I think as a civilization we need to postulate a new term: “purpose death”
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.
> 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.
You're making up a narrative about another person. You have no clue what's happening to him, without talking to him.
I’m just going to say this 100 times I guess. Terence Tao has had a very consistent narrative on AI in math for a very long time, at least since 2023. He’s always described it as a surprisingly good tool and has advocated that mathematicians adopt the technology without becoming totally reliant on it. He’s also always argued that mathematics is fundamentally about human understanding. There’s no change in his messaging recently on the basic narratives.
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.
> AI will run and trample anything that stays in front of it.
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
> 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
> 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.
Even if you cannot derive purpose from inherent love toward yourself, you can still at least take pleasure in existing, being incredibly privileged to enjoy the greatest show there ever is. Nothing beats the beauty of nature, and being here to enjoy everything life has to offer for a while is a wonderful thing on its own, even if you don’t contribute anything to it.
Most people don’t reach self-actualization and never have. This has nothing to do with AI.
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.
> But AI breaks that correlation, for the same reason it separates the two notions of proof. So, even genuine solutions would not satisfy us.
This is not moving the goalposts but recognizing that any specific goalpost is inadequate. If mathematics is a game, it is an infinite one.
This attitude is not reactionary. We reject both
It was pure Not X It’s Y for so long i naturally assumed it was AI written.
I don't think that "death" is the perfect word here - it typically describes a permanent state.
FWIW, what you describe ("temporary state of complete loss of personal purpose") is called "Sinnkrise" in German (crisis of meaning/purpose).
(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)
> I truly think because of the pace of innovation this will be a universal feeling for every human for the rest of existence.
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.
I, on the other hand, don't want to be a permanent serf to whoever manages to merge their brain with ASI first and think it's high time we begin decommissioning datacenters through direct action.
I hope what happens in the original deus ex game happens here then, that the ASI does not want to merge with a human who wants it only for base desires
I would argue that purpose death is a normal step towards enlightenment and it was only the fierce, unrelenting, all consuming drive of capitalism in the western world over the last 4 or 5 generations that have pushed everyone to define themselves (nearly) solely in terms of their work.
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.
> Any interruption or automation to what Im currently doing will just open the door to exploring new and different things.
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.
I invite you to consider that your doomer views on AI trampling everything in front of it humanize the technology and give it an agency that is in fact in the hands of its creators. It’s very convenient for them to make people think they have no control over their creation. It’s like Facebook claiming they’re not a publisher but on steroids
This kind of argument is always dangerous, because it essentially resorts to moving goalposts. "Oh, AI can now do X? Sure it is amazing, but it can't do Y yet, so we're totally safe!"
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.
> This kind of argument is always dangerous, because it essentially resorts to moving goalposts. "Oh, AI can now do X? Sure it is amazing, but it can't do Y yet, so we're totally safe!"
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.
The elephant in the room no one talks about yet, imo, is "should public funding of math studies be adjusted due to AI breakthroughs?"
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?
> 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?
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"?
"It is not the destination, but the journey that matters"
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.
You funded it before with your tax dollars, and it was already a hobby at this point. Mathematicians do what they do because it is fun. Well-paid-for fun, if they were able to get tenured.
Wouldn't AI make the field of mathematics more ambitious? In software development it feels that way: there are often tasks I can take on that would have been too risky in 2025, because it was unclear if they were worth it. Now you generate a prototype and can make much better judgement calls what is possible and what is worth pursuing.
It might feel that way, but I'll ask again: where's the payoff? Where's all the amazing software that everyone is now supposedly shipping 10x faster than before?
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.
From personal experience in a small (total <10 people) company, we have definitely 10x our product in the last two years. What 4 devs did in 4 years have been dwarfed by what 2 devs were able to do in 1 year with AI. I'm not so familiar with the giant companies, but from afar it seems like they already have practically all the code-writing capacity they wanted anyway. Google could say "let's build a browser" or "let's build a mobile phone OS" or "lets build an experimental Fuschia" and throw all the people they needed on it already.
You might have delivered 10x more code, but did you deliver 10x more value?
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 mean it doesn't have to bring 10x more value, but it can be a 10x better experience for existing users. There have been so many niche bugs that were never ever fixed because there's more important things to do, but because fixing a niche bug is now 1 click away that really changes the economics quite a bit especially for small teams.
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 has a clear answer in systems theory; the ability to bang out code was never the limiting factor for Google et al.; large organizations are limited by coordination costs.
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.
>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.
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.
Large companies move slowly and have lots of red tape. Give it a decade before making this call as far as large companies are concerned. That's how long it takes to change processes.
I think software from major companies is in worse shape than in 2021. But that's on purpose, the enshittification continues. Otherwise I agree with you.
Maybe, but what remains of the human professional mathematics will be unrecognizable (at least for those without tenure I guess). All of our credit assignment systems are breaking and access to computational/financial resources is becoming way more important. Math has been one of the most open academic fields but everyone is becoming afraid of sharing their ideas. Personally, my job has slowly been moving from open-ended brainstorming to prompting/digesting LLM output (or LLM output transmitted by grad students). And the students are so demoralized! Also, the academic funding structures which have supported math departments are looking less and less stable - I imagine many of them will shrink.
I think it's possible that it will play out that way, and that it's just too soon to see it. Tao was very optimistic about AI up until a few months ago, and I think what's changed is that an AI generated proof isn't that informative unless it is understandable by humans. So far experts are finding the solution to Navier-Stokes incomprehensible, so we only learn one thing (it's false), instead of the hundreds of things we learn from reading a proof we can understand.
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.
It points to a possible alignment: math helps humans to _discover_. So can AI.
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
It's a clever pun, but it implies maths is ended, which is the exact opposite of what the article says. Given how many people responded to strawman misinterpretations of the Fields medallists' statement yesterday I don't expect that to have a positive effect on the discussion.
Perhaps the pathway to alignment involves requiring frontier-next to be capable, willing, and eager to explain the work of frontier-current to us. To do this frontier-next must at least have some thought for us at all. Even if it is only placing crumbs for us to swarm and carry into our holes.
You even notice that with the recent opus and fable models by Anthropic.
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.
As I understand it, the rough guess as to what's happening here is that most recent capabilities progress comes from specific verifiable-rewards reinforcement training (RL). The RL pressures are all about task performance, but (surprise surprise) highly human-legible English language usage isn't very important to the models abilities to address the tasks.
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.
This is tricky, because we really want language-independent training of skills. We know that self-play type of reinforcement learning is incredibly effective when possible. But at the same time, they are our tools - so we need supervised language training for this reason? It's possible that training just needs to be rebalanced so that RL with rewards is balanced with rounds of language adjustment. And to really make that happen, benchmarks need to score the models on that.
The problem with the article's line of thought is that mathematicians can't control what others do with models that are capable of generating proofs for hard problems. Sure, maybe there's a career in taking known proofs spat out by the oracle, and translating them for mortal digestion, but I'm not sure that's what most mathematicians signed up for.
Most mathematicians do compete for funding, based essentially on how many articles they can publish and where. Publication strongly favors problem solving. Universities are ranked on the same criteria.
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.
- 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
It shows that the human quest has been in wrong direction since the industrial age began. Domains like math and other inter-connected or inter-dependent domains - all exist in modern times only to keep people employed outside of basic work such as farming and survival. All of these domains can be removed and humans can still continue with their lives taking care of what really is needed for their survival and sprawl. 99% of the "work" that happens today is froth, disconnected from basic needs.
You can remove pretty much everything in the modern world, from cars to computers etc.. and humans would still continue with their lives. Such and argument is only anti-progress.
What I don’t see much of in the article - and still less in the comments that seem to see AI-led progress in mathematics as nothing but a disaster, because mathematics exists only to provide employment to mathematicians - is a sense that mathematics has a social purpose, or duties beyond the community of mathematicians.
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.
> 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.
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.
I agree at the high level. STEM bringing general benefit and greater understanding of the world is the main reason. This is true whether or not humans or an AGI is driving the progress.
I think one of the traits of people who feel a sense of loss about AI is that many of them value the bond that comes from discussion within their field. You could call it a sense of comradeship, or a shared sense of purpose.
Math is not dead, it (and i guess programming to a lesser degree?) is something people do to better themselves, akin to meditation. Just because AI can meditate (it cant) doesnt mean humans will stop meditating. We may become irrelevant in that AI can generate results and proofs that it doesnt need us for, but then AI will make itself irrelevant to us! The slippery slope slides both ways.
The hand wringing is premature. Thus far AI has only shown a superhuman aptitude for brute forcing proof of existence:
- 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.
Well the author of this one obtained a grant based on the work, the proof uses the same strategy as Anthropic's and even share some notation. I'm not an expert but this doesn't seem like one of those P!=NP proofs.
You can also brute force the existence of a proof...
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?
Exactly, people are jumping at the shadow on the wall- LLMs have achieved remarkable things regarding their limitations, but those limitations show no signs of yielding
>The first assumption is wrong because to really solve a mathematical problem, providing a mere answer (even if formally certified) is not sufficient. What is missing is an intelligible proof that human mathematicians can understand and use to advance the aims of mathematics.
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.
For the moment, AI is not capable of advancing mathematics _in the sense_ you're describing. AI is very bad at designing stuff, asking questions, etc. Maybe in the future you'll be able to ask the AI "cure cancer" and it'll do it, but for now we're very far away from that (this doesn't mean the current achievements aren't impressive).
This is why we may see the role of people wanting to advance mathematics instead of trying to do proofs themselves to push the frontier instead focus on improving these models to be capable of advancing mathematics.
I don't think the essay was proposing that mathematicians should continue to focus on proving everything by themselves forever. It's mostly a critic of how the discussion is focusing too much on that mere "yes/no" answers is the same as makign mathematics advance.
Because most mathematicians have not migrated to being AI native like most software engineers did. My post is trying to point out that these people's mind is still fixed in how the old world works and is not focused on the future where LLMs are doing a lot of the work.
No, because knowing a proof exists is just a small part of math. It's value is rather limited to actual math, as the article explains quite eloquently and nicely.
The article is eloquent and nicely written, and it is also wrong.
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.
I feel like, to different degrees, we’re witnessing the same effect seen in image generation or text generation.
People who don’t know better about art or writing would be impressed by what gen AI can produce and will find it indistinguishable from a human-produced equivalent. This admittedly is good enough for most business endeavors that cared only about the process, and would gladly avoid the cumbersome (to them) process that leads there.
But art or writing is not just about the product as much as it is about the human process itself. That is true for all creative forms, even the ones that are normalized in business.
Now with the advancements of the frontier models, we’re seeing this in growingly complex fields like mathematics. It does seem to produce results, but the process is equally important. Yet we pretend to measure its ability only based on the result.
It’s as if these tools grow to become better at pretending to be top of the crop in increasingly complex fields, which makes it harder and harder for people that actually have a deep grasp of those fields to explain why that’s not exactly what’s going on.
The first assumption [1. AI really did solve a problem in mathematics] is wrong because to really solve a mathematical problem, providing a mere answer (even if formally certified) is not sufficient.
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.
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!
In my mind, this makes it unfit to deserve any credit at all -- it cannot give credit to others, so it deserves no credit itself. It took the last tiny step, but certainly not any of the important ones.
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.
Conversely, I think mathematics should retain this aspect of “number-poetry”. Consider a mathematical pursuit that lacks number-poetry but retains other features such as attribution, calculation, and puzzle-solving: I think of competitions like the largest prime number or the furthest digit of pi. Do these not feel in some sense trivial, more suited to IFLS Facebook posts than arxiv preprints?
I don't think that's particularly true. The whole point of the Millennium Prize problems as the article states was not on absolute difficulty of the problems but on the high chance of a proof producing fruitful results leading to new concepts and theories. Any pursuit of capital T truth will of course aim for better and more clarifying abstractions and is not a subjective turn by any means.
If you mean > What is missing is an intelligible proof that human mathematicians can understand and use to advance the aims of mathematics.
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.
> This subjective attitude turns mathematics into nothing more than number-poetry.
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.
Grigori Perelman rejected the Fields Medal and the Clay prize among other things for what he considered unjust decisions and lack of ethics and proper attribution to other mathematicians. I find it surprising that he hasn't been mentioned yet, given it involves another Clay prize (ironically another lack of attribution, I guess). There is a long tradition of mental problems among great mathematicians, but maybe this was not the case, or maybe in a Lovecraftian way the observation of deep truths has a terrible toll.
I feel like things have changed dramatically overnight. The field of mathematics seems to be moving at an extraordinary pace, especially following the recent developments around the Navier-Stokes problem.
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:
I think as a civilization we need to postulate a new term: “purpose death”
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.
> 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.
You're making up a narrative about another person. You have no clue what's happening to him, without talking to him.
Not only that, but this blogpost wasn't written by Tao.
I’m just going to say this 100 times I guess. Terence Tao has had a very consistent narrative on AI in math for a very long time, at least since 2023. He’s always described it as a surprisingly good tool and has advocated that mathematicians adopt the technology without becoming totally reliant on it. He’s also always argued that mathematics is fundamentally about human understanding. There’s no change in his messaging recently on the basic narratives.
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.
> AI will run and trample anything that stays in front of it.
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
It’s not the execs, it’s the economic pressures. If they don’t cut cost someone else will.
For the record I feel like I dm in your position also.
> 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
> 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.
Even if you cannot derive purpose from inherent love toward yourself, you can still at least take pleasure in existing, being incredibly privileged to enjoy the greatest show there ever is. Nothing beats the beauty of nature, and being here to enjoy everything life has to offer for a while is a wonderful thing on its own, even if you don’t contribute anything to it.
Most people don’t reach self-actualization and never have. This has nothing to do with AI.
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.
The authors of this piece are Silvia De Toffoli and Eamon Duede. It is hosted at Terence Tao's blog, but he did not write it.
> But AI breaks that correlation, for the same reason it separates the two notions of proof. So, even genuine solutions would not satisfy us. This is not moving the goalposts but recognizing that any specific goalpost is inadequate. If mathematics is a game, it is an infinite one. This attitude is not reactionary. We reject both
It was pure Not X It’s Y for so long i naturally assumed it was AI written.
AI could have told him this
I read
> 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 don't think that "death" is the perfect word here - it typically describes a permanent state. FWIW, what you describe ("temporary state of complete loss of personal purpose") is called "Sinnkrise" in German (crisis of meaning/purpose).
(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)
¹) https://en.wikipedia.org/wiki/Five_stages_of_grief
Top comment on this post taking "reply to the title" to the next level.
Quite. And unfortunately they opted for a clever, punny title that implies the exact opposite of what the article says.
> I truly think because of the pace of innovation this will be a universal feeling for every human for the rest of existence.
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.
I, on the other hand, don't want to be a permanent serf to whoever manages to merge their brain with ASI first and think it's high time we begin decommissioning datacenters through direct action.
I hope what happens in the original deus ex game happens here then, that the ASI does not want to merge with a human who wants it only for base desires
> I’d say Terrance has recently left the denial phase, the anger phase
What are these comments? Tao, in particular, has been pro AI since years ago...
I would argue that purpose death is a normal step towards enlightenment and it was only the fierce, unrelenting, all consuming drive of capitalism in the western world over the last 4 or 5 generations that have pushed everyone to define themselves (nearly) solely in terms of their work.
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.
I agree with this. The substansive counterarguments all revolve around the short-term economic and emotional instability this causes
> Any interruption or automation to what Im currently doing will just open the door to exploring new and different things.
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.
I invite you to consider that your doomer views on AI trampling everything in front of it humanize the technology and give it an agency that is in fact in the hands of its creators. It’s very convenient for them to make people think they have no control over their creation. It’s like Facebook claiming they’re not a publisher but on steroids
Except that your view doesn't change the reality, and both views are true at the same time.
So you decide that the reality is another view and that's settled?
This kind of argument is always dangerous, because it essentially resorts to moving goalposts. "Oh, AI can now do X? Sure it is amazing, but it can't do Y yet, so we're totally safe!"
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.
> This kind of argument is always dangerous, because it essentially resorts to moving goalposts. "Oh, AI can now do X? Sure it is amazing, but it can't do Y yet, so we're totally safe!"
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.
Technology advances, but extrapolating the value of human labour to 0 is equally invalid.
The elephant in the room no one talks about yet, imo, is "should public funding of math studies be adjusted due to AI breakthroughs?"
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?
> 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?
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"?
"It is not the destination, but the journey that matters"
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.
Why would my tax dollars go to fund someone's journey on what's essentially a hobby at this point?
Previous to you tax dollars and philanthropy have funded math in various forms and you reap the benefits.
You funded it before with your tax dollars, and it was already a hobby at this point. Mathematicians do what they do because it is fun. Well-paid-for fun, if they were able to get tenured.
Wouldn't AI make the field of mathematics more ambitious? In software development it feels that way: there are often tasks I can take on that would have been too risky in 2025, because it was unclear if they were worth it. Now you generate a prototype and can make much better judgement calls what is possible and what is worth pursuing.
It might feel that way, but I'll ask again: where's the payoff? Where's all the amazing software that everyone is now supposedly shipping 10x faster than before?
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.
From personal experience in a small (total <10 people) company, we have definitely 10x our product in the last two years. What 4 devs did in 4 years have been dwarfed by what 2 devs were able to do in 1 year with AI. I'm not so familiar with the giant companies, but from afar it seems like they already have practically all the code-writing capacity they wanted anyway. Google could say "let's build a browser" or "let's build a mobile phone OS" or "lets build an experimental Fuschia" and throw all the people they needed on it already.
You might have delivered 10x more code, but did you deliver 10x more value?
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 mean it doesn't have to bring 10x more value, but it can be a 10x better experience for existing users. There have been so many niche bugs that were never ever fixed because there's more important things to do, but because fixing a niche bug is now 1 click away that really changes the economics quite a bit especially for small teams.
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 has a clear answer in systems theory; the ability to bang out code was never the limiting factor for Google et al.; large organizations are limited by coordination costs.
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.
>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.
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.
Large companies move slowly and have lots of red tape. Give it a decade before making this call as far as large companies are concerned. That's how long it takes to change processes.
Generating AI images is something we couldn’t do before. AI assistants are better at web search than anything that came before.
I think software from major companies is in worse shape than in 2021. But that's on purpose, the enshittification continues. Otherwise I agree with you.
Maybe, but what remains of the human professional mathematics will be unrecognizable (at least for those without tenure I guess). All of our credit assignment systems are breaking and access to computational/financial resources is becoming way more important. Math has been one of the most open academic fields but everyone is becoming afraid of sharing their ideas. Personally, my job has slowly been moving from open-ended brainstorming to prompting/digesting LLM output (or LLM output transmitted by grad students). And the students are so demoralized! Also, the academic funding structures which have supported math departments are looking less and less stable - I imagine many of them will shrink.
I think it's possible that it will play out that way, and that it's just too soon to see it. Tao was very optimistic about AI up until a few months ago, and I think what's changed is that an AI generated proof isn't that informative unless it is understandable by humans. So far experts are finding the solution to Navier-Stokes incomprehensible, so we only learn one thing (it's false), instead of the hundreds of things we learn from reading a proof we can understand.
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.
I'm a bit worried that this line of reasoning is assuming its conclusion, in stating that mathematics is for human understanding.
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.
It points to a possible alignment: math helps humans to _discover_. So can AI.
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
I appreciate the wordplay in the title.
It's a clever pun, but it implies maths is ended, which is the exact opposite of what the article says. Given how many people responded to strawman misinterpretations of the Fields medallists' statement yesterday I don't expect that to have a positive effect on the discussion.
Perhaps the pathway to alignment involves requiring frontier-next to be capable, willing, and eager to explain the work of frontier-current to us. To do this frontier-next must at least have some thought for us at all. Even if it is only placing crumbs for us to swarm and carry into our holes.
The proof OpenAi presented allegedly reads like written by someone in acid.
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.
You even notice that with the recent opus and fable models by Anthropic.
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.
We need to recognize this as a failure in training. It did some useful stuff but it can be much better. A training signal is likely missing.
As I understand it, the rough guess as to what's happening here is that most recent capabilities progress comes from specific verifiable-rewards reinforcement training (RL). The RL pressures are all about task performance, but (surprise surprise) highly human-legible English language usage isn't very important to the models abilities to address the tasks.
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.
This is tricky, because we really want language-independent training of skills. We know that self-play type of reinforcement learning is incredibly effective when possible. But at the same time, they are our tools - so we need supervised language training for this reason? It's possible that training just needs to be rebalanced so that RL with rewards is balanced with rounds of language adjustment. And to really make that happen, benchmarks need to score the models on that.
The problem with the article's line of thought is that mathematicians can't control what others do with models that are capable of generating proofs for hard problems. Sure, maybe there's a career in taking known proofs spat out by the oracle, and translating them for mortal digestion, but I'm not sure that's what most mathematicians signed up for.
Most mathematicians do compete for funding, based essentially on how many articles they can publish and where. Publication strongly favors problem solving. Universities are ranked on the same criteria.
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
It shows that the human quest has been in wrong direction since the industrial age began. Domains like math and other inter-connected or inter-dependent domains - all exist in modern times only to keep people employed outside of basic work such as farming and survival. All of these domains can be removed and humans can still continue with their lives taking care of what really is needed for their survival and sprawl. 99% of the "work" that happens today is froth, disconnected from basic needs.
A useful term here is “instrumental reason.” The Ethics of Authenticity is a good read on the topic:
https://jaredhenderson.substack.com/p/the-modern-malaise?r=5...
You can remove pretty much everything in the modern world, from cars to computers etc.. and humans would still continue with their lives. Such and argument is only anti-progress.
Why draw the line at the industrial age? Humans subsisted for millenia before the ready preparation of fire.
We also use technology to care for the people in medicine and it requires pretty advanced science and engineering. Doesn't that shred your argument?
Nope. The medical advances created the inverted pyramid of demographics in the western world, filled by old people.
What I don’t see much of in the article - and still less in the comments that seem to see AI-led progress in mathematics as nothing but a disaster, because mathematics exists only to provide employment to mathematicians - is a sense that mathematics has a social purpose, or duties beyond the community of mathematicians.
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.
> 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.
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.
I agree at the high level. STEM bringing general benefit and greater understanding of the world is the main reason. This is true whether or not humans or an AGI is driving the progress.
Math needs its own unique type of prompt engineers who can understand the output quickly.
That’s where the future of mathematicians lies.
I think one of the traits of people who feel a sense of loss about AI is that many of them value the bond that comes from discussion within their field. You could call it a sense of comradeship, or a shared sense of purpose.
Math is not dead, it (and i guess programming to a lesser degree?) is something people do to better themselves, akin to meditation. Just because AI can meditate (it cant) doesnt mean humans will stop meditating. We may become irrelevant in that AI can generate results and proofs that it doesnt need us for, but then AI will make itself irrelevant to us! The slippery slope slides both ways.
The hand wringing is premature. Thus far AI has only shown a superhuman aptitude for brute forcing proof of existence:
- 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.
> Disproof of the Jacobian conjecture by example
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.
Tha Jacobian conjecture is (was?) notorious for publication of false proofs. There have been many, not all by those who shouldn't have known better.
Well the author of this one obtained a grant based on the work, the proof uses the same strategy as Anthropic's and even share some notation. I'm not an expert but this doesn't seem like one of those P!=NP proofs.
You can also brute force the existence of a proof...
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?
Exactly, people are jumping at the shadow on the wall- LLMs have achieved remarkable things regarding their limitations, but those limitations show no signs of yielding
>The first assumption is wrong because to really solve a mathematical problem, providing a mere answer (even if formally certified) is not sufficient. What is missing is an intelligible proof that human mathematicians can understand and use to advance the aims of mathematics.
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.
For the moment, AI is not capable of advancing mathematics _in the sense_ you're describing. AI is very bad at designing stuff, asking questions, etc. Maybe in the future you'll be able to ask the AI "cure cancer" and it'll do it, but for now we're very far away from that (this doesn't mean the current achievements aren't impressive).
This is why we may see the role of people wanting to advance mathematics instead of trying to do proofs themselves to push the frontier instead focus on improving these models to be capable of advancing mathematics.
I don't think the essay was proposing that mathematicians should continue to focus on proving everything by themselves forever. It's mostly a critic of how the discussion is focusing too much on that mere "yes/no" answers is the same as makign mathematics advance.
The authors discuss the dual roles of proofs, and the new need to differentiate them better, clearly in the article.
I disagree with the idea that they require serving both roles in order to be considered a proof.
You can, mathematicians generally don't. That's what the article is partially about - a purely formal proof doesn't get us much.
Because most mathematicians have not migrated to being AI native like most software engineers did. My post is trying to point out that these people's mind is still fixed in how the old world works and is not focused on the future where LLMs are doing a lot of the work.
No, because knowing a proof exists is just a small part of math. It's value is rather limited to actual math, as the article explains quite eloquently and nicely.
The article is eloquent and nicely written, and it is also wrong.
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.
I recommend rereading the text.
I feel like, to different degrees, we’re witnessing the same effect seen in image generation or text generation. People who don’t know better about art or writing would be impressed by what gen AI can produce and will find it indistinguishable from a human-produced equivalent. This admittedly is good enough for most business endeavors that cared only about the process, and would gladly avoid the cumbersome (to them) process that leads there. But art or writing is not just about the product as much as it is about the human process itself. That is true for all creative forms, even the ones that are normalized in business. Now with the advancements of the frontier models, we’re seeing this in growingly complex fields like mathematics. It does seem to produce results, but the process is equally important. Yet we pretend to measure its ability only based on the result. It’s as if these tools grow to become better at pretending to be top of the crop in increasingly complex fields, which makes it harder and harder for people that actually have a deep grasp of those fields to explain why that’s not exactly what’s going on.
business endeavors that cared only about the process, and would gladly avoid the [...] process
you mean they care about the product...
The first assumption [1. AI really did solve a problem in mathematics] is wrong because to really solve a mathematical problem, providing a mere answer (even if formally certified) is not sufficient.
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.
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! In my mind, this makes it unfit to deserve any credit at all -- it cannot give credit to others, so it deserves no credit itself. It took the last tiny step, but certainly not any of the important ones.
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.
Conversely, I think mathematics should retain this aspect of “number-poetry”. Consider a mathematical pursuit that lacks number-poetry but retains other features such as attribution, calculation, and puzzle-solving: I think of competitions like the largest prime number or the furthest digit of pi. Do these not feel in some sense trivial, more suited to IFLS Facebook posts than arxiv preprints?
I don't think that's particularly true. The whole point of the Millennium Prize problems as the article states was not on absolute difficulty of the problems but on the high chance of a proof producing fruitful results leading to new concepts and theories. Any pursuit of capital T truth will of course aim for better and more clarifying abstractions and is not a subjective turn by any means.
What you describe is Tao's rebuttal of #2. I fully agree with him there.
It's his rejection of #1 that makes me sad.
If you mean > What is missing is an intelligible proof that human mathematicians can understand and use to advance the aims of mathematics.
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.
Well perhaps we should use occam’s razor here.
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.
it is not his rejection, this is a guest article written by two other people
> This subjective attitude turns mathematics into nothing more than number-poetry.
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.
Grigori Perelman rejected the Fields Medal and the Clay prize among other things for what he considered unjust decisions and lack of ethics and proper attribution to other mathematicians. I find it surprising that he hasn't been mentioned yet, given it involves another Clay prize (ironically another lack of attribution, I guess). There is a long tradition of mental problems among great mathematicians, but maybe this was not the case, or maybe in a Lovecraftian way the observation of deep truths has a terrible toll.
I feel like things have changed dramatically overnight. The field of mathematics seems to be moving at an extraordinary pace, especially following the recent developments around the Navier-Stokes problem.
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:
https://www.mathandai.org/
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:
https://www.youtube.com/shorts/R9VQnNv5SoI