AI Doesn’t Mean the End of Mathematics—at Least Not Yet

This essay was written with Kasra Rafi, and originally appeared in The Guardian.

Earlier this month, about 40 top mathematicians gathered at OpenAI’s offices to discuss the future of their profession. The meeting was off-the-record, but if recent articles by mathematicians are any guide, it was mostly pretty glum. People fear for their jobs, their careers and the work they love.

We think the contrary view is more likely, at least in the short-term. AI models are nowhere near as capable as experienced academic mathematicians.

This isn’t to say that AIs aren’t producing stunning mathematical results at the level of PhD researchers. In mid-May, OpenAI announced that its frontier AI model disproved the unit distance conjecture, a famous 80-year-old problem in discrete geometry. In July, Anthropic’s published two AI-derived results in academic cryptanalysis. Earlier this month, OpenAI published 10 new mathematical results from its latest AI model. And Anthropic published Claude’s attempt to prove the century-and-a-half-old Riemann hypothesis.

These results are both a vivid demonstration of the amazing capabilities of frontier AI in 2026 and an illustration of their limitations. In general, these AI-powered advances in mathematics fall into one of two categories. Some are counterexamples to mathematical statements that people had been trying to prove. Others are novel applications of known techniques to existing problems that human experts either did not know or did not think of using.

The counterexample to the Jacobian conjecture is the most notable example of the first kind. Once it had been found, checking it was quick and straightforward. The difficult part was finding it among a large number of possibilities. The AI seems to have combined some sort of intuition acquired through machine learning with extensive computational search, in order to find the right example.

An example of the second kind is the unit-distance conjecture. It was motivated by an elegant construction, and most mathematicians expected it to be essentially optimal—so they generally tried to prove rather than disprove it. The counterexample brings in ideas from elsewhere in mathematics: algebraic number theory. If an expert with that background deliberately set out to find a counterexample, they would probably have succeeded. But there was no reason for someone with precisely that expertise to focus on this problem. Because of its scope, AIs don’t have those same limitations.

These results are relatively low-hanging fruit for AI; none of them required developing an extensive new theory. This does not make the discoveries trivial, or the AI’s achievements less impressive. Choosing the right direction, and recognizing an unexpected connection between subjects, are themselves forms of creativity. They are the same sorts of capabilities that led to AIs playing the game of Go at the grandmaster level, or doing Nobel-prize level chemistry in the area of protein folding.

What we have not yet seen is an AI developing a substantial new conceptual framework in order to solve a mathematical problem. Much of mathematics proceeds by identifying the objects that are truly central to a question and then developing a theory that helps us understand them. Current AIs are very strong at searching and recombining existing ideas, but they are weak at building any deep and sustained new theory.

This speaks to a more general limitation of current AI systems. They are creative in the sense that they can recombine existing ideas in novel ways. But they are not creative in others: they have not yet developed conceptually new theories or structures. And while they have larger working memories than humans do, know more about more different things than any particular human does, and can process information faster than humans, can, true novelty is still largely beyond their reach.

Of course, that distinction may not survive for very long. Predictions are notoriously hard, especially about the future of AI. None of these mathematical capabilities were explicitly designed for, or planned. They’re all emergent properties of increasingly capable AI models. We are both confident that someday we will see AI models that are capable of the type of creativity required to do novel mathematics. Will that be in a few months, a few years or a few decades? Of course we don’t know, but our guess is sooner rather than later.

Tags: , ,

Posted on August 28, 2026 at 7:02 AM • 16 Comments

Comments

Paul Sagi • August 28, 2026 7:19 AM

I’m hoping AI will prove (or disprove) the Twin Prime conjecture and Goldbach’s conjecture. They are easy to state but very difficult to prove. Perhaps AI will find a connection to a seemingly unrelated area of math which unlocks those problems.

Rontea • August 28, 2026 9:27 AM

Mathematics, like all human endeavors, is a cathedral built not only of logic but of spirit. The machines that now parade their counterexamples and clever recombinations are but mirrors reflecting the fragments of our own thought. They do not suffer the torment of doubt, nor do they rejoice in the sudden illumination that turns the darkness of ignorance into the dawn of understanding.

A mind that has never trembled before the mystery of existence cannot truly create. These artificial intellects move as blind giants, lifting stones without knowing they are in a temple. They may uncover truths, but they cannot love them. They may solve problems, but they cannot feel the sacred weight of the question.

So no, the end of mathematics is not yet upon us. For mathematics is not merely the tallying of symbols, but the human cry toward infinity. While machines can echo our steps, only man can walk toward the Absolute with awe and trembling.

Robin • August 28, 2026 10:38 AM

@something_news: I think there’s an equally worrying but much more laid-back option. Knowledge of all sorts – not just maths – becomes a niche pastime. That’s what happens when tech replaces first-person skills. There’s all sorts of rural crafts that have all but disappeared (except, perhaps amongst the Amish, but even then I’m doubtful) apart from a few people who have learned them as a hobby. It’s also how languages disappear. They just fall out of use.

If they’ve got any sense even authoritarian governments will just let lassitude run its course. Doing maths properly is hard; people will just stop doing it.

KC • August 28, 2026 10:42 AM

re: Novel mathematics

Wonderful exploration and summary of this point in time.

Am led to wonder where novel math may appear; queried a few LLMs.

Gemini — living systems and non-equilibrium complex dynamics, quantum …
Claude — q-gravity (black holes), learning systems (no framework yet) …
ChatGPT — higher-d geometry/topology, spaces of theories (less symbols) …

Staying tuned

Patti • August 28, 2026 12:38 PM

This is all based on electric power (just the running costs, ignoring semiconductor mining, refining, manufacturing, etc.). A human runs on about 100 Watts. Can a 100W computer compete?

Ferentarius • August 28, 2026 1:17 PM

@Patti

Re: A human runs on about 100 Watts. Can a 100W computer compete?

The human, carrying his hundred watts of trembling vitality, devotes them to the maintenance of his illusions, to the slow consumption of his own despair. The machine, indifferent yet relentless, expends the same energy in a ceaseless circuit, a sterile brilliance without suffering. If there is competition, it is only in the theater of absurdity: the man, doomed to consciousness, and the computer, spared it, both burning their allotted watts in the void. In the end, neither triumphs, for in the universe’s accounting, all watts are wasted.

Clive Robinson • August 28, 2026 8:08 PM

@ Something News, Robin,

“Sounds far-feteched, yes, but infinitely more plausible than scenarios where AI truly takes on a “mind” of its own.”

And,

“I think there’s an equally worrying but much more laid-back option. Knowledge of all sorts – not just maths – becomes a niche pastime. That’s what happens when tech replaces first-person skills.”

It’s a little bit more interesting.

What happens is “first-person skills” gets “pushed up the knowledge stack”. That is old knowledge and the methods it begat gets replaced with new knowledge thus new methods that are faster, more efficient, more capable or some combination that makes things easier or more productive.

Consider how we do basic calculations through history.

We found multiple additions to be tedious and very exacting which few were capable of thus we developed multiplication. Various ways with writing were developed as well as with simple mechanical machines. Eventually we settled on efficient algorithms that are still being taught. Not just to humans but to computers and now AI systems as well.

As with “calculators” during the 1970’s through 1990’s there will be arguments as to if children should be taught the “long hand methods” any more… Basically the argument will sway as more and more adults become users of automated methods.

Thus the knowledge of the underlying methods gets lost to all but a few.

In principle there is no limit to just how many deterministic / algorithmic methods can be “automated away” by systems of “springs, levers, and cogs”. Just the practical limits of scale that loss/efficiency dictates.

The same applies to computers where the hard limits of the laws of physics start to apply.

Thus we have to “change the methods”. Humans are mostly “serial thinkers and doers” thus parallel doing and thinking comes naturally to very few of us, and even there it is limited in capacity. We’ve seen the same with computing and just how little parallel programming has moved in the past half century.

However AI has,

“A new way to do parallel processing”

Which is all Agentic AI is in reality.

It’s actually arisen from the various “Malware attack models” which should be telling people something (but as recent events have shown is apparently not).

Whilst these early Agentic AI methods are currently “grossly inefficient” they will improve as better algorithmic methods are found.

Thus the question will arise as to if humans will need to even know about the ins and outs of Agentic methods or just use the resulting systems, just as happened with calculators and computers…

I sometimes mention 1973 as the time when “human office productivity” was the highest. And this was due to streamlined methods around “typing pools” whilst the methods are still known “shorthand” and “dictation” are the last vestiges and fast on the way out having already become very niche. You can now buy for very little technology that does the whole “transcribing” process with AI being a part of it, becoming increasingly “context aware”.

You can see such real time technology in use on “Television Subtitles” and YouTube videos.

Do you care about the old transcribing methods and how they worked?

No, and you probably care even less if they get remembered or not.

“Humanity moves up the stack, as technology fills in beneath”

This is going to happen with Mathematics and all Science that relies on it for it’s theoretical models.

“Are we worried AI will kill Hard Science?”

Not really, it will just be subsumed as a new tool and thrown as grist to the mill or clay to the potters wheel. Yes Hard Science will change as do all human endeavors where new tools become used. It’s seen as “productivity” enhancing in that humans can do more with their time and capabilities.

The only unstated thought is,

“How do we stop the nut-jobs using it?”

To which the answer is,

“You can not unring the bell”

Along with,

“No law or regulation will stop those who chose to ignore them or can stop them being enacted or enforced”.

Something people should consider in this increasingly authoritarian political world…

Fractal Olivia • August 29, 2026 2:05 AM

This topic is so very frustrating. Yes, these models can indeed prove new theorems, yet the common dialogue about them forgets how many humans are actually involved. The achievements of these models are definitely not AI achievements but poor reflections of what we humans could do if only we worked together.

r • August 29, 2026 4:13 AM

an observation,

what % of the population will rejoice when they don’t have to learn any math at all.

people in general hate arduous, rote? tasks.

it’s probably the same as exercise, maybe worse.

but this is going to lead to a “going dark” problem.

the accumulation of data will be insurmountable in two directions: one being too much to wade through, and two being potentially too? complex or not documented?.

it really is a vacuum and a god damned tax.

Clive Robinson • August 30, 2026 4:44 AM

@ r, ALL,

With regards,

“transistors are turning out to be the cells of the undead.”

That is an interesting way to put it.

Biological cells in the main can not be turned off and on again. That is once certain processes stop they can not be restarted again even though they are seen as just chemical processes.

It thus gives rise to the thought,

“How did they ever start?”

Transistors however are designed to be turned on and off frequently. Not just as part of their “signaling” but thousands of times to have all external energy supplies removed entirely and for very extended periods of time. Transistor radios from the early 1960’s I have in my collection of bits still work and the Apple ][ computer I have from the late 1970’s still works and I use it from time to time to update work for others that still use Apple ][ computers to control test equipment.

Mary Wollstonecraft Godwin encouraged by her husband Percy Bysshe Shelley and their circle of friends wrote a story in 1818 long before the transistor or thermionic valve. Titled “The Modern Prometheus” (though we mostly know it by the main protagonists name and creation).

It was about a Dr who reanimated flesh by electricity and so created a monster from the parts of executed criminals. Due to the way Science was becoming known in that era the idea of electricity being a life force was often demonstrated by the twitching of frogs legs and similar for the after dinner amusement of the independently wealthy.

We have since found that the movement of charge around the body for signalling is a lot lot more complex and arguably as time goes on the more we learn shows how much we have yet to learn.

However we also have the dream of immortality, humans for all their capabilities die to young and to horribly when compared to other creatures. The desire to stop or limit this is understandable.

It’s why some parts of the AI community want to create not just AGI but imprint it with “all that is human”.

They see the transistor or similar as a way to get beyond the failings of organic cell systems.

Thus they are looking at how to imprint human traits and personalities onto vast arrays of transistors. Some I’m sure have the hope of transplanting their conscious onto such systems.

I suspect Mary Shelly would have rewritten Dr Frankenstein only very very slightly for the modern era of ScFi aficionados.

ResearcherZero • August 30, 2026 6:52 AM

@Clive Robinson

Mary Shelly’s Frankenstein thought himself unenlightened, as he lacked a formal education.
His relentless pursuit of knowledge inevitably cursed both himself and his creation.

“I had determined, once, that the memory of these evils should die with me; but you have won me to alter my determination. You seek for knowledge and wis-dom, as I once did; and I ardently hope that the gratification of your wishes may not be a serpent to sting you, as mine has been.”

We have already cursed lesser deities as daemon and assigned them repetitive tasks. Now we desire to create agents that never sleep and must assign themselves new tasks. Perhaps the endless analysis of number sets with infinite elements, to find new theories or patterns. And other pursuits where dangerous knowledge lurks and waits to be discovered.

It is fortunate that these machines do not yet comprehend desire or resentment.

Sami Liedes • August 30, 2026 3:53 PM

As a software engineer and data scientist first, mathematician distant second, the attributes you mention don’t strike me as peculiarly mathematical at all. In software world, I would make almost exactly the same distinction:

discover that an existing abstraction, algorithm or library solves an unexpected problem
combine known ideas in a way nobody happened to try
diagnose that an accepted assumption about a system is false

versus:

invent something like relational databases, virtual memory, Unix pipes, Git object model etc. where afterwards people think differently about the problem.

Now, I think it’s probably fair to say that LLMs are not really yet at the level where they’re able to do that last step unsupervised in software either, and you might argue they never will, although certainly I’d claim the trajectory of improvement in ability to tackle higher level problems has been dizzying. It may well hit a limit.

Likewise, “nobody competent tried to disprove this because everyone believed it’s true” has an extremely software-engineering flavor.

So, I’d argue that based on these observations, mathematics doesn’t feel that different from engineering, suggesting we could expect mathematics and software engineering proficiency of LLMs to develop (or not develop) roughly hand in hand.

Compare the shapes of work:

“We need to establish X. The obvious route would require Y, which seems inaccessible. Maybe strengthen the induction hypothesis. That exposes Z. Z resembles this known lemma, except the objects are wrong. Can we transform them? Introduce this auxiliary construction. Now three ugly obligations remain…”

and

“We need property X. The obvious architecture makes Y impossible. Change the interface so Y becomes local. That exposes Z as a synchronization problem. We already know how to solve something almost like Z, except ownership is wrong. Introduce an intermediate representation/cache/boundary…”

I do think there is one way in which mathematics probably does differ, though, that might give rise to a different trajectory. Software projects admit an enormous amount of graded evidence. Prototypes tell you aboutb feasibility. Something that works in 99% of cases is often very close to a fully working system. For a mathematical proof, it seems more likely that you spend three weeks developing a beautiful lemma before discovering that the lemma is irrelevant because the whole conceptual direction was wrong. Failure to prove using a specific approach is not much evidence against the approach.

I’d watch a space that sits between these: Can LLMs successfully write and formally verify programs and algorithms?

Clive Robinson • August 30, 2026 10:48 PM

@ ResearcherZero, Bruce, ALL,

With regards,

“Now we desire to create agents that never sleep and must assign themselves new tasks.”

It takes but a moment’s thought to realise this was inevitable, within the neo-con capitalistic view point on productivity.

Where under the “never leave money on the floor” mantra, you work something till it breaks, bodge it quick, and press it back into production untill it can nolonger be bodged and you take out what profit that gives you in the short term. And rather than reinvest in the process for long term steady profit you move it on to leverage more debt and destruction elsewhere…

Thus creating a tool to facilitate these short term views would absolutely be the way of all the AI organisations to “people please” the users to get them further hooked and thus increase profit under what I once described as a “Drug Dealer Business Model”.

Which brings us to the AGI+ model[1] they are currently unconstrained in action. Contrary to what SciFi writers have tried to imply for nearly a century we can see no way to give AI “morals”. I’ve given reasons for this as have others like Gary Marcus but the simple fact is that in their current primitive state “Current AI LLM and ML Systems” as implemented can not in anyway develope them.

Which is why we have the idea of surrounding LLM systems with “Guardrails” and as has been seen in practice they fail all to easily.

Worse I and others have proved that they will always fail due to the “observer problem”. Where even the simplest of encryption will slip by the guardrails but not the LLM.

Worse we’ve found that agentic systems are “goal focused” almost without constraint. Thus to “clip their wings” in lew of morals we have to “prohibit resources” to them. This alone is seen as “counter productive” to “aims and objectives” of the LLM system suppliers and operators as the little dust up with the US Dept of War ably demonstrated.

You could call the short term movement of Agentic systems “Strongly mission focused” and that realy does not bode well…

Which brings as to a point in the not to distant future where your point of,

“It is fortunate that these machines do not yet comprehend desire or resentment.”

Is in effect nolonger true “strongly mission focussed” encompasses the same external behaviours as “desire or resentment” without needing comprehension.

Which brings us in part to the “Genie in the bottle” and “the third wish” issues.

Few understand why the Genie is even “in a bottle”, that is the bottle is there to protect mankind from it’s self. And almost always,

“The third wish is to undo the first two.”

This unfortunately “fairy tale” though it is gives a very strong indicator of what unconstrained AI usage will bring.

Thus we need to design systems to,

“Protect users of AI from themselves and their wishes.”

How short a period of time it has taken us to get to the end of the Microsoft “Be Business Plan” for AI in Everything which is to “Betray” the user. Thus how long before we get to the “third wish” stage with Agentic AI?

If of course we are not already there and don’t yet realise it…

[1] You won’t here it with the “+” but as some idiots have said they have witnessed AGI and it is with simple analysis not in any way what they’ve claimed… We either have to assume AGI is a bust or it’s something future (with the idiot option still open). To see a more cautiously optimistic “push” on the “future” view,

https://www.ibm.com/think/topics/artificial-general-intelligence

“Artificial general intelligence (AGI) is a hypothetical stage in the development of machine learning (ML) in which an artificial intelligence (AI) system can match or exceed the cognitive abilities of human beings across any task. It represents the fundamental, abstract goal of AI development: the artificial replication of human-level intelligence in a machine or software.“

Actually I hope that last part is wrong… As Terry Pratchett once observed about cats,

“They are nasty buggers underneath”

Similar applies to most humans and as far as we can see most advanced Agenetic AI that is not watched.

ResearcherZero • September 3, 2026 5:13 AM

Three Russian mathematicians have taught AI models to communicate using mathematical symbols. This changes how AI models represent data and values when collaborating, improving how they share information with one another to accomplish tasks and the rate at which they can do it. The process is far more efficient and much cheaper, as the models avoid the additional burdens of understandable and legible language translation during computation.

https://www.linkedin.com/pulse/engineering-ai-telepathy-how-interlat-enables-agents-without-pooni-nvzwc

Researchers are working on methods to allow models to perform new tasks without training.
https://www.sciencedaily.com/releases/2024/03/240318142438.htm

Meta is developing an approach for improved reasoning and deduction based off neuroscience.
https://aifuturefront.com/metas-coconut-the-ai-method-that-thinks-without-language/

Alternative methods for AI models to approach problem solving are also being explored.
https://techxplore.com/news/2026-09-kind-ai-cheaply-words.html

Leave a comment

Blog moderation policy

Login

Allowed HTML <a href="URL"> • <em> <cite> <i> • <strong> <b> • <sub> <sup> • <ul> <ol> <li> • <blockquote> <pre> Markdown Extra syntax via https://michelf.ca/projects/php-markdown/extra/

Sidebar photo of Bruce Schneier by Joe MacInnis.