these goals cease to align. The issues the mathematical community faces now are similar to issues that other scientific and creative professions are facing, and indicate issues that all of humanity might face: how to make sure that, as AI changes the way work is done, we do not lose sight of what that work was meant to achieve in the first place. According to many mathematicians, focusing on AI’s prowess in proving challenging theorems misses the point. Bryna Kra wrote in a blog post: A proof is more than a certificate that something is true. Instead, it is a story, a picture, an insight, an explanation. A proof highlights novel ideas and opens new directions for what we should ask next. It becomes part of the toolkit of the community. A deep theorem changes how we think, not because of its statement, but because of what it teaches us. As Bill Thurston wrote on MathOverflow in a 2010 response to a question about what mathematicians do: “The product of mathematics is clarity and understanding. Not theorems, by themselves.” As many mathematicians now recognize, the challenge today is coming up with a way to shift from a world in which proofs (by humans) are rewarded, to one where all the other aspects — the stories, pictures, and insights that are part of a proof — are rewarded as well, or even instead. Terence Tao, one of the mathematicians playing a prominent role in the debate about the future shape of mathematics in the age of AI, has even even said (pdf): 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. Max Weinreich wants to go further: We could replace traditional authorship with co-ownership of mathematical ideas. In this paradigm, any mathematician who demonstrates authoritative understanding of a work – the type you would expect of an author today – would be entitled to claim co-ownership, even after publication. Some papers might have a few co-owners; others would have tens, or even hundreds. Journals would have the exciting, but challenging, role of establishing norms for validating understanding and maintaining the infrastructure of co-ownership. This would take time – immense amounts of it. Explaining an entire paper in full detail to an appropriately skeptical audience often constitutes an entire graduate course. But if mathematicians aren’t writing papers any more, we will have more time. That time should be returned to mathematics, in its most social and human forms. I think it sounds like fun. This would require not only a fundamental re-thinking of how academic journals work — something long overdue anyway — but also how educational institutions structure and reward academic work. In a follow-up to his earlier “The End of Mathematics” post, Daniel Litt has written a more optimistic one entitled “A beginning for mathematics.” In it, he calls for more emphasis to be placed on the human and social aspects of mathematics — the very things lacking from even the most impressive AI proofs: The allocative aspects of our job (hiring, graduate admissions, etc.) are in dire need of reform if we want to retain human mathematical expertise. Broadly speaking I think we should focus on rewarding skill in the parts of our jobs that cannot be automated: the internal (e.g. understanding mathematics) and social-relational parts, and operationalizations that hew as closely to those aspects of the profession as possible. For example, talks and sustained mathematical discussion now demonstrate understanding much better than papers. Once AI systems improve at exposition and “digestion,” this will be even more the case. There seems to be an emerging consensus in the field that, like it or not, mathematics has changed for ever, and that we have entered “The Age of Wonders and Terrors” as the mathematician Scott Aaronson puts it: It seems to me that the Singularity has already started; it’s just wildly unevenly distributed. Yes, I still unload the dishwasher and clip my toenails. On the other hand, in whatever years I have left, I don’t expect that I’ll ever again prove a theorem because I’m actually needed to prove it. If I do, it will only be for my or others’ enjoyment or edification. In this view, mathematicians will still prove theorems and come up with counterexamples, but they will do it because they enjoy it, not because their career depends upon it. Society will benefit from the coming flood of new AI-derived results, as we move from an era of proof scarcity to an era of proof abundance, but there will still be a place for human mathematicians to interpret those results, to pass them on to the wider community, and to build on them, probably using AI to do so. In this respect, they will become what the “technical philosopher” Logan Graves calls “priests”: Yes, it is true — but how? What does it mean? It has been passed down to us from on high and its secrets must be disentangled. Its form may be foreign, perhaps even disgusting, but it is true. It remains to understand it. To attempt to grasp the truth in its full glory and deliver it to the community, the flock, the seekers of truth and the lovers of wisdom — that is the task for the priest. That is the task the we are watching the human mathematical community transition to, at this moment. However, the dizzying pace of AI development brings with it a danger, articulated here by Terence Tao (pdf): We may soon be faced with the very real possibility of a verified proof of a major result that no human understands well enough to explain. Already AI proofs run to hundreds of pages — 166 in the case of Navier-Stokes (pdf); there is no reason why they won’t reach thousands of pages one day. At that point, no human, or even team of humans will ever understand it in detail. What then for mathematics and mathematicians? Follow me @glynmoody on Mastodon and on Bluesky. Filed Under: academic journals, ai, caltech, conjecture, mathathon, mathematics, millennium prize, navier-stokes, singularity, terence tao, theorem, threats Companies: anthropic, openai Comments on “In The Wake Of The Latest Unprecedented AI Proofs, What Now For Mathematics And Mathematicians?” Learning not to use the billionaire funded industrial theft engine as a sounding board for their ideas and developments might be a good start. If we don’t understand it and can’t explain it, is the proof then really complete? Re: Well, the answer to that question is purely a matter of opinion—like, what/who is a proof for, and, more broadly, what is mathematics for?—but one of the quotes Glyn included was “A proof that no human can properly explain should be viewed as incomplete, even if it has been formally verified.” As if mathematicians spend all their time on proving some older conjecture or other. But hey, they can join the AI algo assholes making more AI, surely. Re: I think most people have little idea what mathematicians do, or why society needs them. Similarly with poets, for example. But optimizing society too much toward “useful work” seems like a sure-fire way to slide into dystopia. I suppose it’ll be up to the mathematicians to decide what being a mathematician should mean. This gets into a broader issue with LLM-based tech—the stuff we colloquially call “AI”—that expands beyond this field alone: If “AI” can destroy someone’s career or job, how are they supposed to make enough money to cover the cost of living, and how are companies supposed to survive when enough people can’t afford those costs because machines took their jobs? Re: They can do what anyone else with a “destroyed” job does: find a different one. Where are the telephone operators now? Even in the 1980s and 1990s, one might’ve encountered them for collect or international calls, directory assistance, and such; but they’re mostly gone now, replaced by computers, and people would be quite surprised to stumble upon one. Same for elevator operators (I last saw one circa 1990). These concerns have never stopped us from eliminating jobs before. Ah, that’s the better question. Henry Ford is famously said to have intentionally made cars affordable to the people making cars. I don’t see a similar effort being made by the billionaires running these “A.I.” companies; I think they just want to eliminate the humans making their stuff. The good news is that they’re full of shit, and all the job-replacement they promise is unlikely to happen—like when humans were gonna be replaced by machines in the 1980s (which kind of happened, but not to an especially harmful extent). The bad news is that things are gonna be rough for a while after “the market” really figures that out, like when people realized Enron and Nortel and Wachovia were full of shit. I don’t particularly like the framing about how people are “supposed to” have jobs and be making money. Jobs and money are technologies like any other, invented to solve problems that existed at the time. If money no longer works, it could be adjusted—like via Universal Basic Income—or replaced or eliminated. After all, why should this one technology be sacrosanct, stuck in the 1490s while all other technology “advances”? And jobs have been stuck around 40 hours a week since Ford’s time; we’re overdue for a significant reduction, if not a radical overhaul. (For similar reasons, I don’t think LLMs and such ought to be considered profane; they won’t be controlled by assholes forever.) Re: Re: Here’s the problem with your scenario: If “AI” destroys enough jobs, there will be far more people than there will be jobs for them to fill. How would people who can’t get a job because of that state of affairs cover the costs of living? Even so, the chances of a large dose of job replacement aren’t low. My question still stands: If there are more people looking for jobs than there are jobs available to be filled, how are the people who can’t get jobs supposed to afford the necessities of life? Well, guess what: That’s the world we live in right now. People either get a job and make enough money to live or they don’t and they die. Sucks, doesn’t it? But you don’t hear AI evangelists talking seriously about Universal Basic Income or any sort of plan for what happens if all their dreams of decimating the labor market come true and millions of people are left without any way to afford the costs of living. When those head-up-their-ass fuckheads start talking about how to help the people said fuckheads want to drive out of the labor market, maybe I’ll stop thinking they’re anything but friendless assholes who are using tech to replace human connection. If you were to start from a compass and straightedge, I bet it would take at least a thousand pages to prove the First Fundamental Theorem of Calculus. Just because there may be entire fields of study packed into those proofs that we haven’t understood yet, is no reason to assume that we can’t do so in the future. And, of course, it’s “AI”. There’s no guarantee that it took the most direct route to its destination or that all of those pages are even necessary. (Assuming the proof is even correct, which, I defer.) Re: Imagine, if you will, a goblin, x. Replacing “authorship” with “ownership”, “co” or not, seems like a terrible suggestion. This may have just been poor phrasing, but ideas should not be owned; it’s caused nothing but problems lately. I’m pretty sure that the more important aspect of these proofs is whether companies who, by their own admission, failed to sandbox their hacking bots, also managed to let other bots use private customer data. Money involved The ones promoting a reward for solving the problem should refuse to pay. No human had anything demonstrable to add to the solution. Neither did any corporation. A program did. Ooh, ooh, I know, 42!