After Math
terrytao.wordpress.com
terrytao.wordpress.com
Defined something like: temporary state of complete loss of personal purpose and the experience of existential dread from never achieving self-actualization in spite of the tremendous time commitment towards excellence in a now automated intelligence.
I truly think because of the pace of innovation this will be a universal feeling for every human for the rest of existence.
As a software engineer, I myself have only recently recovered from it. So, it’s really interesting to watch a prominent figure in their industry publicly go through “purpose death” and the related grief. It’ll be a useful case study to re-read his written meditations through this cycle.
I’d say Terrance has recently left the denial phase, the anger phase I’m sure he wisely kept off the Internet, and is currently in the bargaining phase - ie scrambling to change the goal posts. I wonder if he will wisely keep the depression / burnout phases also off the internet.
However, soon as the goalposts keep falling, I think like most humans he will accept, retool, and come out of this grief with renewed purpose with larger expectations of himself and mathematics. This recent post even starts towards some of that - but sadly is slightly off the mark.
“The important question is, therefore, not whether AI will defeat mathematicians, but which mathematical ends we want AI to serve.”
He still thinks there is controlling AI. AI will run and trample anything that stays in front of it. He needs to one day find acceptance in letting AI run while he learns how to suggest it minor course corrections which it may or may not accept, and when it doesn’t accept quickly learn from the AI why he was right or wrong.
I maybe wrong, but I think this is the cycle of “purpose death” we will all have to contend with in our own time.
Maybe reread my comments in both contexts.
Capitalistic acceleration will most definitely keep trampling on anything in front of it. Because, the humans with capital will steer it that way.
Eg OpenAI steering the “AI the tool” at Navier Stokes on a rumor (inaccurate as we found out) of Anthropic having solved it.
It was pure Not X It’s Y for so long i naturally assumed it was AI written.
I do not at all feel purpose death from AI (been a software developer professionally for 20 years, now a founder), but I consider myself a lifelong learner, with infinite curiosity, in a universe with infinite challenges. Any interruption or automation to what Im currently doing will just open the door to exploring new and different things. This doesnt come from an immediate desire to do anything different, but having the confidence that whatever comes along, I will figure it out and have a lot of fun doing so.
But that's the rub: At some point, possibly sooner than most people think, there won't be anything left that isn't automated. Any human trying to do something new will then be worse at achieving it, and likely even detrimental. Just like the monetary value of horse labor today isn't just small, it's actually negative.
Purpose rebirth is the re-finding of play in the thing we choose to partner with AI (likely with a paycheck attached) to do N hours a week.
What are these comments? Tao, in particular, has been pro AI since years ago...
Why would it be? Imagine a man who is a student in a kollel in Kiryas Joel. He spends his day studying the Torah, Tanach, Mishnah, Talmud, the Mishneh Torah, the Shulchan Aruch, the Zohar, etc. Then at night he goes home to his wife and 12 kids.
Do you think he experiences "purpose death"? Do you think his wife does? Do you think his children will? Do you think AI is going to make them start?
If anything, AI might make his lifestyle more economically sustainable than it was before – if nobody works because AI has taken all the jobs, and everyone gets paid universal basic income, he is no longer faced with the arduous struggle of supporting a large family as a full-time student.
And there's nothing specific to Judaism about this – I'm sure in some seminary in Qom, you'll find the Usuli Twelver Shi'a analogue.
> The important question is, therefore, not whether AI will defeat mathematicians, but which mathematical ends we want AI to serve.
and had the opposite interpretation as I see you having. To me, it reads as the authors [1] acknowledging, as you put it, that
> AI will run and trample anything that stays in front of it.
and that mathematicians need to find out where they want to go:
> Rather, it is an opportunity to clarify what mathematics is all about. We should ask again what we are after when we do mathematics.
This surely does involve purpose death, but also purpose rebirth.
[1]: Silvia De Toffoli and Eamon Duede, instead of Terence Tao, although I imagine Terence endorses the message.
I think you’re right that mechanically I’m saying the same thing as they are saying “use AI to build your dreams”.
I think the psychology is the difference. Specially, “oh it can now do that too” has to be met with “finally” rather than “no way! That was supposed to be human only”.
eg “what are we after?” Is also likely a question that will be solved by AI in N years after mathematicians chip at it for all that time.
Any and every singular focus will be eliminated. There will be more tho to work on - that’s where the purpose rebirth is. I think.
Maybe said differently, I think one day these same authors will write the post - “more of math is handled by AI. It doesn’t matter. We will work on the infinite amount of it that’s left. Whatever that happens to be, even when we are surprised it wasn’t what we tried to control it to be.”
That will be acceptance. That we’ll be purpose rebirth.
With regards to technology specifically, new tech has been making high-investment skills useless for the last 500+ years. The printing press, the loom, etc. This is not anything new.
Some of these AI doomers really need to read more than AI Substacks and Twitter feeds. I suggest a book about the history of technology.
It is perfectly reasonable to suggest that we make an effort to direct a technology in certain directions. It is not reasonable to throw all rational thought out the window and operate as if real world AI is synonymous with science fiction.
> this will be a universal feeling for every human for the rest of existence
As someone that has accepted a very long time ago that nihilism is the only self-consistent philosophy, I've made peace with the fact that humanity has no purpose and therefore disagree with your conclusion. I haven't been driven to absurdism, either.
Instead, I derive purpose and self-worth from within.
You're making up a narrative about another person. You have no clue what's happening to him, without talking to him.
I trust the opinions of people that tend to be close to the mark. By shooting wildly, you’ve lost my trust.
(And because not everybody might know: the comment references Kübler-Ross' five stages of grief (when confronted with a tragic outlook)¹: denial → anger → bargaining → depression → acceptance)
FWIW I never enjoy naming things lol. I agree death and temporary seem at odds. “Purpose Loss” didn’t have the same gusto. I mostly riffed the vocabulary and definition off “Ego Death” https://en.wikipedia.org/wiki/Ego_death
Right now that entirely depends on the whims of a few people. We can try to make it not be that but I don't have much hope in that area. See climate change
> As a software engineer, I myself have only recently recovered from it
Can I ask you how you came here? Right now I am getting closer to "purpose death" as you call it. Software engineering is one of the few things I am good at and the only thing that I can rely on to put food on my table. Even if I end up being someone who gives minor suggestions to the AI, it always feels like C level execs can't wait to get rid of me
For the record I feel like I dm in your position also.
Meanwhile, I really don’t believe the narrative about fewer jobs.
As long as you’re curious, as long as you continue learning, the new tools and focus on the parts of your current job that involve judgment you will continue putting food on your table as long as you are alive.
Use any and every tool you have to save the people around you time, and the C level will continually fight to keep you.
When AI gets good at something you are good at - great use the time (accelerated by using AI for self help) to find another thing you’re good at the AI isn’t yet. Make sure you have 3-4 things you’re good at at any given time so as AI gets good you have redundancy.
It is not right to project one’s grief onto others and then characterize all their actions through this lens! (This post is attributed to other people and doesn’t sound like his voice by the way, which highlights the problem with doing so.) And it’s especially wrong to pose what looks like a therapy diagnosis on a stranger in public as a non-expert.
It's also worth pointing out he didn't come up with this viewpoint just to "cope" with the headlines. Thurston articulated this way back in the 90s (https://arxiv.org/pdf/math/9404236) and also more recently (https://mathoverflow.net/questions/43690/whats-a-mathematici...):
> The product of mathematics is clarity and understanding. Not theorems, by themselves. Is there, for example any real reason that even such famous results as Fermat's Last Theorem, or the Poincaré conjecture, really matter? Their real importance is not in their specific statements, but their role in challenging our understanding, presenting challenges that led to mathematical developments that increased our understanding.
There’s a byline at the top of the article that lists the authors of this blog post, neither of which are Terry Tao.
> [This is a guest post by Silvia De Toffoli and Eamon Duede. This blog post was initially written in a different file format and converted using AI. — T.]
The only thing it reveals is how poor humans are predicting and defining things.
The sports comparison is wrong here because general public never paid for the specific match results. The value was always in the show, the advertising and betting around it, the health and educational value of doing sports, etc. And the sport mostly lives on what it earns, not on public funding.
Math, on the other hand, was paid for because people and states believed progress in math might lead to meaningful improvements in other branches of science, and, in turn, in our lives. If this is better served by AI, should we keep paying for the same number of tenure positions? Should we increase their number to handle the speedup brought by AI? Or decrease because they're being replaced? Should we pay more to those using AI to do their research, or to those explaining and exploring the AI-generated results?
That is a pretty huge "if". The ultimate purpose of mathematics and really all scientific inquiry is human understanding of the natural world. It's not at all clear whether large language models can replace that any more than calculators can replace human mastery of arithmetic. Is society ready for engineers to design bridges and airplanes without understanding the underlying mathematics by simply handing off the entire process to a black box "AI architect"? Are people ready to ingest drugs "vibe-designed" by human drones pushing buttons on an "AI drug discovery" machine and just going "meh, seems about right!"
It might also be useful to take a step back from the hype that the frontier labs are obviously incentivised to incite. Before speculating about how "AI math" capabilities might supplant cutting-edge research in mathematics and other sciences, take a look at OpenAI's own job postings (https://openai.com/careers/search/?). Why doesn't OpenAI demonstrate its world-beating AI capabilities by automating more routine roles like "Account Associate", "Systems Architect" or "Android Engineer"?
...is the fact that the people pontificating on AI and math the most are also the people who understand AI and math the least.
Just because humans can't process the proof or see the advancements, it doesn't mean that it will remain that way in the future or that it will not change the field. If you only define yourself by things AI can't do yet, you're about to have a rude awakening. We've gone from high school, to university math, to Euler problems all the way to Millennium problems in a time frame most people couldn't even do a PhD. If you start any math research now with a horizon beyond the next two years, I'd be terrified of the current rate of progress.
It's not about moving goalposts. What you're not understanding is that, even if for you is crystal clear that AI will be 100x smarter tomorrow, those in charge cannot simply bet all on that. Right now AI cannot really replace the core sauce of mathematicians (all the "understanting" and "asking the real questions" stuff), so it'd be unwise for, say, countries to start making decisions as if AI is capable of that.
A lot of people seem to believe that with more time and training, LLMs will surpass human intelligence. But they are fundamentally not like human intelligence. They do not reason, learn, conceptualize, think creatively or abstractly, even though we have some hacks to mimic these behaviors.
I posit that treating LLMs like GPU-powered brains that will eventually surpass us in most fields is pure science fiction if you know anything about how they work. I think they will remain astronomically powerful in some areas (pertaining to fuzzy deep search and recombining existing knowledge) and hilariously bad in others.
Since we lack a definition of all these things or a clear understanding of how they are realized in humans or LLMs, the only thing you can really claim is that LLMs do not look like biological neural networks when it comes to processing information at the fundamental level. Everything else is dogma. But since all this stuff clearly emerges as size of the network grows, we can not ignore the possibility that complexity itself is the only requirement, not architecture (hardware or software). At the most basic levels, we are also just machines that evolved to make more of themselves and learned to interact with the environment to achieve that goal more efficiently. Humans will certainly have to come to terms with the fact that we are not the pinnacle of intelligence in the universe, but that won't stop them from denying it.
Nope, introspecting motivation is useful. There have been similar posts made about people using bots in PvP competitive online gaming: some people play to enjoy the game, and hopefully get better until they reach their skill ceiling. Others play (just)to win, and are open to buying aimbot hardware and software
Not if the essence boils down to moving goalpoasts to the next best thing that will likely be flown past as well within months rather than years. Sure mathematicians who do this stuff purely for the love of it will continue to exist. Just like human chess or the retro-programming scene exist for similar reasons. But if you are a professional who expects to earn a living salary on these things, your intellect will suddenly be outcompeted by orders of magnitude. Ignoring this today by saying LLM's can't do this or that will undoubtedly set you up for a rude awakening. For all we know is that there is still no fundamental limit to what they can do in theory and compute may well be the only true limit.
Which organizations do the most math research? What are their budgets, droves and culture? You may be falling victim to the pre-IPO PR razzle-dazzle by AI companies expending unreasonable resources on problems they know will get them good press and make their models appear superhuman.
A university math department will not spend $20M in one summer to solve 1 problem when it can hire 10 human mathematicians for 10 years.
If you ask it to then prove its assertion, it will then certainly output the tokens that look like a plausible proof. It has to, as it is programmed too.
This "proof" might even be thousands of pages of very technical looking and professional sounding jargon.
It's not a real proof though, and as an artifact it is 100% useless to both the field of mathematics and humanity.
If I look at the software I'm actually using day-to-day, or that my friends are using, all this stuff looks exactly the same as it did in 2021. Not a single product release from Google, Microsoft, or more scrappy companies in the past 6 months made me go "wow, they couldn't have pulled that off before". All the vibecoded "Show HN" projects seem to be half-broken and then abandoned before being finished.
It feels like we've gotten less ambitious, not more. Because yes, you can prototype more easily, but this means less commitment to what we create.
Mathematics is probably the same way. There's a short-term rush when you pull the lever, but there's less desire to get invested in what comes out.
Or put another way, are you making 10x more money?
It's easy to spend excess productivity effectively wasting time. Most companies did it before AI, and will continue doing it after.
It's easy to believe you are not wasting time because you have more bugs fixed or more features delivered, but if it doesn't move the bottom line, what is the point?
I'm personally working on a product solo that would not be possible without a team of 3-4 people that are all knowledgeable in that field and would take at least 1 year to get it started. It only took me 3 months to get a fully working solution with $1200 worth of AI subscriptions, the value multiplication is nearly x100 here.
This is something many firms are struggling to grasp.
Anyway, Sturgeon's law and all that.
It's the small teams you need to pay attention to, the organizations that really were limited by engineering capacity. These are by definition also less visible - for now.
The constraint is taste and vision.
You can’t buy either.
Perhaps it will take until the next generation to come along to really embrace the new AI-assisted way of doing mathematics. The current generation has too many reservations.
Especially the implicit trade off. - if it becomes ‘easy’ to produce little thought and care is put into it.
Personally I saw this coming ages ago.
After all we live off the contributions of the few in relative terms.
No surprise - most humans don’t have much imagination or creativity.
- Nourish is a nutrition coach/food diary that is recommended by dieticians. It's mostly AI generated, but logging food is easy even if it's not correct (it's better to log and have it be slightly off than to not log at all --- I've been doing it for almost 20 years now)
- Cronometer isn't AI generated but its photo logging feature, which I use heavily, definitely uses a vision model to guess at what you're eating. This is SUPER STUPIDLY HELPFUL when I'm out with friends at, let's say a Korean BBQ joint, and don't have the time to log everything that I'm eating as I eat it. (The right thing to do is pre-plan your meal, but this isn't always possible.)
- Feeling Good! is a CBT therapy app that was written mostly by one of the creators of CBT with Claude. My therapist recommended it to me recently. I haven't used it yet but I'll find a way to.
- The creator of SparkPeople is re-releasing it as an almost completely AI-generated platform written by Claude. He's super upfront about this (on the landing page, in the privacy policy, in the onboarding docs), which I highly respect, but I actually hit him up on LinkedIn because the landing page was on Vercel and looked so suspect.
- Boris has said multiple times that Claude Code is increasing writing/maintaining Claude Code, which I deeply respect as a person who's a fan of compilers compiling themselves (like Go, which has since 1.4).
- Have a look at /r/apple on Reddit on a Sunday or even here on "Show HN" threads. Most of the stuff coming out is vibeslop, but some of what's being published looks like solutions to real problems.
So no, we're not advancing, we're in fact regressing. The crash and disillusionment of the post-vibe-code world will be immense.
Maybe this is a one-off, or maybe in a few weeks we'll figure out how to get AI to explain the proof in terms we can understand. Then math research will accelerate. But maybe it's not a one-off, and by this time next year we will have an oracle that just answers all of our questions, but in such a way that we don't even know what questions to ask anymore. Then AI will just mop up the existing and the subject will end.
Regarding AI getting better than humans, in some ways it may be more like the invention of the microscope - instead of worrying about it reducing demand for people with good eyes it may open up new discoveries.
One example I'm kind of excited for is that I've long suspected that the inability to combine quantum mechanics with general relativity is because the maths gets too hard for humans. I mean Navier Stokes on singularities in fluid flow in euclidean space is hard enough. Particles may be singularities in quantum fields in relativistic space time which is probably mathematically far harder. But it would be interesting to know how that works.
you mean they care about the product...
For now, AI will be another tool in the toolbox of mathematicians. With humans driving the conversation to help understand the world better. If/when AGI is reached maybe we won’t be in the driver seat as much. I don’t think that matters. The goal of math is to achieve greater understanding of the world, regardless if humans are driving or if an AI is.
This is not something that's just true on its own; it's on us as a species to make sure that it remains that way, for the sake of our dignity.
However, this position will be extremely difficult to defend, against the economic value of not caring.
You're right though; the economic and emotional impact of OpenAI's executives are much more obvious today. In comparison, the actual implications of Navier-Stokes blow-ups on the epistemological landscape will take at least two years to become obvious to any intelligence, if ever.
I'm optimistic that the Joe in the street will soon be able to make earth shattering discoveries with the help of AI, at least once a week, when universal basic tokens come to pass. Then the nihilists will also keep their existential or otherwise depression to their AI therapists. Hopefully then the puritan billionaires can only helplessly compare the profligate blooming of a thousand new avenues to unrestrained moral depravity
That’s where the future of mathematicians lies.
This makes a bad assumption that humans need to be the one to advance the aims of mathematics. LLMs could be what advances the aims of mathematics and we just have to worry on making it so LLMs can digest these proofs.
>Nevertheless, if it turns out that what OpenAI has provided is a mere answer
It has a proof attached. Saying that it "doesn't provide understanding" does not invalidate that there is a formal proof. It fundamentally is trying to expand the requirements of proof to be something more than is required.
To say that "1. AI really did solve a problem in mathematics." is wrong, is just wrong. No matter how nice and eloquent you try to explain it afterwards.
1. AI really did solve a problem in mathematics.
The first assumption is wrong because to really solve a
mathematical problem, providing a mere answer (even if
formally certified) is not sufficient.
There are, in fact, two notions of proof: a logical
notion and an intelligible notion.
I would be very surprised if Tao couldn't extract a proof that he understands within a few months out of the formal proof. I would compare this to understanding a difficult proof in some math journal.
In the end, all of this forth and back does not matter. A logical proof is a perfectly fine answer, and given that AI can provide that now, this is the future.
Fine, do that if you want. What you cannot do is call a logical and formal proof of a problem not solving the problem in mathematics. That is absurd, and mathematicians that insist on this are absurd, too. Note that Tao didn't say this, but two guests.
I understand that you might want more than a formal proof generated by a machine, just like I want fries with my steak. But saying that the steak is not food, is just ridiculous.
You are not further along in the discussion, you are having the wrong discussion. What is at stake here is nothing less than the question: Is mathematics subjective or objective? This mathematician thinks it is objective.
Formal proof only emerged early in the 20th century, and the standard became that in theory a proof should be formalizable to answer any skepticism, but the real goal in Euclid's time and ours has been to communicate why a theorem is true to your fellow humans. There were a few theorems that are only known via computer proof, like the Four Color Theorem, but this has always been regarded as disappointing or even controversial, and the fact that there hasn't been any conceptual breakthrough has meant that we didn't learn anything other than the sheer fact that the Four Color Theorem is true. Theorems that produce understanding, on the other hand, typically produce many new ideas that lead to more theorems.
The purpose of scholarship is understanding. This is just as true for science as it is for math. If AI produces a unified theory of fundamental physics, but it's just an opaque blob, physicists will find it just as unsatisfying.
However, I would be remiss if I didn't question your historical claim, which seems to me a bit too strong:
> Mathematics predates the idea of formal proof by millenia [...] Formal proof only emerged early in the 20th century [...]
You seem to associate the start of "Mathematics" with Euclid, but (as far as I know) he worked at approximately the same time as Aristotle. Aristotle's syllogisms are perhaps the most famous formal logic system: their correctness relates only to their form, not their content. All deductions of the form "All X are Y, All Z are X, hence All Z are Y" are valid (assuming the premises are), regardless of the meanings of X, Y, and Z. (Outside of Greece, my understanding is that a few hundred years earlier Panini had also developed a system of formal manipulations, but for representing grammars.)
What, to my understanding, "emerged" only the 19th and 20th century was 'merely' a formal logic both expressive and sound enough to properly express modern mathematics (the Beggriffsschrift in the 19th century and FOL+ZFC in the 20th). Between Euclid and the 19th century the development of calculus was probably the biggest advance in mathematics, and my understanding is that Leibniz himself spent significant time working on formal logic.
Perhaps I have the wrong definition in mind of 'formal logic' or 'mathematics,' but I do think the history of formal logic is much more closely tied to the history of mathematics than your post makes it seem on first glance. Though I certainly agree that "mainstream mathematics" has never felt it necessary (or necessarily that useful) to express proofs in a formal logic carefully enough that they could be checked by computers; this was a fringe focus of a minority group of mathematicians and computer scientists that was co-opted as a marketing stunt into 'what mathematics is' for major corporations trying to justify their money burn.
Of course in retrospect we can see that the syllogistic part of Aristotle's logic can be formalized (as can grammar), but it was viewed as part of language or philosophy. I get the impression that a lot of more traditional philosophers of logic hated the formal turn.
Leibniz anticipated the turn towards formalism, but he didn't publish any of it in his life and it wasn't rediscovered until the 20th century.
If you have some sources to read more about this:
> I get the impression that a lot of more traditional philosophers of logic hated the formal turn.
I would be very interested! (Don't doubt it at all, just sounds like an interesting period I don't know as much about as I'd like.)
It will take probably a while before we will get a translation into something that than will actually have a positive impact.
That could be either a second proof or a streamlined version of the AI one.
If you give them a wide open problem statement, they'll start talking a lot of semi intelligible gibberish.
My guess is that this happens because that's not what they are evaluated on anymore for these kinds of tasks. The generated code is evaluated (in this case the lean code). So talking a bit of gibberish in the language part so you have more test time compute is not punished.
Weirdly enough, the pressures are having them drift toward novel dialects of English that work well for their own chains of thought. Open question about whether they'd drift all the way to a new language given enough time.
In a sense, programming (et tu, mathematics) has become a bullshit job. Though, it will take some time (~6 months) before "fulltime" investors or serial SaaS founders start to lose that high of shared purpose when celebrating around the campfire how they each surprised themselves with an artisanal leverage of "network effects" that one time..
Those of us still trying to extract personal purpose by commiserating on quirks of fable or astra in this warm-colored forum..
May want to contemplate the personal (but unrelated to ozempic or pharmaceuticals) meaningfulness of "month-scale network effects" next
There are warning examples in religion and myth, like Ikarus, the Tower of Babel etc that striving for more capability is not always good. Of course these religions are dismissed by modern people. Well, this is the result.
Then there were the brothers Epimetheus and Prometheus. One was a thief and one was implicated in the opening of Pandora's box. Who is an inspiration, who was punished? (With the caveat that it should have been climate-neutral electricity he stole, not fire.)
So the heuristic seems to be: if you come up on the wrong side of AI, you did the right thing. Ikarus was not wrong to try.
In mathematics, it is more natural to treat AI as an assistant rather than as a competitor. As Jeremy Avigad (2026) puts it, “We should keep in mind that AI is nothing more than technology, designed to serve our purposes. It is misguided to think of mathematicians as competing with AI; when we drive a car, we aren’t competing to see who can go faster, and when we use a phone, we aren’t competing to see who can speak louder.”
What we see is a panic reaction to the fact that problem solving is "easy", which affects the future of the management of mathematics, not of mathematics itself.
Mathematicians are not luddites afraid of AI, is the academic publishing industry mixed with management interests speaking here.
If you look at the Leiden Declaration https://leidendeclaration.ai/ then you notice two weird IMO facts:
- that it was stirred by the International Mathematical Union Committee on Publishing
- that is a mixture of the older San Francisco Declaration on Research Assessment DORA https://sfdora.org/read/ and recent fear of commercial AI competition
> If, instead, mathematicians treat AI as a technology for advancing its long-standing and centrally human purposes, the technology may come to contribute to an accelerated flourishing and enrichment of the discipline.
For the longest time, people who conducted research (not just math) ranked and measured their peers (albeit quasi-subjectively) by 1.) their ability to solve difficult problems, 2.) the cleverness of their solutions, and 3.) the clarity of their explanations.
With the advent of these models, we're understandably worried because #1 and maybe #2 may be gone (maybe even #3).
Are researchers going to shift the ranking/measuring of their peers over to ... "so-and-so is a world-class explainer of AI proofs"?
If you go around telling people, "no no no. You don't understand. The important part of our intellectual work is now going to be <something that was never weighted highly>", well, then it feels like you're handing me a participation trophy and telling me I came in 1st place.
I have no idea how the next 5 years will shake out, but I think that's why there's a lot of anxiety at the moment.
- Disproof of the Jacobian conjecture by example
- Construction of a non-sofic group
- Existence of singularity in Navier-Stokes
Mathematical conjectures tend to be universally quantified, especially those conjectures that are used as building blocks (e.g. RH). If anything, AI models are currently performing a useful service by disproving false conjectures, a kind of mathematical weeding.
The good news from the last couple of years of coding agents is that while models have become more persistent and knowledgable, their creativity (defined as being able to escape their training distribution and synthesize completely novel ideas) is improving at a much slower rate.
AI will only become a threat to mathematics if/when it develops the capability for creative big-picture problem solving. If that happens, the impact on mathematics will be a footnote compared to the impacts on society at large, since creativity unlocks a host of new economic capabilities.
It should be noted that there was a manuscript, available online since the beginning of 2025, with a solution to the Jacobian conjecture:
"Adrian Vasiu claims that the 7 page AI paper on the 3D Jacobian conjecture counterexample used notation and concepts from a draft of a paper jointly written with Alexander Borisov and Ofer Gabber, dated to January 14, 2025 and made publicly available on January 16, 2025."
The extract is from wikipedia, where the sources are given.
It is true that the biggest splashes have been from counterexamples and such, but I have seen many smaller examples of proving positive results in my own field.
Perhaps not superhuman yet but sufficient to completely upend the status quo. And I'm supposed to believe that this won't change in a year? or two?
This subjective attitude turns mathematics into nothing more than number-poetry.
That would reduce mathematics to something very pathetic.
Focus instead on attribution. Yes, OpenAI took the last tiny step in the process of solving this problem (= proving it). But it cannot attribute credit to all the mathematicians whose chat logs from the past few months were fed into its training data. Unlike a human, it can't even remember where it learned things from! For many theorems, I can still recall which exposition was the one that "sank in" for me (often not the first one!) a decade after grad school.
In my mind, this makes current LLMs unfit to deserve any credit at all -- they cannot give credit to others, so they and their owners deserve no credit themselves. OpenAI's LLM took the last tiny step, but not any of the important ones.
It's his rejection of #1 that makes me sad.
Then that is not necessarily subjective either if an AI can produce an actually intelligible proof. The problem is that as mathematicians with PDE expertise have mentioned on Twitter the actual solution seems to devolve into an unreadable mess focusing on irrelevant details after a more readable first few pages in the proof. If it wasn't a Lean compiled proof and presented as a human artifact, it would be hard to assess if the deviser of the solution had any actual understanding of the solution.
If it doesn’t seem like the lllm understands the proof, it probably doesn’t.
Just because it generated something that works doesn’t mean it understands it, and the evidence from its proof is that it does not, which is not very surprising given the technology we’re talking about, which generates likely phrases based on a corpus and training.
Yes I fear a lot of doom and gloom around AI is unearned and only really serves to prop up the valuation of AI companies. It's still very much unclear how much work OpenAI actually did versus just copying the nearly complete homework of someone 5 minutes earlier.
I don't think you understood the point. The point applies to all of basic science. You of course want some explanation supporting the raw answer, so you can use that insight in other contexts.
Chess and Go have most definitely not been solved. In AI research when we say that a game (and usually we mean a traditional board game) has been "solved", what we mean is that we can correctly predict the outcome of any game instance from any board position provided both players play perfectly [1]. This can mean we have an algorithm that chooses the perfect play for each player in any position or that we have mapped the entire game tree for that game: every move in every game, from start to finish.
There are games that have been solved this way: tic-tac-toe; connect four; nim; othello, checkers. But not chess and Go.
Rather, what we have for chess and Go are super-human engines, i.e. computer players that no human has ever beaten in the full game. So for example a chess engine like Stockfish can always choose what move to make in order to win against a human player in any board position. However, that is still not the same as knowing how to play perfectly in any board position. Like the old joke with the two guys trying to outrun a bear, a chess engine doesn't have to play perfect chess to be super-human: it just has to play better than any human.
To the extent that chess and Go are games it could be said that winning is the whole point of AI game-playing [2]. But the article above suggests that there is no point of "winning" in maths. In AI chess it is unfortunate that the development of super-human chess engines has ended all research in using AI chess to understand the way that human players play chess (which is to say: not by alpha-beta minimax, or Monte Carlo Tree Search, or anything like that) which was the original reason that luminaries of the field such as Shannon, Minsky, McCarthy, Michie, and even Turing himself were interested in the question in the first place. The concern then for mathematics is that AI "winning" at maths will kill mathematics research and replace it with ... as McCarthy would have it "very fast fruit flies" [3].
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[1] https://en.wikipedia.org/wiki/Solved_game
[2] Then again see John McCarthy's criticism of that idea written in the wake of DeepBlue vs Kasparov: https://www-formal.stanford.edu/jmc/newborn/newborn.html
[3] ibid.
I thought the reason we were encouraging people to go into STEM, and providing clever people with large salaries, was that mathematical results were of practical value to the wider human race. In which case, it’s surely very good news that AI can get those results.
Isn’t all this handwringing just the equivalent of Hackney cab drivers bemoaning the advent of the motor car? Phrased in much fancier language, of course, because the people involved are cleverer and more articulate.
The impact of mathematics on other disciplines goes much further than specific results. Just as important, if not even more so, are the language and ways of thinking that typically come out of the process of establishing those results. Results without the accompanying conceptual understanding are about as useful as a mere oracle for math theorems.
Another point that seems to be frequently missed or mischaracterized is that mathematicians are not opposed to computational tools as a matter of principle and in fact do use them when they help their research. The current controversy is not about a hypothetical future where mathematicians have easy access to open-source, open-weight, auditable natural-language assistants to help them internalize a new result or search for counterexamples. It's primarily about the recent behavior of certain for-profit companies suddenly trying to disrupt mathematics by caricaturing it in the public eye as a game they can "solve" or "beat" for headlines and valuation.
What makes you think so? I can think of lots of specific results that helped e.g. economists - convex optimization or supermodularity, say - but I don’t think the broader way of thinking of mathematicians has had any input into economics for the past fifty years, and arguably nor should it - the discipline can think for itself. Similarly, sociologists can use linear regression, or geneticists can partition variance, without having to think like mathematicians.
https://jaredhenderson.substack.com/p/the-modern-malaise?r=5...
25 Field Medalist and 5000+ mathematicians from leading institutions around the world endorsed an open letter expressing concerns about the impact of AI on mathematics:
More than 1,900+ mathematicians have also shown concern over the Caltech Mathathon:
https://docs.google.com/document/d/1IL0b2oG2KvvSnxn_DuXsNxuH...
James Maynard, a Fields Medalist, has also publicly expressed concerns about the implications of AI for mathematics: