The emerging picture from the last two years of AI in mathematics is not that machines are more creative than humans, it is that they are relentlessly patient, unafraid of dead ends, and willing to try approaches that a human mathematician would dismiss as absurd, tedious, or not worth the risk, they use the same techniques humans have built over centuries but they never get lazy and never worry about wasting a year on a dead end.

As mathematician Thomas Bloom put it in his companion paper to one of the most significant AI math breakthroughs of 2026: AI "often produces the most surprising results by persevering down paths that a human may have dismissed as not worth their time to explore, combining superhuman levels of patience with familiarity with a vast array of technical machinery." (Remarks on the Disproof of the Unit Distance Conjecture, arXiv:2605.20695, May 2026)

Jacob Tsimerman (University of Toronto) made a similar point: "AIs have an edge: It's not just that they can try all known methods. They can play for longer and in more treacherous waters than mathematicians without getting overwhelmed." (Scientific American, May 2026)

This is the key insight: the AI is not inventing new mathematics from thin air. It is taking the enormous "cathedral of mathematics" that humans have built over centuries and exploring it with an endurance and lack of ego that no human can match. It will try a technique that looks hopeless. Then another. Then thirty more. A human researcher would have stopped at attempt three, concluded "this can never work," and moved on. The AI has no such instinct.

The Machine That Has No Beliefs

There is something deeper going on here than just patience and brute force. A human mathematician carries beliefs. They carry a sense of what is "likely to work," what is "elegant," what is "worth trying." These instincts are often right — they are distilled from years of training and experience. But they are also a filter. They silently discard entire branches of possibility before anyone even looks at them.

AI has no such filter. It has no beliefs. It has no sense of what is fashionable, what is respectable, or what a "real mathematician" would consider beneath them. Every method, no matter how absurd it looks, is just another path to explore. Where a human sees a dead end and turns away, the AI sees a door it hasn't opened yet.

This is not a new dynamic in mathematics. History is full of breakthroughs that came precisely from people who ignored the conventional wisdom, who tried the thing that everyone else "knew" wouldn't work.

Georg Cantor showed in the 1870s that there are different sizes of infinity — that the real numbers are "more infinite" than the integers. His contemporaries did not just disagree; they were outraged. Leopold Kronecker called him a "corrupter of youth." Henri Poincaré called his ideas a "disease." Cantor's own colleagues treated the work as absurd, a violation of mathematical common sense. He was right. His set theory became the foundation of modern mathematics.

AI is the ultimate outsider, It carries no assumptions about what Erdős would have believed, It has no reverence for authority, It does not care if an approach looks ugly or inelegant. It does not worry about its reputation if it spends a month exploring a method that turns out to be a dead end. As one critical analysis of the Erdős unit distance breakthrough put it: "The AI has no social biases, no reverence for Paul Erdős, and no fear of wasting a year of grant funding on a dead end."

This is not intelligence in the human sense. But it is something that the history of mathematics tells us is extraordinarily valuable: the willingness to try the thing that nobody else would try.

The Proof Is in the Problems: Real Results

The Erdős Unit Distance Problem (May 2026)

The most dramatic example so far. Paul Erdős posed this problem in 1946: scatter n points on a flat plane, how many pairs of points can be exactly one unit apart? For 80 years, mathematicians assumed Erdős's answer was essentially correct. Nobody seriously tried to disprove it — because Erdős was Erdős, and the community had a shared belief that the conjecture was true.

An internal OpenAI model did the opposite. It assumed the conjecture was wrong and systematically hunted for a counterexample. It found one, using deep tools from algebraic number theory (class field towers, Golod-Shafarevich theory) — tools that existed in mathematics already but that nobody had thought to apply to this problem. Will Sawin at Princeton later refined the result, showing that the improvement is at least n^1.014 — a polynomial factor better than what Erdős thought possible.

Bloom's analysis of why the AI succeeded where humans didn't is revealing. He identified four conditions a human would have needed to meet simultaneously: (1) spend serious time on the problem, (2) bet against Erdős's established opinion, (3) translate the original construction into the language of number fields, and (4) have deep familiarity with class field theory. "The AI met all of these criteria," Bloom wrote. No single human had the combination of stubbornness, breadth, and patience to do all four.

DeepMind's FunSearch and the Cap Set Problem (December 2023)

Google DeepMind's FunSearch tool, built on top of AlphaZero and the Codey LLM, found a new solution to the cap set problem — related to the Erdős-Szemerédi sunflower conjecture, which had been open since 1972. The tool treats math problems like games and generates candidate solutions as code, keeping the good ones and discarding the rest through an evolutionary loop.

DeepMind research VP Pushmeet Kohli told The Guardian: "When we started the project there was no indication that it would produce something genuinely new. As far as we know, this is the first time that a genuine, new scientific discovery has been made by a large language model."

Researcher Alhussein Fawzi was candid about how little they understood the process: "To be very honest with you, we have hypotheses, but we don't know exactly why this works."

The solution was not in any training data. It was genuinely new. But the techniques it used? All known. The FunSearch system combined existing methods in ways that no human had bothered to try.

AlphaProof Nexus and 9 More Erdős Problems (May 2026)

Days after the OpenAI announcement, Google DeepMind revealed that its AlphaProof Nexus system — which pairs LLMs with the Lean formal proof assistant — had autonomously solved 9 out of 353 open Erdős problems and proved 44 conjectures from the Online Encyclopedia of Integer Sequences (OEIS). The cost: a few hundred dollars per problem. Some of these problems had been open for decades.

The system works through "agentic loops": the AI proposes a proof, the Lean checker verifies every logical step, and if it fails, the AI tries again. No human intervention once the problem is given. All proofs have been publicly shared and logged on Terence Tao's wiki on AI contributions.

The Claude-Knuth Episode (Early 2026)

Donald Knuth's colleague Filip Stappers gave Claude (Anthropic's model) a combinatorics problem Knuth had been stuck on for weeks. Over one hour, Claude tried 31 different approaches. It started with simple formulas, tried brute-force search, tried simulated annealing from physics, tried geometric "serpentine patterns." Each failed. At one point it wrote to itself: "Maybe the right framing is: don't think in fibers, think directly about what makes a Hamiltonian cycle." On the 31st attempt, it found the answer. This documented session showed something that looked less like a machine producing an answer and more like watching a researcher think.

What Real Scientists Think

The Enthusiasts

Terence Tao (UCLA, Fields Medalist): Tao described current AI systems as "very good at scouring big lists of problems for low-hanging fruit." In his ICM 2026 presentation, he framed the future as a "self-correcting loop" where AI suggests paths and formal proof checkers enforce the truth. He sees AI as a power tool for the "long tail" of mathematics: "AI tools are already helping us explore the long tail — hundreds of thousands of math problems that rarely receive attention because the number of mathematicians is finite." On Google's AlphaEvolve, Tao noted it "excels at discovering constructions that were already within reach of current mathematics, but had not yet been discovered due to the amount of time and effort required."

Ken Ono (University of Virginia): After witnessing OpenAI's o4-mini at a private demonstration, Ono said: "I've never seen that kind of reasoning before in models. That's what a scientist does. That's frightening." He added that colleagues "literally said these models are approaching mathematical genius."

Melanie Matchett Wood (Harvard): On the unit distance problem, she argued that human progress was limited by the shared belief that Erdős was right. "If all the experts assembled after the fact to parse the LLM's answer had instead spent the same time seeking a counterexample, they would have found one." In other words, the AI's advantage was not intelligence — it was the absence of bias.

The Skeptics

Kevin Buzzard (Imperial College London): Buzzard poses the sharpest challenge: "The bottom line is, have any of these systems ever told us something interesting that we didn't know already? And the answer is no." He argues that AI's math ability is "solidly in the realm of the ordinary, if mathematically talented, human" — the solutions use known techniques applied to neglected problems. He sees genuine utility in AI for formal proof verification, but draws a line between competence and fundamental mathematical creativity.

Ursula Martin: She described the OpenAI unit distance result as "remarkable" and "impressive," but added: "We are not told about the model's failures. If you put vast quantities of human effort into this problem, you likely would've solved it in the same way. But in math, human effort is scarce, and just tends to be spent on different things."

The Problem with the "Blablabla": Transparency and the Erdős Announcements

This is the crucial criticism, and it goes beyond whether AI can or cannot do math. When OpenAI announced the unit distance breakthrough, it said an "internal model" had "one-shotted" the problem — implying pure autonomous reasoning with no human help. But there are serious questions:

What was actually hidden?

Rodrigo Ochigame (Leiden University), an anthropologist and historian of computing, said in the New York Times: "Basic information needed to assess the scientific meaning of the result is kept secret. The company disclosed nothing about the methods, human-written prompts, training data, or computational resources consumed."

OpenAI did not reveal: what prompts were used, how much human guidance was involved, what the model's failure rate was (how many times it tried and failed before succeeding), or what computational resources were consumed.

The Boris Alexeev question

A critical blog post by Anatol Wegner (Medium, June 2026) raised a pointed question. An OpenAI researcher named Boris Alexeev, who works on the math team, had been researching exactly the mathematical machinery used in the proof (bounded degree number fields, Golod-Shafarevich theory) before the model "independently" produced it. As Wegner wrote: "We are supposed to believe that an OpenAI researcher independently had the exact, highly esoteric mathematical intuition required to solve the problem... and then, completely by coincidence, a 'general-purpose reasoning model' woke up one day and 'one-shotted' that exact same highly specific approach without any human guidance? The statistical probability of that happening is zero."

The implication: the prompt or hidden system context almost certainly encoded deep mathematical direction. The model did not stumble onto algebraic number theory by itself — it was pointed there.

A pattern of overclaiming

This was not the first time. A previous OpenAI announcement claimed GPT-5 had "solved" ten Erdős problems. Thomas Bloom, who maintains the Erdős Problems website, called it a "dramatic misrepresentation" — the model had merely surfaced solutions already existing in the literature.

The Leiden Declaration (June 2026)

In response to these concerns, 16 researchers from 15 universities published the Leiden Declaration on Artificial Intelligence and Mathematics. The declaration warns about "flashy headlines publicizing results that haven't yet been vetted by the standard academic process" and urges researchers to disclose which AI tools were used, what prompts were given, and how much human guidance was involved.

Mathematician Michael Harris (Columbia) explained the motivation: "The purpose, from my perspective, is to recover control of the narrative about the values and the goals of mathematics from the AI industry." He described mathematicians' values as "openness, honesty, giving credit where credit is due, sharing, transparency."

The Real Picture: What AI Actually Does Well in Math

Synthesizing the expert opinions and the actual results, a clearer picture emerges:

What AI does: It takes existing mathematical techniques and applies them with inhuman patience and breadth. It tries combinations of tools that no human would bother with. It explores the "negative resolution" — the possibility that a conjecture is wrong — without caring about reputation or wasted time. It connects distant areas of math (like algebraic number theory and discrete geometry) that specialists in one field would never think to combine. As Bloom put it: it reveals "unexpected connections" and pushes "existing technical machinery to its limit."

What AI does NOT do (yet): It does not generate fundamentally new mathematical frameworks. It does not have the intuition to know which problems matter and why. It does not develop new conceptual tools that reshape how we think — the kind of breakthrough that defines a Fields Medal career. As Tao noted, AI might find that a bound is 0.487 instead of 0.5, but it cannot tell you why that matters or what it means for the larger structure of mathematics.

The honest summary: AI is an extraordinarily powerful research assistant that is fearless, tireless, and unbiased. It is not a mathematician. But for a growing class of problems — especially those where the tools exist but human effort is scarce or biased in the wrong direction — it is already producing results that humans could not.

Key References

  1. Romera-Paredes, B. et al. "Mathematical discoveries from program search with large language models." Nature 625.7995 (2024): 468–475. (The FunSearch paper)
  1. OpenAI. "An OpenAI model has disproved a central conjecture in discrete geometry." OpenAI Blog, May 20, 2026.
  1. Alon, N., Bloom, T.F., Gowers, W.T., Litt, D., Sawin, W., Shankar, A. et al. "Remarks on the disproof of the unit distance conjecture." arXiv:2605.20695, May 2026. (The companion paper with mathematician reflections)
  1. Sawin, W. "An explicit lower bound for the unit distance problem." arXiv:2605.20579, May 2026. (Refined the bound to n^1.014)
  1. DeepMind (AlphaProof Nexus). "Advancing Mathematics Research with AI-Driven Formal Proof Search." arXiv:2605.22763v1, May 2026.
  1. Wegner, A. "OpenAI disproves the unit distance conjecture — Act II." Medium, June 2026. (Critical analysis of human guidance)
  1. Leiden Declaration on Artificial Intelligence and Mathematics. Published June 2, 2026, by 16 researchers from 15 universities.
  1. Quanta Magazine. "The AI Revolution in Math Has Arrived." April 2026.
  1. Tao, T. "Mathematics in the Age of AI." ICM 2026 Slides. (On formal proof assistants and AI collaboration)
  1. Buzzard, K. Quoted in Live Science (December 2025) and various lectures. (The skeptical position on AI mathematical creativity)
  1. Scientific American. "AI just solved an 80-year-old Erdős problem, and mathematicians are amazed." May 2026.