Frieren: Beyond Journey’s End calls Frieren “The Last Great Mage”. She is from an era so ancient that she has mastered by herself many forms of magic to a level that she can be called a “living grimoire”. Nobody alive in the present is a Great Mage because the Demon King is already dead and there is no need for humans or elves to master magic at this level any more. Or as Frieren herself puts it, “Ordinary Offensive and Defensive Magic is enough for mages of this era.” There is no need to master spells to construct golems, summon black holes, volcanoes or purple lightning. There is no need to master spells that cannot even be identified as magic.
Just last week, I wrote an article mourning the loss of craftsmanship and mastery of various fields due to the creeping advance of AI capabilities.
On the 8th of September, just two days after I published this article, OpenAI announced that an unreleased internal model, a successor to GPT 6 Astra, had produced a solution to the Navier-Stokes existence and smoothness problem, one of the most difficult mathematical problems in the world. It is one of the seven millennium prize problems that entitles the solving mathematician to a prize of one million USD, awarded by the Clay Mathematics Institute. The problem asks the question: does the Navier-Stokes differential equations that model fluid flow always have smooth solutions in 3D space given some initial conditions? OpenAI’s model constructed a counter-example that blows up in a finite amount of time, showing that the answer is no, they do not remain smooth forever.
After hearing the news, I felt a little bit like Richard Feynman described feeling in the months and years immediately after the Manhattan Project. Returning to civilian life, he recalled looking at people building bridges and roads and thinking about how pointless it all was since nuclear weapons would just destroy everything in the world very soon.
Like nuclear weapons, AI has the potential to both free humanity from scarcity and destroy everything we’ve built over millennia. The millennium prize problems sit at the frontier of mathematics, accessible only those who have built up decades of expertise in the specific subfield of mathematics in which the problem exists. Solving one of them is no small feat. This solution is, in my mind an early sign of things to come. Richard Feynman’s feelings turned out to be wrong and there was no world ending nuclear war in his lifetime. I’m hoping that my feelings are wrong in a similar way.
That being said, the circumstances of the result are hotly debated. NYU professor and mathematician Tristan Buckmaster says that he and Levent Alpöge, a mathematician at Anthropic, had been pursuing an approach that was very similar to that used in OpenAI’s solution and had put drafts and mathematical work into OpenAI’s Codex throughout the project. When OpenAI informed him that its internal model had independently pursued the same unusual route, Buckmaster says he sought clarification on whether the model had been trained on, or had access to, those sessions. He was told that it had not looked up user data, but says his question about training initially went unanswered. Buckmaster nevertheless explicitly cautioned: “I do not know whether our data was used. I am not accusing anyone of anything.”1
OpenAI said they launched an investigation into the matter, and they now say that Buckmaster’s codex prompts “could not have influenced the system in any way, including through training.”2
There is also a parallel dispute over who gets credit for the work. Buckmaster says OpenAI proposed that he present its Navier-Stokes result without Alpöge as a coauthor because Alpöge worked for Anthropic. OpenAI researcher Sébastien Bubeck however, disputes this, saying that he never sought to remove Alpöge from authorship of his own work, but acknowledged saying that “it would be simpler if Levent was not an Anthropic employee” when discussing authorship of a rewrite of OpenAI’s proof. Buckmaster also says that when he threatened to make the dispute public, he was asked, “Why would you ruin your career?” Bubeck has said that this is true and has since apologized for his “extremely poor choice of words.” But he still denies that it was intended as a threat.
While this back and forth plays out in partial view of the public. I think an outsider like me cannot yet confidently say whether the accusations of OpenAI’s model having been fed human mathematicians’ partially completed work are true. It could possibly be the case that the model stumbled upon a similar inspiration from past work as Buckmaster himself did, considering that LLMs assisted his own work. Regardless. This is something of a “Deep Blue-Kasparov moment” for mathematics, as he puts it.
If it is indeed true that Astra’s successor was trained on Buckmaster’s conversations, then it appears that this internal model seemingly only filled in the final few steps of the solution. If the accusations are false, then it is even more mind boggling that the model came up with the solution. Whatever the truth is, AI models have gone from not being able to count the r’s in strawberry to assisting with proofs at the frontier of mathematics, working toe-to-toe with experts in the field who have spent decades building up their expertise.
My prediction is that the next big breakthrough by an AI model will probably not be as hotly debated as this one. And not long afterwards, I suspect we’ll start to see more and more difficult problems on the frontier of mathematics and physics fall to AI assisted human mathematicians. And eventually we’ll start to see these models produce proofs that no human may ever have come up with. I sometimes wonder if we’re heading towards a world like in Asimov’s The Feeling of Power. Will mathematics as a skill, considered to fundamental today, become so completely outsourced to machine thinking that practicing it un-aided will start to seem strange?
Mathematicians are Feeling the Loss
Terrence Tao and other working mathematicians have chipped in with their views in the days that came after. A lot of the conversation centers the on the meaning of mathematics and about the value of the process vs. the final solution.
At the 2026 International Congress of Mathematicians, Tao went on record to say that “We can have these 100,000-line proofs that we have to verify, but no one understands them…” A proof must also be explained, communicated and absorbed into the broader body of mathematics. He also pointed out that mathematics has historically been able to focus on outcomes because the process of producing those outcomes was necessarily human: “This worked until we figured out a way to automate outcomes without process.”
For most mathematicians, solving a problem is often not just about the final answer. This is something that is true in engineering and the sciences as much as it is in the field of mathematics. The struggle that occurs between the identification of a problem and its eventual resolution can sometimes spawn entire fields of study. Ravi Vakil, president of the American Mathematical Society, gives Fermat’s Last Theorem as a prominent example. Andrew Wiles’ proof, while important and significant was not the only result of the centuries of attempts at a solution. Along the way, mathematicians discovered entire new areas of mathematics. Vakil says he “cannot think of a major advance in mathematics” that does not fit this pattern.3
This brings us face to face with a strange possibility. Imagine a future where a mathematical result is needed and someone with relatively little expertise can simply prompt an AI until it produces a correct, formally verified proof. The most important factor in the discovery becomes being in the right place at the right time. If the final answer was the most important thing, then perhaps nothing of importance was lost. But what if the intermediate results a human mathematician might arrive at produced new techniques and unexpected connections to other fields that end up being a critical step in the solution of another important problem? These new techniques may never be discovered by AI driven proofs, which tend to lean heavily on existing mathematics.
The Navier-Stokes solution a few days ago was not an isolated occurrence. Earlier this year in January, Erdös Problem #728 was reported to be fully resolved using GPT 5.2 Pro together with the Aristotle formal theorem prover.4 Erdös Problem #659 was resolved in a similar way by Benjamin Grayzel, a computer-science master’s student using Gemini to identify the missing step.5 Not even a month later, in May, OpenAI reported that they’d used a general purpose reasoning model to disprove a conjecture related to Erdös’ planar unit-distance problem. The result was produced by simply prompting the model to solve the problem without any bespoke mathematical solving harnesses. Mathematicians have subsequently verified the proof.6
There is also the question of how mathematicians themselves are trained. Vakil says that what he teaches his graduate students is not merely how to obtain answers, but how to develop “human understanding” and communicate it through mathematical storytelling. Kontorovich similarly distinguishes between students who use AI to accelerate their understanding and those who simply use it to produce their homework. Students who use AI to answer homework questions possess the answer without having undergone the intellectual struggle that the homework was designed to create.
Fields Medalist Akshay Venkatesh goes even further. He argues that mathematics has a dual identity: it is both a science and a humanity. His hope is that AI might actually force mathematicians to place more value on the latter, the human understanding, taste, beauty and sense of meaning surrounding a result, rather than the bare fact that a theorem is true.
But does any of this matter in the long run?
To me, it feels very possible that this distinction between answer and understanding is only a temporary limitation of present-day AI. I think that as these models get better they may become capable of inventing the concepts, techniques and entire new directions themselves. We may only need to assign a compute budget and a broad problem statement for the AI to work its way through several human-mathematician-years worth of work and produce a solution and the intermediate steps, complete with new subfields for other AI models to explore.
Isaac Asimov imagined a world like this in his short story The Feeling of Power where humanity has become so dependent on computers that people have largely forgotten how to perform arithmetic by hand. The one man who rediscovers the ability to calculate with pencil and paper is treated like a magician.
I wonder whether the future of mathematics may look like this. We may live in a world filled with more mathematical knowledge than ever before, with machines that have near unlimited knowledge and ability to verify, prove or disprove any theorem we want, if it is important enough. But much of that knowledge and skill would live inside the machine, with humans only capable of guiding the machines in limited ways.
How Special is Human Intelligence, Really?
We once thought of mathematics as one of the highest expressions of human intelligence. The last bastion of esoteric expertise into which AI would struggle to make inroads.
It seems that we were wrong.
For a long time, my own opinion has been that human intelligence is not as mysterious and special as most people would like to believe. I’ve worked as an engineer doing research for most of my life. My experience has been that novel ideas are really just remixes of old ideas in new contexts, familiar tools applied to new problems and existing techniques connected together in ways nobody happened to notice before. I’m not dismissing human creativity. Being creative is not trivial or easy, but I am skeptical of any claim that there is some irreducible spark or non-physical essence of novelty inside the human mind that machines could never reproduce. In finding new ideas, I think that our minds are searching through an enormous space of possible connections, guided by experience, intuition, analogy, taste and sometimes pure randomness. We like to celebrate the few ideas that turn out to be useful, elegant or beautiful as discoveries.
If this is even approximately true, then we may need to recalibrate our expectations for the coming decades. It seems increasingly likely to me that a large fraction of any kind of work could eventually be performed by AI systems, with humans providing only goals, constraints, judgement and occasional course correction. There is little reason to assume that the most esoteric parts of mathematics and physics will be exempt from this.
If humanity ends up reconciling quantum mechanics and gravity or producing a grand unified theory of everything, key components or even the entire solution may be discovered by an AI model. Machines may solve long standing questions about why the observable universe contains matter rather than anti-matter, or invent warp drives, and discover entire new fields of mathematical theory. Humans may only be there to try to understand, translate and teach the results.
Frieren’s title of “The Last Great Mage” in Beyond Journey’s End is sad to me because it creates this feeling of something amazing that was lost to time. She is the last great mage because she is the last survivor of an age that demanded that level of mastery of magic. Modern mages don’t try anymore because they don’t live in an environment where such mastery is needed.
I wonder how many future mathematicians would spend twenty or thirty years developing the kind of deep, far-reaching expertise possessed by the great mathematicians and physicists of today? Will we have another Ed Witten or Terrence Tao? Will they have the same mastery of mathematics and physics? Or will AI take over the hardest parts of reasoning and discovery, leaving a dwindling amount of handmade mathematics?
I think that humans will continue doing mathematics. People continued to play chess after Stockfish and Go after AlphaGo. We may be motivated by the meaning we find in understanding, mastery and beauty, even when machines can outperform us. Maybe there will be a split between mathematical answers where the solution is the more important output, so that we can achieve a practical goal like more efficient energy generation or faster interstellar travel, and where meaning is the more important output. Or maybe I’m wrong and future humans will be brought up in a world where meaning, art, culture, and thought are manufactured by alien minds.
And humans will no longer be at the center of the universe.
The last great mathematician may already have been born. And they may live long enough to watch machines dominate the upper echelons of mathematical discovery.
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