8 ms·

After Math

154 points by throwaway81523 20d ago

sp-ti-en-ma-in 20d ago
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/ https://www.mathandai.org/ More than 1,900+ mathematicians have also shown concern over the Caltech Mathathon: https://docs.google.com/document/d/1IL0b2oG2KvvSnxn_DuXsNxuHaCefDJ9hNY1k9BP3kKQ/edit?tab=t.0 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 https://www.youtube.com/shorts/R9VQnNv5SoI
octoberfranklin 20d ago
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.
- traject_ 20d ago
  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.
  - octoberfranklin 20d ago
    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.
    - traject_ 20d ago
      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.
      - grey-area 20d ago
        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.
    - auggierose 19d ago
      it is not his rejection, this is a guest article written by two other people
- deleted 20d ago
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- Pulcinella 20d ago
  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.
  - deleted 20d ago
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- adalacelove 20d ago
  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.
- fwlr 20d ago
  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?
- robotpepi 20d ago
  > 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.
stabbles 20d ago
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.
- matherial 20d ago
  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.
  - tpm 20d ago
    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.
  - doctor_phil 20d ago
    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.
yazaddaruvala 20d ago
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.
- camillomiller 20d ago
  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
  - rnewme 20d ago
    Except that your view doesn't change the reality, and both views are true at the same time.
niemandhier 20d ago
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.
- magimas 20d ago
  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.
  - kzrdude 20d ago
    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.
    - NiloCK 20d ago
      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.
      - kzrdude 20d ago
        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.
camillomiller 20d ago
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.
- em-bee 20d ago
  business endeavors that cared only about the process, and would gladly avoid the [...] process you mean they care about the product...
HedonicEscal8r 20d ago
I appreciate the wordplay in the title.
- omnicognate 20d ago
  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.
zarzavat 20d ago
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.
- robotpepi 20d ago
  > 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.
  - contubernio 20d ago
    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.
    - robotpepi 20d ago
      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.
charcircuit 20d ago
>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.
- robotpepi 20d ago
  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).
  - charcircuit 20d ago
    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.
    - robotpepi 20d ago
      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.
- anonymouz 20d ago
  The authors discuss the dual roles of proofs, and the new need to differentiate them better, clearly in the article.
brador 20d ago
Math needs its own unique type of prompt engineers who can understand the output quickly. That’s where the future of mathematicians lies.
coderenegade 20d ago
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.
- encyclopediai 20d ago
  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/ 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/ https://sfdora.org/read/ and recent fear of commercial AI competition
  - bonoboTP 19d ago
    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.
zkmon 20d ago
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.
- keiferski 20d ago
  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=5dwulj&utm_medium=ios https://jaredhenderson.substack.com/p/the-modern-malaise?r=5...
- pdhborges 20d ago
  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?
  - zkmon 20d ago
    Nope. The medical advances created the inverted pyramid of demographics in the western world, filled by old people.
- alansaber 20d ago
  Why draw the line at the industrial age? Humans subsisted for millenia before the ready preparation of fire.
- peacefullmind 20d ago
  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.
globalnode 20d ago
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.
- bonoboTP 19d ago
  How many people are employed to meditate all day?
jdw64 20d ago
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.
- bonoboTP 19d ago
  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.
  - oliculipolicula 19d ago
    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
    - jdw64 19d ago
le-mark 20d ago
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.
- slickytail 20d ago
  [dead]
svara 20d ago
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.
- oliculipolicula 20d ago
  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
sigmoid10 20d ago
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.
- alansaber 20d ago
  Technology advances, but extrapolating the value of human labour to 0 is equally invalid.
- robotpepi 20d ago
  > 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.
- stabbles 19d ago
  Strongly reminiscent of God of the Gaps [1] [1]: https://en.wikipedia.org/wiki/God_of_the_gaps https://en.wikipedia.org/wiki/God_of_the_gaps
- kaffekaka 19d ago
  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.
- archagon 19d ago
reisse 20d ago
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?
- curt15 20d ago
  > 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/ 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"?
- otabdeveloper4 19d ago
  > 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.
dash2 20d ago
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.
- justinparus 20d ago
  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.
- curt15 19d ago
  > 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.
justinparus 20d ago
"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.
- moralestapia 20d ago
  Why would my tax dollars go to fund someone's journey on what's essentially a hobby at this point?
  - justinparus 19d ago
    Previous to you tax dollars and philanthropy have funded math in various forms and you reap the benefits.
  - auggierose 20d ago
    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.
    - deleted 19d ago
      [deleted]
awr-asjkt 19d ago
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.
- bonoboTP 19d ago
  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.
bbor 19d ago
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?
busyant 19d ago
> 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.
tim333 19d ago
"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.
YeGoblynQueenne 19d ago
>> 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]. _______________ [1] https://en.wikipedia.org/wiki/Solved_game 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 https://www-formal.stanford.edu/jmc/newborn/newborn.html [3] ibid.

154 points by throwaway81523 20d ago

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/ https://www.mathandai.org/ More than 1,900+ mathematicians have also shown concern over the Caltech Mathathon: https://docs.google.com/document/d/1IL0b2oG2KvvSnxn_DuXsNxuHaCefDJ9hNY1k9BP3kKQ/edit?tab=t.0 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 https://www.youtube.com/shorts/R9VQnNv5SoI

octoberfranklin 20d ago

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.

traject_ 20d ago
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.
- octoberfranklin 20d ago
  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.
  - traject_ 20d ago
    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.
    - grey-area 20d ago
      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.
  - auggierose 19d ago
    it is not his rejection, this is a guest article written by two other people
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Pulcinella 20d ago
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.
- deleted 20d ago
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adalacelove 20d ago
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.
fwlr 20d ago
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?
robotpepi 20d ago
> 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.

stabbles 20d ago

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.

matherial 20d ago
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.
- tpm 20d ago
  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.
- doctor_phil 20d ago
  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.

yazaddaruvala 20d ago

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.

camillomiller 20d ago
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
- rnewme 20d ago
  Except that your view doesn't change the reality, and both views are true at the same time.

niemandhier 20d ago

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.

magimas 20d ago
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.
- kzrdude 20d ago
  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.
  - NiloCK 20d ago
    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.
    - kzrdude 20d ago
      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.

camillomiller 20d ago

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.

em-bee 20d ago
business endeavors that cared only about the process, and would gladly avoid the [...] process you mean they care about the product...

HedonicEscal8r 20d ago

I appreciate the wordplay in the title.

omnicognate 20d ago
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.

zarzavat 20d ago

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.

robotpepi 20d ago
> 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.
- contubernio 20d ago
  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.
  - robotpepi 20d ago
    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.

charcircuit 20d ago

>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.

robotpepi 20d ago
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).
- charcircuit 20d ago
  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.
  - robotpepi 20d ago
    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.
anonymouz 20d ago
The authors discuss the dual roles of proofs, and the new need to differentiate them better, clearly in the article.

brador 20d ago

Math needs its own unique type of prompt engineers who can understand the output quickly. That’s where the future of mathematicians lies.

coderenegade 20d ago

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.

encyclopediai 20d ago
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/ 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/ https://sfdora.org/read/ and recent fear of commercial AI competition
- bonoboTP 19d ago
  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.

zkmon 20d ago

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.

keiferski 20d ago
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=5dwulj&utm_medium=ios https://jaredhenderson.substack.com/p/the-modern-malaise?r=5...
pdhborges 20d ago
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?
- zkmon 20d ago
  Nope. The medical advances created the inverted pyramid of demographics in the western world, filled by old people.
alansaber 20d ago
Why draw the line at the industrial age? Humans subsisted for millenia before the ready preparation of fire.
peacefullmind 20d ago
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.

globalnode 20d ago

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.

bonoboTP 19d ago
How many people are employed to meditate all day?

jdw64 20d ago

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.

bonoboTP 19d ago
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.
- oliculipolicula 19d ago
  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
  - jdw64 19d ago

le-mark 20d ago

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.

slickytail 20d ago
[dead]

svara 20d ago

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.

oliculipolicula 20d ago
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

sigmoid10 20d ago

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.

alansaber 20d ago
Technology advances, but extrapolating the value of human labour to 0 is equally invalid.
robotpepi 20d ago
> 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.
stabbles 19d ago
Strongly reminiscent of God of the Gaps [1] [1]: https://en.wikipedia.org/wiki/God_of_the_gaps https://en.wikipedia.org/wiki/God_of_the_gaps
kaffekaka 19d ago
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.
archagon 19d ago

reisse 20d ago

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?

curt15 20d ago
> 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/ 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"?
otabdeveloper4 19d ago
> 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.

dash2 20d ago

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.

justinparus 20d ago
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.
curt15 19d ago
> 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.

justinparus 20d ago

"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.

moralestapia 20d ago
Why would my tax dollars go to fund someone's journey on what's essentially a hobby at this point?
- justinparus 19d ago
  Previous to you tax dollars and philanthropy have funded math in various forms and you reap the benefits.
- auggierose 20d ago
  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.
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awr-asjkt 19d ago

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.

bonoboTP 19d ago
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.

bbor 19d ago

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?

busyant 19d ago

> 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.

tim333 19d ago

"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.

YeGoblynQueenne 19d ago

>> 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]. _______________ [1] https://en.wikipedia.org/wiki/Solved_game 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 https://www-formal.stanford.edu/jmc/newborn/newborn.html [3] ibid.