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.
> 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.
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.
> 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.
I’m glad you found something that works well for you. For me, nihilism is much like ice cream: I don’t want it in my fridge - not because I think it’ll taste bad, but because I think it’s gonna taste real good.
> 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
You might also be going though purpose death which is an emotional concern related to thriving, but I think your bigger emotional concern is related to survival. This answer is the same as ever - race to financial freedom. Live under your means. Invest wisely. Survive till you hit your milestones.
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.
Since OpenAI is undoubtedly also under economic pressure, why is it still hiring humans for Account Associates, Android Engineers, Data Scientists instead of demonstrating its AI prowess by automating those roles?
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)
Thanks for the German word. I can’t read most of the literature, but initial glance it does indeed seem the same.
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
> and that mathematicians need to find out where they want to go
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.
> 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 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.
Literally no person on earth can beat a computer in chess. Practically no person on earth can really teach a computer anything about chess. For some crazy reason, people keep playing
So then I can spend time with my partner, my loved ones, reading more sci-fi, being fitter and working on improving myself and others around me? Sounds horrid
I think you’re likely right about the path to enlightenment. I love the wisdom you’ve found. The rest of us need some time to get there, in a likely rolling wave of realization that keeps getting reset across generations.
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
Look, I could be wrong. That said, I think AI is both in some conversations a tool and in some conversations it’s short hand for “Capitalistic Acceleration”. With each iteration of RSI, this second context is increasingly the more important mental model.
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.
But this is mathematics, modern mathematics had it's purpose death decades if not centuries ago. People got into it over the last 60-100 years for autistic reasons, now that AI has got rid of those reasons you'd have to be even more autistic to entertain the idea. But plenty of people are, and more tech = more mental illness so it'll probably be a wash.
> I’d say Terrance has recently left the denial phase, …
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.]
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
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"?
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.
Fundamentally, the question is this: what is a large language model, what is it capable of, how does it differ from human cognition, and what can humans do that it cannot do?
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.
If you ask an AI if "P=NP", it will certainly give an answer. It has to, because the answer is either "yes" or "no".
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.
That's why this proof and all the other high profile AI proofs were written in lean, which allows you to verify them in a formal deterministic language. You still might not understand it as a human, but any computer with a simple processor can verify the proof's correctness. And from there you will undoubtedly see other people make sense of the proof's key steps using AI too. And the models might even pick up on further details useable for other proofs that humans didn't see. In the end I'm 100% convinced that abstract math will eventually be primarily done by computers, similar to how linear algebra and numerics have been done exclusively done by computers for a while. Noone would even consider multiplying a 100x100 matrix by hand anymore, if only because it is much more likely that you as a human will make a mistake.
That's assuming that "formal deterministic language" is equivalent to mathematical reality, or that it even reflects it correctly. It's the accepted view nowadays but akshually quite the hot take.
> This kind of argument is always dangerous, because it essentially resorts to moving goalposts.
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.
> 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
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.
> 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.
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.
Again, I've heard that countless times on HN. Every time you ask, everyone is 10xing it and building amazing things that will go to market very soon now. But it's been a while; where is it? Somehow, everyone is just sitting on all these revolutionary advances while selling the exact same stuff as they were selling before, with the same warts and the same annoyances. Gmail is the same, Chrome is the same, Photoshop is the same, Slack is the same, Excel is the same... the only thing that seems to have changed is how SWEs feel about their jobs.
To be fair, what revolution were you expecting to see, exactly? You can 10x a "dashboard but with AI" app all you want but it's only ever going to be a "dashboard but with AI" app. And I note that nobody seems to be claiming that the AI helped them create a brand new kind of app.
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.
I'm not a huge fan of AI, but there _are_ examples of mostly-AI generated software that are being used by real people:
- 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.
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.
That "for now" has been going on for ~10 months. That was enough for some startups to make a splash in the pre-AI era; now that they can move 10x faster, shouldn't we be seeing evidence left and right? All the contenders stealing lunch from Big Tech by offering better products or exploring new frontiers?
>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.
"After Math" seems premature. Maths is probably infinite meaning there will always be more to discover.
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.
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.
"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.
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.
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
>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.
This is exactly backwards. Mathematics predates the idea of formal proof by millenia. The purpose of proofs since Euclid is to explain to your fellow human why something is true. The idea that the purpose of math is formal proof alone is a new idea that (some) computer programmers want to impose on the field (for the understandable reason that it makes computers primary).
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.
I broadly agree with your sentiment and am saddened (outraged?) to see the financially motivated cheapening of (destruction of?) what mathematics truly is. I agree that formal logic is merely a model of one aspect of what mathematicians do, in the same sense that a computer simulation of a roller coaster can never bring the same value to us as the real thing.
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.
The idea that logic is a branch of mathematics is itself a modern notion. Aristotle's logic was part of the trivium (grammar, rhetoric, and logic) in classical notions of education, while mathematics made up several parts of the quadrivium (arithmetic, geometry, music, and astronomy).
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.
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.
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.
Yes, it is sufficient. To say otherwise is just goalpost shifting because you don't like the answer.
There are, in fact, two notions of proof: a logical
notion and an intelligible notion.
Wrong again, the only real notion of proof is logical. It is not even possible to say exactly what an "intelligible" notion would be, because that would make it logical.
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.
There is a point in distinguished a formal proof from an intelligible one, notwithstanding the fact that one can, with effort, be extracted from another. That's a key point of the discussion that we mathematicians are having now, and fits into the bigger question as to what it means to gain mathematical understanding. In fact, Tao happens to have some older posts about this himself. The naive formalist reduction that you are making is not really bringing us forward in having a useful reflection about this, as the discussion is already beyond that.
> There is a point in distinguished a formal proof from an intelligible one,
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.
Sad to see such smart people with such small visions for the future.
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.”
Like… surely I don’t need to explain why artificial minds are a unique invention?
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.
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.
Old timey mining, or the more romantic eras of seafaring also provided more camaraderie than what modern jobs provide. There may be some longing for those more pure and valiant days, but we moved on. You can still discuss about AI assisted results with your peers. I guess it's like how people miss the days of heated arguments about factual questions whereas the smartphone era led to someone just pulling out their phone, googlimg and settling the question. Or heated discussion about Maradona's handplay, and today we have slomo VAR replay. Better at the task, takes away from the romance and the spirit. It's like beautiful winged hussars in the military vs people in an office piloting drones in VR and soon just prompting the AI piloting the drones. I get it. But this should be talked about head on, not just about petty academic career stuff, which of course matter to those involved but are not a society wide issue.
This is incredibly related to how people seek meaning in family, friends, or (hardly foolproof) new experiences. Because careers, skills, and other society-wide abstract purpose no longer provide a stable identity based on monthly guarantees of dopamine.
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 examples of foregoing the use of known tech, like the Amish, but it really means a deep break with the western ethos and takes something like a deep committing to a religion to carry out. The default is that you work on things with the honest goal of making it work. Yes, we've become too good at making tech
But a balance of camaraderie in honest effort, but not quite as much to be really successful is a knife's edge. In a way, solving all the problems in the textbook may be a sad moment, despite the fact that working on it hard meant that it would happen if you're competent. If you want to try hard but fail perpetually, like Sysyphus, you have to be intentionally incompetent. Like always try to solve your math problems when drunk. Then you won't get sad that there are no more problems because you will not be able to solve them.
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.
Sisyphus is apt because he was "punished" by a superhuman entity for doing basically the right thing (from a mortal's pov). Doesn't it remind of alpoge, who tried to keep it as traditional as was possible? Nice one!
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.
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 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
The academic system developed in the last ~50 years will crumble. On the broad scale of human intellectual history, this was a blip and not even the most productive phase. But it provided mass office employment and so this is all much more about the general knowledge work job replacement issue than anything specific to math. All desk/computer/office jobs will face the same fate. Math is just easier to verify. But engineering design, architecture, lower levels of lawyers and accountants where the selling point isn't charm and connections, they will all have this moment soon.
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.
It is curious that the academics discuss their own narrow issues and are completely silent on the broader issues of copyright violations, energy usage etc.
What copyright violation? Training is free use. Anthropic was fined for torrenting books, not for training. The price of legally obtaining those books was peanuts too. Energy usage has to be compared to the value of the output.
> Terence Tao (2026) lists many goals of mathematics beyond problem solving. These include developing new theories and techniques, understanding the world, sustaining a community, training the next generation of mathematicians, contributing to cumulative knowledge, and creating works of aesthetic value. Of course, these have been positively correlated with genuine solutions.
> 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.
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.
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
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?
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.
>> If future AI systems will produce genuine proofs, logically correct and intelligible, like those produced by “master” mathematicians, it would still be incorrect to think that mathematics would have been “solved” as some say that chess or Go have been solved.
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].
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.
> 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
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.
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.
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.
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.
Yes, I am making up a narrative. That’s why I used words like “postulate”, “think”, “I wonder”, and “I may be wrong”.
As a reader, it’s very off-putting. Spinning a story about someone, without first verifying whether it is at all close to the truth.
I trust the opinions of people that tend to be close to the mark. By shooting wildly, you’ve lost my trust.
I hope one day I can re-earn it. All the best!
Not only that, but this blogpost wasn't written by Tao.
And he's called Terence, not "Terrance".
Oh dang… sorry that I fucked that up.
The Tao is eternal. The game plays the game.
And the whole stages of grief model was just made up anyway.
[dead]
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.
> He’s also always argued that mathematics is fundamentally about human understanding.
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.
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.
AI could have told him this
> 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.
> 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.
I’m glad you found something that works well for you. For me, nihilism is much like ice cream: I don’t want it in my fridge - not because I think it’ll taste bad, but because I think it’s gonna taste real good.
I'd rather compare it to Linux. You are in charge of your own destiny.
I like that!
> 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
You might also be going though purpose death which is an emotional concern related to thriving, but I think your bigger emotional concern is related to survival. This answer is the same as ever - race to financial freedom. Live under your means. Invest wisely. Survive till you hit your milestones.
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’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.
Since OpenAI is undoubtedly also under economic pressure, why is it still hiring humans for Account Associates, Android Engineers, Data Scientists instead of demonstrating its AI prowess by automating those roles?
https://openai.com/careers/search/
To use the humans to generate training data for the AI to learn from.
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
Thanks for the German word. I can’t read most of the literature, but initial glance it does indeed seem the same.
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
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.
> and that mathematicians need to find out where they want to go
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.
> 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 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.
Literally no person on earth can beat a computer in chess. Practically no person on earth can really teach a computer anything about chess. For some crazy reason, people keep playing
Sure, but play is different from work.
Oh wow, that’s a lovely way to say it!
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.
So then I can spend time with my partner, my loved ones, reading more sci-fi, being fitter and working on improving myself and others around me? Sounds horrid
I think you’re likely right about the path to enlightenment. I love the wisdom you’ve found. The rest of us need some time to get there, in a likely rolling wave of realization that keeps getting reset across generations.
I agree with this. The substansive counterarguments all revolve around the short-term economic and emotional instability this causes
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?
Look, I could be wrong. That said, I think AI is both in some conversations a tool and in some conversations it’s short hand for “Capitalistic Acceleration”. With each iteration of RSI, this second context is increasingly the more important mental model.
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.
> 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...
But this is mathematics, modern mathematics had it's purpose death decades if not centuries ago. People got into it over the last 60-100 years for autistic reasons, now that AI has got rid of those reasons you'd have to be even more autistic to entertain the idea. But plenty of people are, and more tech = more mental illness so it'll probably be a wash.
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’d say Terrance has recently left the denial phase, …
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.]
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
The blog post was not written by Terry Tao :)
That is the definition of an existential crisis, btw. I like the neologism too, tho
This goal post stuff is tiresome.
The only thing it reveals is how poor humans are predicting and defining things.
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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"?
> The elephant in the room
...is the fact that the people pontificating on AI and math the most are also the people who understand AI and math the least.
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.
Fundamentally, the question is this: what is a large language model, what is it capable of, how does it differ from human cognition, and what can humans do that it cannot do?
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.
The authors are absolutely not saying "we are totally safe". They are not denying that AI will affect math deeply in the short and the long run.
If you ask an AI if "P=NP", it will certainly give an answer. It has to, because the answer is either "yes" or "no".
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.
That's why this proof and all the other high profile AI proofs were written in lean, which allows you to verify them in a formal deterministic language. You still might not understand it as a human, but any computer with a simple processor can verify the proof's correctness. And from there you will undoubtedly see other people make sense of the proof's key steps using AI too. And the models might even pick up on further details useable for other proofs that humans didn't see. In the end I'm 100% convinced that abstract math will eventually be primarily done by computers, similar to how linear algebra and numerics have been done exclusively done by computers for a while. Noone would even consider multiplying a 100x100 matrix by hand anymore, if only because it is much more likely that you as a human will make a mistake.
That's assuming that "formal deterministic language" is equivalent to mathematical reality, or that it even reflects it correctly. It's the accepted view nowadays but akshually quite the hot take.
Strongly reminiscent of God of the Gaps [1]
[1]: https://en.wikipedia.org/wiki/God_of_the_gaps
> This kind of argument is always dangerous, because it essentially resorts to moving goalposts.
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
>introspecting motivation is useful
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.
> 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
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.
How is this theoretical university math department funded? Which college sports team or endowment is backing it? What do they want?
> 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.
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.
Again, I've heard that countless times on HN. Every time you ask, everyone is 10xing it and building amazing things that will go to market very soon now. But it's been a while; where is it? Somehow, everyone is just sitting on all these revolutionary advances while selling the exact same stuff as they were selling before, with the same warts and the same annoyances. Gmail is the same, Chrome is the same, Photoshop is the same, Slack is the same, Excel is the same... the only thing that seems to have changed is how SWEs feel about their jobs.
To be fair, what revolution were you expecting to see, exactly? You can 10x a "dashboard but with AI" app all you want but it's only ever going to be a "dashboard but with AI" app. And I note that nobody seems to be claiming that the AI helped them create a brand new kind of app.
Anyway, Sturgeon's law and all that.
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.
With all due respect when you strip away the noise - unless it shows up in the financials it hasn’t done anything for the firm.
This is something many firms are struggling to grasp.
The world does not revolve around money, specifically non profits come to mind where this can be a huge boost in quality for everyone involved.
I'm not a huge fan of AI, but there _are_ examples of mostly-AI generated software that are being used by real people:
- 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.
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.
That "for now" has been going on for ~10 months. That was enough for some startups to make a splash in the pre-AI era; now that they can move 10x faster, shouldn't we be seeing evidence left and right? All the contenders stealing lunch from Big Tech by offering better products or exploring new frontiers?
That’s not true at all.
The constraint is taste and vision.
You can’t buy either.
>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.
Beautiful post.
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.
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.
All vibe-coded software is complete unusable trash. There are no exceptions.
So no, we're not advancing, we're in fact regressing. The crash and disillusionment of the post-vibe-code world will be immense.
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.
"After Math" seems premature. Maths is probably infinite meaning there will always be more to discover.
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.
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.
"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.
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...
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
Math needs its own unique type of prompt engineers who can understand the output quickly.
That’s where the future of mathematicians lies.
>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.
This is exactly backwards. Mathematics predates the idea of formal proof by millenia. The purpose of proofs since Euclid is to explain to your fellow human why something is true. The idea that the purpose of math is formal proof alone is a new idea that (some) computer programmers want to impose on the field (for the understandable reason that it makes computers primary).
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.
I broadly agree with your sentiment and am saddened (outraged?) to see the financially motivated cheapening of (destruction of?) what mathematics truly is. I agree that formal logic is merely a model of one aspect of what mathematicians do, in the same sense that a computer simulation of a roller coaster can never bring the same value to us as the real thing.
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.
The idea that logic is a branch of mathematics is itself a modern notion. Aristotle's logic was part of the trivium (grammar, rhetoric, and logic) in classical notions of education, while mathematics made up several parts of the quadrivium (arithmetic, geometry, music, and astronomy).
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.
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.
It is not that nicely written.
Maybe, but it's not worth discussing the content if you dont comprehend what the authors are actually saying.
Here is what they are saying:
Yes, it is sufficient. To say otherwise is just goalpost shifting because you don't like the answer. Wrong again, the only real notion of proof is logical. It is not even possible to say exactly what an "intelligible" notion would be, because that would make it logical.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.
There is a point in distinguished a formal proof from an intelligible one, notwithstanding the fact that one can, with effort, be extracted from another. That's a key point of the discussion that we mathematicians are having now, and fits into the bigger question as to what it means to gain mathematical understanding. In fact, Tao happens to have some older posts about this himself. The naive formalist reduction that you are making is not really bringing us forward in having a useful reflection about this, as the discussion is already beyond that.
> There is a point in distinguished a formal proof from an intelligible one,
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.
Sad to see such smart people with such small visions for the future.
Like… surely I don’t need to explain why artificial minds are a unique invention?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.
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.
Old timey mining, or the more romantic eras of seafaring also provided more camaraderie than what modern jobs provide. There may be some longing for those more pure and valiant days, but we moved on. You can still discuss about AI assisted results with your peers. I guess it's like how people miss the days of heated arguments about factual questions whereas the smartphone era led to someone just pulling out their phone, googlimg and settling the question. Or heated discussion about Maradona's handplay, and today we have slomo VAR replay. Better at the task, takes away from the romance and the spirit. It's like beautiful winged hussars in the military vs people in an office piloting drones in VR and soon just prompting the AI piloting the drones. I get it. But this should be talked about head on, not just about petty academic career stuff, which of course matter to those involved but are not a society wide issue.
This is incredibly related to how people seek meaning in family, friends, or (hardly foolproof) new experiences. Because careers, skills, and other society-wide abstract purpose no longer provide a stable identity based on monthly guarantees of dopamine.
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 examples of foregoing the use of known tech, like the Amish, but it really means a deep break with the western ethos and takes something like a deep committing to a religion to carry out. The default is that you work on things with the honest goal of making it work. Yes, we've become too good at making tech But a balance of camaraderie in honest effort, but not quite as much to be really successful is a knife's edge. In a way, solving all the problems in the textbook may be a sad moment, despite the fact that working on it hard meant that it would happen if you're competent. If you want to try hard but fail perpetually, like Sysyphus, you have to be intentionally incompetent. Like always try to solve your math problems when drunk. Then you won't get sad that there are no more problems because you will not be able to solve them.
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.
Sisyphus is apt because he was "punished" by a superhuman entity for doing basically the right thing (from a mortal's pov). Doesn't it remind of alpoge, who tried to keep it as traditional as was possible? Nice one!
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.
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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.
How many people are employed to meditate all day?
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
The academic system developed in the last ~50 years will crumble. On the broad scale of human intellectual history, this was a blip and not even the most productive phase. But it provided mass office employment and so this is all much more about the general knowledge work job replacement issue than anything specific to math. All desk/computer/office jobs will face the same fate. Math is just easier to verify. But engineering design, architecture, lower levels of lawyers and accountants where the selling point isn't charm and connections, they will all have this moment soon.
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.
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It is curious that the academics discuss their own narrow issues and are completely silent on the broader issues of copyright violations, energy usage etc.
What copyright violation? Training is free use. Anthropic was fined for torrenting books, not for training. The price of legally obtaining those books was peanuts too. Energy usage has to be compared to the value of the output.
> Terence Tao (2026) lists many goals of mathematics beyond problem solving. These include developing new theories and techniques, understanding the world, sustaining a community, training the next generation of mathematicians, contributing to cumulative knowledge, and creating works of aesthetic value. Of course, these have been positively correlated with genuine solutions.
> 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.
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.
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
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?
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.
>> If future AI systems will produce genuine proofs, logically correct and intelligible, like those produced by “master” mathematicians, it would still be incorrect to think that mathematics would have been “solved” as some say that chess or Go have been solved.
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.
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.
> 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
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.
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.
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.
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.
Why draw the line at the industrial age? Humans subsisted for millenia before the ready preparation of fire.
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