Who Watches the Watchmen?

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Every story has a beginning, a middle, and an end. The difficulty in faithfully recounting a story detailing a rapid succession of events that individuals lived through, is that it is a tale told in a thousand ways: a collective dream that meshes together, blending facts with fiction, living in the never-never land of symbols and portents. And, as is the case with dreams, sometimes there is a monster at the end of it.

While idly scrolling social media app X (formerly Twitter), I come across a tweet by Will Depue, previously an employee at the American superintelligence corporation OpenAI. The tweet says (verbatim) “we’re really living through a historic moment for mathematics. it’s pretty incredible to watch”. Reading the ensuing conversations with mathematician Daniel Litt in the replies, it dawns on me that the tweet alludes to the future impact of LLMs in the field of mathematics.

I must admit that at this point of time I had not really been following the developments happening in the space between mathematics and AI, despite being a mathematician myself, although occasional news about progress (eg, on Olympiad problems, various Erdős problems, …) would slip through the cracks and appear on my news feed. Nevertheless, I had certainly heard much about AI itself, primarily from X The Everything App, which happens to be the typical congregation spot for buzz surrounding all things AI. Much of these conversations are no doubt populated and amplified by individuals, like Depue himself, who are based in the San Francisco Bay Area, a place that is intimately tied to the genetics of the AI boom.

Much has been said about how this unending 24/7 AI-news cycle originating from the Bay Area has to do with Silicon Valley VC startup funds being intimately tied to the great financial pyramid scheme of the century, working tirelessly to ensure that the AI capex breathes while starving every other aspect of the palpable economy and global financial ecosystem of oxygen, all at the cost of leaving world economy hanging in balance, reliant on speculative fortunes of the One Bet. What I had not realized, and could not dream of, was how my relatively quiet and monastic inner life as a mathematician would soon be dragged into the whirlpool that is “Project AI”.

The discussions between Depue and Litt went back-and-forth. Litt firmly made the point that, even though solving problems is a significant part of what mathematicians do, the discipline of mathematics is strictly less directionally bound by the kind of precise questions that one can train AI systems on. On the other hand, Depue kept insisting, almost as if he’s reaffirming his unshakable blind faith in the second coming of Christ in the form of a Machine God, that the same framework of RL that excelled at solving the Olympiad problems will (verbatim) “more slowly solve the rest [of mathematics]”.

At this point, an unlikely character enters the conversation with a response to Litt’s rebuttals: Jacob Tsimerman, who would eventually go on to win the Fields medal in a few months. He was already known within mathematical circles to be one of the possible contenders for the Fields medal this year. I’d like to pause here and let my readers know that at this point of time Tsimerman himself already knew that he had secured the Fields medal; this news was conveyed to him in January by the president of the IMU.

Continuing with the story: Tsimerman responds to Litt by construing the domain of mathematical investigations as one that, eventually, centres around solving specific problems anyway, once they are formulated precisely. Therefore, he argued, can one really say with certainty that the impact of AI in the field would not be so significant and severe?

On May 04, Tsimerman goes on to elaborate on his points by a lengthy tweet.

He makes the following claim (verbatim): “problem-solving is an immense, and pervasive part of modern mathematical research. Consequently, if human problem-solving disappears by virtue of the AIs becoming strictly and substantially better at it, then most of the time currently spent by modern mathematical researchers will have to be spent on an activity that is altogether pretty different. Whether such an activity is viable as a professional endeavour is something I am unsure of …” Overall, the tone of this post seemed rather urgent. Tsimerman spoke like Nostradamus warning of an undeniable impending apocalypse that he had in his visions.

From my vantage, the discourse was rather confusing. Having grown up in the pedigree of Henri Poincaré and Bill Thurston, the claim presented by Depue and Tsimerman that large swaths of mathematics ultimately boils down to problem-solving seemed misplaced at best, and disingenuous at worst. Whether mathematics is problem-solving or not always seemed more a matter of taste than an intrinsic question about the nature of mathematics, to me. But more importantly: there were no major news of AI solving major mathematical problems of general interest at this point of time. So what on Earth were these people even talking about?

I remember my friend, who is also a mathematician, sent me the humorously worded message: “how can a serious mathematician say this “my entire main research field disappears if someone solves the Zilber-Pink Conjecture in its more general form”” quoting (verbatim) a line from Tsimerman’s tweet.

On May 20, two and a half weeks after the exchanges between Depue, Litt and Tsimerman, OpenAI announced that their internal models had disproved the “unit distance conjecture”, posed by Paul Erdős in 1946, (rightly) described in their milestone research announcement as “one of the best-known questions in combinatorial geometry, easy to state and remarkably difficult to resolve”.

At the time, it felt like our Nostradamus’ predictions are coming true before our very eyes. I would eventually come to realize that these predictions were not proclamations of a prophet, nor paranoid ramblings of a delinquent, but something more sinister: a smokescreen— a seemingly plausible reframing of the essence of mathematics and its impact on the real world, something that our generally naive and narrow-minded intellectuals could have sterile debates over, while unnoticed behind the curtains certain entities could continue their project of large-scale institutional capture and economic disenfranchisation of vast sections of human society.

The graphic novel Watchmen, originally a 12-issue comic book series, was written in the 80s by the self-proclaimed English wizard Alan Moore and is now hailed as one of the most influential comic book literature to date. The central theme of Watchmen is a meditation on the concept of power. The book is set in the backdrop of an alternate 80s America, where US won the Vietnam war, Richard Nixon is still the president, and the Cold War rages on. The ones responsible for this divergence in history are the protagonists of the novel, the Watchmen collective, entailing various anti-hero vigilante characters, each with their own conception of power and flawed visions of morality and that of greater good. The main plot of Watchmen follows several of these characters as they slowly uncover their comrade Ozymandias’ plan to unite the world and avert the impending nuclear armageddon by sacrificing millions of lives and blaming it on a fake alien invasion. At its core, Watchmen is a story about the vices of the American society. The world of Watchmen is bleak and violent, seen through the overbearingly nihilistic lens of the masked vigilante Rorschach as he roams the crime-ridden streets and backalleys of a gothic-noir version of New York City. Throughout the novel the reader is catapulted across a jarring barrage of newspaper cut-ups, television media, journals, memoirs, stories-within-stories that emphasize a collapse of the information ecosystem not unlike the present world we live in. In face of these existential crises, the heroes of the novel retreat into sex, drugs, religion, delusion, anything to avoid reality. Watchmen posits that if a superintelligent omniscient being really did exist, the American government would no sooner reduce it to a weapon and a propaganda tool.

The research announcement by OpenAI on the unit distance conjecture is accompanied by a short teaser launch video. It is here that we are introduced to some of our proverbial Watchmen: visionary AI researchers at the frontier of Mathematics and AI, witnessing the “superintelligence” emerge at their terminal as out of nothing, suddenly realizing that they are the ones upon which a greater part of the responsibility is bestowed to chart out a course forward for humanity into the Great Unknown.

Cinematic music reminiscent of Hans Zimmer’s compositions plays in the background. We see OpenAI researchers, previously career mathematicians, Mark Sellke and Mehtaab Sawhney, explaining the unit distance conjecture on an easel chalkboard; camera pans revealing quick flashes of interior design boasting Silicon Valley startup decor. Sellke and Sawhney both recall their initial disbelief upon seeing the model output; Sellke recounts having trouble sleeping upon witnessing the proof. Next, we are introduced to Ozymandias himself: then-head of the OpenAI math research team, Sébastien Bubeck. Sébastien succinctly explains how AI succeeded over humans here: the proof-tree for this approach was a dark-forest, there were too many delicate decisions to make; by searching through all possible approaches, the relentless blind idiot god Yaldabaoth had found the right idea, like a lightning strike to the ground. He ends with the following remarkable comment (verbatim): “And so what this tells us, is that AI is able to make breakthroughs in science, and can enable breakthroughs not only in mathematics, but in engineering, in physics, in biology, in medicine … (pensively) This result is showing glimmers of a Golden Era for mathematics.” At the beginning and the end of the video, Fields medalist Timothy Gowers also makes quick appearances, stating rather matter-of-factly, in his usual British demeanour, the importance of the result that has been proved by AI.

I am far from a combinatorist, but I had heard of the unit distance conjecture before, as it closely relates to the distinct distance problem by Erdős which has been both a source and part of much interesting mathematics. Notably, work due to Larry Guth and Nets Katz on the distinct distance problem used strikingly simple new ideas from polynomial methods and real algebraic geometry. While I did not find, admittedly only upon rather cursory reading, OpenAI’s internal model’s proof of the unit distance conjecture nearly as gripping as the work of Guth-Katz on the distinct distance problem, it was clear that proof had some decently interesting ideas extending Erdős’ original number-theoretic constructions. Furthermore, the experts seemed pretty much unanimous that this was an exciting development. The paper produced by the model was decently written, and it was accompanied by a paper co-authored by several experts summarizing the ideas of the proof very clearly, together with additional commentary by each of the experts — Tsimerman’s comments began: “This is a really impressive piece of work, and I would accept it for any journal without hesitation”. Bubeck’s prediction of the future seemed on the horizon: a Golden Era of mathematics, fuelled by a synergistic collaboration between humans and AI, is now upon us.

Everything flows. All nice things come to an end. The unit distance conjecture would mark the last time OpenAI collaborates with the broader mathematical community in good faith. Since the initial highs, the story of AI corporations’ obsession with mathematical theorems has increasingly felt like coming down an up-staircase… and the encounter at the end was crushing.

In the weeks following, the discourse on X seemed to return time