On chess and mathematics
How AI may impact the two domains differently
This post may seem untimely in the midst of all the urgent discussion of AI hacking scandals and loss-of-control scenarios. Is there really time right now to spend on the slower and more mundane issues such as whether my mathematician colleagues and I will still have jobs in late 2028? I think, however, that we should not let the urgency and importance of the loss-of-control issue lead us to set all other issues aside. Two undesirable consequences of doing so stand out. First, since the current intensity of discourse around crises caused by dangerously capable AI is likely lower now than it will ever be again, we would run the risk of never getting the peace and quiet needed to go back to those other issues. Second, we would turn into monomaniacs.
So here we go: chess and mathematics! Other than more generic activities like reading and spending time with friends and family, those two things are probably the ones I’ve put most effort into over my lifetime (so far).1 Chess was hit by (narrowly) superhuman AI nearly 30 years ago, while for mathematics we are only now on the verge of the corresponding event. For chess (by which I here mean chess as an organized human activity) the transition has arguably gone well, and one may ask whether that gives reason for optimism about how mathematics (as an organized human activity) will do in the era of superhuman AI mathematicians. Here I will gesture at a negative answer to that question, based on a comparison between the two domains.
Three months ago, on May 11, I published my blog post A paradigm shift in mathematics, about the stupendous rate of progress AI capabilities in mathematics had exhibited over the first four months of 2026, along with the mathematical community‘s early and scattered attempts to grapple with this new situation. On May 20, just nine days after its publication, the blog post became obsolete, with OpenAI:s announcement that one of their internal models had independently come up with the most spectacular AI achievement so far within mathematics: it provided a solution to the most famous open problem in the field within mathematics known as combinatorial geometry, by disproving Paul Erdős’ so-called Unit Distance Conjecture from 1946. One of the leading experts in the field, Noga Alon, offered these remarks about the achievement:
This has been one of Erdős’ favorite problems, I have heard him myself mentioning the problem multiple times in his lectures. I believe it would be fair to say that every mathematician working in Combinatorial Geometry thought about this problem, and lots of mathematicians working in other areas spent at least some time thinking about it.
Let me also add that although this problem may look at first as a recreational one this is not the case, it is in fact closely related to other mathematical areas including Number Theory and Algebraic Geometry. The solution of the problem by the internal model of OpenAI is, in my opinion, an outstanding achievement, settling a long-standing open problem.
Over the summer, this breakthrough was followed by announcements of other, similarly spectacular, advances in mathematics achieved by AIs, including a proof of the half-century old cycle double cover conjecture, and a counterexample to the similarly famous even older Jacobian conjecture. On August 1, OpenAI apparently decided that publishing such breakthroughs one at a time was getting boring, and announced their stunning 253-page manuscript Ten advances in mathematics and computer science. We mathematicians stand in awe, scratching our heads, and wondering what to make of all this. Many of us wonder whether there will still be a role to play two years from now for ordinary flesh-and-blood mathematicians.
Not being in possession of a crystal ball, I obviously don’t claim to know the answer to that last question, but I’ve heard others express less humble opinions about this. Those who answer “well yes, not much to see here, things will obviously carry on roughly as before” might be tempted to point to the example of chess, which is very much alive and kicking as a human endeavor, nearly three decades into the era of superhumanly capable chess machines. We symbolically entered that era in 1997, when IBM’s Deep Blue beat Garry Kasparov, who was then reigning world champion and undisputedly the world’s best human chess player, by 3.5-2.5 in their six-game match. From that point, the chess-playing machines have continued to improve, and we all quickly lost interest in human-vs-computer encounters in chess. But interest in chess as a whole remains as great as ever, both as a spectator sport where we amateurs watch (remotely over the Internet) grandmasters playing each other, and as an amateur mass sport. Over-the-board play long dominated, but the covid pandemic saw a huge boom in remote play, and now the two forms of chess flourish side by side on all levels.
The influence of superhumanly capable chess-playing AIs on chess as a human activity has been significant but has left the basic nature of human chess remarkably intact. I would say the effects have mainly been fourfold. First, everyone who wants to has free access through their smartphones to chess programs so strong that for most practical purposes they are regarded as providing the ground truth about chess positions. One change downstream of this is that (sadly) the traditional friendly post-mortems conducted between the two opponents after a tournament game have become less popular, because many players prefer to simply check what the AI says. Second, and relatedly, cheating has become more of a problem, due to various schemes for illicit consultation of AIs during games, but this problem remains relatively contained.
Third, the slowest form of chess known as correspondence chess, where players take days or sometimes even weeks to decide on a move, has, even though tournaments are still being played, in spirit been killed. This is due to the practical impossibility of preventing players from consulting AIs, which is therefore allowed, and as a consequence games at the top level very rarely result in anything other than a draw.2 3
Fourth, these AIs have turned out to be very useful for discovering new approaches to the opening stage of the game, and have led to much new opening theory. They are therefore much-used in preparation for games and tournaments and have had much influence on especially how grandmasters and other elite players approach the opening, and from there the influence propagates down to players on lower levels who take inspiration from the grandmasters.
All in all, chess in 2026 is, with the notable exception of correspondence chess, in excellent condition. Can one generalize from this to the field of mathematics, and expect a similarly happy and healthy future for it in the presence of superhumanly capable AIs?
For the purpose of trying to answer this, a comparison between mathematics and chess is in order, and indeed, there are some striking similarities between the two domains.4 Problem solving is central to both of them, and they are both activities where participants become successful by combining systematic thinking with (on happy occasions) a bit of creativity. Participants also need to find geometric patterns and other heuristics to overcome a combinatorial explosion of mostly irrelevant possibilities. The domains both have clearly defined rules, especially as compared to the messy world of human social interactions, and it seems to me that in both of them a substantial minority of the participants enjoy those clear rules as a kind of refuge from the harder-to-navigate social world out there.
But there are also major differences in how the two domains fit into the larger fabric of society, as can be illustrated by the following anecdote about my esteemed friend and chess-playing team mate Mats Eriksson (1958-2022, RIP).5 In a break between rounds during a team event in the late 1990s, Mats asked to speak with me, and I immediately knew he was up to something more serious than the usual chit-chat that chess players like to engage in to relieve some of the tension built up during games. Here, in slightly condensed form, is how I remember our conversation:
ME: Olle, you obviously have talent for this game. But I can’t help noticing that your tournament results have been stagnant for several years. Isn’t it time that you put some serious time and effort (beyond what you are doing on a routine basis) into improving your play? You have passed 30, and I fear that without a concentrated effort of this kind, your play is likely to remain stagnant forever.
OH: I appreciate you saying this, and I believe your analysis is largely correct. Yet, the kind of dedicated effort you are suggesting is not going to happen.
ME: But why?
OH: Look, Mats, there are two activities that I have put much of my effort into for well over a decade: chess and mathematics. They both give me a similar kind of enjoyment and intellectual satisfaction, so from that perspective it is, in a sense, arbitrary which of them I devote more effort to. But there is a powerful tiebreaker: only mathematics offers a range of other rewards that I consider essential to my overall life satisfaction. It has given me academic positions and a good salary, and it is even beginning to provide me with a broader platform and some amount of respect outside the narrower circle of specialized nerds. Chess gives me none of that, and for that reason I cannot afford to prioritize it beyond the relatively unambitious hobby level I currently give it.
ME: But that is surely a fallacy — can’t you see how it creates a vicious circle?
What makes this conversation so memorable for me is that Mats delivered the last line with his characteristic twinkle in his eye, making it clear that he appreciated the irony: by calling my priorities "vicious", he was playfully elevating his own passionate devotion to chess to the status of objective truth, while fully recognizing that, in the larger scheme of things, it merely reflected his own idiosyncratic set of values.
The difference between mathematics and chess that I pointed out to Mats is, I claim, not only about me personally. Perhaps I have some asymmetry in talent for the two domains, or perhaps at some stage I entered some feedback loop in which greater success in mathematics led me to devote more effort to it, which in turn brought greater success, and so on. Still there remains the broader societal phenomenon that it is much easier to make a good living working in mathematics than playing chess. It is hard to pinpoint the number of elite chess players who earn a decent income from prize money and appearance fees, but it is often said to be just in the double digits. While several thousand more manage to eke out a living by some combination of prize money and other chess-related activities such as writing and coaching, the number of people who earn a good income from mathematics, mainly through teaching and research, is orders of magnitude larger.
This is a reflection of the fact that society pours so much more resources (mainly through taxpayers’ money) into mathematics than into chess. Fundamentally, society’s reason for supporting mathematics is its instrumental usefulness in other areas: engineers designing bridges and electronic circuits, scientists analyzing messy data, economists modelling business cycles, and so on.6 In contrast, chess is motivationally self-contained: the reason why chess exists in society is that chess players like playing chess, and chess fans (a category that overlaps very strongly with chess players) like watching grandmasters play against each other.
In both of these activities (chess-just-for-playing and chess-as-a-spectator-sport), the human element is essential. It would be utterly pointless for chess amateurs to hand over their play to machines: the human involvement is the whole point of the activity. For the case of fans watching elite players the corresponding point is slightly less obvious. We can imagine a world where the advent of superhuman chess machines would have caused chess fans to lose interest in watching human grandmasters, and switch to watching the machines play so much better chess. But this is not at all what has happened: chess fans all over the world are still as engaged as ever in grandmasters facing off against each other and in who is the best, but hardly at all in how various chess AIs score against each other. To chess fans, the human element is essential: they like to see Hikaru Nakamura’s reaction upon an unexpected pawn sacrifice by Magnus Carlsen, and to imagine what at that moment is going through his mind and what he is feeling.
I claim that the absolutely central importance of the human element in how we attach meaning to chess is what has made chess survive and flourish in the era of superhuman chess machines. It is natural to ask whether we can transfer this lesson to the realm of mathematics. Perhaps, by emphasizing the human element in mathematics, we can make it survive and thrive in the new AI era in a similar way as chess has done for decades?
My central claim in this essay is that the predominantly instrumental role of mathematics in society makes this hope very faint. Engineers who design bridges, and people crossing those bridges, want them to be structurally sound and to hold up, rather than collapse under the weight of traffic. Given that goal, they do not much care whether the mathematics used to achieve it was produced by a human mathematician rather than an AI. The story will likely be the same in other applications of mathematics, so in a situation where AI produces better mathematics than humans, at a lower cost, society seems to have little reason to use the human product.
One way of trying to salvage human mathematics at this point is to view it not purely in instrumental terms but also as an art — a discourse that has been around long before the AI issue appeared on the radar. Just like poetry or music or sculpture, mathematical beauty has a value in its own right, and this value can only be realized through the experience of a human. Or so the story goes. A problem with it is that this art form is very elitist. Consider the familiar objection to state support for, say, opera: compared to pop music, its audience is tiny, so why should ordinary taxpayers subsidize the peculiar tastes of a narrow elite? Mathematics is even narrower, because when someone comes up with a beautiful new idea in a proof, this is a beauty that can only be appreciated by mathematicians specializing in the subfield where the proof appears. Getting tax payers to support mathematics in order for mathematicians to enjoy beauty seems to me like a tall order. And trying to broaden the appeal by stressing the beauty of simple classical ideas like Pythagoras’ Theorem does not seem likely to cause a mass following, nor does it seem like a promising source of income for all of us who are presently employed as mathematicians.
The prospects for academic mathematics to go on as before seem bleak. Over the last few months, mathematicians have increasingly begun to wrestle with this issue. What will be the future role of mathematicians? Will it be to direct the AIs at promising areas to work on and problems to solve? Or to teach the results that AIs discover? Or can we serve simply as students, and thereby attain the understanding that is needed to make new results meaningful? Might we perhaps play a role in deciding how to canonize those results? Fernando Borretti, in his brief and well-written recent blog post Mathematics without mathematicians, considers all these suggestions, but finds that none of them has a convincing answer to why we should expect the AIs to not become superhumanly capable at those tasks as well. He considers the various suggestions to be “cope [that] is likely to be refuted by reality”. This view could be mistaken, for instance if AI development hits some unexpected ceiling before attaining some of those abilities,7 and Borretti is open to being wrong. But it seems to me more likely that he is right.8 In which case…
…I’m not sure how to end this essay. Check mate, mathematicians? Or, given the role we mathematicians have played in inventing AI, is it better described as a self-mate?
I’ve been a club player since 1980 (when I was 12) and a national master (a title held by a few hundred Swedish players) since 1987. I very rarely win tournaments, but this summer I came second in the Swedish 50+ veterans championship.
As an almost morbid illustration of this, consider the outcome of the 2022 World Champinonship in correspondence chess, which at the time of writing is the latest one to have been finished. The field consisted consisted of 17 players, each playing one game against each of the others, for a total of 17*16/2=136 games. One of the players, Aleksander Dronov, sadly died during the tournament, and the 10 games he had not finished were declared to be won by his opponents. All the remaining 126 games ended in draws, so the tournament ended as 10-way tie for the World Champion title.
This is similar to the death in the late 2010s of centaur chess (although that was never a prominent part of chess culture).
Last time I wrote about mathematics and chess, in my article Objective Truth versus Human Understanding in Mathematics and in Chess way back in 2007, I exploited some of these similarities, albeit for very different purposes compared to the present essay.
I told the same anecdote (in Swedish) in a 2023 episode of Carl Fredrik Johansson’s podcast Gambit: en podd om schack.
I am not denying that pure mathematics with, at best, unclear relevance to applications is also being funded, but I do believe that this funding would shrink to almost nothing if mathematics as a whole didn’t demonstrate the incredible power in applications that it in fact does.
As an extra bonus, such a turn of events might also save us from the AI apocalypse in which every single human is killed!
By no means do I mean to suggest that mathematics will be the only academic discipline to succumb in this way, but perhaps it will be the first. It will then likely be followed by others, and perhaps even all, but how that might play out is outside the scope of this essay.