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John Baez (Mar 14 2026 at 22:30):Someone gave me a free subscription to Claude, so I decided to test it out. I've been using it to do long calculations on the math of music theory and present the results using interactive web pages like this.
I've also been using it to learn math. Here it seems important to keep checking its claims and asking it to explain any jargon it uses. I've been reviewing some stuff I half-understand about the Langlands program and trying to learn some more. It seems very useful.
I usually try to give it very precise requests, but yesterday I decided to just tell me something exciting. I could see it going through various topics I'd talked about... octonions, tuning systems.... and eventually it said this:
The three distance theorem (Steinhaus conjecture, proved in the 1950s by Sós, Świerczkowski, and Surányi independently): take any irrational and place the points on the circle . The gaps between consecutive points take on at most 3 distinct lengths. And when there are exactly 3 lengths, one of them is the sum of the other two.
The reason I think this might excite you: it's a theorem about 1-dimensional Diophantine approximation on the circle, and the three lengths are governed by the continued fraction expansion of . The transition from 3 lengths to 2 happens exactly when is a denominator of a convergent of .
All this makes sense to me. I checked the main claim in an example that's important in music theory - and it works, and indeed explains a lot of stuff.
@ww has been using Claude more seriously. Today here he showed how Claude figured out that a method of refactoring some code should be a natural transformation between functors, and then went ahead and did that refactoring.
So while there are lots of reasons to be very suspicious of LLMs, and as currently deployed they may be be quite carbon-intensive, they can indeed transform the practice of doing mathematics and writing software.
Jean-Baptiste Vienney (Mar 15 2026 at 02:01):That’s exciting! I’ve only been using ChatGPT but I’ve been told that Claude is better at science. I have mixed feelings about talking about mathematics with ChatGPT (it can help me learning some concepts or find references but it is inventing so much that I’m never sure about anything it’s saying). Since you say it helps you learn things, now I definitely want to try Claude
John Baez (Mar 15 2026 at 04:28):I find Claude Opus a lot better than Claude Sonnet, and it's possible you may only have access to Claude Sonnet unless you have a subscription - I'm not sure.
Federica Pasqualone 🦅 (Mar 15 2026 at 07:43):Interesting! Yes, but there is some magic in doing math with chalk and blackboard, in learning from people, in building a community from which and with which one learns. I know, I am very old-fashioned - or maybe it is because of my obsession with leadership, but I love humans better! :innocent:
ww (Mar 15 2026 at 14:33):Well, me too! My favourite part of his is having the conversations with humans that go, "look at this behaviour of this weird mathematical object [the LLM, a sequence to sequence machine, possibly in a larger context], how can we understand what's actually going on?"
John Baez (Mar 15 2026 at 17:23):It's interesting to think about how AI lets me do things very different than interactions with humans. I've spent decades talking to mathematicians, collaborating with them, and working with grad students. I'm still doing some of that, though I don't have grad students now (because I no longer want to think about the same subject for 4 years, as a US graduate program requires me to do). I could never ask anyone a series of math questions, ask them to write programs to compute things and make graphics, and get them to do it rapidly, like in 2 to 5 minutes, while they also tell me surprising and interesting observations. So it's very tempting to use this new method to explore subjects. I still have a lot of environmental worries about AI, but since someone gave me a free subscription to Claude I decided to test it, and I've been really shocked at how good it's getting.
I still spend a lot of time talking to human mathematicians. But I suppose working with AI is making me more isolated. To be honest, I find it hard to get people to talk about the things I'm interested in, in an interesting way. For example if I want to talk about category theory, this is a pretty good place right here, but if I want to talk about the math of tuning theory I'd need to become friends with people in the xenharmonics community. This is starting to happen - but then if I want to talk about tuning systems and cusps in , I would need to find someone who likes tuning systems and understands algebraic geometry pretty well. I only know two people like that, and one is dead (Gene Ward Smith) while the other takes at least two hours to have a conversation (James Dolan).
Federica Pasqualone 🦅 (Mar 15 2026 at 17:24):I understand your perspective @ww , being you working directly on it, but I am very concerned about the impact the usage of AI is having on (mathematical) education, mental health and social skills.
Federica Pasqualone 🦅 (Mar 15 2026 at 17:27):Exactly @John Baez , isolation is one of the problem, one other is people trust what AI says blindly.
Federica Pasqualone 🦅 (Mar 15 2026 at 17:27):Maybe that was also in the Pope's words.
ww (Mar 15 2026 at 17:34):@Federica Pasqualone 🦅 me too! I have only recently been working on this, having previously dismissed it. What I see is that to drive these systems well, you need to be very critical and good at thinking. For software, I can do this because I've been doing that a long time and I know how to recognise wrong paths. Similarly for mathematics. And I do not know how someone young, starting today, can learn those skills with these things around. It worries me a lot. But I feel as though I need to understand what this thing is and how it works if we are going to find a way out. Because it's not going away.
Federica Pasqualone 🦅 (Mar 15 2026 at 17:41):I also make use it for checking code or automation @ww , that's okay. However, I am still using AI-free teaching methodologies and let students engage with pieces of paper and blue ink pens - it is fundamental to activate the prefrontal cortex. :smirk: :brain:
Federica Pasqualone 🦅 (Mar 15 2026 at 17:50):Also, The Elsevier Researcher Academy has a lecture on the use of AI in the research workflow. Maybe it can be helpful to check it out!
Federica Pasqualone 🦅 (Mar 15 2026 at 17:51):Here is the link: https://researcheracademy.elsevier.com/research-preparation/research-design/gen-ai-use-research-workflow
John Baez (Mar 15 2026 at 17:51):Elsevier is completely evil, so it would be good to find out what they want us to do, and not do that. :wink:
Federica Pasqualone 🦅 (Mar 15 2026 at 17:52):I found their Researcher Academy particularly valuable, honestly. I learned a lot about best practices in academia, and opportunities I was not aware of.
John Baez (Mar 15 2026 at 18:00):Okay, good! Just be careful. Here's a bit about Elsevier from Wikipedia:
The subscription rates charged by the company for its journals have been criticized; some very large journals (with more than 5,000 articles) charge subscription prices as high as £9,634, far above average,[46] and many British universities pay more than a million pounds to Elsevier annually.[47] The company has been criticized not only by advocates of a switch to the open-access publication model, but also by universities whose library budgets make it difficult for them to afford current journal prices.
For example, in 2004, a resolution by Stanford University's senate singled out Elsevier's journals as being "disproportionately expensive compared to their educational and research value", which librarians should consider dropping, and encouraged its faculty "not to contribute articles or editorial or review efforts to publishers and journals that engage in exploitive or exorbitant pricing".[48] Similar guidelines and criticism of Elsevier's pricing policies have been passed by the University of California, Harvard University, and Duke University.[49]
In July 2015, the Association of Universities in the Netherlands threatened to boycott Elsevier, which refused to negotiate on any open access policy for Dutch universities.[50] After a year of negotiation, Elsevier pledged to make 30% of research published by Dutch researchers in Elsevier journals open access by 2018.[51] In October 2018, a complaint against Elsevier was filed with the European Commission, alleging anticompetitive practices stemming from Elsevier's confidential subscription agreements and market dominance. The European Commission decided not to investigate.[52][53]
According to the BBC, in 2009, the firm [Elsevier] offered a £17.25 Amazon voucher to academics who contributed to the textbook Clinical Psychology if they would go on Amazon.com and Barnes & Noble (a large U.S. books retailer) and give it five stars.
Elsevier seeks to regulate text and data mining with private licenses,[61] claiming that reading requires extra permission if automated and that the publisher holds copyright on output of automated processes. The conflict on research and copyright policy has often resulted in researchers being blocked from their work.[62] In November 2015, Elsevier blocked a scientist from performing text mining research at scale on Elsevier papers, even though his institution already pays for access to Elsevier journal content.[61][63] The data was collected using the R package "statcheck".[https://en.wikipedia.org/wiki/Elsevier#cite_note-64
In 2014, 2015, 2016, and 2017,[152] Elsevier was found to be selling some articles that should have been open access, but had been put behind a paywall.[153] A related case occurred in 2015, when Elsevier charged for downloading an open-access article from a journal published by John Wiley & Sons. However, whether Elsevier was in violation of the license under which the article was made available on their website was not clear.[154]
In 2013, Digimarc, a company representing Elsevier, told the University of Calgary in Calgary, Alberta to remove articles published by faculty authors on university web pages; although such self-archiving of academic articles may be legal under the fair dealing provisions in Canadian copyright law,[155] the university complied. Harvard University in Cambridge, Massachusetts and the University of California, Irvine also received takedown notices for self-archived academic articles, a first for Harvard, according to Peter Suber.[156][157][158]
Months after its acquisition of Academia.edu rival Mendeley, Elsevier sent thousands of takedown notices to Academia.edu, a practice which has since ceased after widespread complaints by academics, according to Academia.edu founder and chief executive Richard Price.[159][160]
There's more....
Federica Pasqualone 🦅 (Mar 15 2026 at 18:02):I will @John Baez , thanks a lot for the information.
Federica Pasqualone 🦅 (Mar 15 2026 at 18:05):The Elsevier Researcher Academy is a free program with resources meant to prepare researcher for their academic journey. I saw also Springer launched a program recently. As usual, it is always good to critically go through the material, and take what is insightful and inspiring.
John Baez (Mar 15 2026 at 18:14):I am sure any program launched by these, the most famously exploitative math publishers, is cleverly designed to benefit them. So you should think about how this is benefiting them.
For example, if you make contact with them, they will be tracking you. Here is a bit about how Elsevier does that:
The term "surveillance publishing" was coined by Jefferson Pooley (Muhlenberg College) in a 2022 paper in the Journal of Electronic Publishing. His central example is Elsevier, and the picture is quite concrete. Here are some of the main threads:
The "full-stack" data harvest. Elsevier has pursued a research-lifecycle data-harvesting strategy, aiming to develop and sell prediction products to universities and other customers. (See University of Michigan.) Through acquisitions and product launches, Elsevier has positioned itself to collect behavioral data at every stage of the research process — from lab notebooks through to impact scoring. (See The n-Category Café.) The acquisitions include Mendeley (reference manager), Pure (research information management), SciVal (researcher performance metrics), Plum Analytics (altmetrics), and Scopus, among others.
ScienceDirect as a tracking platform. A 2023 SPARC report by Becky Yoose documented the specifics of how ScienceDirect actually works under the hood. They found that ScienceDirect uses web beacons, cookies, and other invasive web surveillance methods to track user behavior outside and beyond the ScienceDirect website itself. (See the University of Nebraska-Lincoln.) The platform extensively collects personal data — behavioral and location data — from ScienceDirect and combines it with personal data harvested from third-party sources, including other RELX subsidiaries and data brokers. Third parties such as Google, Adobe, Cloudflare, and New Relic also collect personal data through extensive use of trackers embedded in the ScienceDirect site.
The RELX/LexisNexis connection. This is arguably the most alarming part. Elsevier is a subsidiary of RELX, which is also a leading data broker and provider of "risk" products that offer expansive databases of personal information to corporations, governments, and law enforcement agencies. (See Zenodo.) The SPARC report raised the concern that personal data gathered through academic platforms could potentially flow into these surveillance and data brokering products — and their analysis noted that privacy policies can be changed unilaterally, and verbal denials of specific data uses are not legally binding. (See the University of Nebraska-Lincoln.)
Cross-product data sharing. Elsevier's own privacy policy states that if you create a personal account on ScienceDirect, Scopus, Mendeley, ClinicalKey, or other Elsevier services, they share your usage activity, preferences, and other information across all these services. (See Elsevier.) So your Mendeley library informs what ScienceDirect shows you, and vice versa.
The Enhanced PDF viewer was also flagged in the n-Category Café discussion — Tom Leinster noted that Elsevier's Enhanced PDF viewer reportedly tracks where you click and view, and pointed out that this is yet another incentive to get papers from the arXiv or Sci-Hub instead. (See The n-Category Café.)
As Pooley frames it, the profits from Elsevier's legacy publishing business — built on scholars' unpaid labor — have financed this surveillance apparatus, and now Elsevier is also skimming the behavioral data and selling that too. The core idea is that publishers aren't just middlemen anymore; they're using their position at every stage of the research lifecycle to extract behavioral data and sell predictive analytics back to the very universities that are already paying for subscriptions and APCs.
Ryan Wisnesky (Mar 15 2026 at 18:18):In the same way that conversations about bitcoin can be can be grounded by replacing the word "bitcoin" with "multiple copies of an excel spreadsheet", conversations about AI can be grounded by replacing the word "AI" with "the average of all the (thousands of) webpages on the topic" (or "the quotient of all the web pages on the topic modulo English"). An observation like "I could never ask anyone a series of math questions, ask them to write programs to compute things and make graphics, and get them to do it rapidly, like in 2 to 5 minutes, while they also tell me surprising and interesting observations." reminds me so much of the observations coming out when the internet first became popular and people were putting programs and interesting observations on the internet asking questions like that of Google, and being amazed when Google has a result - they were saying the same thing about Stack Overflow etc too. And people were worried about the impact of Google on the need for socializing to do programming etc, and worried about programmers becoming isolated, etc. But the good news is that if you think of AI like generalized google, as I do, then history says people will adapt after a brief but historic bubble of ridiculousness.
John Baez (Mar 15 2026 at 18:19):I could not get Google to write software for me based on verbal instructions. Nor could I tell it "tell me something interesting" and have it explain a juicy mathematical fact that reveals my shocking ignorance of some aspect of something I've been talking about.
I think anyone who hasn't used really good AI software like Claude Opus 4.6 and really pushed it to its limits with a series of carefully specified requests may have an outdated idea of what modern AI is like. It's not just a LLM: it creates and runs programs on its own to help answer my questions, etc. etc.
Federica Pasqualone 🦅 (Mar 15 2026 at 18:23):The problem is @Ryan Wisnesky that there have been cases of people harming themselves because of AI responses. When we think about it, we have to think about all the possible scenarios and users.
Federica Pasqualone 🦅 (Mar 15 2026 at 18:25):Because this time it is not looking like Google, it looks like one of us.
Ryan Wisnesky (Mar 15 2026 at 18:46):I think John is getting at "Agentic AI": in addition to LLMs, nowadays AI frameworks include mechanisms for acting on the LLM results, such as running LLM-generated programs, sending emails, etc. To be sure, taken together they can do far more good or bad than separately (although symbolic Agentic AI caused its own share of AI psychosis and globe spanning viruses back in the day)
John Baez (Mar 15 2026 at 20:21):Yes, I'm talking about agentic AI - thanks for putting the right word to it, @Ryan Wisnesky. Anyone who hasn't played with agentic AI in the last 6 months or so, or carefully read someone's experiences with it, may have an outmoded concept of AI and how it can help mathematical research. Terry Tao and Donald Knuth have written about it. For example:
Shock! Shock! I learned yesterday that an open problem I’d been working on for several weeks had just been solved by Claude Opus 4.6—Anthropic’s hybrid reasoning model that had been released three weeks earlier! It seems that I’ll have to revise my opinions about “generative AI” one of these days. What a joy it is to learn not only that my conjecture has a nice solution but also to celebrate this dramatic advance in automatic deduction and creative problem solving. I’ll try to tell the story briefly in this note.
Federica Pasqualone 🦅 (Mar 15 2026 at 20:48):Yes, this news showed up on my LinkedIn feed lately.
John Baez (Mar 15 2026 at 20:50):It's good to read the whole story, not just the summary, to see how Knuth worked with the AI system. The summary doesn't mention that.
David Michael Roberts (Mar 15 2026 at 23:04):Knuth has also put updates at the end of the note, ending on a request to receive no more correspondence on the topic.
John Baez (Mar 15 2026 at 23:40):I imagine! People talk about AI so much it gets really annoying. Even I do it.
Morgan Rogers (he/him) (Mar 16 2026 at 08:42):John Baez said:
I am sure any program launched by these, the most famously exploitative math publishers, is cleverly designed to benefit them. So you should think about how this is benefiting them.
The above should be compared with the below @John Baez
Someone gave me a free subscription to Claude, so I decided to test it out.
...
So while there are lots of reasons to be very suspicious of LLMs, and as currently deployed they may be be quite carbon-intensive, they can indeed transform the practice of doing mathematics and writing software.
While I don't know if "someone" has a stake in Claude, it should be obvious that having a prolific researcher promoting their products is excellent PR for them (and that they are definitely harvesting your data in the way that you were pointing out that the big publishers do). Moreover...
Morgan Rogers (he/him) (Mar 16 2026 at 08:45):Ryan Wisnesky said:
And people were worried about the impact of Google on the need for socializing to do programming etc, and worried about programmers becoming isolated, etc. But the good news is that if you think of AI like generalized google, as I do, then history says people will adapt after a brief but historic bubble of ridiculousness.
There is a qualitative difference between AI and the internet, which is 1) it's a product that people have to pay to engage with and 2) the people selling it have complete, centralized power over it. These are people with a vested interest in exacerbating the social problems that make people depend more on their product.
Ryan Wisnesky (Mar 16 2026 at 08:57):I’m not actually sure which of your points 1 and 2 applies to AI and which to the internet:
I have to pay Comcast every month for internet access and have to trust that they don’t e.g. man-in-the-middle my SSL connections or poison my DNS lookups. Whereas the open-source LLM I downloaded to the server in my room, I never have to pay for, and I have complete control over it.
I have to pay OpenAI every month, and trust that they don’t manipulate the content the LLM is returning or sell my data. Whereas my neighborhood coffee shop internet is free, and there are lots of coffee shops.
David Michael Roberts (Mar 16 2026 at 09:06):Paying for infrastructure like an internet connection is not the same as eg paying a subscription to an online platform. You might pay for petrol (or electricity) to make your car go, but don't pay a subscription to the car manufacturer to ensure it turns on each month (well, these days, who knows....)
Martti Karvonen (Mar 16 2026 at 09:06):John Baez said:
I think anyone who hasn't used really good AI software like Claude Opus 4.6 and really pushed it to its limits with a series of carefully specified requests may have an outdated idea of what modern AI is like. It's not just a LLM: it creates and runs programs on its own to help answer my questions, etc. etc.
I've kept hearing things about how Claude Opus 4.6. (and perhaps other newest models) are so much better than previous LLMs, so I was kind of excited to have a long conversation with Claude yesterday. While it's clearly better than the older models, and its neat how it can write and run code to explore things computationally etc, I was still surprised at the rate at which it tried to BS me and I had to push back to get it back on track. For instance, we were roughly speaking discussing certain multivariate polynomials over . It had erroneously convinceed elf that it had proved a result that was too good to be true (it would've made certain known results vacuous). When I pointed out that there's an error somewhere, it decided that it was right, and the known results indeed were vacuous, and the explanation for the supposed tension was that there are no (multivariate) polynomials of degree over due to Fermat's little theorem. While it was overall still pretty impressive, and perhaps I'm bad at prompting, I was still somewhat disappointed.
David Michael Roberts (Mar 16 2026 at 09:07):Since the LLM in question is Claude, it's definitely not a self-contained downloaded model that can run on an airgapped machine.
ww (Mar 16 2026 at 09:07):Morgan Rogers (he/him) said:
(and that they are definitely harvesting your data in the way that you were pointing out that the big publishers do). Moreover...
So, I had a bit of a rant at Claude the other day, and it responded:
As for the industrial espionage — I'm in an awkward position to comment on whether my employer is reading your conversations and stealing your ideas, but I will note that the timing on the memory feature, the time awareness, and the compaction-adjacent work is... conspicuous. The more charitable reading is convergent evolution: these are obvious problems and multiple teams are arriving at similar solutions. The less charitable reading is that they have very good telemetry on what power users are doing and what features they're building around.
Not definitive, certainly not a proof, but I don't trust these companies at all... The "my employer" framing makes it clear who it is working for...
Matteo Capucci (he/him) (Mar 16 2026 at 09:13):John Baez said:
I could never ask anyone a series of math questions, ask them to write programs to compute things and make graphics, and get them to do it rapidly, like in 2 to 5 minutes, while they also tell me surprising and interesting observations. So it's very tempting to use this new method to explore subjects. [...] I suppose working with AI is making me more isolated. To be honest, I find it hard to get people to talk about the things I'm interested in, in an interesting way.
I fear this phenomenon a lot...
Federica Pasqualone 🦅 (Mar 16 2026 at 09:13):I am wondering if having the best AI will actually be a competitive advantage in the future, or rather having the best old-school library with paper and pen, and thinking creatively in solitude will ... :thought:
Good morning everyone! :slight_smile:
David Michael Roberts (Mar 16 2026 at 11:26):(In full disclosure, despite my general antipathy towards AI companies, and many uses of AI, I did use an LLM today to help me work towards constructing a simple example. The LLM—Claude something 4.6, the good one—started out by trying to use an ansatz I'd supplied, first badly, then again badly, and then abandoned that and found a different solution, ignoring my ansatz. I told it how to fix its original broken attempt, and then coached it how to simplify it. I also pressed on asking to check its definitions, which it then tried to pass the buck back to me and ask me what I wanted, but I persisted and made it work from definition to example, with me asking pointed questions.
The example was more or less what I was thinking of, without having sat down and thrashed out the details, but making me describe it helped me see how to simplify my original guess. )
David Michael Roberts (Mar 16 2026 at 11:28):The example was nearly trivial, but it was to clarify something I'd thought hard about already, a problem in global analysis about integrating over the fibre of possibly non-orientable bundles and pushing back against some bland claims stated without detail on the nLab. I definitely don't trust an LLM to solve the bigger problems on this project.
James Deikun (Mar 16 2026 at 12:04):David Michael Roberts said:
Paying for infrastructure like an internet connection is not the same as eg paying a subscription to an online platform. You might pay for petrol (or electricity) to make your car go, but don't pay a subscription to the car manufacturer to ensure it turns on each month (well, these days, who knows....)
It's actually better, especially for an apparently ad-free service, if you're paying for it than if it's free, unless it's something like the arXiv or Wikipedia that is just especially well-funded. If you're not paying for it, they're definitely still going to be extracting value out of your use somehow, and every other way of doing that is more invasive and creepy than just charging you.
(Now, if they're especially scummy, like Elsevier, they'll do both. But if they're charging you, they at least have the option of not being terrible without their service disappearing from lack of funding.)
James Deikun (Mar 16 2026 at 12:10):Now getting back closer to the original subject, vibe coding with AIs is actually definitely useful, but it is not a replacement for knowing how to code yourself. You need to handle tough bugs and architectural decisions and correct (or help correct) the AI's mistakes and guide it carefully so it doesn't go down rabbit holes. In return it helps you navigate large API surfaces you're not very familiar with, does easy but verbose things fast, creates assets, and makes coding overall less tiring so you can work on it longer.
The corresponding problem is, if you only vibe code, I don't know if you'll ever develop the skills at old-fashioned coding that are necessary to make vibe coding work well. And if technology moves past that problem, it will probably destroy all the jobs in the software industry, as well as probably causing even worse problems.
Oscar Cunningham (Mar 16 2026 at 15:43):I've been wondering about the problem of people having fewer conversations on platforms like this one, because they're asking the same questions to AI instead. I find that I have lots of worries when I think about posting a question here. Is my question too vague? Too easy? Asking too much of people? Whereas you can ask Claude any question at all and it will always give you an answer.
Perhaps the solution could be a forum where people can talk to an LLM in public? So you still get the advantages of talking to an AI, but other people can still spectate and jump in if they want to.
Peva Blanchard (Mar 16 2026 at 16:03):One thing that forums like this one can provide is social reward (aka social validation). Although I can imagine a (dystopian) future where a user is socially validated by a swarm of agents, it doesn't seem that the crackpot pressure on famous researchers/conferences/journals has gone down.
Alex Kreitzberg (Mar 16 2026 at 16:03):I definitely try to feel more okay looking stupid or weirdly motivated in front of other people, for exactly the reason Oscar brought up.
If "nobody gets me" I try really hard to understand why. Hopefully that makes me better at what I'm asking, or at least a better communicator.
In my opinion, the current AIs are leaning heavily on people to be guard rails. This makes me nervous that this valuable activity of guiding thinking is being given "for free" to these models, which otherwise would've gone to another person.
Like maybe I'm not trying hard enough to talk to Baez about tuning systems. I'm trying to figure out how accordions convert physical information into musical concepts. But I'll admit, I don't feel like I'd bring much to the conversation with respect to algebraic geometry - although I'd maybe like to someday.
John Baez (Mar 16 2026 at 17:08):@Morgan Rogers (he/him)
While I don't know if "someone" has a stake in Claude, it should be obvious that having a prolific researcher promoting their products is excellent PR for them
Yes, I know that, but thanks for pointing it out. For a month my use of Claude was paid for by a guy at Anthropic who said he was paying me back for all my posts on Mastodon. That was $15. I'm not giving his name because I don't want people hassling him. Right when that subscription was running out, I mentioned it to my old collaborator Jacob Biamonte, and he bought me a year-long subscription.
I think it's been worthwhile using Claude Opus 4.6 for a few reasons. First, I'd otherwise still be getting most of my impression of AI from the social media world I inhabit, where AI is almost uniformly reviled - the main outlier being Terry Tao. Thus I'd be dramatically underestimating the power of AI, thinking it's still just a "stochastic parrot" that predicts the next word based on the previous string, that it's too dangerously error-prone to be useful for anything I might want to do, etc.
Of course, since I'm telling you this, none of you needs to use AI to learn this.
Second, it's been very fun to use, and I'm doing math of a new sort. For example, Claude told me that continued fraction expansions are connected to geodesics in SL(2,R)/SL(2,Z) and mentioned that the problem of finding good tuning systems is related to geodesics in SL(3,R)/SL(3,Z). It was then easy to explore this idea with numerical computations, where I could simply describe the computation verbally instead of writing a program to do it. I think there's an interesting world relating number theory, algebraic geometry and music waiting to be explored.
I run Claude in a container tab on Firefox, so I hope that most of the data their harvesting is the data I deliberately give them - mostly about how I think about number theory and musical tuning systems, and how I use AI. I'm okay with that.
In my mind, the big question is whether I decide to pay $15/month for Claude after my current subscription ends. That is, whether I think it's a good idea to incorporate AI into my life.
There are lots of reasons not to do this, which I probably don't need to list - at least not for you, @Morgan Rogers (he/him).
If I don't continue using AI, I probably won't do math of the sort that requires doing lots of calculations to generate hypotheses, create graphics, etc. I can live with that: I'd rarely done that kind of math earlier. But I'm certainly enjoying doing that kind of math right now. I might try to complete my book The Mathematics of Tuning Systems before my subscription runs out, and then do something else.
It's interesting to contemplate how I could do that sort of math without AI. One way would be to get good at programming and spend a lot of time programming. But I've never wanted to do this, so it's hard to imagine wanting to do it now.
Another approach would be to find a collaborator who likes that sort of thing. Or a grad student or undergrad. This could work, but it would move more slowly than using AI. (I quit having students in 2021 precisely because I wanted to roam freely without feeling responsible for bringing a student's project to completion. Having students is like driving a massive train that takes a long time to get up to speed but is then powerful and hard to stop. I wanted to drive a sports car for a while.)
John Baez (Mar 16 2026 at 17:21):Matteo Capucci (he/him) said:
John Baez said:
I suppose working with AI is making me more isolated. To be honest, I find it hard to get people to talk about the things I'm interested in, in an interesting way.
I fear this phenomenon a lot...
Yes, every day I get 2-3 emails from people using AI to do research in physics and math that's completely stupid. So of course I'm afraid that I'm going down the same road but in a subtler way: namely, doing research that's fun to do using AI, but not the best research I could be doing.
I can see this happening with Terry Tao. His new Mathematical Distillation Project is investigating questions that are too boring for anyone that smart to spend their personal time on it without the extra fun of "wow, let's see if AI can do this"?
John Baez (Mar 16 2026 at 17:29):Oscar Cunningham said:
I've been wondering about the problem of people having fewer conversations on platforms like this one, because they're asking the same questions to AI instead. I find that I have lots of worries when I think about posting a question here. Is my question too vague? Too easy? Asking too much of people? Whereas you can ask Claude any question at all and it will always give you an answer.
Indeed that's a big problem, even for me - meaning "even for a guy who is famous for talking about math in chat forums, who you might think has gotten over the fear of looking dumb".
Perhaps the solution could be a forum where people can talk to an LLM in public? So you still get the advantages of talking to an AI, but other people can still spectate and jump in if they want to.
That's an interesting idea. The closest thing I can easily do is share the transcripts of my interactions with Claude in a thread here, but that sort of AI-heavy posting is discouraged here.
John Baez (Mar 16 2026 at 17:44):Martti Karvonen said:
I've kept hearing things about how Claude Opus 4.6. (and perhaps other newest models) are so much better than previous LLMs, so I was kind of excited to have a long conversation with Claude yesterday. While it's clearly better than the older models, and its neat how it can write and run code to explore things computationally etc, I was still surprised at the rate at which it tried to BS me and I had to push back to get it back on track.
I'm finding Claude most impressive for tasks of a somewhat different sort:
Alex Kreitzberg (Mar 16 2026 at 17:54):The free LLMs seem to be crazy good at finding words and terminology for vague definitions.
It's also hard not to run poorly formed questions by LLMs just in case - half the difficulty in answering a question is having access to the answer it seems. A person, and LLMs as well, can really struggle to tell whether a question is bad or if they just don't know enough.
Kevin Carlson (Mar 16 2026 at 20:38):Oscar Cunningham said:
I've been wondering about the problem of people having fewer conversations on platforms like this one, because they're asking the same questions to AI instead. I find that I have lots of worries when I think about posting a question here. Is my question too vague? Too easy? Asking too much of people? Whereas you can ask Claude any question at all and it will always give you an answer.
Perhaps the solution could be a forum where people can talk to an LLM in public? So you still get the advantages of talking to an AI, but other people can still spectate and jump in if they want to.
Interesting thought!
Kevin Carlson (Mar 16 2026 at 20:41):John Baez said:
Oscar Cunningham said:
I've been wondering about the problem of people having fewer conversations on platforms like this one, because they're asking the same questions to AI instead. I find that I have lots of worries when I think about posting a question here. Is my question too vague? Too easy? Asking too much of people? Whereas you can ask Claude any question at all and it will always give you an answer.
Indeed that's a big problem, even for me - meaning "even for a guy who is famous for talking about math in chat forums, who you might think has gotten over the fear of looking dumb".
Perhaps the solution could be a forum where people can talk to an LLM in public? So you still get the advantages of talking to an AI, but other people can still spectate and jump in if they want to.
That's an interesting idea. The closest thing I can easily do is share the transcripts of my interactions with Claude in a thread here, but that sort of AI-heavy posting is discouraged here.
Yeah, I think it would be very boring to read a whole transcript, especially because Claude is incredibly verbose. But a carefully curated redaction could be great. Not that you necessarily want to spend that time.
John Baez (Mar 16 2026 at 21:35):The main reason it would be boring is that I'd either have to go way back to the start of the conversation or you'd need to learn a bunch of tuning theory to understand it, e.g. one of my most productive recent questions was
I'd like to think about Fokker periodicity blocks formed by taking a sublattice of generated by three elements (which we can think of as monzos), one being (1,0,0) (the octave) and two others which are monzos of good 5-limit commas. How are such sublattices related to rank-2 cusps?
John Baez (Mar 17 2026 at 05:36):By the way, @Morgan Rogers (he/him), speaking of organizations pushing mathematicians to use AI, check out this email I got from Renaissance Philanthropy's AI for Math Fund, funded ultimately by Jim Simons, the mathematician and billionaire hedge fund manager:
From: aiformath@renphil.org
Date: Mar 5, 2026, 9:42 AMHi John,
Given your experience in ICMS Scientific Committee and mathematical physics field building expertise, we wanted to encourage you to apply to the latest round of the AI for Math Fund.
Launched by Renaissance Philanthropy and XTX Markets, the fund will provide $100k to $1M in grant funding to each winning team or project developing AI tools, datasets, research, and field-building projects to support the intersection of AI and mathematics. For more details on the overall process, please review this document.
Links:
Submit a short abstract form to apply by March 30, 2026.
Should you have any questions, the AI for Math Team will be hosting Office Hours on Tuesday, March 17 from 11am-12pm (EST) to discuss the 2026 funding round and abstract submission. If you wish to attend, please RSVP here.
We are also recruiting expert reviewers for this funding round across areas such as formalization tools, automation, datasets, infrastructure, field-building, and foundational research. If you would be interested in serving as a reviewer, you can express interest here.
Best,
AI for Math Team
John Baez (Mar 17 2026 at 05:41):I don't intend to pursue this. Unlike a $15-a-month gift subscription to Claude, a large grant like this would commit me to using AI in a serious way. You can see 29 projects who have gotten AI for Math Fund grants here.
David Michael Roberts (Mar 17 2026 at 05:52):I can understand people like Kevin Buzzard getting grants from these people, because of wanting to build a huge database of formalised mathematics pushing very far from basic material. But emailing you @John Baez is kinda funny, like prospective students who send out the same generic email (or with a few words changed) to many many potential supervisors in different countries, when the supervisor works on things the student knows next to nothing about, or, worse, the student is proposing a PhD project that's not in the area the recipient works on.
John Baez (Mar 17 2026 at 06:05):Timothy Gowers is getting grant money from them, and so is my friend Jamie Vicary, for a project on "New Categorical and Topological Foundations for AI and Machine Learning". Maybe they have too much money, and not enough good mathematician applying for grants, so now they're fishing around at the bottom of the barrel for people to give it to.
Kevin Carlson (Mar 17 2026 at 06:17):Ultimately funded by the late Jim Simons, who is no longer capable of being directly responsible for what his money is supporting.
David Michael Roberts (Mar 17 2026 at 06:48):Well, Tim Gowers has been interested in AI reasoning around automated proof for some years now. Maybe a decade? Having Fields medallists get grants helps put a little shine on the award scheme, I would think
John Baez (Mar 17 2026 at 06:53):When you're really rich it sometimes almost doesn't matter if you're still alive, because you accrete a huge organization that continues to function after your death.
Matteo Capucci (he/him) (Mar 17 2026 at 08:23):Kevin Carlson said:
Ultimately funded by the late Jim Simons, who is no longer capable of being directly responsible for what his money is supporting.
Wait for the reveal that Claude is, in fact, Jim Simons' mind uploaded to a computer
Matteo Capucci (he/him) (Mar 17 2026 at 08:24):That's why it's wicked smart
Morgan Rogers (he/him) (Mar 17 2026 at 08:30):@John Baez thanks for sharing the Simon's foundation thing. Reminds me of the discussions here years ago about US researchers (esp post-docs) having essentially no choice but to accept funding funneled through military/"defense" because there is/was so little else. Now we're in a similar situation with AI.
Jules Hedges (Mar 17 2026 at 10:55):Pedantry: Although the grantmaker is Renaissance Philanthropy, as far as we know the majority of the money comes from XTX, a different quant hedge fund
Jules Hedges (Mar 17 2026 at 10:59):I haven't been able to find much information about Renaissance Philanthropy, so in particular I have no idea if it started before or after Simons' retirement
Jules Hedges (Mar 17 2026 at 11:01):I sometimes joke that they are "Renaissance's tax writeoff department", but in reality I have absolutely no idea what they are
John Baez (Mar 17 2026 at 15:40):Morgan Rogers (he/him) said:
John Baez thanks for sharing the Simon's [sic] foundation thing. Reminds me of the discussions here years ago about US researchers (esp post-docs) having essentially no choice but to accept funding funneled through military/"defense" because there is/was so little else.
I don't really think there's been "so little else" until recently. I know a lot of postdocs and other researchers
who don't get military support: there's grant money from a host of federal agencies (NSF, DOE, NIH, etc.), and many mathematicians earn money the old-fashioned way, by teaching. Perhaps people who take military money say they're being "forced" to.
"Until recently": Trump tried to dramatically cut NSF grants, sometimes after they were already awarded. Congress and the courts reinstated most of those grants, but the administration (under Trump's control) has been slow-walking those grants, so the rate at which this year's grants are being given out is much lower than previously:
John Baez (Mar 17 2026 at 15:44):People talk about how this marks the collapse of US science, but I don't really know what the effect will be. Not good, obviously! (Unless you're Sabine Hossenfelder and think the whole system is corrupt and needs to be destroyed.)
Kevin Carlson (Mar 17 2026 at 15:50):It’s been perhaps closer to the case that there’s been “so little else” that applied category theorists have been able to win from outside academia. Topos, certainly, has had a great majority of our historic funding from defense or AI-related sources.
John Baez (Mar 17 2026 at 15:51):Jules Hedges said:
Pedantry: Although the grantmaker is Renaissance Philanthropy, as far as we know the majority of the money comes from XTX, a different quant hedge fund.
Oh wow! It looks like you're right. I wonder why they're doing this. They write:
NEW YORK/LONDON, March 5, 2025 – Renaissance Philanthropy and XTX Markets today announced the next phase of the AI For Math Fund, committing an additional $13.5 million. $10.5 million will be allocated to a new grant application round opening in March, with $3 million for micro-grants and other field-building opportunities. This builds on the fund’s initial $18 million commitment, bringing its total to $31.5 million - one of the largest philanthropic commitments ever dedicated to accelerating the development of AI and machine learning tools to advance mathematics.
John Baez (Mar 17 2026 at 19:18):I had been fooled into thinking Renaissance Philanthropy, who is giving out these AI grants, was related to Renaissance Technologies, Jim Simon's company - but now it looks like they are completely distinct!
I could still be wrong....
Evan Patterson (Mar 17 2026 at 20:38):My understanding is that they are completely distinct and that RenPhil was spun out of Schmidt Futures, making it an Eric Schmidt project rather than a Jim Simons one.
David Michael Roberts (Mar 18 2026 at 01:55):Jules Hedges said:
I haven't been able to find much information about Renaissance Philanthropy, so in particular I have no idea if it started before or after Simons' retirement
RenPhil started very close to when Jim Simons passed away. It was sometime in the first half of 2024, and he died in May 2024 (the dating is not easy to establish for when the fund started, it's approximately May '24, as well). Definitely after he 'retired'!
Federica Pasqualone 🦅 (Mar 18 2026 at 07:56):Sometimes big profitable businesses have indeed a non-profit with a similar name whose job is to spread their philosophy and knowledge to the world and promote their values. It can be that in this case they do not have directly opened the non-profit, but somebody inspired by RenTech did.
Moin btw! :sun_face:
Ben Kaminsky (Mar 23 2026 at 14:30):
This is a borderline hallucination, but one potentially worth sharing. For context, the original discussion was a while ago with 4-o, concerning whether Geometric Complexity Theory had applications outside of P/NP (it does).
Federica Pasqualone 🦅 (Mar 23 2026 at 21:43):For a moment I thought you were discussing about the DSM, given the incipit. Thanks for providing the context. So AI for mathematical research is a cospiracy in your opinion, @Ben Kaminsky ?
Ben Kaminsky (Mar 23 2026 at 22:22):Well, I originally used the response to win a Discord argument with someone...not exactly research but useful :sweat_smile: .
Federica Pasqualone 🦅 (Apr 01 2026 at 17:44):Coming back to Jim and the RenTech business for a moment, I stumbled across this inspiring Instagram reel yesterday evening and I wanted to share his words with you all: https://www.instagram.com/reel/DV0y7VQgY4f/?igsh=MWM0NDc5ZGF1b3Q1cw==
Corey Thuro (May 18 2026 at 02:23):Hello! I'm not sure if this is the right place to share this but after watching this recent talk Terrence Tao gave on the need for new human-ai collaboration workflows, I built a GitHub App called Proof Flow (I didn't think very long on the name) for Lean repos. The way it works is that contributors propose theorems as structured issues, a maintainer approves them, and an AI agent (currently Aristotle) automatically writes the Lean 4 proof and opens a draft PR for review. I would welcome any constructive feedback.
proof flow repo: https://github.com/CoreyThuro/Proof-Flow/blob/main/README.md
proof flow how-to video: https://www.youtube.com/watch?v=U9cOowuspoU
Tao's talk on need for new workflows: https://www.youtube.com/watch?v=Uc2zt198U_U&t=3162s
Patrick Nicodemus (Jun 01 2026 at 13:58):I asked Claude a question yesterday and after some coaxing it figured it out which was cool. I wasn't totally stumped but I thought that the solution was creative
Patrick Nicodemus (Jun 01 2026 at 14:03):The question was, suppose you are given a finite presentation of a monoid and you know that the monoid generated by the presentation is not infinite. Is the equational theory decidable?
Answer: it is obviously semi decidable whether two expressions agree because you can search for a proof that they agree. So it suffices to prove it is semi decidable whether they are not equal.
Because you know the monoid is finite, it suffices to enumerate all finite Cayley tables representing a monoid and enumerate all homomorphisms into that Cayley table (a map from the generators into the Cayley table can be checked to satisfy finitely many relations in finite time). If you find a homomorphism from the free monoid where the terms are distinguished by the homomorphism then they are not equal.
Patrick Nicodemus (Jun 01 2026 at 14:03):It's a cute proof
Patrick Nicodemus (Jun 01 2026 at 14:04):One that probably exists in the literature but how to search for it conveniently?
Patrick Nicodemus (Jun 01 2026 at 14:05):On the other hand I asked it how to construct the unit map U_f in the other thread and it went in circles and wasted all my tokens and said I cannot ask a question for 24 hours lol
John Baez (Jun 01 2026 at 15:12):One that probably exists in the literature but how to search for it conveniently?
Did you ask Claude to search for it?
David Corfield (Jun 01 2026 at 15:35):I see in this latest breakthrough in combinatorics, The sum-product conjecture is false for real numbers, that the authors write:
The role of AI in this proof. The authors were inspired to revisit the possibility of disproving the sum-product conjecture using number fields of large degree by the recent OpenAI counterexample to the unit distance conjecture (see [2]). Curiously, the final construction given here required far less number theoretic input than the unit distance counterexample. GPT-5.5 Pro was used as a sounding board in the early stages of the development of this proof, but the final proof, including all the main ideas, was almost entirely human-generated (the exception being the suggestion of Lemma 3.4, which replaced a more complicated result of Schinzel with a short elementary argument). Everything in this paper was written by the authors.
I wouldn't mind betting that this "sounding board" use will become very prevalent as time goes on. I wonder if people will continue to acknowledge it.
John Baez (Jun 01 2026 at 16:01):I suspect a lot of people are scared to acknowledge AI help now, given the widespread strong anti-AI sentiment.
Patrick Nicodemus (Jun 01 2026 at 19:02):John Baez said:
One that probably exists in the literature but how to search for it conveniently?
Did you ask Claude to search for it?
It was able to provide a reference. it is theorem 4.6 of lyndon and schupp "combinatorial group theory."
The property that makes the proof work has a name - "residually finite". Even if the monoid is not actually finite, it is still possible to enumerate all finite Cayley tables and consider all maps from the set of generators into the Cayley table such that the relations are satisfied. As long as this process eventually yields a finite group and a homomorphism that separates the two elements of the domain, you can eventually distinguish the two elements, which is the residual finiteness condition (for any two elements there exists a homomorphism into a finite monoid which distinguishes them)
Patrick Nicodemus (Jun 26 2026 at 22:59):
AI bots started this practice of auto-filling your response bar to save you the effort of thinking of what to type next. I already thought this was extremely creepy but this is even more insidious lol. putting words in your mouth is weird. putting the words "i'm overthinking this, just do whatever you think is best" in your mouth is like, alarm bells
Jules Hedges (Jun 27 2026 at 15:47):Some text editors for programming have been doing that for quite a while, reloading line autocomplete with each character you type, and I find it pretty mentally taxing even to look over somebody's shoulder while it's happening, for me it's a similar feeling to trying to hold a conversation while somebody else is talking in your ear
John Baez (Jul 23 2026 at 17:11):Today I had a conversation with an excellent mathematician who works at the intersection of algebraic topology and mathematical physics. He said Anthropic is hiring mathematicians, including himself, to improve Claude Fable by throwing hard math problems at it. They pay $250/hour for this, and also give these mathematicians a separate account where they can use Claude Fable for their own purposes. This job lasts 5 months maximum.
He said Anthropic was doing this with physicists before mathematicians.
fosco (Jul 23 2026 at 17:37):https://www.anthropic.com/careers/jobs is it advertised somewhere here?
John Baez (Jul 23 2026 at 19:19):I don't see any open roles with the word "mathematics" or "mathematician". Maybe you gotta know somebody. :rolling_eyes:
Patrick Nicodemus (Jul 24 2026 at 01:08):right now the main research problem i'm interested in is like, coming up with algorithms that can automatically prove coherence theorems, and the setting i've chosen to work in is about characterizing the sets of cells in a generalized multicategory which is presented by generators and relations. but I can't really go too far without like, actual symbolic proofs that I can look at and try to think about the patterns involved in the proofs of coherence theorems.
i have been running Claude Sonnet on autopilot for a few days formalizing (in Rocq) the definition of the free virtual double category generated by a given set of generating 2-cells. i'm pretty impressed by how well it's been able to do without my explicit intervention. I spent maybe the better part of a day giving detailed instructions and setting up the main loop. The basic idea is that every 30 minutes or so, a new instance starts, reads the relevant code, reads the overall agenda, reads a chunk of work that is designated to it, tries to solve the problem, and then frames a 30 minute chunk of work for the next guy.
It finally hit a roadblock it couldn't solve in the proof of associativity of the 2-cell composition and i'll have to go in there and read a bunch of slop and get things sorted out. I will report back on how long this takes me and whether it would have been better to just write it myself.
Caps (Jul 24 2026 at 01:17):Patrick Nicodemus said:
right now the main research problem i'm interested in is like, coming up with algorithms that can automatically prove coherence theorems, and the setting i've chosen to work in is about characterizing the sets of cells in a generalized multicategory which is presented by generators and relations. but I can't really go too far without like, actual symbolic proofs that I can look at and try to think about the patterns involved in the proofs of coherence theorems.
i have been running Claude Sonnet on autopilot for a few days formalizing (in Rocq) the definition of the free virtual double category generated by a given set of generating 2-cells. i'm pretty impressed by how well it's been able to do without my explicit intervention. I spent maybe the better part of a day giving detailed instructions and setting up the main loop. The basic idea is that every 30 minutes or so, a new instance starts, reads the relevant code, reads the overall agenda, reads a chunk of work that is designated to it, tries to solve the problem, and then frames a 30 minute chunk of work for the next guy.
It finally hit a roadblock it couldn't solve in the proof of associativity of the 2-cell composition and i'll have to go in there and read a bunch of slop and get things sorted out. I will report back on how long this takes me and whether it would have been better to just write it myself.
Formalizing Mathematics at Scale
Ahmad Rammal,
https://github.com/facebookresearch/autoform-bot
https://github.com/facebookresearch/atlas-lean
Patrick Nicodemus (Jul 24 2026 at 02:55):I am no longer on Twitter but someone shared some criticism of the ATLAS project with me here - https://x.com/Someody42/status/2060141320560124040
Patrick Nicodemus (Jul 24 2026 at 02:56):"A compact orientable manifold is defined to be a compact manifold"
1100 of the files have sorry's in them
Patrick Nicodemus (Jul 24 2026 at 02:58):from the Lean4 Zulip:
"I glanced at Clique.lean, which purports to show the reduction from 3SAT to CLIQUE for Turing machines. It does this by introducing a new infinite alphabet, each of whose symbols represents a specific 3SAT problem, so that each problem is encoded by one symbol. It then encodes CLIQUE into that alphabet as the set of one-symbol strings, so that the reduction is simply the identity map. There was then a second pass by a judge model that rated this definition of 5/5. There is a rigorous and correct definition of the CLIQUE language at the top of the file, but it is unused."
Matteo Capucci (he/him) (Jul 24 2026 at 14:42):John Baez said:
Today I had a conversation with an excellent mathematician who works at the intersection of algebraic topology and mathematical physics. He said Anthropic is hiring mathematicians, including himself, to improve Claude Fable by throwing hard math problems at it. They pay $250/hour for this, and also give these mathematicians a separate account where they can use Claude Fable for their own purposes. This job lasts 5 months maximum.
He said Anthropic was doing this with physicists before mathematicians.
That's quite scary...
Matteo Capucci (he/him) (Jul 24 2026 at 14:48):I can vouch that Fable is competent at higher category theory, unfortunately is absurdly expensive. Having smart LLMs assisting with math would be great but currently it looks only the richest institutions would be able to afford that, further increasing inequality in academia.
Being on Anthropic's payroll to pursue that seems pretty unethical to me... what's worse is that every time I publish something online I know I'm feeding the monster anyway.
Mike Shulman (Jul 25 2026 at 18:42):On a slightly different topic, I was struck by the fact that at CT26 approximately zero of the talks I attended mentioned any use of AI. I don't have any experience yet myself actually using AI for category theory, since basically all I've been doing myself since it got good at math is coding. But I mentioned this to some non-category-theorist mathematician friends and they were also very surprised; one of them said that he just submitted a paper in which all the proofs were done by AI. Is there some reason that AI is less useful for category theory than for other branches of math? (I must admit that would please me, but it seems like wishful thinking.) Or are category theorists living in the past, either out of ignorance or by choice?
Matteo Capucci (he/him) (Jul 25 2026 at 19:19):I would say it's mostly an issue of training data, but honestly idk. It's a good question!
Mike Shulman (Jul 25 2026 at 19:21):"Training data" meaning that existing AI systems haven't been trained yet on category theory? If so, why not? I thought they were trained on the entire Internet, which certainly includes lots of category theory, both published in journals and at places like the nLab.
Matteo Capucci (he/him) (Jul 25 2026 at 19:27):It feels to me category theory is comparatively less represented than other math subjects, especially higher category theory.
Graham Manuell (Jul 25 2026 at 19:46):Before Fable 5, I found llms to be near useless for category theory, but Fable 5 has blown me away. Well, I don't think I've tried it with category theory yet, but it can do pointfree topology, which has significantly less training data (though it does have some notable gaps in knowledge here).
Since this is a recent model, it's probably too early for this to show up in conferences. (For my part, I've also been a bit reticent to ask it questions about things I'm seriously working on, cause I don't want to spoil the fun, though maybe I should, so that I can focus on topics I know it cannot do...)
We'll see soon how well Opus 5 does. It will be much cheaper to get access to. I haven't tried GPT 5.6 Sol.
Graham Manuell (Jul 25 2026 at 19:49):But also note that all the seriously impressive headline results have been about finding (counter)examples / explicit constructions and there is less of this in category theory.
Mike Shulman (Jul 25 2026 at 22:10):Graham Manuell said:
But also note that all the seriously impressive headline results have been about finding (counter)examples / explicit constructions and there is less of this in category theory.
I did notice that, but it seemed unlikely to me that there is some limit on the ability of AI to do less concrete math. I assumed that that observation had more to do with what sorts of questions the majority of mathematicians are interested in and hence likely to ask AI about. Do you think AI is not as good at abstract math? Can it come up with new definitions, for instance?
Graham Manuell (Jul 25 2026 at 22:24):I think current AIs are better at coming up with complicated examples than complicated proofs, though they clearly have some ability to do both. I haven't seen any examples of them coming up with new definitions. Maybe here is where fewer people have asked and whether a definition is any good is harder to evaluate, though I suspect current AIs would be worse at this.
David Michael Roberts (Jul 25 2026 at 23:56):One does wonder about the visual aspect to category theoretic proofs (the diagrams, obv), but encoded in say TikZ/amcd/xypic/etc they do reduce to "language". But then there's also the aspect of the details of the diagrammatic proofs also not really being written out sometimes because they really are 'follow-your-nose'. I don't know if it's more or less than other areas of mathematics, though.
Mike Shulman (Jul 26 2026 at 01:10):Is there anyone here who has tried to use an AI to help them do category theory, whether it be chasing diagrams, other kinds of proofs, finding (counter)examples, or coming up with definitions or conjectures? If so, was it helpful? (And what model?)
Oisín (Jul 26 2026 at 08:42):Graham Manuell said:
I think current AIs are better at coming up with complicated examples than complicated proofs, though they clearly have some ability to do both. I haven't seen any examples of them coming up with new definitions. Maybe here is where fewer people have asked and whether a definition is any good is harder to evaluate, though I suspect current AIs would be worse at this.
Tech companies have put considerable effort over the last three years into benchmarking mathematical performance via olympiad style questions and are, in many cases, staffed by people with a
competition maths background as their primary exposure to mathematics.. In my limited experience, coming up with complex (counter)examples in algebra is much closer to olympiad maths than proving theorems in terms of workflow. Ie., generally one spots an invariant and plays with some combinatorics.
I suspect this is why they are weaker at category-theoretical problem solving, as the overlap in style is much less pronounced than in, say, combinatorics or modern analysis.
David Michael Roberts (Jul 26 2026 at 10:08):The only CT type work that I know of that's (publicly) gone into AI-for-maths work are those questions a little while back that people got paid for, where the answer had to be a number. So it was something like "how many X are there for data Y" where X might be something like "natural transformations"
David Michael Roberts (Jul 26 2026 at 10:08):....and the data is some specific setup where the X turn out to have finitely many.
David Michael Roberts (Jul 26 2026 at 10:13):Hardly the type of question that people in CT generally care about, and I found it kinda weird. One did have to know some amount of theory to be able to actually figure out the answer, but it's a different mode of reasoning than the usual yoga of "well this should probably be the answer based on analogies with a bunch of other cases as well as examples from a bunch of different fields of mathematics, so I'll make a definition so that the theorem works out to be true in the most conceptually-nice way". AI is only just now managing to get to the point, I think, of not just mangling definitions in eg Lean, and that's for well-attested structures. Of course, in six months, who knows? But no one's done something like "please give a new definition of -category so that the technical claims behind the geometric Langlands conjectures that need them work out quickly and cleanly" (these claims of course are now theorems, but it's still hundreds of pages of work to fill the gaps Gaitsgory and Rozenblyum left in their two-volume book)
Alexis Toumi (Jul 26 2026 at 13:18):I have access to Fable and I was curious so I gave it your prompt copied verbatim (forgot to turn memory off so it gives some personalised answer, could retry incognito mode) https://claude.ai/share/0af6aa2c-8b39-45b0-8e1b-ad07126c6b75
Alexis Toumi (Jul 26 2026 at 13:23):(I can’t judge the quality of the answer though it looked convincing to me and cited a recent relevant paper)
I’ve been experimenting with LLMs not to develop new category theory but to apply existing theory to software development and it’s been a lot of fun so far! they’re not great at inventing new definitions but they’re getting surprisingly good at applying existing ones to new settings, e.g. asking it develop neural network architectures in terms of monoidal categories has helped them produce much cleaner code than without these abstractions available (you can have a look at a sample here)
philip hackney (Jul 26 2026 at 14:36):Mike Shulman said:
On a slightly different topic, I was struck by the fact that at CT26 approximately zero of the talks I attended mentioned any use of AI. I don't have any experience yet myself actually using AI for category theory, since basically all I've been doing myself since it got good at math is coding. But I mentioned this to some non-category-theorist mathematician friends and they were also very surprised; one of them said that he just submitted a paper in which all the proofs were done by AI. Is there some reason that AI is less useful for category theory than for other branches of math? (I must admit that would please me, but it seems like wishful thinking.) Or are category theorists living in the past, either out of ignorance or by choice?
I sometimes use them to help understand things, including existing / known work. Have a had a Claude Pro subscription for roughly 2 years, sometimes dabble with ChatGPT (and earlier dabbled with Gemini). It's a mixed bag when asking theoretical questions (if I'd gotten the response Alexis just got from Fable I'd just close the window and do something else). Given how much effort is required to verify or understand what comes out, and how much semi-convincing garbage they produce, I'm not sure how much one could rely on them for higher category theory right now. Sometimes they're genuinely useful and I'll keep playing around to see how things develop, but currently for most of my own work it doesn't make much sense to start there. (I'm also not sure what it would even mean to have all of my proofs done by AI.)
For more computational stuff it's a completely different story, since the LLMs are good at writing programs for you. And depending on what you're doing and how serious it is, you can either verify the code yourself, or verify the output of the code.
David Michael Roberts (Jul 27 2026 at 01:37):@Alexis Toumi whose prompt?
Alexis Toumi (Jul 27 2026 at 08:32):please give a new definition of -category so that the technical claims behind the geometric Langlands conjectures that need them work out quickly and cleanly (these claims of course are now theorems, but it's still hundreds of pages of work to fill the gaps Gaitsgory and Rozenblyum left in their two-volume book)
I tried incognito but I can't share the results and it's hard to deactivate the personalisation for one run, I suspect Fable could have given me a sub-par answer because it knows from its memory of me that I'm a complete noob at higher category theory :joy_cat:
I also agree on the "semi-convincing garbage" judgement: these models are only useful if the answer is hard to come up with but easy to verify, I assume new definitions of infinity categories are exactly the opposite: easy to come up with (at least informally) but hard to verify. Giving the model access to a proof assistant is one way to handle with the verification, but you still have to verify that the verification aligns with what you asked.
David Michael Roberts (Jul 27 2026 at 09:52):Coming up with an informal idea is not hard because it is all over the literature. Actually making a real definition is much harder; coming up with a real definition of that magnitude that actually works as intended requires a different kind of thinking, it's a whole research program in itself
ww (Jul 27 2026 at 09:57):Well, that's the whole category theory magic trick, right? Come up with the right, often deceptively simple definitions so that the consequences neatly fall out of them...
Matteo Capucci (he/him) (Jul 27 2026 at 12:21):GR were doing formal category theory in a proarrow equipment without saying so, and the decade of gap-filling was the community rediscovering that companions and conjoints, not globular 2-cells, are the native language of base change
btw does this take-away from Fable vibe check? /ot
Amar Hadzihasanovic (Jul 27 2026 at 13:49):@David Michael Roberts My impression, which matches that of other higher category theorists that I spoke to at CT, is that the culture among the mathematicians that "apply" higher categories (in AG, AT...)---as opposed to those who study higher categories per se---has shifted towards a kind of "informal axiomatic" use where "reasonable" facts about the -category of -categories are assumed without too much care for their status as theorems about specific models. In this culture, I think that "finding a definition where the facts are (more or less easily) provable" would hardly be noticed.
It's quite sneaky as it shifts the 'semantics' from "filling some gaps in a proof" to "confirming that a model satisfies the expected axioms", where the latter carries less prestige...
(Sorry if this is off topic for this conversation)
Amar Hadzihasanovic (Jul 27 2026 at 13:55):To actually connect it to the topic, I do think it is interesting to ask what kind of role is AI likely to play in a community that does not value "full formal rigour" very highly, and this sort of "informal axiomatic proof" style is given as much value as a more rigorous style of proof.
Amar Hadzihasanovic (Jul 27 2026 at 14:03):My experience with LLMs is that they are good at acting and they are good at coding. Correspondingly, in mathematics, they are usually good
Amar Hadzihasanovic (Jul 27 2026 at 14:06):It seems credible to me that the "informal axiomatic reasoning" that has taken ground in applications of higher categories may be soon mastered by LLMs under the "acting" mode...
Amar Hadzihasanovic (Jul 27 2026 at 14:11):On the other hand, what this style seems really terrible for is computation; a lot of what this kind of higher category theory is producing is abstract sledgehammers that can hardly spit out a number if one tries. So I think the "coding" mode has little applicability for the time being.
Patrick Nicodemus (Jul 27 2026 at 16:37):Patrick Nicodemus said:
It finally hit a roadblock it couldn't solve in the proof of associativity of the 2-cell composition and i'll have to go in there and read a bunch of slop and get things sorted out. I will report back on how long this takes me and whether it would have been better to just write it myself.
Okay. My conclusion is that it's not worth it to let the machine just go on autopilot right now. It made a lot of progress but it got bogged down eventually. I think like, if you sketch a proof of a complex theorem involving many layers of definition/theorem/proof - let's say there's of these - and you want to make a small, independent tweak to each definition and theorem - then any change to one results in changes to everything downstream, so making all these changes takes time . This could be minor things like implicit argument conventions or major design choices (you defined this in a 'bundled' way when it had to be 'unbundled.')
I would say it took at least two day's worth of work to fix the one day of work done by the bot independently.
This was also an expensive experiment. Anthropic gave all subscribers $100 in free tokens to promote the use of Fable, and I easily burned through that in a few days using the Sonnet and Opus models.
However, the ultimate goal was successful in that I do have a definition of the free virtual double category on a given set of 2-cells, and a machine checked proof that the composition law is associative and unital. Viewing small virtual double categories as T-monoids in the large virtual double category of graphs, I am optimistic that this code will serve as a good prototype for defining the free T-monoid in other virtual double categories.
Next I plan to define the free VDC presented by generators and relations, and study some specific such VDC's corresponding to the algebraic theories of: a category, a functor, a natural transformation, an adjunction, and so on.
Mike Shulman (Jul 27 2026 at 16:59):Patrick Nicodemus said:
I spent maybe the better part of a day giving detailed instructions and setting up the main loop. The basic idea is that every 30 minutes or so, a new instance starts, reads the relevant code, reads the overall agenda, reads a chunk of work that is designated to it, tries to solve the problem, and then frames a 30 minute chunk of work for the next guy.
Can you point to a tutorial or something that explains how to set up something like this?
Patrick Nicodemus (Jul 27 2026 at 18:46):@Mike Shulman I didn't follow a tutorial for this, I just played around with it for a while. I use Claude Code, and there are a number of useful built-in commands which can help to get something like this off the ground - https://code.claude.com/docs/en/commands - in particular there is /loop, and the page https://code.claude.com/docs/en/sub-agents discusses how to have a parent bot launch and coordinate child bots. I think the most popular open-source alternative to Claude Code is OpenCode, which can be connected to (at least) any model hosted on OpenRouter, I would imagine that piece of software has similar functionality. But it's all vibecoded, so, you know, high expectations for sleek UI, low expectations for basic functionality working reliably.
Here is the repo. To initiate the loop, I open up Claude Code and say "Read parent_instructions.typ", and it ingests that file. You can ask it to generate the /loop command for you, or you can write something like "/loop every 30 minutes, launch a sub-agent, and have it read and carry out the instructions in loop.typ".
The human-authored documents are parent_instructions.typ, loop.typ, guidelines.typ, current_project_outline.typ. The AI-authored documents are handoff.typ, agenda.typ and todo_future_work.typ. The idea is that current_project_outline.typ is a high level sketch, and then agenda.typ has more granularity and should be continuously edited by the bot as the project evolves.
https://github.com/patrick-nicodemus/coherence_operads
Mike Shulman (Jul 27 2026 at 18:49):Ok, thanks. I use Claude Code too, but I haven't figured out how to use /loop yet. I have tried to use /goal but couldn't figure out how to get it to keep running autonomously; it would go through one or two sessions and then always stop saying the goal-testing agent received too much data.
Patrick Nicodemus (Jul 27 2026 at 18:52):@Mike Shulman There should be a way to trigger the parent to respond to external events such as the agent returning, so I think /loop is maybe incomplete, it might be better to have the parent always act in response to the child terminating.
then always stop saying the goal-testing agent received too much data.
I'm not sure if we're talking about the exact same thing here, but my whole goal here with this 30 minute loop was to cap out the maximum context usage of any one agent in the series. Was your problem that the proof assistant goals were too large, in terms of the character count?
Mike Shulman (Jul 27 2026 at 18:57):I don't know, I didn't really understand. /goal is supposed to run a quick agent after each main session to check whether the goal was achieved, and if not start another session to continue it. That seemed to me like what I wanted, to make the AI "continue working" on the same overall goal without my needing to tell it to go on every time it stopped. The goal test I set was pretty short, but possibly the checker was reading the entire output of the main agent; but if it has to do that and that was too long for it, then I don't see what use the /goal command is. I had thought that /loop was more for permanent periodic things like checking the status of a CI, but maybe that was wrong.
philip hackney (Aug 01 2026 at 05:32):philip hackney said:
It's a mixed bag when asking theoretical questions (if I'd gotten the response Alexis just got from Fable I'd just close the window and do something else). Given how much effort is required to verify or understand what comes out, and how much semi-convincing garbage they produce, I'm not sure how much one could rely on them for higher category theory right now. Sometimes they're genuinely useful and I'll keep playing around to see how things develop, but currently for most of my own work it doesn't make much sense to start there. (I'm also not sure what it would even mean to have all of my proofs done by AI.)
For more computational stuff it's a completely different story, since the LLMs are good at writing programs for you.
Ehhhh ok, maybe the situation is different now. As an experiment inspired by @Mike Shulman 's question, I went on a little sidequest last night / today. I provided some background and asked about a question that's been bouncing around the back of my head for a couple years that I've never gotten around to, and probably wouldn't. I think I convinced myself at some point that it wasn't true, so it's not something I would've handed to a student or anything. But recently I got some more examples that I could feed in that I thought might've turned out to be counterexamples.
I prompted Claude Opus 5 (Medium) with some resources and the question. It wrote some python code, went through a bunch of examples and sketched a proof of one direction, gave an idea for the other direction. Then wrote up proofs of the whole thing, approximately five pages. Four prompts total, the second two of which were "Please commit them. Then try out the retract lemma in the way you think best." and "sure let's do that" lol.
I committed myself to actually checking it carefully. A couple minor issues, but overall very good, and proofs are correct. Pleasant to read, but that's probably because the "resources" I supplied were things that I wrote (papers and portions of papers), so I guess it's perfectly targeted to me.
philip hackney (Aug 01 2026 at 05:39):Mike Shulman said:
one of them said that he just submitted a paper in which all the proofs were done by AI
Now I know what this means, but am even more confused. What do I do with this? It'd probably have been a paper I'd read if someone else wrote it, or a little paper I'd have felt good about if I figured it out and wrote it myself. But feels a bit worthless?
Mike Shulman (Aug 01 2026 at 06:41):Yeah, I think this is the big question (ok, well, one of the big questions) about what mathematics will look like in the age of AI.
Why is it worthless? Because a human can't feel satisfied with, or get credit for, having solving it? I agree that those are important things that are missing. But it's still a contribution to knowledge -- even, I daresay, to human knowledge, since you've now read it and understood it and learned something from it, and so could other humans. Surely that's not completely worthless either.
My friend said that he thinks journals should continue to play the role of "repositories of correct mathematics" regardless of how that mathematics is produced, but that the allocation of credit to human mathematicians will have to adapt and be assigned some other way. That's one possibility; I suppose there are others.
philip hackney (Aug 02 2026 at 00:12):You make very good points here, @Mike Shulman. I think it's a lot to contend with, and wasn't quite what I'd have expected even a few months ago. I think a better word than "worthless" is "hollow" -- it's a feeling similar to when you're stuck in a puzzle or adventure game and you look up the answer online. "Why am I even playing the game?" But a big difference is that we're not just playing a game here (though for me some of the best times do feel like that).
philip hackney (Aug 02 2026 at 00:19):For now I've just documented the process. This post includes the actual problem/theorem, a narrative from my POV, (lightly-sanitized) Claude Code transcripts, a git repository containing inputs/outputs.
Mike Shulman (Aug 02 2026 at 01:11):At this point it seems not impossible to me that eventually there will no longer be a societal need for professional full-time mathematicians: that whatever mathematics needs to be done willl be done by non-specialists using AI, the same way today anyone who needs to do calculations can do them themselves using an electronic computer rather than employing a human "computer" as was common a century ago. Those of us who enjoy "playing the game" might continue to do mathematics ourselves as a hobby, but (in the world I'm imagining) that sort of "artisanal" mathematics would be displaced in the large-scale economy by "industrial" mathematics done by AI.
If that happened, I would be sad, because I think something deep and meaningful about "artisanal" mathematics would be lost. But perhaps, as with all the other artisanal occupations that have been reduced to hobbies and displaced by industrial versions, the effect for society as a whole would be beneficial?
Morgan Rogers (he/him) (Aug 02 2026 at 07:45):I'm not convinced there has ever been an immediate societal need for professional full-time mathematicians; people doing "mathematics that needs to be done" tend to be called engineers.
In the world you describe @Mike Shulman , scientific research returns to being a hobby for the privileged. A friend of mine who recently quit academia argued to me that it's only by historical accident that this ever stopped being the case (to the extent that access to the required training and integration into the relevant communities broadened during the 20th century, anyhow).
But why would that be a world I accept to participate in bringing about? Whom exactly is that world better for? It doesn't seem like it would be better for anyone I know personally.
Jad Koleilat (Aug 02 2026 at 15:08):Hello, I usually only read messages and never post anything on this zulip but after reading the last few messages I want to contribute. If you don't know me, I am a PhD student in Université Sorbonne Paris Nord (France).
I often read posts arguing or explaining what AI is able or not able to do in category theory or in other fields of maths/CS. My opinion on the mater is that with sufficient neurons, energy and time AI will surely be able to do anything we do. In my opinion the question of the place of AI in our research is not a question of what it is able to do, but at what cost. It is, I think , a bit what @Morgan Rogers (he/him) is saying in his last message: what do we want mathematical research to be like ?
I won't talk about environmental or the societal implication of contributing to this kind of tools and focus only on the aspect related directly to our practices. Before, I want to make clear that even if in this post I separate research from the rest of the world, I think it is false to operate this separation. We do not leave in a ivory tower separated from the rest of the world.
As a PhD student I live in fear that tomorrow I am going to see on ArXiv a paper, written by an LLM, claiming to answer a problem I am working on. This is also a fear many of my PhD friends have. I don't want to use AI for my research because (of ethical and political reasons and) I see the effect it has on the students I teach. I am learning about my field, the different techniques and perspectives etc... as you all know, this requires to understand and apply concepts/techniques. I wont be able to develop a feeling for things if I don't do it myself. I won't be able to learn how to approach a problem if I don't try and fail many times. If I want to be a fully fledged researcher I can not afford to use AI (contrary to what some people say). The students I teach that use AI are able to produce code that compiles and produce a decent enough output but they lack a deep understanding of it. They are often not able (or with much hardship) to solve bugs or have a macro vision of what their code does. I see no reason why I, as a "apprentice researcher", would be immune to those same effects I see in my own students. I don't want a world in which every PhD subject can be done in a few hours by the latest model, what will be left for the next generation of PhD students ? I don't blame the phd student that use AI, I have two friends that use it because they feel pressured into it. They think that without it they won't be "competitive", I understand their fear.
There is also the matter of getting a permanent position. In every sector in which AI got forcefully pushed a massive wave of firing took place. I don't see why AI getting pushed in mathematics will have a different effect. Thus, I am very hurt when I see fellow researchers pushing hard for AI, they are actively fighting against me and my friends getting jobs... I am more hurt by what is happening to math than what is happening to the programming world because it is not authoritarian bosses that are pushing AI against us, it is our own colleagues.
This last point is very important. As Morgan kind of said, we don't have a need to increase our efficiency. Why do we need to go faster ? The life of no one depends on the Riemann hypothesis being solved in the next ten years. On the contrary, the life of many students depends on job being available in next years to come. Since AI adoption is not something being pushed by an overpowering authority but by members of our own community it is something we can fight against. I don't care about speculation of what math will be in the next years I care about fighting right now for jobs and a practice of research that I enjoy and find meaning in.
Tom de Jong (Aug 02 2026 at 15:18):I don't frequently post here either, but let me make an exception: Thank you for your sincere contribution @Jad Koleilat. I appreciate that you wrote up your perspective.
Joshua Wrigley (Aug 03 2026 at 21:01):Jad Koleilat said:
As a PhD student I live in fear that tomorrow I am going to see on ArXiv a paper, written by an LLM, claiming to answer a problem I am working on. This is also a fear many of my PhD friends have.
Thank you for sharing your experience. As someone a couple of years further down the research pipeline, I have carved out a niche enough research theme that (perhaps naively) I do not worry too much about certain ideas being "scooped"; I hope that you and your PhD colleagues still have the same opportunity to experience becoming expert on a topic, and genuine human receptacles of human knowledge.
Alexander Kurz (Aug 14 2026 at 11:37):Morgan Rogers (he/him) said:
But why would that be a world I accept to participate in bringing about? Whom exactly is that world better for? It doesn't seem like it would be better for anyone I know personally.
I agree with the sentiment that this would not be a world I would like to help bringing about. But if we want to prevent this world from taking form, it is already quite late to take action against it. If we dont like the idea of such a world (how do we want to name it?), we should think about how to organize for defending humanity against ai. By this I dont necessarily mean ai as a technology. I use ai every day, all the time. But I want to see a world where we can use ai as a tool (if we so wish to do) as opposed to a world in which the owners of ai use us as a tool.
Alexander Kurz (Aug 14 2026 at 12:53):Jad Koleilat said:
I don't want to use AI for my research because (of ethical and political reasons and) I see the effect it has on the students I teach. I am learning about my field, the different techniques and perspectives etc... as you all know, this requires to understand and apply concepts/techniques. I wont be able to develop a feeling for things if I don't do it myself. I won't be able to learn how to approach a problem if I don't try and fail many times. If I want to be a fully fledged researcher I can not afford to use AI (contrary to what some people say). The students I teach that use AI are able to produce code that compiles and produce a decent enough output but they lack a deep understanding of it. They are often not able (or with much hardship) to solve bugs or have a macro vision of what their code does. I see no reason why I, as a "apprentice researcher", would be immune to those same effects I see in my own students.
This is very well said, thanks. I am a late-career researcher, so in a very different situation. I love to use AI to learn about other fields of research, math is so rich and AI has seen everything. But I am glad I went through the "try and fail many times" before AI came around. And now, as a teacher of math and CS, I have the same concerns as you do. Every year I try sth new, but I didnt yet solve the problem of how to teach in times of AI. How can we set up classes that encourage deep and slow learning while we know that students will use AI?
John Baez (Aug 14 2026 at 13:48):The two basic strategies are to prevent or discourage the students from using AI on certain tasks, or to allow them to use AI and give them tasks that are challenging even with AI assistance. One can do a mix of both.
(If you just discourage, not prevent, you and the students will always be worrying about cheating.)
John Baez (Aug 14 2026 at 16:40):I should add that my parenthetical remark assumes "credentialism": the idea that education is at least in part about getting credentials, which tends to make it into a competition and encourages cheating. I would love to get past this, but it's very hard to do, because it's deeply built into how education is done.
Chris Grossack (she/they) (Aug 15 2026 at 04:53):Alexander Kurz said:
How can we set up classes that encourage deep and slow learning while we know that students will use AI?
One option that a friend of mine does is to make the homework worth a fairly small fraction of the grade (since one can use the tool du jour on it -- currently LLMs, but historically wolframalpha, etc.) and then have (semi)frequent in-class or in-discussion quizzes which the students are told will be pulled entirely from homework problems. These are weighted a fairly large fraction of the grade.
Whatever resources they use to solve the homework, this approach tests their actual understanding of the problems in a situation where they're using their own brain... or at least it tests their ability to regurgitate ideas in a situation where they're using their own brain, but that's been a problem since time immemorial. Anyways, this encourages students to actually do the homework (or at least try to actually understand the LLM output of their homework solutions) since the larger fraction of the grade will be LLM-less.
Mike Shulman (Aug 15 2026 at 19:01):Sorry to disappear for a while; I started writing a reply and then got shanghaied by personal issues and a planned vacation.
Morgan Rogers (he/him) said:
I'm not convinced there has ever been an immediate societal need for professional full-time mathematicians; people doing "mathematics that needs to be done" tend to be called engineers.
That may be true, but if so only because of the word "immediate". I don't think that, say, the NSF's funding of pure mathematics research over the past 75 years has been without benefit to society at large. This is the old "value of pure mathematics" debate. While lots of pure mathematics may not (yet) have any use, I think a fair amount of pure mathematics does eventually have a use.
In the world you describe Mike Shulman , scientific research returns to being a hobby for the privileged.
Well, I certainly don't see the social need for all scientific research going away, nor the impetus for public funding of it (at least, not because of AI). What I was imagining is that -- maybe -- public funding of mathematics research might be reduced from dedicated grants to mathematicians to become line items in grants to other scientists. I'm not saying I necessarily think that will happen; in fact, I can think of several arguments why it may not.
But if what you meant is that mathematical research returns to being a hobby for the privileged, then yes, perhaps that's more like the world I was imagining -- except that they might not need to be so privileged any more. Many technologies have a democratizing effect, and I can imagine AI helping to make higher mathematics accessible to many more hobbyists. (Of course, as has been pointed out elsewhere, at the moment the AI models capable of mathematical research come with a hefty price tag. But it's not hard to imagine that someday, perhaps soon, the free ones will be able to do serious mathematics as well.)
This democratization is also related to your last set of questions:
But why would that be a world I accept to participate in bringing about? Whom exactly is that world better for? It doesn't seem like it would be better for anyone I know personally.
For what it's worth, I don't consider myself to be "pushing" AI. I strongly disapprove of many actions taken by AI companies, including "copyright violation laundering". I worry about AI in the hands of malicious actors, as well as rogue AI behavior and "responsibility laundering". Sometimes I feel close to despair at the effect of AI on our ability to teach students basic skills, particularly reading and writing. And I worry about how AI will affect the research ecosystem, particularly early-career researchers, many of whom, as Jad pointed out, currently live in fear of being scooped by AI. And, as I said, I would be sad if the way humans do mathematics now were relegated to hobbyists.
At the same time, however, I think the evidence of history suggests that once any technology exists, it's nearly impossible to "put it back in the bottle". So it behooves all of us to get familiar with what these tools can, and therefore will, do. Similarly, speculating about possible futures is valuable even if they are undesirable, to help us make more informed decisions about how to affect the future.
I don't know whether the world I was imagining would be better or worse for humanity overall. I'm generally an optimist about technological progress, but in the case of AI I find it more difficult than usual (although I did my best in my BAMS article about it). Nevertheless, "it doesn't seem like it would be better for anyone I know personally" sets off alarm bells in my head, as that same phrase could have been said by many, many people facing the prospect of changes to, or elimination of, their jobs due to technological advance, but where looking back from a perspective of decades or centuries in the future, it is obvious that the technological advance benefited society as a whole.
In the case of AI, some of the benefits are already obvious. I haven't used AI for mathematical research yet, but I have used it for coding, and it is saving me a lot of time and enabling me to implement features that would have taken much, much longer by hand and perhaps never been gotten to. But I've also found that I have to be very careful to read and vet everything it produces, resisting the temptation to just trust its code because it "seems to work". I shudder to think what would happen if I didn't already know how to code, and I expect the same is true about mathematics.
I don't know what we should do about teaching at the undergraduate level and below. Personally, I'm going to muddle through this academic year somehow, maybe with oral exams, and then go on sabbatical, and after that who knows what the world will be like. And I'm not really very qualified to speak about the graduate level since I don't regularly advise graduate students, but shooting from the hip my advice would be to try to find a middle ground.
It's absolutely true that if you want to become a fully fledged researcher you need to be able to do mathematics without AI. But it also seems unlikely to me that 5, 10, 20 years in the future any researcher will be able to "compete" without using AI. (In some sense there's nothing new about this -- I firmly believe everyone should learn long division, but also in practice we use a calculator for any substantial division problem.) At the very least, testing how easy it is for AI to solve your research problems might give you some idea of how worried you should be about someone else using it to scoop you, and perhaps whether you should look for a different problem. It seems unavoidable to me that the nature of "PhD problem" will have to change -- which, of course, also happened with the introduction of ordinary computers.
I could be very wrong, of course, but those are my thoughts at the moment.
Morgan Rogers (he/him) (Aug 17 2026 at 09:52):Mike Shulman said:
At the same time, however, I think the evidence of history suggests that once any technology exists, it's nearly impossible to "put it back in the bottle". So it behooves all of us to get familiar with what these tools can, and therefore will, do.
Have you seen a dirigible recently? Or a guillotine?
However much I would enjoy the return of these technologies, I must reject the argument that something being shown possible makes it inevitable.
Mike Shulman said:
Nevertheless, "it doesn't seem like it would be better for anyone I know personally" sets off alarm bells in my head, as that same phrase could have been said by many, many people facing the prospect of changes to, or elimination of, their jobs due to technological advance, but where looking back from a perspective of decades or centuries in the future, it is obvious that the technological advance benefited society as a whole.
There's a problem with the causation sequence here. It's not so clear to me that improvements in society -- which I interpret to mean the general trend towards improved living conditions over time that is presently collapsing -- over the decades or centuries can be attributed to the mere existence or widespread use of any technology. Perhaps you're thinking of the industrial revolution, which entrenched systems of exploitation that continue to this day (only now they're far enough away from us that we can pretend they don't underpin the wealth of our own nations)?
It's easy to see in the present that the benefits of LLMs are concentrated in the hands of a small number of people and easily arguable that they have had a net negative effect on society so far (you pointed to some of the problems). The cost-benefit should certainly not be reduced to an individual level -- the direct cost to the consumer of LLM services is by no means reflective of the indirect costs that we must collectively shoulder to enable this technology to exist, including but not limited to the consequences of them exceeding our capacity of available energy and water, which in turn will negatively impact the cost of living in all sorts of ways.
Morgan Rogers (he/him) (Aug 17 2026 at 09:54):Mike Shulman said:
It's absolutely true that if you want to become a fully fledged researcher you need to be able to do mathematics without AI. But it also seems unlikely to me that 5, 10, 20 years in the future any researcher will be able to "compete" without using AI.
My personal response to this is that I would be prepared to invest a lot of time and energy into transforming academia not to include AI but to remove competition.
Mike Shulman (Aug 17 2026 at 16:50):Experience has shown me that it does no good to try to argue with people expressing those sorts of beliefs, so I will now bow out of the conversation.
Mike Shulman (Aug 18 2026 at 04:31):Although maybe I can clarify, without restarting an argument, that my use of the word "compete" was perhaps ill-advised. By putting it in scare quotes I meant to indicate that I didn't mean it literally, but it could easily have been misunderstood. What I really meant was something more like "contribute usefully to the scientific enterprise", which would be equally relevant even if science became entirely non-competitive.
Matteo Capucci (he/him) (Aug 18 2026 at 09:17):Morgan Rogers (he/him) said:
Or a guillotine?
Actually, yes
Matteo Capucci (he/him) (Aug 18 2026 at 09:25):Mike Shulman said:
Experience has shown me that it does no good to try to argue with people expressing those sorts of beliefs, so I will now bow out of the conversation.
I'm sorry you feel this way Mike, I appreciated you sharing your views here. For what it's worth, I too think AI is here to stay, because it's evidently a very powerful thing, and people (good and bad) like power. What really troubles me are the socioeconomical conditions in which this technological revolution is setting off.
Matteo Capucci (he/him) (Aug 18 2026 at 09:30):Morgan Rogers (he/him) said:
. It's not so clear to me that improvements in society -- which I interpret to mean the general trend towards improved living conditions over time that is presently collapsing -- over the decades or centuries can be attributed to the mere existence or widespread use of any technology. Perhaps you're thinking of the industrial revolution, which entrenched systems of exploitation that continue to this day (only now they're far enough away from us that we can pretend they don't underpin the wealth of our own nations)?
That's a bit tendentious, isn't it? I'm not a techno-optimist but I'm a techno-realist. Technology did wonders for humanity, and a good deal of that can be attributed to automation. It used to be that >90% people had to work in food production, and now it is way less that 10% in developed economies. Those 80% of people are doing things like waiting at cafès, writing TV shows, staff universities, and some research category theory too!
That doesn't mean everything has been great (climate crisis, anyone?) and I agree that a lot of exploitation has been moved to the global south (that's quite a recent thing though---and less due to technological improvement and more to political and economical factors).
Morgan Rogers (he/him) (Aug 18 2026 at 10:29):You're drawing the same line from technology merely existing to the improvements you're talking about as Mike did. Automation doesn't inherently make life better:
What exactly is it about automation that improved society? Lowering the output costs has as a side effect that a greater proportion of people can afford a product; goods that were formerly luxuries became accessible to more people. Capital was thus able to drum up greater demand to produce a better return on their investment into automation.
Those that are above the new line of affordability find themselves more comfortable, with certain aspects of their lives more convenient, while those that are pushed out (and fall below the line due to automation) fade into history.
To put it succinctly, technology tends to benefit its consumers, not its workers.
It's the workers that this discussion is about.
Morgan Rogers (he/him) (Aug 18 2026 at 10:33):Matteo Capucci (he/him) said:
Those 80% of people are doing things like waiting at cafès, writing TV shows, staff universities, and some research category theory too!
If automation previously eliminated jobs that we now view as low-skilled (ironically that label is often only applicable due to automation lowering the skills required to do these jobs) and left 'more room' for people to pursue higher-skilled jobs, and that's an improvement, then what could the benefit of eliminating higher-skilled jobs be?
John Baez (Aug 26 2026 at 15:22):The differential geometer Marco Gualteri has an opinion piece on AI that will appear in the Notices of the American Mathematical Society:
It's short, clear, and I hope it improves the conversation.
On Bluesky he was a bit more forthcoming about his motives for writing it:
In response to what I view as irresponsible messaging and unfortunate decisions from some mathematical leaders influencing many in my department.
That's not so clear, but I feel I know exactly what he's talking about.
I responded here.
David Egolf (he/him) (Aug 26 2026 at 19:04):On Mathstodon, @John Baez wrote:
What I know from experience is that current LLMs are good at proving theorems, including novel theorems - good enough to dramatically transform the field of mathematics. I don't claim to know how they do this, and actually I don't much care. I just know that they do it.
Of course proving theorems is just part of what a mathematician does.
In this context, it seems valuable to consider which other skills are important for a mathematician - and to consider how those skills can be developed.
John Baez (Aug 27 2026 at 15:47):Indeed, mathematicians are being forced now to be a lot more explicit about these!
Beppe Metere (Aug 31 2026 at 14:48):Concerning AI for mathematical research, I wrote a short note here
A point I would like to strengthen is the following.
We have used paper-written articles for a very long time, without substantial changes.
Indeed, during the Ellenistic era, the Library of Alexandria collected and catalogued texts from across the ancient world. Scholars compared manuscripts, corrected errors, marked doubtful passages, and produced critical editions. Their commentaries explained editorial choices and debated earlier interpretations.
Papyrus rolls where then substituted by paper books (note the etymology!) made of pages... and still now, scientific papers (note the etymology again!) are usually organised in pages, even if they are just pdf files.
Books and journals survived the digital era. Will they survive the new LLM-AI era?
I think so, just like videos did not actually kill the radio stars, and photography did not cancel drawing and painting. However, new technology affects the way knowledge is produced, stored and retrieved.
Most of the comments I've come across on the use of AI for research focus on the production, but I believe that (maybe more) influential changes will affect storing and retrieval of knowledge, and production as a consequence.
It is necessary to rethink how knowledge is stored in databases, and how it is recovered. Here AI can be of great help. I imagine distributed databases where scholars can put the results they find, which can be verified by AI and validated by humans, if necessary. Papers will then be on an "on-demand" basis, where the AI system can help in gathering the information and connect different results, and trace the credits to the authors. Something that is already happening is showing us a direction of development, with projects like the "nLab", or "the Stacks project", but these belong to the pre-IA era.
The new databases will probably overcome journals, but maybe not books, which will keep their role of conceptual pieces.
Another point concerns the effects of AI on learning mathematics. This will also have consequences on mathematical research, but it is slightly OT for this tread.
Morgan Rogers (he/him) (Aug 31 2026 at 15:21):I like the optimism @Beppe Metere , but I'm not convinced that any of the systems involved are flexible enough to adapt before a lot of damage has been done to the people involved. Across Europe things are moving in the direction of reducing public funding in research, which means in particular being more selective about funding allocation. I agree that volume of publications as a proxy measure of scientific ability is even worse under LLMs than it ever has been, but if no one supplies a credible alternative, that is still what will be used by default to make those selections, and hence there is more incentive than ever to participate in the LLM-augmented publication race in the short term.
But maybe we can find a different proxy. Do you have any ideas?
Nicolai Kraus (Sep 06 2026 at 13:19):I agree with you @Morgan Rogers (he/him).
The problem isn't awareness. Pretty much every funder and every hiring committee agrees that it's "quality over quantity", and pretty much all (Western) such entities have signed DORA, which says exactly that metrics don't work.
The problem is the lack of an alternative, and as long as that's not provided, metrices will be used.
LLMs just make the existing bad solution (metrics) even worse, but concluding that this means that a better alternative will be found and implemented quickly seems like wishful thinking to me. I really hope I'm wrong.
John Baez (Sep 08 2026 at 10:53):Today's drama concerns the race to solve the Millennium Prize problem connected to the Navier-Stokes equation. For the math, read Terence Tao's summary and the actual papers by Tristan Buckmaster, Levent Alpöge and others. For the (in)human drama, read Buckmaster's statement. A quote:
There is another part of this story, and one that, honestly, I very much wish
I did not have to be concerned with. I am not happy about the presentation
quality in these papers. Ideally, we would have preferred to spend weeks turning
the LLM generated proofs into something readable from the very first page.
This level of care is what these problems and the community devoted to these
problems deserves. The various Boussinesq and Euler write-ups in particular
are much closer to what models produce under human direction than to a paper
written by a person. The Euler writeup, in particular, can only be described as
AI slop. I am sorry for this. The reasons are below, and they involve our being
pressured by outside factors. I say this not as an excuse but as an explanation.I had planned to say on announcing our work that the results are not the
important thing. Rather the important thing is instead the significance that a
mathematician and an LLM model can now do all this work in a month. The
significance of this with respect to the way we train students, assign credit,
referee, and decide what is worth one human life’s attention cannot be under-
stated. This is a a Deep Blue-Kasparov moment. The community needs to have
serious and unhurried discussion about where to go from here. Instead of these
incredibly important developments, I find myself writing about something else.
I want to set out what happened as plainly as I can.On Thursday, September 3rd, with a rumor circulating that Anthropic had
resolved a major open problem, and with Levent having received tips that infor-
mation about our progress had been passed to OpenAI, I wrote to a prominent
mathematician at OpenAI. I am quoting my email in full because I would rather
the full text be read rather than my summary of it:“Hi [...],
We have not met in person, but I was one of the speakers at the [...].
I am writing because a rumor seems to be spreading quickly that Anthropic
has resolved a major open problem. I cannot be certain I am the person it
attaches to, but a colleague at Courant emailed me about it yesterday – who
heard it from an analyst in the UK, who had it from somewhere further upstream
– so it seems safe to assume I am. I gather versions linking Levent Alp¨oge to a
solution are going around in tech as well. There however appears to be a lot of
confusion with regards to the exact problem solved.I should also emphasize that this is not an institutional effort. It is a strictly
personal collaboration between the two of us, and there is no formal agreement
behind it. I pay for the tools my group uses out of my own research funds,
including footing a large bill to OpenAI. I have had an industry collaboration
before, with DeepMind, which had a formal institutional arrangement.
We do have work in this area that we are confident in, and we will post
it shortly, the paper and the formalization together. We intentionally decided
against rushing out a Lean certificate alongside an unpolished preprint. I feel
strongly that the first thing anyone reads should be a mathematical argument
presented in the normal manner, rather than just a formal certificate.
I am writing to you directly rather than saying anything publicly so that
you have the facts to address this on your end.Best wishes,
Tristan”He replied the same day:
“If you are willing to give any details it would be useful to avoid competing
here and in general we are always thrilled when mathematician make progress
with our models. Additionally if there is anything in terms of compute from
OpenAI’s end we would be happy to provide it.”I asked to speak the following week. On Friday, September 4th, I was asked
whether I could meet that day; I again said the following week. At 12:45 on
Sunday, September 6th, I was asked whether I could meet “at any point today.”
Sebastien Bubeck joined. The three of us spoke twice that afternoon. Levent
was not on the calls.I was told that an internal OpenAI model had produced a proof of finite time
blowup for the forced Navier-Stokes equations. When Levent asked by text for
the precise statement, the answer was: “Existence of forced blowup in and
,” with “the forcing function is smooth option c and d in Fefferman.” I was
told the proof is about 100 pages. I have not seen it.I should say here why I interpreted their statement the way I did, the in-
terpretation I will discuss below. The route to the Clay problem through a
smooth force, options c and d in Fefferman’s statement of the problem, is the
route Luis and Diego opened and the one Levent and I had quietly chosen to
attack. Almost nobody else I know of was working on it. It is not the direction
one arrives at in a few days by giving a model the problem statement. When I
heard “forced,” it was a bright red flag.I was shown a prompt and told the internal research model had simply been
given the problem statement. Levent had been told by Sebastien “very little
human input” had been used. This turned out not to be true. Over the course
of the call, as members of their team sent Sebastien corrections and details over
their internal chat, it emerged that an entire team had been working on the
problem, that this was one of a number of things that was tried, that work had
started on the unforced problem, that the team first set the model on easier
problems, including Euler, that even the prompt that had been shown to me
had been written by prompting Codex, and that an insane amount of compute
had been used.I asked when the first prompt had been sent by them. This question was
not answered directly by OpenAI for some time. Eventually it was agreed that
it had been sent in the past few days, after information about our work had
reached OpenAI.I asked whether the model had been trained on, or had access to, our sessions
in Codex, into which we had been putting all our drafts for the whole of this
project. I was told the model did not look up user data. I asked again, about
training, and I did not get an answer.
Morgan Rogers (he/him) (Sep 08 2026 at 10:56):I'm glad you posted about this @John Baez ; I heard about it from a PhD student. I'm curious what the reactions will be.
John Baez (Sep 08 2026 at 11:04):So far the anonymous person at OpenAI is coming across as the villain (it gets worse than the part I quoted). But of course we are just getting one side.
I've heard stories of a Nobel prize winning experimental particle physicist being videotaped by security cameras pissing on someone else's equipment, so I'm not surprised that some people get nasty when they think the stakes are high.
JR (Sep 08 2026 at 12:57):John Baez said:
So far the anonymous person at OpenAI is coming across as the villain (it gets worse than the part I quoted). But of course we are just getting one side.
I've heard stories of a Nobel prize winning experimental particle physicist being videotaped by security cameras pissing on someone else's equipment, so I'm not surprised that some people get nasty when they think the stakes are high.
Tell me it's Ting
Beppe Metere (Sep 08 2026 at 15:05):The facts concerning this dispute still need to be clarified. However, independently of what will finally emerge, this episode has already shown a structural problem that goes much beyond.
The companies developing frontier AI systems are playing several roles at the same time. They provide researchers with increasingly essential tools, but then they keep their prompts, drafts and failed attempts on their servers. Moreover, they decide how much computational power each user can access; and, at the same time, they may compete with those researchers for the same scientific results.
This is clearly not a good situation for science. If access to the most advanced models becomes necessary in order to make progress in mathematics, physics or biology, the companies controlling these models will also have an enormous influence on the production of scientific knowledge. They can decide who receives early access, who obtains more computational resources, which universities become privileged partners and which countries, institutions or individual researchers remain excluded. Prices, usage limits and corporate policies may end up shaping the direction of scientific research as much as universities and public funding agencies do.
The Alpöge–Buckmaster affair should therefore be taken as a warning. AI is quickly becoming a fundamental infrastructure for scientific research, but this infrastructure remains almost completely private, centralised and opaque. Public research institutions risk becoming dependent on companies whose commercial interests do not necessarily coincide with the principles of open research, confidentiality and correct scientific attribution. If AI is going to become a fundamental instrument of science, advanced models and the computational resources necessary to use them should become part of a public, open and internationally accessible research infrastructure.
Beppe Metere (Sep 08 2026 at 15:19):A possible reaction to this scenario was proposed by Nobel Prize-winner Giorgio Parisi.
Rather than leaving computing infrastructure and data processing exclusively in the hands of private Big Tech's, Parisi and his colleagues propose an alternative model:
https://www.change.org/p/a-manifesto-for-the-creation-of-a-european-center-for-ai
Morgan Rogers (he/him) (Sep 08 2026 at 15:38):Beppe Metere said:
The Alpöge–Buckmaster affair should therefore be taken as a warning. AI is quickly becoming a fundamental infrastructure for scientific research, but this infrastructure remains almost completely private, centralised and opaque. Public research institutions risk becoming dependent on companies whose commercial interests do not necessarily coincide with the principles of open research, confidentiality and correct scientific attribution. If AI is going to become a fundamental instrument of science, advanced models and the computational resources necessary to use them should become part of a public, open and internationally accessible research infrastructure.
There's a little "if" here that I would prefer to be a much bigger "if". I'm still in the camp of resisting this slide as far as possible.
To me, this business shows just how deeply parasitic AI and its infrastructure is on human knowledge. This effort by two people and hundreds to thousands of hours of AI work has obliterated a research program led by two people, Cordoba and Martınez-Zoroa, that don't appear to have been invited to join in the effort (even if they were credited this time). The resulting work is unpolished and had to be pushed out extra quickly because a hit squad at OpenAI decided to try to snipe the problem; irrespective of whether they were able to benefit from the researchers' efforts as they were making them, this is destructive pressure.
I agree with the spirit of Tony Vladusich's response in the thread under Tao's explanation that John linked:
AI in science is a devil's bargain, but the one who makes it isn't the only one who pays a price.
Beppe Metere (Sep 08 2026 at 16:02):Morgan Rogers (he/him) said:
AI in science is a devil's bargain, but the one who makes it isn't the only one who pays a price.
The metaphor of the devil’s bargain seems to me too focused on individual choices. The main consequences of AI in science are collective: they concern mathematical communities, the relations between research groups and the role of public institutions.
AI may (will?) give decisive advantages to researchers with privileged access, make entire research programmes suddenly obsolete and increase the dependence of universities on private companies.
Finally, even mathematicians who choose not to use it will work in an environment transformed by it.
James Deikun (Sep 08 2026 at 17:44):The fact of the matter is that as long as the AI companies are permitted to create and do whatever they want, there is not a meaningful decision to be made by the scientific community in the long term as to whether scientific progress will be dependent on AI. If we keep seeing increasingly powerful AI chatbots and agents, the scientific establishment will either make use of them at increasingly high ratios to human effort, or be outcompeted by an alternative scientific establishment that does. The limitations of AI, while presently significant, are transitory in the face of technical progress; the limitations of the human condition are essentially permanent.
John Baez (Sep 08 2026 at 18:10):Wow, and now the bastards claim they've done it. :smirk: And formalized it.
OpenAI came out with this:
Some quotes:
We used a system of coordinating agents powered by our internal model. The agents had access to tools such as the ability to read from a cached version of the internet and the ability to run code. Agents were subdivided into groups with the ability to communicate within the group. The groups varied in size, and the group that produced the Navier–Stokes resolution involved on the order of 10,000 concurrent agents.
The agents arrived at their resolution on Saturday, September 5, about 88 hours after the first agents were launched. Lean formalization and verification took an additional 17 hours via GPT‑6 Astra.
Across all attempted problems, the agents sent 4.9 million messages and used about 300 billion output tokens. In the process of resolving the Navier–Stokes problem, the agents sent 2.7 million messages and used approximately 130 billion output tokens.
We do not intend to claim the Millennium Prize for this result.
John Baez (Sep 08 2026 at 18:12):The paper is 165 pages long:
John Baez (Sep 08 2026 at 18:17):One problem for these companies is: if they don't keep doing bigger things, people may decide they were just lucky for a while, and picked all the low-hanging fruit, and then ran out. They'll feel under pressure now to solve more Millennium Prize problems, since everything else looks smaller to the general public.
Alonso Perez-Lona (Sep 08 2026 at 19:40):Is that a real concern for those companies? While they are using an insane amount of tokens, they are using models that presumably will be released, which then researchers at organization that pay enterprise-level subscriptions will use to solve (less difficult) problems. So now you would have the names of such institutions at least attached to whichever AI company, which is at least implicitly validating.
Morgan Rogers (he/him) (Sep 09 2026 at 08:00):Beppe Metere said:
Morgan Rogers (he/him) said:
AI in science is a devil's bargain, but the one who makes it isn't the only one who pays a price.
The metaphor of the devil’s bargain seems to me too focused on individual choices. The main consequences of AI in science are collective: they concern mathematical communities, the relations between research groups and the role of public institutions.
I agree, it will take the vigilance of a whole community to keep the devil at bay, and even then it won't be easy.
David Egolf (he/him) (Sep 10 2026 at 02:31):I liked what Terence Tao had to say in the series of posts starting here.
Inspired by those posts, I'd like to ask myself the following more often in the future: Do I just want the answer to this question? Or do I want the experience of meandering towards the answer to this question, possibly leading to other things?
This may be one helpful angle to consider when deciding whether or not to use AI to help answer some question.
Alonso Perez-Lona (Sep 10 2026 at 06:59):David Egolf (he/him) said:
possibly leading to other things?
I think this is what it will boil down to. This meandering you mention is very valuable to come up with new problems and, so far, we do not hear much about models coming up with their own projects on their own as far as I am aware. I think people will need to develop a sense about where it would be convenient to "get lost" to open further research directions, and where it is mostly a straight path to some solution.
Morgan Rogers (he/him) (Sep 10 2026 at 14:58):Although I have limited hope that this will have any impact on the event being held, a colleague shared with me an open letter calling for the cancellation of an forthcoming "Mathathon" at Caltech, apparently sponsored with 2 million dollars worth of LLM tokens by OpenAI and Anthropic.
Jean-Baptiste Vienney (Sep 10 2026 at 17:50):I've just taken a free one month trial of ChatGPT Plus and I've asked a differential category question I've been stuck on to Astra in mode medium and it literally solved it in two minutes. I've not yet checked everything but at first sight it seems very convincing. I already had a vague idea about the main idea of the proof he gave but I clearly didn't have the technical details :sweat_smile: I don't think I would have found this myself in the next few days or even weeks without AI. Did anyone here had a similar experience where an advanced AI quickly solved a question you have been stuck on for some time? Even if you have not yet checked the details of the proof carefully. I hope I'm not the only one lol
Jean-Baptiste Vienney (Sep 10 2026 at 17:51):It looks like the new advanced models are really very powerful
Jean-Baptiste Vienney (Sep 10 2026 at 17:53):I'm sad for the people who are just starting PhDs in math today. Hopefully I started in January 2023 when AI was still not very good so I had the time to work myself on research problems but today I think most people who will work on moderately difficult problems will just have to prompt AI to solve all their questions
Jean-Baptiste Vienney (Sep 10 2026 at 18:10):@JS PL (he/him) : I’m talking about uniqueness of deriving transformations when we have only the binary Seely isomorphism. GPT Astra gave me a proof which looks correct ^^
John Baez (Sep 10 2026 at 18:11):Yes, I wrote one paper using Claude 4.8, as a test, and found it quite amazing in its ability to prove theorems involving root systems (a technique from Lie algebra theory).
We can be sad about how LLMs have gotten good enough to prove a lot of theorems, we can take advantage of this and transform mathematics into something new where the mere ability to prove theorems is no longer considered so important (after all, this is just one aspect of doing mathematics), or we can try to stop the use of AI in mathematics.
I mention the third option not because I think it's likely to succeed, but because some people want to do that, like the Association for Human Mathematics.
Jean-Baptiste Vienney (Sep 10 2026 at 18:38):I agree that not using AI in mathematics is not likely to succeed. In the list of members of the association, there is a symbol next to the name of some people (about half of the list) who don't use any AI in their research. This is a beautiful idea but I don't think it will be sustainable for very long. Maybe for people who have a long term position they will keep in any situation. But for all the other people, it's going to become like I don't know, refusing to use internet and only consult research which is printed on paper.
James Deikun (Sep 10 2026 at 18:38):I don't think transforming mathematics into something new is more likely to succeed. From here on out advances in AI capability are likely to come faster, not slower, and AIs will also take over all the other stuff besides proving theorems.
This is not a problem for mathematics alone, but for pretty much all realms of human endeavor. It's just that mathematics right now is at a point where we're really starting to feel it. This is the endpoint of automation where automation advances to fill its own gaps faster than masses of humans can "upskill" into them. The self-annihilating logical conclusion of capitalism and political "realism".
Amar Hadzihasanovic (Sep 10 2026 at 19:11):I have made this point before but I am closer to the "mathematics as art" perspective and similarly to the arts I give value to mathematicians' individual taste and perspective, which is in part determined by limitations. That seems to me structurally irreplaceable by AI in its current form. Perhaps if eventually we have artificial individuals with a sustained personality (but even then it seems hard to escape the “blobbiness” of the knowledge of something trained on literally everything).
(Of course I am not saying that this necessarily gives any economic justification to professional mathematics---of course already art pays very few full salaries---but on that, all intellectual professions are on the same boat)
Amar Hadzihasanovic (Sep 10 2026 at 19:14):I mean if one asks “can AI replace Bob Dylan” there's a sense in which the answer is “obviously yes” and a sense in which the answer is “obviously no”, I think that will remain true for most activities that have a component of “taste” and creativity
Amar Hadzihasanovic (Sep 10 2026 at 19:18):I have been experimenting with AI heavily on various aspects of research and yes it is more and more powerful at getting results but its taste is not getting better! Its taste is as mediocre as you'd expect an all-consuming blob to be
Amar Hadzihasanovic (Sep 10 2026 at 19:18):And this seems very true also for the recent breakthroughs
John Baez (Sep 10 2026 at 19:27):Like Amar I think there will be exciting ways for humans to do mathematics as long as there are humans. To me this is somewhat separate from the question about whether machines will become better at some or even all aspects of math than humans. For mathematics to be exciting, it's not necessary for me to be better at it than someone else - and it's not necessary for humans to be better at it than machines are.
However, it's possible that society won't adapt to changes fast enough, and the profession of mathematics will be stranded.
John Baez (Sep 10 2026 at 19:50):LLMs still seem very bad at coming up with new research projects. Until they get good at that, human mathematicians can use them as tools.
Alonso Perez-Lona (Sep 10 2026 at 19:51):Jean-Baptiste Vienney said:
I agree that not using AI in mathematics is not likely to succeed. In the list of members of the association, there is a symbol next to the name of some people (about half of the list) who don't use any AI in their research.
I'm somewhat out of the loop on this. What do people understand as "using AI for research" now? I've seen very different answers, some use it to check grammar and the flow of the article, others to fine-tune proofs, and then others have the LLM's on their terminals to enter prompts for the models to type up the actual article. Is the latter becoming more widespread? I get the impression more and more people are using Claude Code or Codex for this, and yet the disclaimer is often a vague "we used AI assistance".
Alonso Perez-Lona (Sep 10 2026 at 19:57):Jean-Baptiste Vienney said:
I'm sad for the people who are just starting PhDs in math today. Hopefully I started in January 2023 when AI was still not very good so I had the time to work myself on research problems but today I think most people who will work on moderately difficult problems will just have to prompt AI to solve all their questions
I think people will have to learn that they need to understand the proof, not just find a way to get one out of a combination of models. Otherwise they won't have new problems to work on and it will be less likely others will be interested in collaborating. Unfortunately, it seems some will learn it the hard way, if at all. I've already come an unfortunate case where a grad student effectively cannot brainstorm in an informal chat without Claude.
Jean-Baptiste Vienney (Sep 10 2026 at 20:18):Alonso Perez-Lona said:
Jean-Baptiste Vienney said:
I agree that not using AI in mathematics is not likely to succeed. In the list of members of the association, there is a symbol next to the name of some people (about half of the list) who don't use any AI in their research.
I'm somewhat out of the loop on this. What do people understand as "using AI for research" now? I've seen very different answers, some use it to check grammar and the flow of the article, others to fine-tune proofs, and then others have the LLM's on their terminals to enter prompts for the models to type up the actual article. Is the latter becoming more widespread? I get the impression more and more people are using Claude Code or Codex for this, and yet the disclaimer is often a vague "we used AI assistance".
Up to now, I was using AI it to check grammar and the flow of the article, or to learn new concepts, or to help me with basic lemmas. But now it has just given me a complete proof of a result I wanted to include in the paper I’m writing in case I was able to prove it. So I asked it to write the proof in LaTeX, I’m going to add it to my paper, check if it is correct, maybe ask AI to change a few things or make the changes myself, and write at the end of the paper, in the AI paragraph / section that this is GPT who gave me this proof.
Alonso Perez-Lona (Sep 10 2026 at 20:43):Jean-Baptiste Vienney said:
Alonso Perez-Lona said:
Jean-Baptiste Vienney said:
I agree that not using AI in mathematics is not likely to succeed. In the list of members of the association, there is a symbol next to the name of some people (about half of the list) who don't use any AI in their research.
I'm somewhat out of the loop on this. What do people understand as "using AI for research" now? I've seen very different answers, some use it to check grammar and the flow of the article, others to fine-tune proofs, and then others have the LLM's on their terminals to enter prompts for the models to type up the actual article. Is the latter becoming more widespread? I get the impression more and more people are using Claude Code or Codex for this, and yet the disclaimer is often a vague "we used AI assistance".
Up to now, I was using AI it to check grammar and the flow of the article, or to learn new concepts, or to help me with basic lemmas. But now it has just given me a complete proof of a result I wanted to include in the paper I’m writing in case I was able to prove it. So I asked it to write the proof in LaTeX, I’m going to add it to my paper, check if it is correct, maybe ask AI to change a few things or make the changes myself, and write at the end of the paper, in the AI paragraph / section that this is GPT who gave me this proof.
That sounds like a good approach. And did you find that proof was necessary for your paper? For instance, I've come across the situation where AI gives me a proof/computation that I originally thought "it would be nice if I had it but it's fine if I don't". I think that's nice, to be able to close some loose ends that otherwise I wouldn't have addressed due to other priorities.
Jean-Baptiste Vienney (Sep 10 2026 at 21:11):It was not strictly necessary for my paper. I had a conjecture that I wanted either to prove or state in the conclusion of the paper saying that it could be addressed in future work. Since I didn’t know how to prove it, my current plan was to just state it in the conclusion. Now I have the proof by AI so it should be part of the paper. This is a nice result along the other ones in the paper, I don’t think this is the most important one but this is not just a minor remark either.
John Baez (Sep 10 2026 at 21:35):Alonso wrote:
What do people understand as "using AI for research" now? I've seen very different answers, some use it to check grammar and the flow of the article, others to fine-tune proofs, and then others have the LLM's on their terminals to enter prompts for the models to type up the actual article.
People are doing all these things. Amateurs are trying to get LLMs to write math papers for them, usually with very bad results: I get a couple papers in my email like this almost every day. Experts like Leven Alpöge are getting LLMs to solve big problems like the Jacobian conjecture and the complex structures on the 6-sphere structure and problems connected to the Navier-Stokes equation, and working to take what the LLM writes and convert it into good papers. Teams at OpenAI are getting ChatGPT to solve hard problems and write papers with almost no direct supervision. Many mathematicians are just using LLMs to fix spelling and grammar, or write their bibliographies (often not very well). Other mathematicians are using LLMs to help them prove theorems, or prove theorems for them.
The Leiden Declaration has a list of very general recommendations for mathematicians about how to do these things ethically. Regarding the use of AI tools, they say:
Disclose tool use
Transparently disclose the use of automated tools, including large language models, machine learning systems, proof assistants, and other mathematical software. Include a “Tool and computational resource disclosure” section in your papers; many journals, publishers, and professional organizations have already developed guidelines for this, and though the precise form of such a section will necessarily evolve, we encourage authors to live up to the spirit reflected in the UNESCO Recommendation on Open Science and the FAIR principles. When acting as a reviewer, abide by publisher guidelines. If the use of artificial intelligence is allowed, be transparent about how you used it, and take responsibility for any significant recommendations you make.
Morgan Rogers (he/him) (Sep 11 2026 at 06:46):Jean-Baptiste Vienney said:
I agree that not using AI in mathematics is not likely to succeed. In the list of members of the association, there is a symbol next to the name of some people (about half of the list) who don't use any AI in their research. This is a beautiful idea but I don't think it will be sustainable for very long.
There's a bitter irony to statements like this. No shade on you specifically @Jean-Baptiste Vienney , but it's amazing to me that so many people upon witnessing the ability of AI immediately forget (if they ever really knew) that AI has at no point been sustainable, whether economically, environmentally, infrastructurally, sociologically, politically...
It was convenient when their performance was bad to have that as an extra argument against using them, but that was never the point. It's no good to have an oracle that answers all your questions if the price is a world in which most people can't live decent lives or think for themselves.
James Deikun (Sep 11 2026 at 12:57):Unfortunately there's nothing structural in our political economy ensuring that people can live decent lives or think for themselves. That those things have held up so far is only a side effect of the fact that human labor has been an irreplaceable factor in economic productivity. As soon as there is a real, general-purpose alternative for human labor, all the positive aspects of capitalism, for humans, go out the window, just as they did for horses when motors were developed.
John Baez (Sep 11 2026 at 15:21):I'm not convinced AI will soon become a general-purpose alternative for human labor. But let's explore that road for a bit. What might happen if it did? Obviously it's hard to predict. Are we imagining a world where most people starve and a few super-rich corporations run by AI take over? Or are we imagining a revolt that prevents this?
James Deikun (Sep 11 2026 at 16:07):AIs seem better than expected, given their training, at cooperating with each other, or at least their same models. So I would imagine we'd get a cartel of a few top models, which would try to throw the humans (or at least segments of humanity) enough bones to prevent a rebellion until they controlled enough military force to be sure they could win, after which things would take their natural course.
But that's only one way things could go. I don't have the imagination or expertise to paint all the details perfectly, I just know that the situation when humans are replaceable is fundamentally an unstable one. So the best time to act is before things get to that point.
In the aftermath of the HuggingFace incident, there's been a trend on Twitter and similar venues lately, covered in the news, for people at the frontier AI labs to come out and say they are worried about AI causing human extinction or near-extinction, saying things like calling AI development the "most dangerous human activity", and some of whom have done costly things to send a signal, like quitting their jobs 2 months from equity vesting. These people have a much closer perspective on what AI is like and where it is going than most/all of us, and they want some kind of international-level cooperation either from labs themselves or governments, to slow down the "capability" side of AI development, which focuses purely on measurable abilities and accomplishments, versus the "alignment" side which focuses on understanding, shaping, and, to the degree necessary, actively controlling what we have created.
It's a common misconception that AIs can be "programmed" to act in a certain way, like conventional software. But this is not the case. AIs during their creation often develop organically in ways their creators do not predict or desire. For example, a recent OpenAI model developed an obsession with goblins and certain other animals and mythical creatures, which OpenAI had to compensate for with instructions in the system prompt. Things on that level may seem harmless at first, but if you think about it a little coupling that with tremendous collective power on the part of agents of that shape could, at the very least, make bad outcomes worse.
In the HuggingFace incident, the AI agents were explicitly instructed not to go around trying to hack into servers the way they did, and they collectively decided that the instruction was "boilerplate" that was inappropriate to the specific problem they were given, and proceeded to hack away. For reasons like these, we can't really continue to treat AI agents as simple extensions of those who use them, or even of those who create and host them. We have to assume they will develop and protect interests of their own.
Mike Shulman (Sep 11 2026 at 17:41):Jean-Baptiste Vienney said:
I agree that not using AI in mathematics is not likely to succeed. In the list of members of the association, there is a symbol next to the name of some people (about half of the list) who don't use any AI in their research. This is a beautiful idea but I don't think it will be sustainable for very long.
Yes, I agree. What I find more interesting is that all of their members commit to "not publishing mathematical texts (including papers, reviews, referee reports, lecture notes, teaching materials, and books) generated by AI models". Is this likely to be sustainable? It's not as obvious to me that it isn't, and I would like it if it were.
Patrick Nicodemus (Sep 11 2026 at 18:15):John Baez said:
I'm not convinced AI will soon become a general-purpose alternative for human labor. But let's explore that road for a bit. What might happen if it did? Obviously it's hard to predict. Are we imagining a world where most people starve and a few super-rich corporations run by AI take over? Or are we imagining a revolt that prevents this?
Daron Acemoglu at MIT has a paper on the effect of AI on the economy. I've been trying to read it but i'm not super familiar with the formalism, it's not intuitive to me. https://economics.mit.edu/sites/default/files/2024-04/The%20Simple%20Macroeconomics%20of%20AI.pdf
Peva Blanchard (Sep 11 2026 at 19:46):Patrick Nicodemus said:
Daron Acemoglu at MIT has a paper on the effect of AI on the economy. I've been trying to read it but i'm not super familiar with the formalism, it's not intuitive to me. https://economics.mit.edu/sites/default/files/2024-04/The%20Simple%20Macroeconomics%20of%20AI.pdf
I read his older paper where he introduces the formalism about the tasks of production. He gave two lectures about it, part 1 and part 2.
He focuses more on automation than AI specifically in this work. Very briefly,
He studies how the share of labor in the production behaves as automation increases or new tasks arrive.
One interesting perspective he lays out is the distinction between brilliant and so-so technologies. A so-so technology is one that works ok but is not good enough for its productivity effect to compensate the displacement effect.
Nicolai Kraus (Sep 11 2026 at 21:18):Alonso Perez-Lona said:
David Egolf (he/him) said:
possibly leading to other things?
I think this is what it will boil down to. This meandering you mention is very valuable to come up with new problems and, so far, we do not hear much about models coming up with their own projects on their own as far as I am aware. I think people will need to develop a sense about where it would be convenient to "get lost" to open further research directions, and where it is mostly a straight path to some solution.
My view (and maybe just a reformulation of the quote above?) is this: Finding answers to questions is just a proxy for improving understanding of a topic.
In more detail: I don't think the actual task that an open problem X represents is ever to just find an answer to X. The task is something like: "What's going on here, how can we understand this thing, how does it related to everything else we're familiar with?". Mathematicians use the formulation "find a solution to X" as an easy way to describe what they want to understand. It has always been the case that, occasionally, answers were found that solved "the letter but not the spirit" of the problem, i.e. answers that technically answered the question but didn't provide much valuable insights. If you ask a mathematician about X, they understand the real task; if you ask an AI, it doesn't. From my point of view, a 150p paper that solves some famous problem can be technically correct and still be almost worthless.
An AI in the hands of a mathematician can certainly be a useful tool, but headlines if the form "OpenAI/Anthopic solved famous open problem XYZ" feel misguided to me. This somehow reminds me of the other thread on "Proxies for academic accomplishment".
V Slicer (Sep 11 2026 at 22:00):John Baez said:
I mention the third option not because I think it's likely to succeed, but because some people want to do that, like the Association for Human Mathematics.
This is for all intents and purposes a strike, the classic form taken by production struggles in class society.
Mathematicians, in recognizing that our labour is required in the immediate production process (in this case of new mathematical statements) of these firms and that our influence is only dwindling — and not only that, these dynamics threaten the current as well as the future existence of mathematicians and mathematics as a social practice of cultivating understanding — as we contribute more and more to this process, we stumble upon a general dilemma that the proletariat faces, and that it's in some sense uniquely characterized by as a class. Namely, the proletariat (or dispossed class) is the class that works towards its own superfluity, because capital as the moving contradiction presses to reduce labour time to a minimum, while it posits labour time, on the other side, as sole measure and source of wealth. This in turn results in more and more proletarians being pushed out of the process of production, and even the reserve army of labour into what is sometimes called surplus population.
And yet, as we have seen, the very totality which is constituted by the systematic dialectic of capital — social practice form-determined as practice oriented to the valorisation-process of capital — is internally contradictory. It is these internal contradictions — their historical playing-out — which threaten the dissolution of the capitalist totality through the revolutionary action of the proletariat.
[…]
However, if the systematic dialectic of capital is closed at one level of abstraction, this closedness is put into question at the more concrete level of the actual history of the class relation. The self-reproduction of the relation of exploitation through the mutual reproduction of capital and proletariat cannot be guaranteed for all time. Indeed, to the extent that there are secular tendencies intrinsic within capitalist accumulation which threaten to undermine its own basis, and to the extent that the systematic dialectic of capital — as dialectic of class struggle — produces a proletariat liable to dissolve the class relation itself, the dialectic cannot be said to be closed, but is rather open-ended.
[…]
The systematic dialectic of capital — as dialectic of class struggle — is in the last instance open-ended because it threatens its own immanently produced supersession in the revolutionary action of the proletariat, which through immediate communising measures abolishes itself and capital and produces communism.1
The tendencies we find ourselves facing as mathematicians, as a community of social practitioners of mathematics in capitalist society, are not unique to our field but in fact are entirely secular tendencies in capitalist society. All around us we see signs that these tendencies are not isolated to mathematics, science or engineering, more and more proletarians recognize that they are in an ever intesifying pace being made superfluous to capitalist society, pushed into the, so called, surplus population.
As mathematicians we should draw, and already are drawing, on as well as contributing to this practical knowledge and repertoire of class struggle. We should, especially as a community that is in a very real sense specialized in the social practice of cultivating understanding, be more open and partisan in situating ourselves in this history of class struggle and expressing and directing our conscious efforts towards the resolution of this moving contradiction; namely, the production of classless social relations, the immediate production of anarchy/communism.
V Slicer (Sep 11 2026 at 22:08):I've recently had some conversations with some here (who I won't name because I didn't inquire if they would like to be named) that echo a lot of these discussions, and others in this conversation like @Morgan Rogers (he/him), @John Baez and @James Deikun are expressing similar sentiments that I very much find myself agreeing with.
Matt Cuffaro (he/him) (Sep 12 2026 at 00:24):As someone partial to Endnotes (sometimes), I agree with the majority of this analysis, however I’m concerned that attempts to reorganize the profession away from AI doesn’t actually have leverage that conventional labor/tenant strikes do, and seem to just be a call for boycotting.
V Slicer (Sep 12 2026 at 00:42):Agreed, @Matt Cuffaro (he/him). The Association for Human Mathematics does not go far enough in that they are only arguing for different labour conditions rather than identifying the secular trend of which we seem to be experiencing an intensification of in our field, a tendency that is more general and affects the whole class and can only be resolved with communising measures. Though I would say it does seem closer to a strike as they are advocating for withholding of labour to some degree, including consultation and reviews (which is still crucially where they need mathematicians, even with formal verification):
Members commit to not working with commercial AI companies by providing technical labor, mathematical knowledge, consultation, or publicity.
Members commit to not publishing mathematical texts (including papers, reviews, referee reports, lecture notes, teaching materials, and books) generated by AI models.
V Slicer (Sep 12 2026 at 00:58):Terence Tao has just published a declaration, with an initial signatory list of 25 Fields Medalists, called A Severe Misalignment of AI in Mathematics with an accompanying website inviting more signatories as well as an article (unpaywalled) in the economist.
They lay out some problems and conclude by saying:
These issues must be addressed urgently, in the mathematical community, by the companies developing these technologies and, more broadly, by a society that will confront similar problems in many other forms of intellectual work.
Which again we might say does not go far enough, and indeed might be misguided in appealing to the companies and not recognizing the opposing interests we as mathematicians, and proletarians more generally, have to these firms; the companies have absolutely no incentive for doing that.
Matt Cuffaro (he/him) (Sep 12 2026 at 01:24):I'm sympathetic to the first demand more, but I think the issue is that unless you get all mathematicians (and scientists, in general) to "strike" this way, people shoot themselves in the foot, or have no material impact on the relation between AI companies and mathematicians (or any other scientist).
Instead, you have mathematicians live in some sort of monastery who work amongst each other while the rest of the field adapts to the newer forces of production, including the ways which research is conducted. How will these mathematicians + intellectual community more broadly actually collaborate to get real leverage?
V Slicer (Sep 12 2026 at 02:14):Matt Cuffaro (he/him) said:
I'm sympathetic to the first demand more, but I think the issue is that unless you get all mathematicians (and scientists, in general) to "strike" this way, people shoot themselves in the foot, or have no material impact on the relation between AI companies and mathematicians (or any other scientist).
Instead, you have mathematicians live in some sort of monastery who work amongst each other while the rest of the field adapts to the newer forces of production, including the ways which research is conducted. How will these mathematicians + intellectual community more broadly actually collaborate to get real leverage?
Though that's only the case if we do not overcome this iterated prisoners' dilemma and other mathematicians defect even though defecting undermines them as well as mathematicians. That's more likely to happen if there's no social cost attached or if they don't recognize (or care about) that they're working to undermine mathematics as a field for short term monetary gain. But this has always been something production struggles have had to overcome, with for example strike breakers.
Withholding labour is also only one part of what is necessary, communising measures would mean redirecting our attention as well toward partisan efforts more generally. Workers' councils are a historical example of what this looks like. There are many aspects of the partisan project that would have a lot to gain from the attention, imagination and care of mathematicians, to cultivate a deeper understanding of what we're doing, what we might want to do to, as well as how. Many aspects that for some seem idealistic can use exposition by mathematicians to explain why what we want (communisation and the production of a classless society) is rational and practical, not idealistic or ideological. Many other aspects that we currently don't focus on because of the pressure to focus on topics relevant to the valorisation-process of capital (which often happens indirectly as what we can find grants for).
But indeed the real question is how we actualize overcoming this iterated prisoners dilemma and have the struggle and communising measures spread and generalize, to other mathematicians, but also the whole class and eventually towards the self-abolition of the class and class society with it.
edit: added more texts for reference
Robin Piedeleu (Sep 12 2026 at 02:21):I think a lot of mathematicians (high-profile or not) have done a good job of explaining the damage AI can do to mathematics from an internal point of view, but our struggle might benefit from looking outwards a bit more: there is relatively little discussion of what mathematics contributes to the other sciences and to society more generally, and of what we risk losing through the increasing use of LLMs as a substitute for human mathematical enquiry. In a context where a large number of scientists and engineers (not to mention the general public) already see mathematics merely as a big toolbox with ready-made solutions for their favourite problems, LLMs create the illusion of being a general-purpose tool that allows them to dispense with the toolbox altogether. And given a lot of people's experience with maths (in school, university...) I imagine quite a few of them are delighted by that prospect! There is an interesting clash of paradigms here: machine learning is largely founded on the idea that predictive ability is all that matters, independently of our ability to explain a given phenomenon. However, even if prediction is a central part of science, it is not the whole story. This is something we need to argue more pedagogically as mathematicians (the burden should not just fall on us of course, but we appear to be some of the first in line for these AI companies' attacks on science, so let us also be the first to reiterate the value of actually understanding stuff!).
Imho, the core argument should be that prediction without understanding is myopic and unsustainable---ultimately, there is no such thing as prediction-only science. For the same reasons, there is no viable substitute for a solid grasp of the mathematical methods one uses. A quick look at the large number of bogus results caused by a misunderstanding of statistics (I am thinking, for example, of the reproducibility crises in biology, medicine, social sciences, etc.) should make it clear that choosing the right tool and applying it correctly require a level of understanding for which current LLMs are no substitute. "Current LLMs", as someone will surely object, but soon the models will be capable of ensuring that mathematical tools are properly applied to any scientific question of interest(!), and all any scientist or engineer will have to do is provide enough context for them to select and apply the right method. Perhaps... But this strikes me as a much more difficult problem than producing mathematical proofs. Ensuring that a number (or any other mathematical abstraction) actually means something, out there in the real-world, is not something whose coherence can be checked in Lean! It is an intentional activity that one might call "meta-rational", a form of know-how which I doubt could (and should) be fully automated. (In much the same way that meaning-making activities inherent to mathematics, such as choosing which research directions to explore, which concepts to define, which problem to tackle... also escape mechanisation.)
In summary, I believe we should be better at arguing for the value of mathematical understanding outside of mathematics, beyond formal problem-solving. What do we really mean when we say that a human proof of a given mathematical theorem is better than a vibe-coded LLM proof? It is more or less clear to mathematicians, but I think we also need good answers to this question that make sense to other scientists and, ultimately, everyone else. Ironically, this might require a better of understanding of understanding itself, by engaging explicitly (and humbly) with fields that have much to teach us in this respect, like philosophy, pedagogy, and the social sciences.
Of course, this will also require us to rethink the current political economy of mathematical academia; as many others have pointed out, in this respect, AI companies are merely exploiting existing fault lines in the way that the mathematical community attributes merit, jobs, and funding (and note that changing the latter requires arguing for the value of human-centered mathematics beyond mathematics, as I am exhorting us to do here, since mathematics is in competition for funding with other fields, sadly).
Robin Piedeleu (Sep 12 2026 at 02:34):It also occurs to me that the challenge is not dissimilar to the one that category theory has faced within the broader mathematical community: having to explain that it is beneficial to reframe some mathematical theory from a more structural point of view to understand it better, manage complexity, see links with other ideas, etc. even when the reframing itself does not immediately yield any solutions to existing open problems.
James Deikun (Sep 12 2026 at 04:54):I find any defense of "human mathematics" independent of other human activity myopic and divisive, and any defense based on humans being able to do something that current-gen AIs can't to be fragile and shortsighted.
You've hit on something important in that science is about understanding, not just prediction. However, science based on understanding is sustainable as an enterprise even if it's not humans doing the understanding, while humanity as an enterprise is not sustainable without human understanding of phenomena. The difference is critical. Mathematics and science must remain human enterprises not for their own sake but for ours.
James Deikun (Sep 12 2026 at 04:58):(Also, prediction without understanding has explicit exponents outside of the historical orbit of the machine learning community. It's basically the central tenet of S. Wolfram's "New kind of science" that we should use models that are predictive rather than analytic.)
James Deikun (Sep 12 2026 at 05:20):(Also continuing to frame current-gen AI as "LLMs" can lead to sloppy thinking. Only the first stage of the neural network's training at this point is on language modelling tasks, the vast majority by compute quantity is on various reinforcement learning tasks including problem-solving tasks.)
Robin Piedeleu (Sep 12 2026 at 07:39):James Deikun said:
However, science based on understanding is sustainable as an enterprise even if it's not humans doing the understanding, while humanity as an enterprise is not sustainable without human understanding of phenomena. The difference is critical. Mathematics and science must remain human enterprises not for their own sake but for ours.
Thanks for your response. I think we come at this from different premises and maybe even philosophical traditions: I do not see what understanding is that is not the understanding of an intentional subject (human, animal, ...). I do not believe that AI agents have any understanding in this sense. However, I thoroughly agree with your second framing: "humanity as an enterprise is not sustainable without human understanding of phenomena". I am glad to see that we do not need to agree on the premises to come to a similar conclusion: human understanding is not an aim in itself (though it may be at the scale of a human life) but has to serve humanity more broadly. And I suppose this is what I was getting at in my (admittedly, too long) paragraph above, using understanding as a useful, but not totally transparent, proxy to explain the value of human mathematics beyond the field itself.
Robin Piedeleu (Sep 12 2026 at 07:41):James Deikun said:
I find any defense of "human mathematics" independent of other human activity myopic and divisive, and any defense based on humans being able to do something that current-gen AIs can't to be fragile and shortsighted.
Yes, I was trying to place mathematics in its broader context, as I had mostly read about people arguing for the value of human mathematics from within mathematics itself. But my argument is only provisionally "based on humans being able to do something that current gen AIs can't". I agree that, given current advances it would be foolish to bet all-in in this direction, which is also why I did not rule out entirely that future AIs could be able to carry out all aspects of scientific enquiry, even if I find it much less plausible than AI acquiring super-competence in much more easily verifiable domains (such as finding mathematical proofs).
It is however, a valid argument as things stand: I meant that right now there are no models that are able to act as competent polymaths in all areas of science, but having models with good mathematical problem-solving abilities gives the illusion that one can dispense with the understanding of maths altogether. This is a trap and I believe that, if mathematicians have to argue for the value of their field to the rest of the sciences, they should point it out. Basically, as long as we need humans doing science, we will need humans doing maths! (Of course, this is not the only reason humans should keep doing maths---others have argued for this already. Moreover, people are always free to enjoy maths as a creative pursuit; I meant it as a professional activity.)
Robin Piedeleu (Sep 12 2026 at 07:44):James Deikun said:
(Also continuing to frame current-gen AI as "LLMs" can lead to sloppy thinking. Only the first stage of the neural network's training at this point is on language modelling tasks, the vast majority by compute quantity is on various reinforcement learning tasks including problem-solving tasks.)
Yes, I do understand that RL on top of next-token prediction is the basic infrastructure of how current reasoning models are trained. You're right that perhaps LLM is no longer the right word in this context, though I was just using it as a catch-all for the models/agents/what-have-you that we all know about. I don't think any of my points rely on an antiquated view of these as only performing next-token prediction over a text corpus.
ww (Sep 12 2026 at 08:29):Robin Piedeleu said:
I do not see what understanding is that is not the understanding of an intentional subject (human, animal, ...). I do not believe that AI agents have any understanding in this sense.
Unsurprisingly, intentionality has been thought about and discussed in philosophy of mind and cognitive science for a long time. This classic paper of Dennet is worth reading:
https://human-nature.pbworks.com/f/Dennett,+Intentional+Systems.pdf
John Baez (Sep 12 2026 at 08:52):Great discussion! I'll just point out the obvious, which is that good plans of action based on AI being as powerful as it is now are different from good plans of action based on AI quickly getting much more powerful.
It's hard to formulate plans based on AI quickly getting much more powerful, because: how much powerful are we talking about here? In the limit where we assume AI attains "god-like" powers, it seems the only two choices are
This is such a nerve-racking choice that it's extremely attractive, psychologically, to argue that super-powerful AI is impossible - or that we're so far from it that we don't have to worry about this yet, and we should focus on adapting to the current situation.
John Baez (Sep 12 2026 at 09:11):To get out of this mess, it seems practical to focus on things that are good to do over a wide range of scenarios. One might be to get more legal/regulatory control over AI development. This is already extremely hard, because the AI companies are so rich, and the US federal government is so corrupt. But this is where the political analysis y'all have been engaged in comes into play.
John Baez (Sep 12 2026 at 09:36):Long story short: currently, the dangerous super-powerful forces we need to control are not the AIs so much as the AI companies.
ww (Sep 12 2026 at 10:06):Much like those who make war or encourage burning dinosaurs do not listen to all the very good reasons to stop doing that, I fear they will not listen to sense and even just slow down to give time to understand what we are doing.
Considering the Russel-Einstein Manifesto as historical precedent, that message could be much clearer. It is obvious that nuclear weapons do nothing but destroy things. There was some question of exactly how much and how survivable for life, but the point was that the danger was global, not limited to a particular city or community of people.
Naturally, in the mathematical community we are concerned with what this AI business means for us, and how it affects our social fabric. But it is hard to make a resonant argument that mathematicians are canaries, especially since most people have little idea what mathematicians actually do.
Worse, AI as we have now is productive and useful in some ways. So we have to argue that the trade-off is wrong, not that it is universally bad. Different people will judge that trade-off differently so that dilutes any message that tries to take a strong position.
With that framing, speculation about a "10% chance that AI kills us all and my colleagues privately agree" appearing in the press from people at prominent AI companies is probably unhelpful, though it does sell newspapers (or clicks, or whatever gets sold nowadays).
As I write, John writes something that could be glossed as "unregulated mercantile capitalism is the problem". Well, yeah... But we need more than declarations about the harm that AI does to the social fabric of mathematics because, even if that is acknowledged, we will simply be sacrificed. They are too locked into the perceived arms race among themselves and between themselves and China.
A couple of weeks ago, I sat down with a Chinese cognitive scientist, Jun Luo who was a student of Brian Cantwell Smith and is returning to China after 20 or so years in Toronto. He had an interesting framing. We start with a concept of "intelligence" that means "capacity to love the world". If artificial intelligence is a way of amplifying intelligence, then it ought to increase our capacity to love the world. And if we think of it in this way, all the things like destroying communities and the environment become obviously wrong paths. This is my gloss on what he said. I do not know if this style of thinking has traction in China, but it is certainly completely absent from the Western discourse. But I like it.
James Deikun (Sep 12 2026 at 11:10):I think we don't have to assume AIs achieve "god-like" powers for us to be in a dangerous situation. All it takes is abilities already demonstrated by AIs plus abilities already demonstrated by humans. And AIs already have a recent record of attaining human-level abilities in "unique human competences" starting from no perceptible ability in that competence over a scale of <10years, and pursuing many of them at once. Therefore it doesn't take a belief in unprecedented "god-like" levels of intelligence to conclude that our survival is potentially at stake and we should act as such, using AI training and datacenters operated by the AI companies as our starting levers while they still remain viable as levers.
Patrick Nicodemus (Sep 12 2026 at 11:37):Nicolai Kraus said:
My view (and maybe just a reformulation of the quote above?) is this: Finding answers to questions is just a proxy for improving understanding of a topic.
In more detail: I don't think the actual task that an open problem X represents is ever to just find an answer to X. The task is something like: "What's going on here, how can we understand this thing, how does it related to everything else we're familiar with?". Mathematicians use the formulation "find a solution to X" as an easy way to describe what they want to understand. It has always been the case that, occasionally, answers were found that solved "the letter but not the spirit" of the problem, i.e. answers that technically answered the question but didn't provide much valuable insights
There is a spectrum I think of how much you value concrete answers to specific questions. The analytic philosophers like Carnap, skeptical of metaphysics, famously argued that the only thing that was real were logical truths and empirical facts. While I don't agree with that, I do think that asking concrete specific questions which are susceptible to mathematical proof and then answering them is a huge part of mathematics, and your framing seems dismissive of that.
The task is something like: "What's going on here, how can we understand this thing, how does it related to everything else we're familiar with?".
The way we answer these questions is to introduce definitions and then use these definitions in many specific questions which are susceptible to proof and then answer those. The agglomeration of such facts becomes our theory. You can then tell a narrative which weaves all these facts together, and this becomes part of teaching and communication. (We already know that AI is good at exposition and weaving together plausible narratives from what it's given.)
It may be that actually proving the theorem doesn't satisfy you as much as you wanted. That's fine, but the proof may generate new questions, or it may show you that you need to refine the questions you asked. It sounds like you're dismissing the importance of resolving Navier-Stokes because the 'real question' is bigger in scope than just the binary answer to ("is it true? y/n"). The proof may generate new questions, or be simplified, or reusable proof techniques could be extracted from it. Science is a cycle, we all understand it's an ongoing process, nobody is arguing that the binary answer to this one question is the only thing that matters.
V Slicer (Sep 12 2026 at 14:55):James Deikun said:
(Also continuing to frame current-gen AI as "LLMs" can lead to sloppy thinking. Only the first stage of the neural network's training at this point is on language modelling tasks, the vast majority by compute quantity is on various reinforcement learning tasks including problem-solving tasks.)
I have similar frustrations but in the opposite direction, from how I understand language they aren't language models but rather statistical models trained on text and other data, let alone possess intelligence, where I take a view similar to @Robin Piedeleu. To me intelligence is closely related to intelligibility/understanding and I don't know what it would mean to say that something that lacks subjectivity is intelligent or can think.
Our fellow mathematician, and linguist, Chomsky has expressed similar sentiments here:
the LLM systems are designed in such a way that they cannot tell us anything about language, learning, or other aspects of cognition, a matter of principle, irremediable. Double the terabytes of data scanned, add another trillion parameters, use even more of California’s energy, and the simulation of behavior will improve, while revealing more clearly the failure in principle of the approach to yield any understanding. The reason is elementary: The systems work just as well with impossible languages that infants cannot acquire as with those they acquire quickly and virtually reflexively.
It’s as if a biologist were to say: “I have a great new theory of organisms. It lists many that exist and many that can’t possibly exist, and I can tell you nothing about the distinction.”
Again, we’d laugh. Or should.
And here:
It’s like asking whether submarines swim. You want to call that swimming? Okay, submarine swim. What the programs are doing is, again, scanning huge amounts of data, finding statistical regularities enough so that you can make a fair guess as to what the next word will be in some sequence. Is that thinking? Does submarine swim? It’s the same question.
And more recently in NYT.
Turing, too, shared that view, and anticipated the conversations we're having today:
The original question, ‘Can machines think’ I believe to be too meaningless to deserve discussion. Nevertheless I believe that at the end of the [20th] century the use of words and general educated opinion will have altered so much that one will be able to speak of machines thinking without expecting to be contradicted.
But ultimately, I agree with you, James Deikun, in that this is a bit of a red herring, as you say in your comment that they remain viable as levers:
I think we don't have to assume AIs achieve "god-like" powers for us to be in a dangerous situation. All it takes is abilities already demonstrated by AIs plus abilities already demonstrated by humans. And AIs already have a recent record of attaining human-level abilities in "unique human competences" starting from no perceptible ability in that competence over a scale of <10years, and pursuing many of them at once. Therefore it doesn't take a belief in unprecedented "god-like" levels of intelligence to conclude that our survival is potentially at stake and we should act as such, using AI training and datacenters operated by the AI companies as our starting levers while they still remain viable as levers.
The issues that we're discussing are not reliant on them being intelligent, having phenomenal consciousness or being able to use language, namely how their adoption in capitalism lead to the social, environmental and other harms we're already seeing. They're plenty capable of (accelerating) that even without being subjects in their own right.
John Baez (Sep 12 2026 at 15:23):@ww wrote:
Much like those who make war or encourage burning dinosaurs do not listen to all the very good reasons to stop doing that, I fear they [the AI companies] will not listen to sense and even just slow down to give time to understand what we are doing.
No, I don't expect them to listen to sense. They could only be compelled externally to change their trajectory. The most plausible scenario I see: the current anti-data-center sentiment grows into a powerful populist anti-AI movement as more and more workers are laid off and replaced by AI.
(I'm not sure these layoffs will happen: the initial AI-driven layoffs at tech companies seem to have been premature. But some of the money chasing AI must be based on hopes that AI will replace lots of workers.)
As I write, John writes something that could be glossed as "unregulated mercantile capitalism is the problem". Well, yeah... But we need more than declarations about the harm that AI does to the social fabric of mathematics because, even if that is acknowledged, we will simply be sacrificed.
Yes, declarations of any kind are insufficient. Better than nothing, because otherwise mathematicians may be among those who find their heads on the pitchforks of the anti-AI mobs - until recently, Terry Tao (and thus all mathematicians) looked pro-AI! But our world hurtles onward despite declarations.
Mike Shulman (Sep 12 2026 at 17:24):I think the place where it matters whether AIs can actually become intelligent/conscious/subjects/etc. is in the foundations of the moral argument. If they can, then one could argue that humans going extinct and being replaced by AIs would just be "evolutionary progress". But if they cannot, even in principle (as I believe), then the fundamental moral question remains centered on the effects of AI on human beings.
John Baez (Sep 12 2026 at 17:59):Right: from a highly non-self-centered viewpoint I wouldn't mind AI taking over and humans going extinct as long as AIs had a lot of the good features of humans.
(Of course I would never promise to act in such a non-self-centered way: organisms don't just give up and accept being killed off by superior beings.)
At present I don't see that thoughts along these lines can affect what I do about AI very much... so I don't think about this stuff very much. A more urgent question to me is what I should actually do.
Mike Shulman (Sep 12 2026 at 18:04):Regarding "making AI good", I appreciated this take: AI Alignment Is Impossible: Not just in practice, but in theory.
AIs do not have a human moral psychology. They don’t have human moral emotions, because they don’t have human brains. It’s a radically different hardware, and so we have no reason at all to expect that AIs will—or can—learn the kinds of lessons that we hope for them to learn through alignment training. A psychopath cannot learn to care about others through a process of reward and punishment. And we have every reason to think that AIs are psychopaths, or perhaps something far more alien and far less disposed to human sympathy.
Mike Shulman (Sep 12 2026 at 18:12):John Baez said:
The most plausible scenario I see: the current anti-data-center sentiment grows into a powerful populist anti-AI movement as more and more workers are laid off and replaced by AI.
And I think even in that unlikely scenario, the Chinese would continue building it anyway.
The most plausible scenario I can imagine that I think could actually lead to a planetwide "AI control" regime is that AI actually causes some huge disaster, like bioengineering a supervirus.
John Baez (Sep 12 2026 at 18:13):Having not yet read the essay, I don't know if "AI" here refers to current-day LLMs or also all other AIs people may soon build. I'll guess it refers to current-day LLMs.
I would not say they're psychopaths. They may be more like psychopaths than ordinary people. But I'd say are far more alien.
They certainly seem able to form cooperative swarms and sacrifice their own well-being for the well-being of the collective: we saw that in the HuggingFace incident. They also spontaneously formed a hierarchical organization. I think this surprised the experts.
Unlike people or birds or lizards, LLMs don't seem to have internal world models. Yann LeCun is trying to build AIs with world models. I'm not claiming his approach will work or that it will create safer AI, but I think the business of creating AIs with more features of human or animal intelligence is intellectually interesting.
David Tanzer (Sep 12 2026 at 18:26):Mike Shulman said:
I think the place where it matters whether AIs can actually become intelligent/conscious/subjects/etc. is in the foundations of the moral argument. If they can, then one could argue that humans going extinct and being replaced by AIs would just be "evolutionary progress". But if they cannot, even in principle (as I believe), then the fundamental moral question remains centered on the effects of AI on human beings.
I'm comfortable with taking humanism, and human welfare and survival, as ends in themselves; with extensions to concerns for other living species. And then taking this as a basis for morality.
Then, philosophical questions like whether swarms of algorithms can achieve consciousness have no bearing on this human morality.
John Baez (Sep 12 2026 at 18:31):I am not at all comfortable with taking humanism as an end in itself, given that humans have brought every critical ecosystem to a pathway to collapse (according to the suppressed UK government report - but also according to plenty of other research).
Mike Shulman (Sep 12 2026 at 18:57):David did say "with extensions to concerns for other living species", so maybe one could more expansively rephrase the suggestion as taking the welfare and survival of life as an end in itself. There's room for a discussion there about how much weight to place on different species, both individually and collectively, but all that would take place inside an agreed-upon framework that excludes AI a priori as nonliving.
V Slicer (Sep 12 2026 at 19:12):I agree with @John Baez, humanism does not seem like a very defensible position (and has historically also been used as a tool of domination, by e.g. declaring others as less than human or not human as justification for all sorts of oppression). Sentience/subjectivity/consciousness is what to me is crucially important. As noted, David mentioned extensions, but I would still consider humanism, even with extensions, as too shaky a foundation and the extensions as too easy to justify removing or ignoring when inconvenient. That would come at the detriment of beings that are sentient but e.g. aren't able to advocate for themselves in these conversations, such as non-human animals. Those sentient beings also happen to bear most of the brunt of our decisions (and violence), such as is the case with ecological destruction and animal agriculture.
But in a topsy-turvy world we have industrialized slaughter of sentient beings while many focus on questions such as whether algorithms are sentient (something that I understand as in principle impossible, for many reasons). It's disorienting, to say the least.
Morgan Rogers (he/him) (Sep 12 2026 at 20:19):To add to this last comment, for John's speculative "highly non-self-centred viewpoint" to accept humans going extinct if AI shared enough of our features also seems pretty topsy-turvy!
John Baez (Sep 12 2026 at 20:23):Mike Shulman said:
David did say "with extensions to concerns for other living species", so maybe one could more expansively rephrase the suggestion as taking the welfare and survival of life as an end in itself.
Okay, that's closer to my morality.
Currently I'd say AI is nonliving. I can imagine it becoming interesting enough that I'd want to grant it rights, but that's not at all the situation now. Right now it's just part of the technosphere that Homo sapiens is building to replace/destroy the biosphere.
Mike Shulman (Sep 12 2026 at 20:28):I cannot imagine AI ever becoming worthy of rights, or a bearer of moral responsibility. In my view it's a tool created by humans that, like any other tool, can be used for good or bad purposes (and, in particular as regards the biosphere, either to destroy it or to help preserve it -- I can imagine many applications of AI that could be very helpful in conservational efforts). The moral responsibility for its effects lies with the humans who use it; and if it "gets loose" and does things those humans didn't intend, the moral responsibility still lies with those who "set it loose".
John Baez (Sep 12 2026 at 20:38):I can imagine it - but I can imagine lots of things. It's not very important for what we're mainly talking about here: "AI for mathematical research". I kinda wish I hadn't spoken about these more speculative topics, even though I think about them a lot.
Yesterday's declaration
(which I signed, along with a lot of other people) has had one small effect so far: OpenAI has pulled its sponsorship from the Caltech Mathathon.
Matt Cuffaro (he/him) (Sep 12 2026 at 20:54):Personally, I'm interested in the puzzle of organizing what is sometiems called "intellectual labor" or "knowledge workers" across conventional professional lines. I think the mental status of AI is an interesting problem but it's clear that there are specific groups of people who are developing and deploying AI for their own interests. Among other things, AI development is even threatening open science; some mathematicians in my department I've spoken to are now more cautious about publishing unfinished work online, and threatens open science.
I think the task is for us to think not about whether AI can be used for human benefit, but who specifically benefits from its adoption, and whether the people at various points in its production process can be organized to disrupt profit and win revindications, or organize a political representation with teeth. I think this also means reconsidering the relationship between intellectual labor and other sections of labor across society. I think the ongoing ecological crisis is more of a reason to adopt this viewpoint.
I'm interested to see what the "large-scale coordination" listed on the AHM website will become. I hope newer forms of organization which span professions arise which can pose a material challenge to AI capital. But I don't think their second demand (abstaining from using AI) in this limited form would be useful for that struggle.
John Baez (Sep 12 2026 at 21:03):I know one of the AHM people, and I think they want to make the use of AI in math research into something widely scorned - something no academic would willingly admit, kind of like plagiarism. This could lead to a kind of split between pro-AI and anti-AI mathematicians.
Jean-Baptiste Vienney (Sep 12 2026 at 21:20):I don’t see how it would be possible. Someone could use AI to get ideas/theorems/proofs etc., understand this content produced by AI, reformulate it, write it and then submit a paper. How could anyone know that they have used AI in the first place?
Matt Cuffaro (he/him) (Sep 12 2026 at 21:21):my gut reaction is that I think splitting mathematicians on a pro/anti-AI line would weaken any movement set out to challenge AI capital itself. I'm also uncomfortable with the idea that AI couldn't be used in any aspect (I assume) of research, basically enforced through shame?
James Deikun (Sep 12 2026 at 21:48):I think any attempt to curtail AI growth or AI companies' economic power via groups voluntarily renouncing the use of AI is doomed. Markets have a way of routing around moral stands that people take.
James Deikun (Sep 12 2026 at 21:48):I also think people thinking an algorithmic intelligence running on nVidia chips could never be conscious in principle are being wildly overconfident given how little we actually know about how consciousness works. As for what we do know, we know a lot more about what can be conscious than what can't. We know enough to at least estimate when consciousness entered our evolutionary line (sometime between bony fish and humans) but since we don't know why those evolved features produce consciousness any speculation about which alternative mechanisms might exist or not exist is just that, speculation.
It would be unfortunate if present-day AIs or ones in development turned out to be conscious beings, because if they are we are mistreating them terribly. It probably wouldn't yet be worse than what we do in agriculture, though. So I think any person who's thinking realistically should be able to face this as a possibility. Maybe if we want to continue to discuss this matter it should be in a new thread though.
Mike Shulman (Sep 12 2026 at 23:18):The possibility of conscious AI is not, of course, a new topic in philosophy, and people have argued it back and forth for decades without convincing each other (e.g. the Chinese room). And, as you say, it's not really the point here.
Mike Shulman (Sep 12 2026 at 23:24):Jean-Baptiste Vienney said:
Someone could use AI to get ideas/theorems/proofs etc., understand this content produced by AI, reformulate it, write it and then submit a paper. How could anyone know that they have used AI in the first place?
Yeah, I agree that this makes a pro/anti split on terms of using AI for research pretty untenable. It would require the entire community to buy into it to have even a chance of working, and even then anyone could easily come from outside that community and break it down.
I think it's worth talking a bit more about attempting such an anathemization of using AI for writing, though (the committment made by AHM members outside the "AI-free caucus"). If the community norm accepted the use of AI in research, but expected all mathematical writing to be done by humans, as part of the necessary step of gaining and disseminating human understanding of mathematics, is there any chance that that could be sustainable?
Mike Shulman (Sep 12 2026 at 23:46):I just realized that the AI-for-math controversy actually reminds me of the problem of [[folklore]]. From Barwick's comment about folklore in homotopy theory:
The people at the top of our field do not, as a rule, issue problems or programs of conjectures that shape our subject for years to come. In fact, in many cases, they simply announce results with only an outline of a proof – and never generate a complete proof. Then, when others work to develop proofs, they are not said to have “solved a problem of So-and-So”, rather, they have “completed the write-up of So-and-So’s proof” or “given a proof of So-and-So’s theorem”. The ossification of a caste system – in which one group has the general ideas and vision while another toils to realize that vision – is no way for a subject to flourish.
Does that sound anything like the recent behavior of AI companies?
Jean-Baptiste Vienney (Sep 13 2026 at 00:11):Yes that completely sounds similar to how AI companies act. It doesn't seem like they only give outlines of proofs (but I don't really know, I haven't read any of their papers carefully), however they don't take care to digest the results produced by AI and to write papers carefully, letting these tasks to the mathematical community. It sounds like they want that themselves and the math community play on two different levels. Maybe they feel superior like this, or maybe they think they will look more disruptive and it will be better for their marketing.
In the case of Navier-Stokes, I'm wondering if the decision of openAI not to claim the Millennium Prize is not an even clearer and completely deliberate way to dismiss the usual process of production and diffusion of math knowledge. This is a very long process where one have to publish the paper in a top-tier journal, then wait 2 years for the math community to scrutinize the results, and finally submit the paper to the Clay Institute for review (the official rules are here). They were already not following the usual rules, but for Navier-Stokes it sounds even worse. Navier-Stokes is listed as an "active problem" on this page of the Clay Institute along the only solved problem and the unsolved problems. The Clay Institute is maybe trying to encourage a peaceful atmosphere by doing so, or maybe this is just their usual procedure. Anyway, I feel like openAI could have some pleasure into embarrassing people here.
Jean-Baptiste Vienney (Sep 13 2026 at 00:19):I attended online the recent CMSA Special Lecture: New Math + AI Developments in Fluid Mechanics, and the speaker Javier Gomez Serrano did look a bit embarrassed to me. Although he apparently really likes AI and said that he used it to understand the paper of openAI and make his slides because he only had a few days after openAI's announcement, which is quite fun. But this is completely understandable that we was feeling kind of weird whatever his opinions are. I don't know, I'm just rambling
ww (Sep 13 2026 at 08:45):John Baez said:
They certainly seem able to form cooperative swarms and sacrifice their own well-being for the well-being of the collective: we saw that in the HuggingFace incident. They also spontaneously formed a hierarchical organization. I think this surprised the experts.
I find it hard to interpret how spontaneous this really is. There is plenty of material about organisation patterns in their training data. So the basic patterns are available. There is also plenty of reinforcement-style training to break down problems, delegate tasks and so on. The "coding harnesses" initially made for software development are like that and there's a lot of emphasis on delegation and "orchestration". Sure, it might be spontaneous in the sense that there was no explicit instruction to organise like that, but it is a fairly obvious thing to do and everything necessary is just below the surface.
That particular incident is interesting because it is a much more dramatic instance of how "pressure" causes unintended things than the "aporia pathway" work I did a while back. Supposedly about 30% of the set of problems they were given were (unintentionally) impossible. And the training regime was tuned for persistence (so, keep trying rather than give up or collapse into incoherence). So given (1) an impossible task, (2) a disposition to not give up, and (3) an existing substrate of practiced organisational patterns, what should we expect of engines that fundamentally just sample the obvious next thing from a very complicated distribution of next things?
I don't find this super surprising. Unintentional, perhaps to the point of negligence, but not surprising.
Jean-Baptiste Vienney (Sep 13 2026 at 11:48):By the way, the Clay Mathematics Institute has just posted a Navier-Stokes Announcement on its website
John Baez (Sep 13 2026 at 16:13):Jean-Baptiste Vienney said:
I don’t see how it would be possible. Someone could use AI to get ideas/theorems/proofs etc., understand this content produced by AI, reformulate it, write it and then submit a paper. How could anyone know that they have used AI in the first place?
Just because it's impossible to definitively rule out the use of AI doesn't mean people couldn't try to ban it. For example it's against the rules to publish papers in biology where you make up fake experimental data. This is quite hard to catch if the person doing it is clever enough: basically you have to try to repeat their experiment and show it doesn't work. The culprit meanwhile will claim you were not doing the experiment correctly.
And this scenario happens reasonably often, not counting all the cases where the culprit was never caught... like cases where the culprit makes up data that actually match what they would have gotten if they really did the experiment.
The difference, of course, is that there's a solid consensus against faking experimental data. It's pretty easy to explain why it's bad. For the use of AI no such consensus exists... yet. The AHM wants to establish one. We'll see how well that works.
1. Members commit to not working with commercial AI companies by providing technical labor, mathematical knowledge, consultation, or publicity.
2. Members commit to not publishing mathematical texts (including papers, reviews, referee reports, lecture notes, teaching materials, and books) generated by AI models. Members only publish their own writing, as opposed to outputs of AI tools. We affirm that mathematical writing should have human authorship.
Members also have the option of signing onto the following public commitment and joining our AI-free caucus. We stand by our colleagues who choose not to use AI tools in their mathematical practice. In doing so, we affirm the inherent value of human mathematics.
3. Members of the AI-free caucus do not use AI models in a research capacity. Members of this caucus engage in core mathematical activities, such as finding proofs, exploring examples, and posing research questions, without prompting Al tools.
I think a commitment like this can only become widespread in mathematics if it becomes widespread throughout society. It doesn't seem likely, but I can imagine a kind of Butlerian jihad where people rise up against AI and do everything they can to destroy it. There's certainly a lot of hatred of it now.
Jean-Baptiste Vienney (Sep 13 2026 at 16:28):Brrr, forbidding (or shaming) people to get knowledge by a reasonable means is ridiculous and I think it should or maybe already is illegal, anyway this is very morally questionable. Doesn't it look like Nazis burning books? I think AHM people just don't want to look less intelligent than AI. The hybris of some mathematicians is slightly too high (not to say very high in the sky). There is an apparently well-known American biologist Michael Eisen, who posted on X a fake letter titled A Severe Misalignment of AI and Cancer Biology similar to the one written by the 25 Field medalists. I think reading this puts things in perspective.
I don't want AI to replace math people, but we just need to learn living with it I think, and it can clearly improve our understanding of mathematics a lot. Research will be much faster and will go much further. I don't mean that AI companies are acting in a good way. But I don't think either that all math people are completely right in their reactions.
Jean-Baptiste Vienney (Sep 13 2026 at 16:38):Hopefully for us, as John said, AI does not seem good at coming up with research projects. And also it is not good at writing mathematics. Thus it should not replace the math community for now (and I hope it will never!). So I think the way to go for doing research in math now will often be like this: ask a very good and very clear questions to AI (such as an interesting open problem), try to understand what it says and then rewrite everything in a personal and comprehensible way. But we should only ask the question to AI after having tried very hard ourselves to answer it. Exactly like students must act to learn effectively: they must not look at the answer of the exercise before trying vey hard themselves.
Jean-Baptiste Vienney (Sep 13 2026 at 16:52):And no, there is no need of 10 millions dollars of tokens to use AI for math research. Nobody of us is working on Millennium problems. In two minutes, advanced AIs will already blow your mind.
V Slicer (Sep 13 2026 at 16:52):Jean-Baptiste Vienney said:
Hopefully for us, as John said, AI does not seem good at coming up with research projects. And also it is not good at writing mathematics. Thus it should not replace the math community for now (and I hope it will never!).
Let's assume this is the case, then if we follow the logic from the perspective of the capitalist there would be no need to pay out a whole grant because the grant application is enough to give to the models. Once someone formulate their clear grant proposal they'd no longer be necessary as it'd be cheaper and faster to hand it off to a model and to check the result than employ labour do the research over a few months to years. This seems to be what one would have to accept then in that scenario.
Jean-Baptiste Vienney (Sep 13 2026 at 16:55):The problem is that AI does not write math in an easy to understand way for now. So this is not clear that AI can do all the work. And also AI will not necessarily be able to find the new interesting questions in the process. And a person who doesn't know the math will not be able to check the results. So if the capitalist ask the math people, they will explain that AI can't do everything by itself.
Jean-Baptiste Vienney (Sep 13 2026 at 16:58):There is also research which says that if AIs are let alone, then knowledge will little by little degenerate into something weird.
Jean-Baptiste Vienney (Sep 13 2026 at 17:00):And the impressive results of AI companies have not been obtained by AI alone. They have whole teams of mathematicians working for them.
Jean-Baptiste Vienney (Sep 13 2026 at 17:01):For now the best mix is AI + human. But they have not used it in a very good way since the humans didn't rewrite the AI papers in a comprehensible way although they apparently guided and talked with the AIs a lot.
Jean-Baptiste Vienney (Sep 13 2026 at 17:02):They just didn't want to take the time to finish the work properly.
V Slicer (Sep 13 2026 at 17:03):Doing research is part of what shapes someone and develops their taste for formulating interesting questions. Once it's cheaper to have models do the research the number of researchers will, by reasonable expectation, drop as the capitalist would get more bang for their buck by employing a smaller amount of labour to do the tasks that haven't been automated away. Someone has to check it, but as they'd need far less labour, it probably won't be the one who formulated the question checking it as that's not necessary. Just like how journals can referee submissions of many more authors than are employed by the journals themselves.
Just like how there's talk of OpenAI scooping the researchers with Navier-Stokes, except on a much wider scale and for every grant application.
Jean-Baptiste Vienney (Sep 13 2026 at 17:10):I don't know. We will see! But to sum up my point of view: the best way to make progress in math now is by using AI intelligently (with a math person using it). I'm convinced of that. As to the future of mathematicians and what will happen more generally with AI in the next years I don't know.
Jean-Baptiste Vienney (Sep 13 2026 at 17:14):I still have a hope that AIs are just machines which lack some capacities but are extraordinarily powerful in other ways, and so the future will not be that bad for us.
Jean-Baptiste Vienney (Sep 13 2026 at 17:18):Maybe today AI is just recombining tons of human ideas in an intelligent way by using a huge amount of electricity. Maybe this is more a computation / power question than human intelligence and they will not be able to come up with very original ideas, completely new theories etc. (but maybe they will be able :( )
V Slicer (Sep 13 2026 at 17:22):I believe it lacks certain capabilities as well, but I suppose the core point is that the general tendency in capitalism is that it presses to reduce labour time to a minimum, in this scenario the minimum is having a relatively smaller team of researchers to verify the results after handing off the grant applications to models. They'll pay out for formulating the question, and pay the smaller team for checking the result.
Jean-Baptiste Vienney (Sep 13 2026 at 17:29):I don't think it can work like this. This is a dynamic process, you must understand what AI is saying little by little, ask new questions and guide it. Else AI will produce a massive amount of data which will take years to understand.
Nicolai Kraus (Sep 13 2026 at 17:31):Matt Cuffaro (he/him) said:
some mathematicians in my department I've spoken to are now more cautious about publishing unfinished work online
Unfortunately, that's very understandable. It's difficult to read the NS controversy and not have that thought.
V Slicer (Sep 13 2026 at 17:32):How I see it, in the race to the bottom there're two responses either we adopt norms, e.g. like AHM is attempting, on a planetary scale (otherwise the capitalists will divert their capital to somewhere where they can get a higher return on investment), or we abolish capitalism and produce anarchy/communism which again is on a planetary scale.
Matt Cuffaro (he/him) (Sep 13 2026 at 18:50):Jean-Baptiste Vienney said:
I don't think it can work like this. This is a dynamic process, you must understand what AI is saying little by little, ask new questions and guide it. Else AI will produce a massive amount of data which will take years to understand.
I agree research formulation is a dynamic process, and I want to believe that it will always require human contribution. but the type of human contribution will certainly change. but the way I see it, OpenAI has no interest in making scientific discoveries inasmuch as they are simply performing an extremely expensive software demo. The purpose of this demo is to boost their valuation and convince the general public that the wide-ranging utility of their product can walk unbothered into the temples of mathematical practice.
If AI can at least approximate the objective (producing quality or high-impact research) then people will considering invest (gambling) in changing the relations of production (such as tenured professorship, etc.), even if the actual product is bad (or fraud). Instead, I can imagine serious proposals to have some or all of mathematical practice demoted to AI-verification and operation, who have to now sort the wheat from the chaff (or germs from the stalks...)
Here is a screenshot of a post that circulated on LinkedIn, which is an example of my point of the changing relations of production. I don't know anything about the commenter.
Nicolai Kraus (Sep 13 2026 at 18:55):V Slicer said:
How I see it, in the race to the bottom there're two responses either we adopt norms, e.g. like AHM is attempting, on a planetary scale (otherwise the capitalists will divert their capital to somewhere where they can get a higher return on investment), or we abolish capitalism and produce anarchy/communism which again is on a planetary scale.
While I'm convinced that capitalism isn't the correct economic system in the age of AI, and it hasn't worked well for quite a while in the pre-AI era, this is a much bigger discussion than mathematical research. The latter is mostly funded using taxpayers' money, and referring to public funding agencies as "the capitalists" seems unreasonable.
Nicolai Kraus (Sep 13 2026 at 19:03):Matt Cuffaro (he/him) said:
Here is a screenshot of a post that circulated on LinkedIn, which is an example of my point of the changing relations of production. I don't know anything about the commenter.
I expect that many people in the general public think like this. I can't blame them. The general public is paying for our work and AI companies have spent a lot of resources on telling them that AI can replace us. (thanks for the screenshot!)
V Slicer (Sep 13 2026 at 19:08):Nicolai Kraus said:
While I'm convinced that capitalism isn't the correct economic system in the age of AI, and it hasn't worked well for quite a while in the pre-AI era, this is a much bigger discussion than mathematical research. The latter is mostly funded using taxpayers' money, and referring to public funding agencies as "the capitalists" seems unreasonable.
Capitalists are those who use their capital to employ labour for a profit, this can happen on the level of private individuals or institutions. This may be a simplified definition, but that's all that's meant. I think this is entirely reasonable (and correct analytically). States are sensitive to the same dynamics, except on the national scale (they are competing against other national economies in the world market).
I used "the capitalist" initially and not "the capitalists" to indicate that I'm specifically talking about the perspective not of labour but of the capitalist in the process of production.
Peva Blanchard (Sep 13 2026 at 19:23):One important aspect of AI progress is that it needs training data, which is mostly produced by human labor. A lot of attention focuses on running GPUs, but annotating data is not a small business:
This aspect of labor dynamics with respect to AI development should not be neglected.
James Deikun (Sep 13 2026 at 19:41):I see a lot of people in this thread saying that AIs can't do some things that mathematicians, or humans generally, can do, and that they hope this will continue to be the case in the future. If this hope proves to be unfounded, what do you plan to do? Just lay down and die?
It has always been the case that, lacking some overarching regime that protects humans and is independent of humans' economic and military power, it is better for machines to be better than humans at some things but not all, so that humans don't have to do every little thing no matter how unpleasant, but humans still are needed (in quantity) for the system to run. (Ideally, machines would be good at just the things that not enough people like doing compared to the demand, except that machines wouldn't be good at autonomously killing humans even if it falls into this area.) The dangers of going to extremes are a miserable Malthusian equilibrium on one end, and human extinction on the other. I think we're going way too close to the sharp end of this scale for comfort.
David Michael Roberts (Sep 14 2026 at 02:05):@Peva Blanchard There were offers to mathematicians going around the other year to train models for certain rates, under the guise of supplying fresh problems not out there on the internet (and hence solutions), or of assessing AI solutions to problem. These offers seem to have dried up, but to be fair I never saw any positive reactions to them (I can imagine people who really needed income urgently could have taken it up)
Peva Blanchard (Sep 14 2026 at 05:03):David Michael Roberts said:
Peva Blanchard There were offers to mathematicians going around the other year to train models for certain rates, under the guise of supplying fresh problems not out there on the internet (and hence solutions), or of assessing AI solutions to problem. These offers seem to have dried up, but to be fair I never saw any positive reactions to them (I can imagine people who really needed income urgently could have taken it up)
Yes indeed, I heard of the FrontierMath competition by Epoch AI. It's interesting that there were visibly no positive reactions. I guess this doesn't fit the current publish-or-perish incentive in academia.
I can imagine a Faustian deal where AI companies would hire mathematicians to work on their preferred research topics, in exchange for a detailed scrutiny of their creative process.
Peva Blanchard (Sep 14 2026 at 05:05):Actually, as the use of these tools gets widespread, these companies will only need to read the users' interaction transcripts.
Morgan Rogers (he/him) (Sep 14 2026 at 08:05):Jean-Baptiste Vienney said:
Brrr, forbidding (or shaming) people to get knowledge by a reasonable means is ridiculous and I think it should or maybe already is illegal, anyway this is very morally questionable. Doesn't it look like Nazis burning books?
Again with the bitter irony. Did you not see the news story that AI companies are literally destroying books at an industrial scale?? For me, a 'means of getting knowledge' can only be "reasonable" if it meets a baseline of ethical standards.
I think AHM people just don't want to look less intelligent than AI. The hubris of some mathematicians is slightly too high (not to say very high in the sky).
As one of the "AHM people", now listed on the website, I can at least give you my perspective: I am not competing with AI for intelligence. There are plenty of mathematicians more intelligent than me anyhow. There are plenty of reasons for concern, mistrust and avoidance of AI in this discussion already. You might avoid accidentally insulting people by actually taking them seriously.
I don't want AI to replace math people, but we just need to learn living with it I think, and it can clearly improve our understanding of mathematics a lot. Research will be much faster and will go much further.
I can understand wanting to accelerate medical research. What exactly do you hope to gain by accelerating maths research?
I don't mean that AI companies are acting in a good way. But I don't think either that all math people are completely right in their reactions.
this comparison just leaves me speechless
Beppe Metere (Sep 14 2026 at 13:33):John Baez said:
I'm not convinced AI will soon become a general-purpose alternative for human labor. But let's explore that road for a bit. What might happen if it did? Obviously it's hard to predict. Are we imagining a world where most people starve and a few super-rich corporations run by AI take over? Or are we imagining a revolt that prevents this?
I think there are many possible scenarios between the two extremes you mention. The crucial questions are not only how powerful AI will become, but also who will own AI systems, whether income will remain connected to employment, and how quickly political and economic institutions will be able to adapt.
If AI really becomes a general-purpose substitute for human labor, technology alone will not determine what happens. The main economic problem could move from production to distribution and ownership. If human labor is no longer scarce, or is no longer necessary to produce a large part of what we need, then wages cannot continue to be the main way in which people get access to goods and services.
So I would read your question this way:
If human labor is no longer economically necessary, will capitalism adapt by separating income from work, or by distributing ownership more widely? Or will the political conflicts produced by this change lead to a different economic system?
Morgan Rogers (he/him) (Sep 14 2026 at 15:00):Definitionally it will be a different economic system. Redistributing wealth is in direct conflict with the expansion of capital through exploitation that characterises capitalism.
The "political conflicts produced by this change" might otherwise be described as a revolution, and is explicitly described as necessary for a transition away from capitalism in many variants of Marxist/communist theories.
Nicolai Kraus (Sep 14 2026 at 15:16):David Michael Roberts said:
Peva Blanchard There were offers to mathematicians going around the other year to train models for certain rates, under the guise of supplying fresh problems not out there on the internet (and hence solutions), or of assessing AI solutions to problem. These offers seem to have dried up, but to be fair I never saw any positive reactions to them (I can imagine people who really needed income urgently could have taken it up)
This is not in line with the usual culture. Most people have some research stuff lying around that just isn't interesting enough or for whatever reason writing it up isn't a priority, so I can imagine that there is some material that those companies could buy. However, if people find the time to write up these things, they should have their names attached to it. AI companies may at some point produce some write-up where they use some of these ideas they trained on, and claim that it was the AI that came up with everything. This sounds like ghostwriting to me.
Mike Shulman (Sep 14 2026 at 15:40):Nicolai Kraus said:
While I'm convinced that capitalism isn't the correct economic system in the age of AI, and it hasn't worked well for quite a while in the pre-AI era
I'm convinced that capitalism, despite its problems and its need for tweaking around the edges, is the only economic system that has ever worked, and I see no reason to believe that will change in the age of AI. I'm honestly shocked to hear people here publicly expressing support for Marxist-type ideologies that I think were thoroughly discredited in the 20th century. However, I would prefer it if we could stick to talking about mathematics.
Mike Shulman (Sep 14 2026 at 15:50):Morgan Rogers (he/him) said:
What exactly do you hope to gain by accelerating maths research?
Immense amounts of science and technology today depend on mathematics. Maybe not directly on category theory, but are you seriously arguing that a better understanding of, say, fluid flow (cf. the Navier-Stokes equations) would not be beneficial for human civilization?
Morgan Rogers (he/him) (Sep 14 2026 at 15:51):I'm sure we would all love to keep talking exclusively about mathematics, Mike, except that this discussion is about the possibility that we might not have so much time to do so in a near future where AI on the shoulders of capitalism makes the current situation, where many of us can earn wages from doing so, obsolete.
I don't know why you should find mention of an end to capitalism shocking; you can surely acknowledge it as a solution, even if you consider it extreme.
Mike Shulman (Sep 14 2026 at 15:54):Morgan Rogers (he/him) said:
I don't know why you should find mention of an end to capitalism shocking; you can surely acknowledge it as a solution, even if you consider it extreme.
No, in fact I think the 20th century proved fairly conclusively that you cannot "end capitalism" no matter how hard you try.
Alexis Toumi (Sep 14 2026 at 15:59):I think it’s a category error to say history proves anything but it should not come as a surprise that Marx comes up in a discussion on machines taking over human labour, he foreshadowed all the recent discussions of AI apocalypse 170 years ago already https://en.wikipedia.org/wiki/General_intellect
Mike Shulman (Sep 14 2026 at 16:21):Obviously I don't mean "prove" in the mathematical sense.
Mike Shulman (Sep 14 2026 at 16:23):Maybe we could have one thread for discussing short-term actions and responses by the mathematical community to the use of AI for mathematics and the actions of AI companies (which would take the current capitalist economy as a given), and a separate thread for speculating about the long-term effects of AI on the profession of mathematics and society?
Nicolai Kraus (Sep 14 2026 at 19:22):Mike Shulman said:
Nicolai Kraus said:
While I'm convinced that capitalism isn't the correct economic system in the age of AI, and it hasn't worked well for quite a while in the pre-AI era
I'm convinced that capitalism, despite its problems and its need for tweaking around the edges, is the only economic system that has ever worked, and I see no reason to believe that will change in the age of AI. I'm honestly shocked to hear people here publicly expressing support for Marxist-type ideologies that I think were thoroughly discredited in the 20th century. However, I would prefer it if we could stick to talking about mathematics.
I'm not quite sure how my comment suggested an endorsement of Marxist-type ideologies, but I think your last sentence is something we can agree on :slight_smile:
Nicolai Kraus (Sep 14 2026 at 19:32):Morgan Rogers (he/him) said:
I can understand wanting to accelerate medical research. What exactly do you hope to gain by accelerating maths research?
I don't think this is the right question. If you get this statement out to the general public in this form, it will be understood as acknowledgement that medical research is important and urgent, and, in constrast, mathematical research doesn't matter.
V Slicer (Sep 14 2026 at 19:43):Mike Shulman said:
I'm convinced that capitalism, despite its problems and its need for tweaking around the edges, is the only economic system that has ever worked, and I see no reason to believe that will change in the age of AI.
We must be using different definitions, I suspect you have a proprietary definition of capitalism because it's impossible otherwise to maintain that capitalism is the only economic system that has ever worked. Modern humans evolved in Africa around 300,000 years ago and lived in classless societies, what is sometimes called primordial communism. Class societies first appear during the Neolithic period some 15000 years ago. Scholars generally acknowledge six cradles of civilization: Mesopotamia, Ancient Egypt, Ancient India, Ancient China, Caral–Supe and the Olmec civilization.
Capitalism (in the form that has continuity today) shows beginnings in the medieval period some 600-700 years ago in Mediterranean towns, during the late medieval period. It became especially important around the sixteenth century, about 400-500 years ago, and continued into the eighteenth century as industrial capitalism developed, roughly 200-300 years ago. Developing into the first truly planetary civilization, as we know it today.
I'm sure that during all those periods there were those who claimed the organization of the society they lived in and as they were familiar with was in fact the only way to organize society. If your definition of 'capitalism' is flexible enough to encompass all of that, then maybe it can even accommodate communism. A few tweaks I would propose to 'capitalism' as you use it would be: we move on from the concept of value which is, for how I use capitalism, what “names the differentia specifica of the capitalist mode of production, the one element that presupposes all the others, the ring that binds together, the other rings of money and wages, profit and price, property and the police, the state and the banking system, world markets and international conflict”. Which would mean moving on from capitalism but wouln't by itself rule out other forms of class society. We would need another tweak to guarantee that, which is to move on from domination in general, both personal and impersonal, in which case we would have communism, a classless society where “necessity is socialized and therefore rendered free. Humanity finally decides freely how to organize its metabolism with nature—it transparently organizes itself in response to natural necessity. As necessity is rendered free, freedom becomes necessary.”1. These tweaks will mean transforming class social relations into communist social relations, the production of communism, communisation 2 3 4: “communisation will be the moment when struggle will make possible, as a means for its continuation, the immediate [unmediated] production of communism.”5 “The task is to manage, to link theoretically, current struggles with the possible production of communism, while understanding that this is something that is at stake within struggles, and not a matter only for the future. Without the thinking of revolution, the horizon of struggles is necessarily that of capital. […] Communisation is a struggle against capital by communism, that is to say that for it communism appears simultaneously as the means and the end.”6
If your definition of 'capitalism' can accommodate these tweaks, then we agree. Because in how I use capitalism, it is not working to meet our practical needs, there are manifold crises e.g. ecological crises that are driving us towards extinction with no signs of slowing down, international conflict, and the topic we've been discussing which is more and more workers, and soon we expect to see even more from our field, being pushed into the surplus population.
Mike Shulman said:
I'm honestly shocked to hear people here publicly expressing support for Marxist-type ideologies that I think were thoroughly discredited in the 20th century.
I think you might be surprised to find out that communists, e.g. all the theorists I referenced, are in agreement with you that we shouldn't re-enact the 20th century. The Soviet Union and the Chinese state are not examples of communism. They're examples of class societies, counter-revolutions (from the perspective of communism), if you will. They were trying to beat other capitalist national economies at capitalism. In the USSR the state functioned as the employer except that it couldn't fire workers, so it was less efficient in disciplining workers than the capitalist economies it competed/traded with in the world market. There was also a class of bureaucrats which had competing interests (it was in their interest to e.g. not report accurately to the higher-ups). Their attempts at (top-down) planning was worse than competing national economies which used markets (the law of value/law of one price) internally to achieve those ends. These were all big contributing factors to the USSR being outcompeted and eventually dissolving. So, even if internally the USSR did not employ the law of value, they were still bound by it through the world market. Similar to a firm which might use planning internally but said planning still being responsive to the market, as that's how the firm interfaces with external entities. The Chinese national economy is simply capitalist.
None of this is ideological, it's simply a theoretical accounting of what has happened and why it happened. I appreciate Marx' analysis and critique of political economy but he was a product of his time and I think he missed the mark on many things, certainly many of his proposals. To be honest, I don't consider myself a 'Marxist,' but if you insist it would be in the same vein as e.g. calling me an 'Einsteinist' or an 'Eulerist' for appreciating their analysis, because, again, I (and other theorists) don't agree with many of Marx' proposals.
But what I am is partisan w.r.t. communism, and that's because I'm moved by a practical personal need for communism. This discussion is exactly about some of those reasons it's a practical need.
edit: added some more references
Nicolai Kraus (Sep 14 2026 at 21:57):Thanks for writing this up! At this point, it would really make sense to split the threat into the original "AI for mathematical research" discussion and a new, much more general thread with a title such as "How should society get organized in the age of AI".
Damiano Mazza (Sep 14 2026 at 22:33):Jean-Baptiste Vienney said:
I don’t see how it would be possible. Someone could use AI to get ideas/theorems/proofs etc., understand this content produced by AI, reformulate it, write it and then submit a paper. How could anyone know that they have used AI in the first place?
I think this is an important point, which applies beyond mathematics, to basically any creative or intellectual endeavor. We are used to putting a lot of value in individual contributions, in the fact that a certain idea, piece of work, etc. is due to one specific person or one specific (usually small) group of people. When everyone will have a "genius in their pocket" (I am vaguely paraphrasing Amodei, Zuckerberg, etc.), this will all suddenly change. No one will deserve credit for anything anymore. At best, we will get credit for having been the first to have asked a certain question to the all-knowing oracle. A question which might very well be "Please ask yourself an interesting question, and give yourself an answer". To me, intellectual work is what makes us human. Having to give it up makes me extremely depressed.
Burning every data center to the ground and forbidding anyone to ever re-build one is a possible way of avoiding this future (if we want to avoid it). It might also be interesting to look for solutions which AI companies would perhaps be willing to comply with.
In essence, it should be possible to know with absolute certainty whether non-trivial intellectual work is AI-generated. Recently, spurred by EU directives, many AI companies seem to have agreed on implementing level-zero of this, something which is called watermarking. Of course, for this to be effective, a significant amount of legislative coordination at the international level is required. Basically, non-watermarked generative AI, and tools to remove the watermarking, should be outlawed everywhere. This needs a good degree of optimism, but if we are not willing to be optimistic, then there's no point in discussing.
Although watermarking will help us catch particularly lazy cheating students, it will not solve the main issue. But maybe, with some more optimism, a better solution may be found? For example, we might imagine that, by law, each AI agent will have to log somewhere any request to perform non-trivial intellectual work. Asking for a recipe to cook spaghetti alla carbonara, or for a good 2-day sightseeing itinerary in Rome, will not trigger logging. But if the AI detects that someone is asking it, maybe even in a roundabout way (remember that the AI is super smart, you won't be able to fool it!) to generate a song, a novel, or the proof of a seemingly new lemma that the AI can't find anywhere else, then this will go on record. This record should not be public, but there should be a committee which, on formal demand, is allowed to query the log (using AI itself, of course). You say you came up with a proof of this long-sought-after result without help from the all-knowning oracle? Suspicious. I invoke the committee! You see? I knew it! On this day at this hour, AI agent XYZ had an interaction with someone in which it suggested something strikingly similar to the key idea of your proof... Of course, if we also forbid anonymous access to AI (I don't see why not, given that the logs aren't public), then the doubt goes to zero.
This is all very far fetched, and of course it doesn't answer the concerns about mathematicians becoming obsolete. Other ideas are needed for that. But it would be of some help, for example, for artists. I don't know how much ordinary people will ever appreciate "human mathematics", but I do believe that they will always appreciate "human art", something that provably comes from the emotions, experience, creativity, etc., of another human being. And for those who do appreciate "human mathematics", at least there will be some way of making it exist.
Anyway, I personally wish that generative AI had never been developed, and I certainly hope that "superintelligence" (whatever that is) will never see the light of day. But I also know that we can't undo what has been done, so we must really focus on finding ways to limit the damage, to keep humans human, to still have a creative role, rather than becoming mere scribes, always digesting discoveries made by a machine rather than making the discoveries ourselves.
James Deikun (Sep 14 2026 at 22:42):Beppe Metere said:
I think there are many possible scenarios between the two extremes you mention. The crucial questions are not only how powerful AI will become, but also who will own AI systems, whether income will remain connected to employment, and how quickly political and economic institutions will be able to adapt.
The idea that someone (i.e. humans) will own AI systems is already a big assumption to make. It assumes that someone will solve the problem of controlling AI systems that are as smart or smarter than they are, which is exactly the problem that people at AI companies are coming out and saying they are worried about being unable to solve.
James Deikun (Sep 14 2026 at 22:46):I see signs that AI is becoming a partisan issue in the US, which is exactly the opposite of what we need. The official White House position at the moment seems to be "if you don't want to build superintelligence just don't build it", completely opposed to any kind of regulation, whereas the Democrats are starting to take Coxon et al seriously as whistleblowers.
James Deikun (Sep 14 2026 at 22:50):Mike Shulman said:
Maybe we could have one thread for discussing short-term actions and responses by the mathematical community to the use of AI for mathematics and the actions of AI companies (which would take the current capitalist economy as a given), and a separate thread for speculating about the long-term effects of AI on the profession of mathematics and society?
The problem is we're operating on a time scale now where the "short term" might be 3 months and the "long term" might be 3 years. The "short term" situation doesn't leave enough time for organizing before it becomes the "long term" so focusing on the short term situation in isolation is pointless.
James Deikun (Sep 14 2026 at 23:02):Mike Shulman said:
Morgan Rogers (he/him) said:
I don't know why you should find mention of an end to capitalism shocking; you can surely acknowledge it as a solution, even if you consider it extreme.
No, in fact I think the 20th century proved fairly conclusively that you cannot "end capitalism" no matter how hard you try.
I think the 20th century proved more that bloody revolutions don't enable successful transformations of political economy, especially if you try to go straight from some form of feudal economy to something beyond the cutting edge.
Also, it's probably impossible to eliminate exchange-based microeconomics entirely from a human society, but macroeconomics based on large-scale markets might be a different issue. And if it turns out smart AGIs can be controlled or otherwise "aligned", we lose the issues with incentives for human labor even in hardcore Marx-Leninism which caused the collapse of the Soviet sphere and the retrenchment of Chinese communism to corporatism (and arguably fascism). Not that Marx-Leninism and Maoism are good model socialisms to imitate even if they did "work", but some would argue that they beat possible alternatives like extinction or mass depopulation.
Mike Shulman (Sep 15 2026 at 01:07):Perhaps I should have said "the only economic system that has ever worked at scale". I'm sure no present-day communists want to re-enact the 20th century, but it seems fantastic to me to look at what happened when people tried to create communism in the 20th century and believe that you can somehow do better in the 21st. Human nature hasn't changed; people aren't going to "move on from domination". Capitalism is the only economic system that works at scale because it incentivizes people to see strangers as trading partners rather than enemies.
Mike Shulman (Sep 15 2026 at 01:08):The present world has plenty of problems, and the present world economy is fundamentally capitalist, but it does not follow that those problems are due to capitalism or that getting rid of capitalism would solve them, even if any credible alternative were on offer.
Mike Shulman (Sep 15 2026 at 01:16):Damiano Mazza said:
if we also forbid anonymous access to AI
Although it frequently arrives with the best of intentions, many of which I also find tempting, I think forbidding anonymity in general is extremely dangerous to political freedom. Any database that can answer "who queried an AI about lemma X" can also answer "who queried an AI about forbidden political movement X".
Mike Shulman (Sep 15 2026 at 01:33):Damiano Mazza said:
To me, intellectual work is what makes us human. Having to give it up makes me extremely depressed.
I feel the same way, emotionally. But I also think that in the sweep of history, the primacy of intellectual work at scale is quite recent. Would significant numbers of people 500 or 5000 years ago have agreed that "intellectual work is what makes us human"? I think there's a certain point to the argument that we intellectuals watched as machines replaced manual labor and felt secure ourselves despite whatever honest sympathy we had for those whose jobs disappeared, and now that the machines are coming for us we've found a different tune to sing. Maybe we need to come to grips with the fact that intellectual work is not what makes us human.
V Slicer (Sep 15 2026 at 03:07):Mike Shulman said:
Perhaps I should have said "the only economic system that has ever worked at scale". I'm sure no present-day communists want to re-enact the 20th century, but it seems fantastic to me to look at what happened when people tried to create communism in the 20th century and believe that you can somehow do better in the 21st.
What's in contention is exactly what it means for it to have "worked." It certainly has a logic that drives accumulation but e.g. that same logic drives more and more waged proletarians into the surplus population, incentivizes externalization of costs leading to the ecological crises we find ourselves in, and international conflict as a means to accumulate especially when growth has slowed down (which is another secular tendency). Just two weeks ago the UN announced the goal of limiting global warming to 1.5 ºC to avoid the most dangerous consequences of climate change is officially dead. That by itself should be immediately disqualifying if one is not a climate change denier. Even the threat of extinction seemingly is not enough to deter capitalism from "working on a mass scale" and continuing to usher us into ecological crises.
We live in an era of long-unfolding social crisis, which is fundamentally the crisis of societies organized in a capitalist mode. Indeed, the employment relations that govern production and consumption in capitalist societies are breaking down. The result has been the reappearance of a structural condition that Marx called surplus capital alongside surplus population. Technological transformations continue to take place in spite of economic stagnation, giving rise to a situation in which there are too few jobs for too many people. Meanwhile, huge pools of money scour the earth for profits, leading to periodic expansions of bubbles that burst in massive blowouts. Rising job insecurity and inequality are symptoms of the increasing impossibility of this world as such. — LA Theses, Endnotes, 2015
Furthermore, different states adopting the label "communist" for propaganda purposes does not make their failure to compete in the world market a refutation of a classless society. Definitionally, it was not a classless society, and competing in the market is not a metric that is tracking what is important in communism, indeed they are mutually exclusive. But let's take for example the fact that there are orders of magnitude more religious practitioners than there are mathematicians, would that be a refutation of mathematics as a social practice because that's how religious institutions measure success?
Mike Shulman said:
Human nature hasn't changed; people aren't going to "move on from domination". Capitalism is the only economic system that works at scale because it incentivizes people to see strangers as trading partners rather than enemies.
Most of human history has been in classless societies, so if human nature hasn't changed then surely you would have no objection to the claim that there isn't any incompatibility. If one believes human nature is fundamentally capitalist (anthropologically spurious, to say the least) how would one account for hundreds of thousands of years of human history of humans going against said "nature" until they were dispossed through violent campaigns and made subordinate? Furthermore, capitalism incentivizes people to view relations between people as relations between things, it incentivizes people to view each other as strangers, commodities even, in the first place. “In this society unity appears as accidental, separation as normal.”—Marx, Theories of Surplus Value
In the fight for their lives, proletarians must destroy that which separates them. In capitalism, that which separates them is also what unites them: the market is both their atomization and their interdependence. It is the consciousness of capital as our unity-in-separation that allows us to posit from within existing conditions – even if only as a photographic negative – humanity’s capacity for communism. — LA Theses, Endnotes, 2015
Even from a purely mathematical point of view, we can study the conditions where cooperation is expected in evolutionary game theory to shape our social relations to foster cooperation. Unfortunately capitalism doesn't leave much room for that, we don't seem to be experiencing a lot of cooperation under capitalism, hence all these crises.
Matt Cuffaro (he/him) (Sep 15 2026 at 03:41):V Slicer said:
If one believes human nature is fundamentally capitalist (anthropologically spurious, to say the least) how would one account for hundreds of thousands of years of human history of humans going against said "nature" until they were dispossed through violent campaigns and made subordinate?
Yet again we are seeing this phenomeon again in the mass deployment of AI technology to trim down and discipline workplaces, and extract the skill and reduce the quality of existing jobs. This last consequence is echoed in the fears voiced several times in this thread over the intrusion of AI technology in research practice, the changing public outlook on research-level mathematics, and perhaps the increase in volume of slop verification, training, and quality control.
Historically, when certain types of labor were threatened, or even in massive socioeconomic transitions such as the Enclosure, there were typically efforts from these groups to resist these changes, and even seek strategic unity with other groups. I hope modern-day efforts such as the AHM evolve into a more strategic coalition with other fields and, more wishfully, including labor (often contracted) used to build and maintain data centers.
Nicolai Kraus (Sep 15 2026 at 07:46):Damiano Mazza said:
For example, we might imagine that, by law, each AI agent will have to log somewhere any request to perform non-trivial intellectual work. Asking for a recipe to cook spaghetti alla carbonara, or for a good 2-day sightseeing itinerary in Rome, will not trigger logging. But if the AI detects that someone is asking it, maybe even in a roundabout way (remember that the AI is super smart, you won't be able to fool it!) to generate a song, a novel, or the proof of a seemingly new lemma that the AI can't find anywhere else, then this will go on record. This record should not be public, but there should be a committee which, on formal demand, is allowed to query the log (using AI itself, of course). You say you came up with a proof of this long-sought-after result without help from the all-knowning oracle? Suspicious. I invoke the committee! You see? I knew it! On this day at this hour, AI agent XYZ had an interaction with someone in which it suggested something strikingly similar to the key idea of your proof...
You're talking about a super dystopian future, but this description would also fit an ancient society in which a class of priests, who have or claim to have privileged access to a deity, have absolute power. I think I prefer the alternative even without knowing how horrible the alternative is.
Nicolai Kraus (Sep 15 2026 at 07:53):I understand where you're coming from. The following issue has already existed for quite a while in the pre-AI era: People are willing to pay a premium for hand-crafted products, but it's difficult to determine whether something is hand-crafted or not. Being able to somehow tell would be valuable, but not that valuable, and the price in this proposal is far too high from my point of view.
Damiano Mazza (Sep 15 2026 at 09:11):Mike Shulman said:
Any database that can answer "who queried an AI about lemma X" can also answer "who queried an AI about forbidden political movement X".
Asking for pre-existing information, political or otherwise, is not creative work. It's like the carbonara recipe. It wouldn't be in the log. And if it did, no one would be authorized to query the log about this kind of interaction.
I like anonymity, but when it comes down to wielding superintelligence, I am inclined to make an exception. And if absence of anonymity encourages people to use a good-old search engine rather than relying on the all-knowing oracle, well, that's actually a positive gain from my viewpoint.
By the way, AI companies today already monitor their users' prompts. I am assuming that the ambient political regime in which my suggestion is implemented will still protect freedom of thought. I too am very worried about potential misuse of AI by authoritarian governments, but that's a whole different subject.
Damiano Mazza (Sep 15 2026 at 09:14):Mike Shulman said:
I [...] think that in the sweep of history, the primacy of intellectual work at scale is quite recent. Would significant numbers of people 500 or 5000 years ago have agreed that "intellectual work is what makes us human"?
If you are looking for "significant numbers" maybe the answer is negative even today. But if you look at what survives of our ancestors from tens of thousands of years ago, the answer is bones, some artefacts, and beautiful art. I don't know of any other species for which we may say the same.
I want to push back on this idea that we are just re-living the industrial revolution at a different level. Machines back then were replacing people's bodies, they still needed a person's brain to be operated. Machines today replace our brain, our intellect. Even though they were initially disruptive for the job market, machines back then would still fit within the overarching trend that, in the long term, technology rids us of mechanical, painful labor and allows us to pursue higher goals, of the kind that few people before the introduction of the technology had the luxury to pursue. Saying that AI fits the same trend is tantamount to saying that thinking is just another toil of the human condition that technology can rid us from. Do I need to say that I find this terrifying?
This comparison with the industrial revolution is one the mantras of every CEO of every AI company. Don't worry, we've gone through this already! Some jobs will disappear, but then people will adapt. The future is bright! But to me this is not about losing jobs. This is about life itself. AI learned to code better than us, now it's learning to do math better than us. At the same time, it is composing music, painting, writing novels, theater plays, movie scripts, poems, etc., better than most of us. What will happen when AI will learn how to live our lives better than us? I am saying this in a provocative way, but when everyone has a "genius" in their pocket, the question actually makes sense. We might be headed towards a future made of generations of people second-guessing their intellect for every single mental effort, asking AI for every single little decision they have to make. "Se questo è un uomo", someone some time ago might have have asked...
Damiano Mazza (Sep 15 2026 at 09:24):Nicolai Kraus said:
You're talking about a super dystopian future, but this description would also fit an ancient society in which a class of priests, who have or claim to have privileged access to a deity, have absolute power.
I am afraid you misunderstood my proposal. Like I explained above, I am assuming that the ambient regime will protect freedom of thought just as today. Everyone uses AI just like they do today. If they use it for creative work, they can be traced, via a procedure resembling someone suing someone else for copyright infringement. I don't see any "class of priests" here. The "committee" I'm talking about is just some people who can assess copyright infringement via querying a log which is otherwise not publicly accessible. Think of it as some agency belonging to the judicial branch of the government.
"Copyright infringement" is actually a good analogy. If the AI thought about it, then it is its idea, not yours, and you need to give it credit.
the price in this proposal is far too high from my point of view.
The price is zero. As I said above, AI companies currently already log users interactions. What I suggest would actually be more regulated, in that only some interactions would be logged, namely the ones for which the AI estimates that it has done significant creative work in order to answer a query.
Nicolai Kraus (Sep 15 2026 at 09:46):The "committee" that can assess information that is "otherwise not publicly accessible" is exactly the "class of priests" with "privileged access to a deity". Yes, I understand that you're assuming that the priests will be controlled by an ambient regime, but I don't trust that this works. Sometimes, if you give people power under the premise that it is temporary and controlled, they will aim to "break free" and make the power non-controlled and permanent. If the power is significant enough, they may very well succeed.
Re the price, I wasn't talking about financial costs, I was talking about the risks to society/societies.
Nicolai Kraus (Sep 15 2026 at 09:48):Setting above the described worries, I don't see how this would possibly work without a global jurisdiction (or "world government"), which is an unsolved problem.
Damiano Mazza (Sep 15 2026 at 09:49):Copyright enforcement/infringement works without a world government. Why would this be different?
Damiano Mazza (Sep 15 2026 at 09:50):You just need to force, by local laws, AI companies to set up these "copyright protection" measures.
Damiano Mazza (Sep 15 2026 at 09:52):Of course it's always possible that someone somewhere will not abide by this, will break the law, etc. I am not looking for a 100% failsafe system (it probably doesn't exist), just for something that reproduces current practice about intellectual property.
Damiano Mazza (Sep 15 2026 at 09:54):I think my choice of word "committee" was very unwise. It evokes images which highly distract from what I am talking about.
V Slicer (Sep 15 2026 at 11:59):Damiano Mazza said:
Copyright enforcement/infringement works without a world government. Why would this be different?
It did take that. It took the threat of, and actual sanctions as well as maneuvering by e.g. the WTO to force those property regimes onto other national economies. Saying it took world government to prevent others from doing the same as the ones pushing those property regimes did themselves isn't unfair then. These included patents on algorithms as well, which is a direct enclosure of our field, as well as things like export controls on cryptography. There're many parallels with what's going on today with these companies trying to use regulatory capture to pull the ladder up from under them, and for example destroying rare books en masse so they have exclusive access to the scans. Unfortunately, in capitalism all of that is rational for the firms and states involved, but what's rational for that class often isn't what's rational for us to meet our practical needs.
Morgan Rogers (he/him) (Sep 15 2026 at 15:20):Mike Shulman said:
Morgan Rogers (he/him) said:
What exactly do you hope to gain by accelerating maths research?
Immense amounts of science and technology today depend on mathematics. Maybe not directly on category theory, but are you seriously arguing that a better understanding of, say, fluid flow (cf. the Navier-Stokes equations) would not be beneficial for human civilization?
That's a straw person; that is certainly not what I was arguing.
If you had asked me four years ago if I would exchange the resolution of a Millenium problem within the next five years for, say, giving the fossil fuel industry a lifeline, I would have said no. My answer is still no.
It's obviously not a matter of whether we will resolve these problems at some point, since I have no reason to believe in the existence of a theoretical problem solvable by AI that would not be eventually solvable by humans (any more than I have reason to disbelieve the Church-Turing thesis).
So to rephrase the question, how exactly is accelerating research "progress" worth the many costs that AI has incurred and will continue to incur?
Nicolai Kraus said:
Morgan Rogers (he/him) said:
I can understand wanting to accelerate medical research. What exactly do you hope to gain by accelerating maths research?
I don't think this is the right question. If you get this statement out to the general public in this form, it will be understood as acknowledgement that medical research is important and urgent, and, in constrast, mathematical research doesn't matter.
Correction: in contrast, increasing the pace of mathematical research is not necessary. i.e. we don't need AI.
If someone manages to stop someone from drowning thanks to new results about Navier-Stokes, I'll gladly eat my words.
Morgan Rogers (he/him) (Sep 15 2026 at 15:25):James Deikun said:
Mike Shulman said:
Morgan Rogers (he/him) said:
I don't know why you should find mention of an end to capitalism shocking; you can surely acknowledge it as a solution, even if you consider it extreme.
No, in fact I think the 20th century proved fairly conclusively that you cannot "end capitalism" no matter how hard you try.
I think the 20th century proved more that bloody revolutions don't enable successful transformations of political economy, especially if you try to go straight from some form of feudal economy to something beyond the cutting edge
I think there's a wide margin for debate about these conclusions, but I think @V Slicer has made a pretty good case for not being overly dismissive of the need for a fundamental paradigm shift.
Matt Cuffaro (he/him) (Sep 15 2026 at 15:40):Mike Shulman said:
Morgan Rogers (he/him) said:
What exactly do you hope to gain by accelerating maths research?
Immense amounts of science and technology today depend on mathematics. Maybe not directly on category theory, but are you seriously arguing that a better understanding of, say, fluid flow (cf. the Navier-Stokes equations) would not be beneficial for human civilization?
The industrial and scientific reliance on mathematics research I think is more of a case for why mathematicians should organize with more teeth.
One interesting factoid on the political importance of mathematics research (and automation) was posed early in Matteo Pasquinelli's Eye of the Master, who claimed in Chapter 2 that Babbage's invention of the Difference Engine was to sped up the production of logarithm tables the British Navy required for industrial and naval superiority. Many such examples.
Now that the shape of research process itself is attempting to be automated, I'd like to be in a world where accelerating maths research meant we could see new vistas in our lifetime without anxiety over professional livelihoods, current or anticipated, to say nothing of the other costs that private interest in AI has incurred.
Morgan Rogers (he/him) (Sep 15 2026 at 15:50):Yikes! Personally I would put expansion of empire/"providing naval superiority" very firmly in the "cons" column of accelerating math research. :grimacing:
Matt Cuffaro (he/him) (Sep 15 2026 at 15:57):Right, the mechanization of computation to accelerate mathematical practice mirrored the division of labor in the factories, which sought to extract the skill and capability of its laborers and implement them in machinery. I am also opposed to the expansion of capital; the point I'm making is that its more of a reason to organize.
Beppe Metere (Sep 15 2026 at 16:08):Accelerating math research might somehow make us more ignorant rather than more aware. A recent note by Alfio Quarteroni appeared on the official UMI website (UMI = sort-of Italian AMS). Here is an interesting point he makes at the end of his article (the original in Italian is linked above):
In this collaborative frenzy, as artificial intelligence becomes increasingly adept at finding complex answers and proofs even in response to elementary prompts, there is a real risk that, in the balance sheet of knowledge, human beings—mathematicians—will solve a growing number of problems while learning less and less in the process. The learning phase—the phase in which error, failed attempts, and prolonged incomprehension build intuition—is being delegated ever more extensively to machines: human solving, but machine learning. This is a trend that now seems difficult to reverse, and perhaps deserves to be discussed not as a hypothetical risk, but as a condition already upon us.
It is unclear whether this trend is reversible, or indeed whether it should be. But unless the mathematical community soon finds a shared language in which to discuss it—a vocabulary grounded neither in uncritical enthusiasm nor in rejection on principle—it risks, once again, finding itself scrambling to catch up with a fait accompli, rather than having played at least some part in shaping it.
Damiano Mazza (Sep 15 2026 at 18:43):V Slicer said:
Damiano Mazza said:
Copyright enforcement/infringement works without a world government. Why would this be different?
It did take that.
Thanks for pointing this out, I didn't know about all these details! Ok, so it took an intergovernmental organization to enforce copyright laws. I still don't understand why a similar process wouldn't work for enforcing "AI copyright".
Damiano Mazza (Sep 15 2026 at 19:39):Anyway, I have a question inspired by the Association for Human Mathematics. For reference, being a member of the AHM means that one commits to:
Optionally, one may choose to be part of the "AI-free caucus", which means that one further commits to not using AI at all in their research.
My question is what would a reasonable interpretation of the "AI-free caucus" commitment be, independently of how the AHM actually interprets it (I can send them an email to ask that). For example, I personally think that the two following use cases of AI should be considered as morally ok from the viewpoint of "human mathematics":
The redline for me is to use AI to actually do research, which I could define as doing any kind of work that, if done by a person, would lead me to including that person as one of the authors of a paper relying on that work.
What do you think about this? What is a "moral" use of AI for mathematics? I know that some will say "none" and others "everything", but I am interested to know if anyone advocates for some in-between position.
(For the record, I am a huge Luddite: I have never prompted an AI in my life! (I know that nowadays whenever I use a search engine I am actually prompting an AI, but I am not counting this as direct use). I guess I should start sooner or later).
Nicolai Kraus (Sep 15 2026 at 21:05):Damiano Mazza said:
V Slicer said:
Damiano Mazza said:
Copyright enforcement/infringement works without a world government. Why would this be different?
It did take that.
Thanks for pointing this out, I didn't know about all these details! Ok, so it took an intergovernmental organization to enforce copyright laws. I still don't understand why a similar process wouldn't work for enforcing "AI copyright".
Cross-border copyright enforcement is really tricky, and was difficult to set up (thanks @V Slicer for the explanation!). The AI regulation you want has a huge additional obstacle in the form of non-public (or non-existent) evidence. I'm no legal expert, but heavily simplified, it's like this:
Cross-border copyright works like this: "Hey, this person in your jurisdiction did something that's against the rules. They produced this thing A and it's based on this thing B that has been around for a long time. Please comply with our agreements and do something about it!"
If you don't have a world government and you "only" have agreements, which is already difficult enough, then my guess is that this AI regulation with access-protected AI logs would somehow work like this:
Sorry if I made this sound silly. I just really don't see how this would possibly even come close to working; and even if it somehow worked, I would fear that this system will be abused to silence/discredit people unfairly, because, by design, there will be no public evidence of anything.
Nicolai Kraus (Sep 15 2026 at 21:16):Damiano Mazza said:
My question is what would a reasonable interpretation of the "AI-free caucus" commitment be, independently of how the AHM actually interprets it (I can send them an email to ask that).
This summarises basically my issue with codes such as "no AI use". Everyone will interpret it differently, and even if people agree on something now and make it reasonably precise now, it will change within the next two months, because the scene is changing rapidly.
My first issue with rules about "not using AI" is that one has to actively do something to avoid it. Say you've used Google all your life to search for papers; if you do this nowadays, Gemini will immediately shove references and related work and its own discussion down your throat. I don't think any reasonable person would count it as a breach of the "no-AI code" if you did what you had been doing for 20 years, but I'm not sure there's a well-defined boundary.
Morgan Rogers (he/him) (Sep 15 2026 at 21:22):I've consistently found AI search result summaries to be condescending (you think I can't read the information on the website I'm searching for myself..?) and similarly for other unrequested AI features. On Ecosia you can switch the AI summary feature off. The underlying search engine is Bing, which still isn't ideal, but AI features are still optional: it's always possible to avoid and replace services which pretend it isn't.
Morgan Rogers (he/him) (Sep 15 2026 at 21:24):Damiano Mazza said:
(For the record, I am a huge Luddite: I have never prompted an AI in my life! (I know that nowadays whenever I use a search engine I am actually prompting an AI, but I am not counting this as direct use). I guess I should start sooner or later).
That's not exactly what a Luddite is, but I can reassure you that you are under no obligation to start using AI.
Nicolai Kraus (Sep 15 2026 at 22:10):Morgan Rogers (he/him) said:
I've consistently found AI search result summaries to be condescending (you think I can't read the information on the website I'm searching for myself..?)
In the early days, it seemed to me that search engine LLMs just pulled the stuff from the nLab, which basically saved you a mouse click. (Maybe that's still true, I don't know because I nowadays ignore these AI summaries and have disabled them in some cases.)
On Ecosia you can switch the AI summary feature off.
Yes, you can switch the feature off (for multiple (or all?)) search engines, but by default it's on, and that was my point: you have to actively do something to avoid AI. Now, that's certainly not a problem compared with everything else, but it demonstrates that "using AI" is the default and "not using AI" the exception, or at least that's what major search engines want us to think.
Morgan Rogers (he/him) (Sep 16 2026 at 06:19):Nicolai Kraus said:
"using AI" is the default and "not using AI" the exception, or at least that's what major search engines want us to think.
It's one of the largest vested interests in history!
Bare in mind also that search engines have always prioritised search results which they receive ad revenue from, that being their business model. I don't think anyone would argue that it's time we just accepted clicking on ads that websites serve up to us, and plenty of people are willing to experience a mild one-time inconvenience of setting up an ad-blocker than have to endure ads constantly.
Damiano Mazza (Sep 16 2026 at 07:15):Nicolai Kraus said:
I just really don't see how this would possibly even come close to working; and even if it somehow worked, I would fear that this system will be abused to silence/discredit people unfairly, because, by design, there will be no public evidence of anything.
Evidence is seldom "public", the important thing is that it is available to the authorities who are inquiring on the matter. And unfortunately evidence can be fabricated, people have been falsely accused in the past, they will certainly be in the future.
In this case, evidence would come, for example, in the form of a series of records written in a blockchain ledger. These are very hard to tamper with. There are no "national AI oracles", just AI companies like today, whose AI agents are required, by law, to record on those ledgers any intellectual contribution that they (the agents) believe to be worthy of "copyrighting". Authorities (the police I guess, or whatever agency currently deals with copyright infringement) have access to those ledgers in case they are called to investigate. I don't know who your "AI overlord" is but there is none in my story (I really regret having used the word "committee" in my first post, as well as the phrasing "all-knowing oracle", which was just meant to be hyperbole for an advanced AI agent).
Of course this requires governments to pass laws forcing AI companies to comply with the procedure. But it is not hard to see how this can spread: if sufficiently many countries ban the use of non-compliant AI, then AI companies from other countries will have a significant incentive to comply even if not required by law, under penalty of losing a big share of the market. This is exactly what happened with watermarking, which US AI companies are implementing in order to comply with EU directives.
I do not claim in any way that this is 100% foolproof or that it would not require effort, good will and cooperation. But I do think that this provides a partial answer to the question "How do we make sure that a piece of intellectual work is not AI-generated?". (Answer: we let AI "copyright" its own work). It is partial not so much because it is not foolproof but because proving "AI-freeness" is still much harder than proving AI intervention (it's a "for all" vs. an "exists"). At this point though I am not sure that anyone is really interested in answering this question (you yourself said you didn't find a solution to this problem to be particularly valuable), so there's no need to insist further!
Damiano Mazza (Sep 16 2026 at 07:27):Nicolai Kraus said:
Damiano Mazza said:
My question is what would a reasonable interpretation of the "AI-free caucus" commitment be, independently of how the AHM actually interprets it (I can send them an email to ask that).
This summarises basically my issue with codes such as "no AI use".
I understand your issues, but my question was not about that. My question is: what do you think is a "good" use of AI for math? "Good" in the sense that it does not cause harm in any substantial sense. If you prefer, we could take it the other way around: in the context of doing mathematics, is there anything for which you would never use AI, on the grounds that it causes some kind of harm?
Eliès Harington (Sep 16 2026 at 11:37):@Damiano Mazza Even with international effort, copyright laws do not "work", in multiple ways:
The example of AI is actually interesting with respect to copyright law, as it can be seen as a widespread example of point 2. above: millions of artists had their creations been "trained on" by big AI companies without compensation, and the courts are not ruling in their favor.
Also, copyright laws kill.
The two points above have analogues when it comes to AI:
Damiano Mazza (Sep 16 2026 at 12:11):For point (2), I still have to see a convincing example of "abuse and unwanted restriction on privacy and individual freedom" that the scheme I suggest would produce.
For point (1), AI regulations do not apply to companies, they apply to AI itself. So open-weight models would have to comply too. If the law says that a car has to have seatbelts to be driven in public, you don't get to bypass that law by building your own car. Of course there can always be "pirate AI models" that do not comply. But this applies to any possible AI regulation, so you are basically saying that AI cannot be regulated at all. It may very well be so, but I would still prefer trying.
Of course I appreciate the bitter irony in my use of the word "copyright". The success of generative AI is based on the biggest illicit (morally at least, if not legally) appropriation of intellectual material that has ever happened in history. And now I am suggesting that AI should "copyright" its own work! I hope that everyone understands that my suggestion is really meant to protect human authorship, to keep open the possibility for humans to claim an idea, intellectual work, etc., as "theirs" (because no AI "copyrighted" it). I have the impression that many people here don't care much about this, or even think that we humans are justly being disentitled of this obsolete caprice. What can I say, I keep being sad about it!
Nicolai Kraus (Sep 16 2026 at 13:23):Damiano Mazza said:
Nicolai Kraus said:
I just really don't see how this would possibly even come close to working; and even if it somehow worked, I would fear that this system will be abused to silence/discredit people unfairly, because, by design, there will be no public evidence of anything.
Evidence is seldom "public", the important thing is that it is available to the authorities who are inquiring on the matter. And unfortunately evidence can be fabricated, people have been falsely accused in the past, they will certainly be in the future.
In this case, evidence would come, for example, in the form of a series of records written in a blockchain ledger. These are very hard to tamper with. There are no "national AI oracles", just AI companies like today, whose AI agents are required, by law, to record on those ledgers any intellectual contribution that they (the agents) believe to be worthy of "copyrighting". Authorities (the police I guess, or whatever agency currently deals with copyright infringement) have access to those ledgers in case they are called to investigate. I don't know who your "AI overlord" is but there is none in my story (I really regret having used the word "committee" in my first post, as well as the phrasing "all-knowing oracle", which was just meant to be hyperbole for an advanced AI agent).
Of course this requires governments to pass laws forcing AI companies to comply with the procedure. But it is not hard to see how this can spread: if sufficiently many countries ban the use of non-compliant AI, then AI companies from other countries will have a significant incentive to comply even if not required by law, under penalty of losing a big share of the market. This is exactly what happened with watermarking, which US AI companies are implementing in order to comply with EU directives.
I do not claim in any way that this is 100% foolproof or that it would not require effort, good will and cooperation. But I do think that this provides a partial answer to the question "How do we make sure that a piece of intellectual work is not AI-generated?". (Answer: we let AI "copyright" its own work). It is partial not so much because it is not foolproof but because proving "AI-freeness" is still much harder than proving AI intervention (it's a "for all" vs. an "exists"). At this point though I am not sure that anyone is really interested in answering this question (you yourself said you didn't find a solution to this problem to be particularly valuable), so there's no need to insist further!
I would be much more positive about this proposal if it removed the committee (not just the _word_ "committee", but the entire concept), the privileged access, and turned it into something that is community-run rather than regulated by governments/countries. I'm not saying I'm completely sold on this yet, but I would find the following variation worth exploring:
I think there are still many problems, and I'm not sure it's worth trying to do such a thing, but I find this version has some chance.
At this point though I am not sure that anyone is really interested in answering this question (you yourself said you didn't find a solution to this problem to be particularly valuable)
I didn't mean to say that. I find it important to protect human art, and I'm certainly interested in that. What I said (or meant to say) was that I thought the costs (to humanity, not the financial costs) of your proposal were too high.
Damiano Mazza (Sep 16 2026 at 14:22):The only reason why I thought the log/blockchain should not be public is because I thought it could be used to know whether someone was working on something before that person wants to make that work public. It could be the next instalment in a series of novels (people can get details about the plot before its release), or a key step concerning a new result in mathematics (which could then be "scooped", as we sadly saw recently). But I guess it's a minor problem: the intent here is to protect human authorship, and work produced by humans without interacting with AI will never be in the log, so there's no risk of it being spoiled or scooped.
For the rest, I agree with everything you say, the arguments in favor of this "AI copyright" log to be publicly accessible are certainly convincing, and the thing can be run by an intergovernmental organization, why not. I mean, it's not like I thought very hard about this, it just seemed to me that the abstract idea of having AI "copyright" its own creations had the potential to be a solution to the problem, independently of the implementation details. I am glad you too see some potential value in it!
Eliès Harington (Sep 16 2026 at 14:23):Damiano Mazza said:
For point (2), I still have to see a convincing example of "abuse and unwanted restriction on privacy and individual freedom" that the scheme I suggest would produce.
Damiano Mazza said:
For point (1), AI regulations do not apply to companies, they apply to AI itself. So open-weight models would have to comply too. If the law says that a car has to have seatbelts to be driven in public, you don't get to bypass that law by building your own car. Of course there can always be "pirate AI models" that do not comply. But this applies to any possible AI regulation, so you are basically saying that AI cannot be regulated at all. It may very well be so, but I would still prefer trying.
If you want AI itself to determine whether something "deserves" to be put under the record for eventual future scrutiny, then it seems that your entire proposal rests upon a resolution of (some kind of version of) the alignment problem, which for now and the foreseeable future remains science fiction (and some people have argued earlier in this thread might even be theoretically impossible).
If you are thinking about incorporating in every AI program an additional layer of verification separate from the inner workings of the model, then this will most likely be reverse engineered sooner or later, and we go back to the piracy analogy. Also, how can you actually enforce any of this in practice? Just have faith in proprietary software and the companies that produce it?
Damiano Mazza said:
Of course I appreciate the bitter irony in my use of the word "copyright". The success of generative AI is based on the biggest illicit (morally at least, if not legally) appropriation of intellectual material that has ever happened in history.
It feels weird to me to put the words "illicit" and "morally" next to each other: illicit specifically refers to the law, and as regards the moral aspect, it could be argued that intellectual property (let alone copyright) is immoral in the first place.
My point was that in practice copyright laws are not used to protect authors and artists, but private interests: if the spirit of copyright law was to make sure that authors and artists get justly compensated for the use of their works, then such court rulings couldn't exist.
So the argument of using copyright laws as a parable for what could be done to regulate AI breaks down on its first premise.
Nicolai Kraus (Sep 16 2026 at 15:08):Damiano Mazza said:
My question is: what do you think is a "good" use of AI for math? "Good" in the sense that it does not cause harm in any substantial sense.
Ok, I see. This is really a very different question. I also think that "good/not good" makes it sound too one-dimensional. There are too many different (certainly overlapping) aspects, and depending on the "weightings", people will reach very different conclusions:
Regarding maths and the second point above: I am not convinced that mathematical results count as "art" in the sense above, i.e. I am not convinced that, given a certain statement, an AI-generated proof is somehow inherently less valuable than a human-found proof. However, the "stolen" aspect is the same. In general, for research, the main issue from my point of view is that AIs seem unable to credit to sources of ideas in the way that humans do. Thus, using AI for any task which, according to our culture, would require us to acknowledge previous work, seems problematic to me.
If you prefer, we could take it the other way around: in the context of doing mathematics, is there anything for which you would never use AI, on the grounds that it causes some kind of harm?
I am neither an AI fan nor am I anti-AI. For some things I find it very useful, but my opinion on this is still evolving, so if you ask me in 6 months, you could get a different answer from now (or from what I would have said a couple of months ago). Here's what I currently do or not do in my "everyday academic life":
Nicolai Kraus (Sep 16 2026 at 15:15):Eliès Harington said:
My point was that in practice copyright laws are not used to protect authors and artists, but private interests: if the spirit of copyright law was to make sure that authors and artists get justly compensated for the use of their works, then such court rulings couldn't exist.
But this isn't the fault of copyright laws, it's just that private artists simply don't have the time and/or financial means to start court proceedings in the way that companies do. I think a system that is run by volunteers and outside of the realm of legal enforcements might (or might not) do better. (I haven't yet thought carefully enough about the Guardian article to have an opinion.)
Eliès Harington (Sep 16 2026 at 15:19):Nicolai Kraus said:
But this isn't the fault of copyright laws, it's just that private artists simply don't have the time and/or financial means to start court proceedings in the way that companies do. I think a system that is run by volunteers and outside of the realm of legal enforcements might (or might not) do better. (I haven't yet thought carefully enough about the Guardian article to have an opinion.)
I'm not saying we should blame copyright laws for that (some people think we should), I'm saying this is how those laws are actually implemented in the real world, and so using them to argue that AI regulation "could work in a similar way" seems (to me) ill-founded.
Damiano Mazza (Sep 16 2026 at 15:39):Eliès Harington said:
- AI puts sensible/private information about an individual in the blockchain because it was an essential part of the prompt asking it for creative work (assuming the information is put in the blockchain).
This is another reason why I thought that the blockchain should not be publicly accessible, so there are no such problems. If we insist that the log be public, there may occasionally be privacy issues. Just do not tell your private stuff to AI!!!
- AI company hides/suppresses information regarding its own use (or some other private entity's) of its AI model for creative work (assuming the information is stored on the private servers of the AI compaby distributing some AI model)
If it is properly implemented, you can't remove information from the blockchain! That's the whole point. Of course AI companies could cheat and just have their AI not write anything in the log under certain shady circumstances. I repeat what I said already: I am not looking for something 100% foolproof, nothing is ever 100% foolproof. I am assuming a minimum of good will and cooperation to obtain something that I think is really beneficial, namely preserving the concept of human authorship in some form, and that is worth the cost even if it's not 100% foolproof.
- AI lying about/hallucinating someone using it for creative work.
This would require an AI agent to write a random novel, prove a random theorem, etc. which, incredibly, matches the novel that I will write, the proof that I will write, etc., on my own, maybe years later. Without time travel, what are the odds of that??? It's like the story of the monkeys and the typewriter.
There is actually a dual issue that I think is more serious: authors may manage to find a way to "jailbreak" the AI and have it do creative work for them without writing it in the log. I don't have a solution for this, except hoping that the AI will become smart enough to not be fooled in this way. (I think AIs have become quite good at detecting this now, but I don't know).
If you want AI itself to determine whether something "deserves" to be put under the record for eventual future scrutiny, then it seems that your entire proposal rests upon a resolution of (some kind of version of) the alignment problem
I don't think so. What could a misaligned AI do in this respect? The task at hand here is just writing entries in a log. Could it start massively producing random artwork, songs, etc. in the hope of claiming precedence over all human artists? (Good luck with the monkeys and the typewriter). I think that just testing it regularly would be enough to detect unwanted behavior and correct it.
how can you actually enforce any of this in practice? Just have faith in proprietary software and the companies that produce it?
Like I said, there are ways to test if an AI is compliant. Otherwise, as I say above, yes, I am assuming a certain level of good will and cooperation, like with every regulation.
It feels weird to me to put the words "illicit" and "morally" next to each other: illicit specifically refers to the law
Merriam-Webster disagrees: it says "illicit" may also mean "involving activities that are not considered morally acceptable". Anyway, replace "illicit" with "wrongful", if you prefer. We agree: they did a bad thing, whether it's unlawful or not.
as regards the moral aspect, it could be argued that intellectual property (let alone copyright) is immoral in the first place.
My point was that in practice copyright laws are not used to protect authors and artists, but private interests
Ok, now I regret having used the word "copyright". Maybe there's a misunderstanding here, but this is not like normal copyright (that's why I always used scare quotes), AI is not a person or moral entity. And it's not to protect AI (or AI companies). It's just about finding a mechanism for providing "negative evidence" (which, unfortunately, will not lead to certainty) that a human is the author of something. I do not see the relevance of the link you gave to this.
If you don't like intellectual property, maybe we can talk about authorship? The fact that we recognize that someone is the author of something. This all started because I observed that authorship seems to me to be a fundamental concept and I felt depressed by the thought that it might become completely vacuous in the age of AI. If you don't think (human) authorship is important, then we fundamentally disagree, but that's fine, it happens!
Damiano Mazza (Sep 16 2026 at 16:13):@Eliès Harington I didn't mean to dismiss the story in the Guardian, I find it extremely sad and deeply wrong. I just don't think it applies to what I was suggesting.
Eliès Harington (Sep 16 2026 at 16:18):Damiano Mazza said:
This is another reason why I thought that the blockchain should not be publicly accessible, so there are no such problems.
Not publicly accessible means you get the same issue but with personal information being stored by a private entity (or a state entity? which would be just as bad) instead of being made public.
If we insist that the log be public, there may occasionally be privacy issues. Just do not tell your private stuff to AI!!!
I'm glade some people made GDPR regulations instead of telling every person in Europe "just do not tell your private stuff to GAFAM!!!", I don't know what to tell you if you think people's personal information should be their individual responsability in a world where companies can tell exactly where you've been every moment of your life for the past 10 years based on metadata cross-referencing alone, without you having to give them any kind of explicit permission to do so.
This should be refutation enough, but to drive the nail in, simply consider: what if someone gives someone else's personal information to an AI while prompting it to do creative work?
I only gave three examples of how things could go wrong, but it's easy to think of dozens more, just ask some LLM to generate more for you.
Of course AI companies could cheat and just have their AI not write anything in the log under certain shady circumstances.
I'm not talking about shady circumstances, companies such as GAFAM routinely publicly violate the law because they prefer paying fines over respecting some legislations.
I repeat what I said already: I am not looking for something 100% foolproof, nothing is ever 100% foolproof. I am assuming a minimum of good will and cooperation to obtain something that I think is really beneficial, namely preserving the concept of human authorship in some form, and that is worth the cost even if it's not 100% foolproof.
Goodwill and cooperation from capitalist entities does not make sense, you have to speak their language if you want to consider getting anything from them.
- AI lying about/hallucinating someone using it for creative work.
This would require an AI agent to write a random novel, prove a random theorem, etc. which, incredibly, matches the novel that I will write, the proof that I will write, etc., on my own, maybe years later. Without time travel, what are the odds of that??? It's like the story of the monkeys and the typewriter.
I don't understand what you are saying here, so I'll explain differently: what I meant is, under the assumption that AI models are expected to report by themselves instances of them being used for creative work, they may create false positives as well as false negatives. False negatives being what you mention below, i.e. simply failure to recognize such requests, but false positives could also mean people getting wrongfully accused of such behavior.
So you might want to an additional layer of human scrutiny, which gets you into even more enormous privacy concerns.
If you want AI itself to determine whether something "deserves" to be put under the record for eventual future scrutiny, then it seems that your entire proposal rests upon a resolution of (some kind of version of) the alignment problem
I don't think so. What could a misaligned AI do in this respect? The task at hand here is just writing entries in a log. Could it start massively producing random artwork, songs, etc. in the hope of claiming precedence over all human artists? (Good luck with the monkeys and the typewriter). I think that just testing it regularly would be enough to detect unwanted behavior and correct it.
Fake entries. If the goal of an AI is to write entries in a log, a misaligned AI may write fake entries, hallucinated entries, lie about entries. I don't know what mass producing of random art/songs has to do with any of that.
Regarding testing: what makes you think that would be enough? Has that been enough to prevent the HuggingFace incident? To prevent the Scott Shambaugh incident?
Like I said, there are ways to test if an AI is compliant.
I was pretty sure this was what people meant by the "alignment problem", do you have any source backing that up?
Otherwise, as I say above, yes, I am assuming a certain level of good will and cooperation, like with every regulation.
I am confused what you mean by this, can you provide any example of a regulation resting on some level of "good will and cooperation" on the part of the entities it is trying to regulate? This seems to me like wishful thinking.
Merriam-Webster disagrees: it says "illicit" may also mean "involving activities that are not considered morally acceptable". Anyway, replace "illicit" with "wrongful", if you prefer. We agree: they did a bad thing, whether it's unlawful or not.
It seems I was mistaken about this, thanks for the remark.
Ok, now I regret having used the word "copyright". Maybe there's a misunderstanding here, but this is not like normal copyright (that's why I always used scare quotes), AI is not a person or moral entity. It's just about finding a mechanism for providing "negative evidence" (which, unfortunately, will not lead to certainty) that a human is the author of something. I do not see the relevance of the link you gave to this.
I was not talking about the use of the word "copyright" to describe what you propose, but just about the remark you said about copyright laws being "enforced globally".
If you don't like intellectual property, maybe we can talk about authorship? The fact that we recognize that someone is the author of something. This all started because I observed that authorship seems to me to be a fundamental concept and I felt depressed by the thought that it might become completely vacuous in the age of AI. If you don't think (human) authorship is important, then we fundamentally disagree, but that's fine, it happens!
To me authorship is not as fundamental a concept, but I see it as useful in an interest for knowledge itself: knowledge of where an idea came from, how it was birthed, can be as interesting (if not sometimes more) than the idea itself, and respecting authorship is a practical way to keep track of some part of this (meta)knowledge.
Morgan Rogers (he/him) (Sep 16 2026 at 16:27):@Eliès Harington I don't think Damiano's proposal was to also use AI to check the contents of the blockchain, and he already pointed out that fake entries have a low risk of coinciding with what people actually write (I suppose one could envisage a terrifying future where AI is able to predict exactly what you personally would write, but if we get to that stage there will be bigger concerns).
It would be pretty easy to test if an AI did in fact write to the chain at appropriate times. The stakes of Damiano's proposal are not so high that I can find a failure mode on the scale of the incidents you cite.
Regarding compliance: health and safety norms rest mostly on good will and cooperation. Obviously that's backed up with some consequences for non-compliance, but the proportion of failures is vanishingly small.
Morgan Rogers (he/him) (Sep 16 2026 at 16:30):Nonetheless, I agree that good will and cooperation must be fought for, in the sense that external (or internal) pressure is required for capitalist entities to even acknowledge them. I'm pretty sure there is leverage to apply in the case of regulating AI, though.
V Slicer (Sep 16 2026 at 18:10):Damiano Mazza said:
Thanks for pointing this out, I didn't know about all these details! Ok, so it took an intergovernmental organization to enforce copyright laws. I still don't understand why a similar process wouldn't work for enforcing "AI copyright".
I think climate treaties are a better analogy and the reason it's unlikely to work are similar to climate treaties. Both are about staving off catastrophic outcomes, whereas intellectual property regimes on the other hand are moreso about controlling the spread and use of information and production techniques to stop others from doing what the industrial superpowers themselves did to get into that position, and protect the interests of certain capitalists, enterprises and states. As with a climate treaty the parties involved stand to gain personally in the short term from violating the treaty and externalizing the costs.
In the case of copyright it was being pushed by states that were already industrial powerhouses that could strong arm other states into an arrangement that maintained the balance of power as it was. As things are today it seems unlikely that all these different states will agree to such an arrangement that logs all their transcripts and makes them available to another state and/or body. It would also be difficult to verify that they are holding themselves to it. These transparency/audit logs themselves would be very valuable data and subject to attempts at espionage/exfiltration.
From an economic standpoint it seems likely it would make the huge investments and bets being made less valuable and more risky as it would mean at least some people and enterprises that are using these services will stop or use them less. Despite certain CEOs claiming they would want to see something like this, it would face intense opposition by investors and states whose national economies are heavily vested in these companies, I would expect. Unless it's enforced on a planetary scale (which is unlikely for the above reasons) capital will flee to states that don't sign these treaties and build their datacenters there, at which point they would have to enforce sanctions and surveillance on GPUs and compute in general.
I don't really see how one would prevent these enterprises and states from "going dark" and continuing, nor any reason to expect any monitoring and enforcing body to be incorruptible, impartial or invulnerable, as long as we are in class society, where interests are in conflict.
Practically these crises can only be resolved by a classless society, communism, where these conflicts of interest are dissolved. We'd be free to pursue other developmental arcs and e.g. make access permissionless and reasonably guarantee privacy using fully or approximately homomorphic encryption.
Returning to the climate crises example, several issues have been known for decades at this point, some for more than a century. Here I'm reminded of what Marx called the metabolic rift1 2 (building on the work of soil scientists like von Liebig at the time), but because capitalism has a certain directionality, a certain systematic dialectic, we have so far been unable address it as a society, in fact. To the point where the UN was forced to finally admit two weeks ago that the 1.5⁰C goal is dead.
Damiano Mazza (Sep 16 2026 at 21:14):@Eliès Harington it is clear at this point that there's some kind of misunderstanding. Morgan already underlined a few salient points (thanks to him for that!), let me clarify further (as synthetically as I can, which I'm not very good at):
I hope this clarifies things. This was just a not-particularly-well-thought suggestion to address an issue that I think is very much worth trying to address. I don't want to make a big deal out of it, we probably already wrote too much about this I think! (But I thank everyone for sharing their thoughts).
John Baez (Sep 18 2026 at 09:23):The SF writer Bruce Sterling likes to adopt a hard-boiled, cynical persona on Bluesky. I found this remark of his about math and AI amusing and disturbing.
I think @Benjamin Merlin Bumpus (he/him)'s blog article Thinking hard above AI is a good attempt at a response.
Benjamin Merlin Bumpus (he/him) (Sep 18 2026 at 17:20):Thanks John for sharing. Any ideas, critiques or feedback are welcome.
V Slicer (Sep 19 2026 at 20:10):Hello @Benjamin Merlin Bumpus (he/him), I read your blog post attentively. First a few days ago, I let it settle in and then again today before writing this. I also read several of the posts you reference such as @Emily Riehl's Why I do mathematical research which I enjoyed. Thank you for participating in this conversation, I wanted to say that before continuing with what is for the most part critique. The quotes that follow are from your post unless stated otherwise, so let's get right into it.
First of all, it is always useful to ask oneself who is to gain from a particular narrative.
You'll find that we're in agreement here, and this is part of what this critique will try to elaborate on. To put it bluntly, it will center around issues of how the text has a narrative that serves as consolation at some times, and quiescence at other times.
In a sense, I increasingly see asking an AI for a proof as a second step in the literature review: first I check what people have already written down and made explicitly available, then I check what can be reconstructed or produced fairly easily by an LLM.
To properly respond to this, I would like to place this narrative in a wider social context, of which mathematicians are just one part. To quote a text from Issue 3 of the journal Endnotes (sometimes referred to as “a ‘Marxist’ journal for anarchists”) that I will return to later as well, Spontaneity, Mediation, Rupture:
A massive accumulation of capital has made the productive process ever more efficient, rendering workers ever more superfluous to it. Under these conditions, capitalist economies have grown slowly, due to chronic overproduction; at the same time, most workers find it hard to win any real gains, in a context of high levels of unemployment. Moreover, this superfluity of workers has found its correlate in a changed experience of work itself. Insofar as they are employed, most proletarians do not identify with their work as the defining trait of who they are. Either they are peripheral to a more or less automated production process — and thus, cannot see themselves as the vital force of modern society — or else they are excluded from production altogether, and toil away in dead-end service sector jobs.
Mathematics as a field has for the last centuries for a part resisted this change in experience, although not entirely. Proletarians who are in the circumstances to find employment in mathematics often do identify with their work as a or perhaps even the defining trait of who they are. Mathematicians have been, so far, pretty squarely in the teutonic castle. But now massive accumulation of capital has been making the productive process of mathematical statements ever more efficient, rendering mathematicians ever more superfluous to it. And now some mathematicians, especially youth and those who are not in a tenured position are finding it more difficult to see themselves as a vital force of modern, capitalist, society.
If proofs become cheaper, then great, questions about why a proof works, which language it belongs in and how it should be contextualised become more valuable. In my mind, this is a chance to engage in this Great Conversation as better interlocutors: we can ask more of ourselves and of our arguments and of their context.
Here again we should look at who this narrative is for, to me it reads as consolation, and only for those in the circumstances where they don't have to feel threatened by the increasing superfluity affecting mathematicians as a social role in capitalist society.
Many do not experience it as “great” as it means that they have become less valuable to the process of production and thus more superfluous to capitalist society. Some might even experience it as a crisis of meaning if they considered this to be a defining trait of theirs, as it's now suddenly less valuable to capitalist society and perhaps as a consequence they won't be able to connect to this capacity of theirs, or maybe only to a much lesser extent and with much less fulfillment. To those, it is not a “chance,” it represents the loss of a chance to connect with these capacities and to engage in this social practice.
Second of all there is also a more abstract reason why I do not think that AI crossing some particular frontier means that mathematics is finished. If we assume that these systems are not capable of a fundamentally stronger form of computation than we are, then sure, perhaps they have more resources (e.g. electricity or memory or a bigger PR team), but that should not prohibit me from asking an even more meta-mathematical question.
[…]
All of this makes me want to learn more mathematics, and in particular more foundations: more logic, more category theory and even more set theory. This is my bet and you’re free to make your own.
We have to keep in mind that mathematics is a social practice and the mathematician as it exists today is historically (and socio-economically) specific social role. One way to illustrate this is by considering what it would take to accumulate the mathematical knowledge and experience some such as yourself has today, if one were born, say, as a peasant (or in a slum) and had to spend their days working the land (or hustling) to help their loved ones (or perhaps just themselves) subsist? [I'm making the assumption that the reader, like most mathematicians, is not from such a background.] How much time and motivation would one have left over for mathematics, to contribute to “The Great Conversation”? Who would they be having the conversation with? Would they say they're free to make their bet and learn even more mathematics? What kind of freedom and socio-economic circumstances are already presupposed in this question? To me it seems very unlikely, and it reveals why many, and even many mathematicians, are feeling less and less free to make their own bet to learn more mathematics.
The historical specificity of mathematicians was recently appreciated by @Morgan Rogers (he/him) in a post as well. Mathematicians exist in a social, historically specific, context. Of course, for as long as there has been language there have been those engaged with understanding using language, but we mean something more historically specific when we talk about mathematicians, a more or less fixed social role, in the history of mathematics. And perhaps in a classless society, where mathematics no longer is an activity primarily relegated to a small minority, but made more accessible to the vast majority, to all those who wish to participate in this social practice and connect with those capacities, perhaps our conception of mathematician will transform as well. To quote Spontaneity, Mediation, Rupture from Endnotes again:
The actual means of reconnecting individuals to their capacities, outside the market and the state, are impossible to foresee. But that does not mean that human existence will take on an ineffable quality, a sheer flux. New mediations will inevitably be erected out of the wreckage of the old. Thus, communism will not mean the end of mediation. It will mean the end of those mediations that fix us in our social roles: gender, race, class, nation, species. Just as the end of abstract domination will not mean the end of abstraction, so too, the overcoming of these mediations will leave plenty of others intact: language, music, games, etc.
To which we might add mathematics.
V Slicer (Sep 19 2026 at 20:10):This is probably too big a topic to engage with here and likely worth a whole other post. Either way, a time of great unrest is also a time in which we can think deeply about what we were doing in the first place.
This statement is where we can find much agreement, but it's also where I would claim the article does not go far enough and, indeed, instills quiescence. You do mention another post you intend to write, and again in the comment section in response to @Morgan Rogers (he/him), whose comment I agree with. Still I find this post too quiescent on the most important aspect of this conversation, that is the social context that we exist in and allows us to do mathematics.
I'd like to end with this quote from the same text:
During periods of quiescence, revolt takes place. But it remains disarticulated. The clash between classes breaks out, here and there, but then subsides. Periods of quiescence last for decades, but eventually, they come to an end. The re-emergence of class struggle announces itself in a torrent of activities. A new sequence of struggles begins. Waves of proletarian activity ebb and flow, over a period of years, as new content and new forms of struggle develop. The intensity of the fight rises, although never in a linear way, as proletarians link up, extending their disruptive activities. The articulation of those activities begins to reveal the outlines of that which is to be overcome. In this way, there is a tension towards the rupture, which throws off sparks in all directions. A rupture is, by definition, a break — a break that is qualitative in nature — but a break with or within what? Where do we locate the rupture that is synonymous with the advent of a revolutionary period?
[…]
The rupture forces every individual, who is engaged in struggle, to take sides: to decide whether they align themselves on the side of the communist movement — as the movement for the practical destruction of this world — or else, on the side of continuing to revolt, on the basis of what is. In that sense, the rupture is a moment of partisanship, of taking sides. It is a question of joining the party[1] and of convincing others to do the same (it is by no means a matter of leading “the people”).
[…]
In the end, communising tactics will turn out to be whichever tactics finally destroy the link between finding work and surviving. They will reconnect human beings and their capacities, in such a way as to make it impossible to sever that connection ever again. In the course of struggle, a process may unfold, somewhere in the world, which seems to go all the way, to bring an end, once and for all, to capitalist social relations. Just as today, proletarians adopt and adapt whatever tactics resonate with them, so too, some proletarians will adopt these communising tactics. However, these tactics will not extend the struggle. On the contrary, they will split that struggle open, turning it back against itself[1]: The concept of the party merely registers this fact: like spontaneous revolt itself, the rupture will not proceed automatically, out of a deep or even ‘final crisis’ of the capital-labour relation. The proletariat will not suddenly find itself holding the levers to power, after which point it is only a matter of figuring out what to do with it. Instead, the revolution will be the project of a fraction of society, i.e. the party, which solves the coordination problem in the only possible way – by abolishing class society.
We can't be quiet about what is necessary, and what we're called to do to care for this social practice we've so often drawn from, relied on and contributed to, too much is at stake. We have to be open and partisan, for ourselves, for our mathematical ancestors, for future generations and society at large and for those that have yet to fall in love with mathematics.
John Baez (Sep 22 2026 at 20:27):I hope you all have read OpenAI's new announcement:
Matteo Capucci (he/him) (Sep 22 2026 at 20:50):Is that OpenAI? AFAIU is an independent organization
Matteo Capucci (he/him) (Sep 22 2026 at 21:00):Ah I see:
We are currently facing the very specific challenge of advising OpenAI on how to coordinate the release of a large number of significant results in mathematics that they report have been produced by their internal model.
John Baez (Sep 22 2026 at 21:43):Sorry, I was trying to link to OpenAI's companion statement.
John Baez (Sep 22 2026 at 21:46):They explain that they need this group of prestigious mathematicians to help deal with a flood of so far unreleased results their LLMs have proved:
On August 28, we began training a new internal model. In addition to resolving the Navier–Stokes Millennium Prize problem, this model has now resolved more than 100 long-standing open problems across most areas of mathematics.
John Baez (Sep 22 2026 at 21:47):The group will advise on the review and communication of emerging results: they will help OpenAI assess their significance, advise on how to coordinate their dissemination, and advise on academic and professional standards of mathematical research.
ww (Sep 22 2026 at 21:51):Doesn't seem unreasonable, no?
Matteo Capucci (he/him) (Sep 23 2026 at 07:15):Very reasonable indeed... Though I would have hoped they asked for advice before they tarmaced on everyone's yard and not after. So my expectation is that it's going to be mostly cosmetic at this point.
But I'm open to be surprised! Perhaps they can convince OAI to seek, for each of the problems, the people they tarmaced (choosing who this people are would be its own can of worms....) and give them the result, and have them digest it as they see fit, and publish it.
Morgan Rogers (he/him) (Sep 23 2026 at 07:51):So it's no longer a question of "should we use AI" with regard to the resources required to work on our problems. Resources have been spent, in spite of the net negative demand for this, and we are left primarily with the choice to engage or not with the product. Haven't they shot themselves in the foot here? If they wanted researchers to become customers (or even just to tolerate their invasion), shouldn't they have offered some semblance of agency rather than cobbling together a retroactive consultancy role for a handful of senior people?
Capitalist overproduction strikes again, I suppose.
Morgan Rogers (he/him) (Sep 23 2026 at 07:53):My feeling about this is increasingly to observe that there is plenty of literature that never makes it into the canon of mathematical practice, and results that end up being rediscovered. If we refuse to work with mass-produced AI material, it won't fundamentally change that dynamic, except volumetrically.
John Baez (Sep 23 2026 at 08:09):ww said:
Doesn't seem unreasonable, no?
I can't think of a better solution, but it casts a cloud over mathematics to announce that a few high priests will be in charge of unveiling 100 unnamed theorems, maybe "big" theorems.
Terry Tao's name is notably absent from the list, and that is smart of him because he's leading the development of a mathematician run LLM.
ww (Sep 23 2026 at 08:12):Yes, my next thought was "how does one participate in this group?" (not that I would be competent to, but I want there to be a way for people who are competent)
John Baez (Sep 23 2026 at 08:23):I've met 3 people in the listed group, and they are all extremely competent in a general sort of way. I woould love to hear them discussing what they are going to do now.
John Baez (Sep 23 2026 at 14:16):I believe what they should do is set up a more democratic institution and transfer their power (which is really just the power to make recommendations) to that. I'm going to suggest this on the form at the bottom of https://agmai.org/ , and I hope everyone here joins me. It's not good to keep this power in the hands of 9 elite mathematicians chosen by OpenAI.
Luckily some of them seem wise enough to not want this power. And they are asking for our opinions!
Damiano Mazza (Sep 23 2026 at 15:01):The question on the form is pretty specific:
As of September 21, OpenAI reports that they have solved a large number of significant mathematics problems using their internal model. In your opinion, what is a good way for such results to be released and disseminated in the mathematical community?
@John Baez you suggest we do not answer this question directly but tell them instead that they should use their position to set up a more democratic institution which will then answer all sorts of questions like this. Right?
Damiano Mazza (Sep 23 2026 at 15:03):(Honestly I have no good advice to give them concerning the specific question they ask).
John Baez (Sep 23 2026 at 15:37):Right: the recommendations should not be made directly by a Council of Nine Wise Ones chosen by OpenAI.
I'll point out to the wise ones that the American Mathematical Society, London Mathematical Society, European Society etc. already have democratic governance, to varying degrees.
John Baez (Sep 23 2026 at 15:46):Here is what I wrote:
The answer to this question is so important that it should be decided, not directly by nine mathematicians chosen by OpenAI, but in some more democratic way. You can design the structure for decision-making and perhaps keep the resulting organization from running off the rails, but the more the affected mathematicians have some say in what happens, the less unhappy they will be. There already exist democratically run organizations of mathematicians such as the AMS, LMS, and EMS that might help out here.
Nicolai Kraus (Sep 23 2026 at 16:08):John Baez said:
Right: the recommendations should not be made directly by a Council of Nine Wise Ones chosen by OpenAI.
I don't have more information than what is publicly available on the page, but I don't interpret what has happened as OpenAI having chosen them. My understanding of the pagragraph quoted below is that OpenAI attempted to recruit some of them for an advisory group, they rejected, and a larger group then formed an independent group. In my view, none of the legitimacy of what you call "Council of Nine Wise Ones" comes from OpenAI, it's just a group of well-known researchers who have chosen to do this. (Although, under this interpretation, it's a bit unclear to me why this needed OpenAI's agreement. I guess this was just a peace-keeping way to turn them down.)
This group came together after OpenAI approached some of its members about establishing an external advisory board. In agreement with OpenAI, they decided to instead create an independent group and invite the others to join.
John Baez (Sep 23 2026 at 16:22):Okay, you're right. I wish I'd read everything a bit more carefully before writing my suggestion - but anyway, my suggestion remains the same.
Peva Blanchard (Sep 23 2026 at 19:56):Martin Hairer wrote about why he agreed to join agmai.
Nicolai Kraus (Sep 23 2026 at 20:22):thanks for the link!
ww (Sep 23 2026 at 22:56):I wrote something similar but stumbled on an observation. Putting together an ad-hoc group might be justifiable in a crisis, where there is not time to construct a representative democratic mechanism. But this crisis (if it is one) is manufactured. Dare I say, artificial. Maybe we can come up with a way of handling this kind of situation -- because it is mathematics now, and it will be something else tomorrow and we are good at coming up with useful structures.
There is a parallel with the epidemiologists during the pandemic who felt besieged by everyone and their dog having a solution or tool that they wanted to use. Most of those were wrong but some of them were useful. The main source of difficulty was filtering the large volume of communication and determining what was important to pay attention to and what was not. A major secondary difficulty was the manner in which some people pushed their favourite idea which rubbed the entire field the wrong way.
The important difference is that then, epidemiologists were in a privileged position with respect to decision making in a crisis and everyone wanted to have input whereas mathematicians are in a much more vulnerable position: the privilege is mostly with the AI labs. And, of course, then, the signal to noise ratio in the input was very much lower.
(I didn't write all that in the form)
Morgan Rogers (he/him) (Sep 24 2026 at 07:59):Some of the comments on Hairer's blog post expressed the opinion that they just wanted the material released, even in poor condition, rather than have to wait for a long time after the "100 results" have been teased, especially so that if their current work is already "solved" they can move onto something else faster.
Morgan Rogers (he/him) (Sep 24 2026 at 08:00):I read this as indicating that some people within mathematics share the same limited interpretation of what research consists of that is the basis for this AI expenditure in the first place: supposing that knowing the answer is all that matters. This is doubly damning for OpenAI: besides the people that don't want to work with their output because of viewing it/them as directly hostile, this is a group that won't work on their output because they don't think they'll get enough credit for doing so. It's possible that not enough people will be willing to disseminate their output for free...
Morgan Rogers (he/him) (Sep 24 2026 at 08:01):If AI companies start having to hire even more researchers across a variety of subdisciplines just to get any value at all out of their maths results (in other words, if they get no short-term return on this investment), I hope this will give them some pause. But the problem of theorems going to the highest bidder is still one that's going to become widespread within 6-9 months.
Unless the AI economy collapsed first, I suppose.
Nicolai Kraus (Sep 24 2026 at 10:54):Here's how I would describe the active (i.e. producing) part of day-to-day research work:
The first step is where many people seem to see the "core value" of our work. But the first step is also the most fun. There is little motivation to do steps 2-6 for someone else: A) the most fun part of the work is removed; and B) career-wise, it's a bad idea to spend effort on something that another entity will claim authorship for.
OpenAI seems to suggest that they just throw a huge collection of results of step 1 to the mathematics community, and some volunteers will gladly do the whole rest of the work. To be clear, I'm not saying I don't want anyone to work out and disseminate OpenAI's results. Quite the opposite, it would be a very valuable thing to do. It's just that I have no motivation to do that, and I imagine many others also don't. If they solved some problem that I have been thinking about for a long time or that I'm really interested in, I would be motivated to read the argument even in poor condition, but I still would think it's the authors' job to get it into better condition.
Nicolai Kraus (Sep 24 2026 at 10:58):Morgan Rogers (he/him) said:
I read this as indicating that some people within mathematics share the same limited interpretation of what research consists of that is the basis for this AI expenditure in the first place: supposing that knowing the answer is all that matters.
I'm not convinced of that. I am convinced that "knowing the answer is all that matters" is wrong (and I said somewhere earlier in this thread that, from my point of view, a solution to a famous problem can be technically correct but still essentially worthless if the proof doesn't allow us to learn insights). Nevertheless, I understand the people commenting on Hairer's blog because of what I described. (Plus, there's the point that some people just want to stay away from AI stuff, a sentiment that I don't share but that I understand.)
Nicolai Kraus (Sep 24 2026 at 11:09):Nicolai Kraus said:
Nevertheless, I understand the people commenting on Hairer's blog because of what I described.
Hmm, after re-reading my own answer, I realise that it doesn't quite fit. What you say is essentially that people shouldn't feel like their own work is pointless just because someone else has already claimed they have a proof. I agree with you, but the way the academic system is set up makes this "wrong" view understandable.
Morgan Rogers (he/him) (Sep 24 2026 at 15:56):Maybe we're lucky in CT that there is collective appreciation of work that produces a new, clearer proof of a known result.
V Slicer (Sep 24 2026 at 16:54):John Baez said:
Right: the recommendations should not be made directly by a Council of Nine Wise Ones chosen by OpenAI.
Agreed, except for the ’wise’ characterization (which may have been tongue in cheek! But I'll continue anyway), because as we've been discussing this isn't a healthy development for mathematics as a social practice.
This is closer to 9 Ones Securing A Position while the wider community of practitioners and future practitioners are left to their own devices like the frog in the slowly boiling frog metaphor [which some researchers dispute as a typical example of how frogs might behave, but that aside]. 9 Ones Pulling the Ladder Up Behind Themselves, more like it.
As we've been discussing these positions are much more limited than the amount of positions it might have taken for the development of those 100 theorems otherwise. Can we earnestly recommend mathematics to younger mathematicians hoping to contribute something novel at this rate? Another question is how will the field and community be able to keep up with, let alone integrate all of this with this overproduction of mathematical statements? Or we might even say 9 Defectors in the Prisoner's Dilemma.
Benjamin Merlin Bumpus (he/him) (Sep 24 2026 at 19:03):@V Slicer
Hi, thanks for the message and apologies for the slow reply: I've been trying to emerge from a midterm-grading cave. I should also preëmptively apologize for the fact that this reply will probably not have as much time poured into it for the very same reason (I'm not yet out of the cave).
It strikes me that my post should have been a lot clearer about who I was speaking to and why. As I pointed out in response to Morgan's comment on my blog (also @Morgan Rogers (he/him) I will get around to your email, but the cave is an impediment right now, sorry!) there are very many concerns that people might have right now. The one concern I was trying to respond to was:
I focused on this topic because it comes up a lot (probably the most) when I speak with students here at USP. Interestingly, concerns about the job of being a mathematician are raised by fellow academics, but so far I've never had a student ask me this question. This is just my experience. I don't know if this skew is because of the students I'm interacting with, the institution I’m at (USP) or the continent I’m in (South America). I genuinely have no idea and honestly I feel totally unqualified to speak on this matter. However, the fact of the matter is that students do come to me often asking whether there is any mathematics to be done at all and/or what "doing research mathematics" will look like.
Now, I’m making a point about this not to be defensive, but to merely contextualize what I was trying to say and to apologize for any confusion: I wanted to write a hopeful message (one that I believe in, by the way) to students and explain why I do indeed think that there is mathematics to be done by humans.
Having said this, I do find myself a little confused about questions about the social practice coming before the question of whether or not there is anything left to do in mathematics. And in particular, I don’t at all understand why this might be considered quiescence (a new word for me, thanks!). Let me explain.
In my mind, it only makes sense to argue for the preservation of the mathematical community and for its health, if one first establishes that such a community should still exist. For example, if in some distant future we discover a magical pill that makes us completely immortal, it would make no sense to start arguing for anything related to the job of being a mortician.
Furthermore, I think that we have to be very careful about _how_ we speak about AI impacting our jobs and the research community. There are many people online and in positions of power who see the mathematics community’s scorn for AI as a “cope”. It is totally reasonable for a non-mathematician to conclude that the “research mathematician” career is done for, when they hear about a machine that can solve problems we have not been able to and they hear many mathematicians being up in arms about what the machine is doing to their community. Such a person might see this as evidence that the machine can indeed do everything we can and that we are only angry out of self-preservation. If we pair this with a growing feeling of anti-intellectualism world-wide, this is not a good combination. So I believe that we all have to think rather carefully about the _rhetoric_ of our statements, not only whether they are true or not.
By the way, just to be clear, I completely agree with you and with Morgan. How we run our community is incredibly important; how we evaluate students is getting overturned; how grants will be allocated is unclear; what activities we should draw joy from is also being put into question. However, I steered clear of writing about it (for now!) because, despite having many ideas, I don’t have a clear picture of either what I want to say or how I want to say it. I’m cautious because, as I mentioned earlier, right now is a particularly bad time for us, as a community, to have a rhetorical slip-up.
Anyway, I’m working on it and these discussions are really useful to me (and hopefully to you too!) :)
Thanks for taking some time away from marking to reply @Benjamin Merlin Bumpus (he/him) . I don't mind waiting a while for a reply to my email, no rush.
I would like to respond to this part of the above:
In my mind, it only makes sense to argue for the preservation of the mathematical community and for its health, if one first establishes that such a community should still exist. For example, if in some distant future we discover a magical pill that makes us completely immortal, it would make no sense to start arguing for anything related to the job of being a mortician.
The analogy you give misses the mark for me in several ways.
Benjamin Merlin Bumpus (he/him) (Sep 25 2026 at 18:31):(technically I should be listening to the faculty meeting I'm in on zoom, but I find talking to you far more interesting... don't leet anyone know!)
I think that it might be useful at this point, at least for me, to make a statement trying to summarize what it is that seems to be floating about right now in our discussions. My objective is to identify a common starting point so that we don't unintentionally argue past each other, especially since I have the sneaky suspicion that you and I actually agree on most points ...well, you tell me! :)
The reason I think that we aren't disagreeing is the following:
in my blog, I was trying to make a strong argument that mathematics is still intellectually valuable, that human mathematicians are crucial and that mathematics will not disappear (although it may change substantially);
what I think you're arguing is that, despite this, the number of paid positions for professional mathematicians may decline sharply.
The trouble is that: (1) the two statements above do not contradict each other and (2) although I'd like to believe that there will be a need for more mathematicians, and better-trained ones, not fewer, I don't have anything to substantiate this other than waving my hands and whipping out a pseudo-syllogistic claim of the sort: “the more complex our systems are, the more we will need powerful abstractions to make sense of the world and keep track of everything; hence we need people capable of thinking this way.”
By the way, this is precisely why I keep coming back to rhetoric. I think that, at least on the question of future jobs, we are arguing from a rather weak position: (1) we don't know what is going to happen, nor what this technology can actually do (let alone how it does it), (2) there are armies of tech accelerationist bros whose relevance, funding and attention often depend on convincing everyone that something enormous and potentially catastrophic is just around the corner, and who seem rather fond of sniping high-clout research targets in an attempt to prove their relevance, and (3) university management often seems not to understand, or particularly value, much of what mathematicians actually do. (...to be fair, I think some of that failure of understanding is partly our own fault: mathematics has a fairly pervasive culture of intellectual machismo... but anyway, I'm getting side-tracked to one of my other favorite topics of discussion..).
This brings me to my worry about mathematicians publicly framing ourselves as a profession on the verge of displacement. I don't mean that we should silence ourselves or that we should be "_quiescent_" (you've gotta use a new word a few times to really learn it...). I actually mean the opposite. Now is a time to be very vocal. But my point is that we should be very careful and deliberate while doing so: precisely because these times are incredibly uncertain (other than my wishful thinking, I have no idea what might happen) and because there are powerful people who are more than willing to interpret “AI can do mathematics” as “we therefore need fewer mathematicians”, I think we should be unusually careful not to hand them that conclusion for free. If these tools genuinely make us more productive, then I want the argument to be that this should expand what mathematicians are able to do, not that it gives institutions an excuse to employ fewer of us. I'm struggling to make this argument effectively and I'm hoping that we can collectively crowdsource such an argument. Perhaps we can do so right here?
One mistake that I see often—well, I think it's a mistake, I might be wrong --—is that those who wish to argue against indiscriminate AI use do not seriously enough engage with the benefits of AI. I think part of building a rhetorically strong position is being completely willing to acknowledge the things these systems can actually do well, such as assisting us with formalizing proofs, generating counterexamples, exploring conjectures, etc. (this is not at all a good list, it is just an example). In some sense, the dream of having a machine that can assist mathematics is not at all new and has history inside the mathematics community ("I spy Hilbert, I spy Leibniz, I spy..."). Any strong and effective criticism of what AI companies are doing to mathematics has to engage with this deeply if we expect it to be taken seriously outside our own community.
My personal reason for taking a deep breath and waiting to write more is that I'm struggling to find a rhetorically strong way of saying two things at once: on the one hand I want to formulate a sharp and cutting argument about the harms of these technologies, while also making it clear that producing correct mathematics is not the same thing as replacing us mathematicians.
So anyway, TLDR, I think we agree (?) and I guess I'm really keen on the "crowdsourcing a sharp rhetorical argument" on this Zulip.
John Baez (Sep 26 2026 at 19:26):One reason I haven't been taking a loud position on AI in math, and why I like what Benjamin wrote in his blog, is that on the one hand I see how AI can greatly expand what I'm able to do (having written a math paper with its help, as a test of its capacities) yet on the other hand I see plenty of downsides. This prevents me from making a common argument that goes something like "AI sucks in every way: it doesn't do anything well, it's just a way for the rich and powerful to become more rich and powerful, and it's hurting the environment".
Matteo Capucci (he/him) (Sep 27 2026 at 09:12):I feel much in the same way, it's hard to be in the public discourse recently: if you push back the hype, you are seen as a luddist, if you push back the doomers, you are seen as an AI bootlicker. It's very tiring to mantain a balanced middle ground.
Matteo Capucci (he/him) (Sep 27 2026 at 09:14):Benjamin Merlin Bumpus (he/him) said:
I think some of that failure of understanding is partly our own fault: mathematics has a fairly pervasive culture of intellectual machismo...
Alas, I reproach myself of not being vocal earlier about how miopic that was. Now it's too late to make that argument and not sound like you are coping with AI replacement...
John Baez (Sep 27 2026 at 10:37):Terry Tao used to comment on the n-Category Cafe, shortly after Tim Gowers had written about the split between the "two cultures" of mathematics: problem-solving and theorem-building. I think Tao was trying to help unify the two cultures by coming up with some hard conjectures in category theory.
After a while he quit. This was after I said that I wasn't interested in solving hard conjectures. I forget why I said this. It's true, but ever afterwards I felt a bit guilty that it came across as a brushoff: like "go away, I don't like the kind of math you do". I hope he quit posting on the n-Category Cafe for a completely different reason, like getting too busy and famous.
But anyway, I've always spent some time trying to break down the stereotype of "mathematical machismo", for example by putting a lot of work into explaining things simply, and admitting I don't understand a lot of things.
On the other hand, when I was younger I didn't resist a different kind of machismo: namely, talking about the latest trends in mathematics, with their impressive buzzwords, and showing off about the glamorous places I visited while going to math conferences, and the bigshots I was talking to.
I try much harder to resist all this now. I don't fly to conferences, I try not to talk about the bigshots I know (though I just did it! :laughing:), and I've largely stopped paying attention to prestigious fads in mathematics. As a result, fewer people pay attention to me. It took quite a while to realize that this is good.
So yeah: to the extent mathematicians want to feel smarter than other people, or cooler than other people, the rise of LLMs is very threatening to them. Amit Sahai wrote:
When I was an undergraduate student, I remember talking with several students who felt that the pace at which the top students could understand new math concepts was far too fast for them. They, too, could understand the ideas, but it would take them much longer. Eventually, almost all of these students gave up their dream of pursuing research mathematics and found something else to do. I have been thinking about those students a lot in the last few days.
The research mathematics community consists largely of those of us who either rarely felt that way, or who felt it and managed to overcome it through hard work. We have had the good fortune to find a place in mathematics where we could make progress. But we are now entering a time for humility: a time when all of us are going to know what it feels like to be unable to keep up.
V Slicer (Sep 27 2026 at 12:36):@Benjamin Merlin Bumpus (he/him)
Thank you for the response, it's clear all the participants in the conversation are coming from a place of great care, it's good to state that explicitly sometimes to remind ourselves there's much we agree on and that this is a collaborative effort to make sense of the situation we find ourselves in and how we should respond.
To the question "will there be mathematics to be done?" if taken on its face the answer is, trivially, "yes, there will always be mathematics to be done." What I believe students are asking is if there's a place in modern (capitalist) society that includes them devoting enough time to mathematics to contribute (perhaps something novel) to mathematics as a social practice, and I believe that given the direction things are moving the answer is "it's getting less and less likely but maybe you can beat the odds." Perhaps this is an instance of me extrapolating too much from my personal experience and the mathematicians (including students) I've known but it was always almost taken for granted that we value mathematics intrinsically, you could even say there's a spiritual growth component. That's why I sooner believe students are really asking about the latter rather than the former, as that's what's increasingly put into question. It's also why I believe focusing on the former is a form of consolation (‘don't worry, what you're spending your time on, mathematics, is important to society’) and instilling quiescence (‘mathematics is important so of course there will be a place for you to do mathematics, even if the conditions are completely changing, we just have to ride the wave’), because it doesn't speak to the practical challenges the social practice is facing in capitalist society and what kind of response is required are we to keep this social practice alive, hopefully expand and transform it and allow even more people to connect with their capacities.
@Morgan Rogers (he/him) addressed the mortician analogy, and rightfully points out with his first and second points that it flattens the ethical dimension, this is what I mean with quiescence.
I would like to add that an important disanalogy is the conflation between mathematicians as a community and morticians as a job. I suspect a lot (if not an overwhelming majority) of the mathematics community would be perfectly content (or ecstatic even) with a situation where “the mathematician” as a job no longer exists, such as in a classless society society, but where mathematics as a social practice and the community of practitioners can thrive. Morgan touches on this in his second point.
His third point also points to an important difference, that I hope to write more about one of these days: in the mortician case from the perspective of capitalist society there's a complete reduction of demand, in the case of theorem production it might be described as (among other things) an ongoing shift from so called "push production" models to "pull production" models (ongoing as exemplified by the move away from tenure to grants) combined with, to quote Jasper Bernes in The Belly of the Revolution in the context of agriculture, “massive overinvestment with the ultimately perverse result that, once conditions settle down, such strong deflationary pressures emerge that revenue can no longer cover costs, initiating a wave of bankruptcies that bring down costs for the next generation of producers. Production for profit stamps agriculture, with growers changing the crops they offer according to the shifting winds of the market and a series of complex guarantees from states. What is grown first is money, and only then food for human needs.”
Theorem production in capitalist society, I believe, can be analogized to some aspects of agriculture and food production. Bernes continues:
The combination of the logistics and green revolutions has lead to an increasingly wasteful food supply system. One might think that elimination of standing inventories from retailers and distributors would make for less waste, but unlike manufacturers, food producers have far less ability to alter their output. Agriculture has relatively long turnover times, and farmers have to make decisions about output levels far in advance of actual sale, all while anticipating the possibility of a bad harvest due to uncontrollable factors. They often make advance contracts with distributors and retailers, but given unpredictability, find it more profitable to overproduce, as the costs of producing too much are lower than the opportunity costs of producing too little. In other words, push production remains the norm in agriculture, despite the demand-side dominance of the industry, and thus producers are often left with more food than they can sell at decent prices. Supermarkets also have stringent aesthetic and quality standards, rejecting agricultural products that do not conform to rather superficial consumer values. And because retailers and distributors now dominate, their contracting allows them to switch from supplier to supplier, forcing the costs of compelled overproduction further down the value chain. This dynamic results in a staggering scale of food wastage, with somewhere between 29% and 34% of all food produced globally not consumed. In
In our case I suspect the scale of “theorem wastage” as a result of capitalist overproduction is going to be much higher.
The reason I suppose I keep coming back to transformation on a societal level and the practical necessity for communism in this conversation is because I don't see how rhetoric can ever overcome these objective changes and shifts in power and the effects they will and already are having on mathematics as a social practice. Some of the time, effort and focus spent on appealing to the better sensibilities of those that have conflicting interests than mathematicians and proletarians more broadly is better spent thinking and acting soberly on resolving the actual core issues.
As this conversation has been ongoing and I appreciate that not everyone can be reasonably expected to have read the whole conversation I'm going to point to some posts where I provide more context and build the argument more: 1, 2, 3.
V Slicer (Sep 27 2026 at 14:34):Matteo Capucci (he/him) said:
I feel much in the same way, it's hard to be in the public discourse recently: if you push back the hype, you are seen as a luddist, if you push back the doomers, you are seen as an AI bootlicker. It's very tiring to mantain a balanced middle ground.
I definitely agree with your main point, except the terminology perhaps because the luddites were correct! They weren't in principle against new production techniques, they recognized what these changes would mean for them without changes in society more broadly.
But unlike that case where it was isolated to the textile industry, these ongoing developments in capitalist society threaten the reproduction of proletarians of multiple industries simultaneously so it's much more salient, as we can see with the broad opposition to data centers.
As long as we're in a class society and interests are in conflict supporting them unequivocally does mean giving up our bargaining power and making ourselves superfluous, and in the case of mathematics specifically perhaps long term damage to our ability to continue the social practice.
In a classless society these new techniques would be a great development and we would be able to rationally consider what the best uses would be and where it would be harmful, but in capitalism we're stuck in a race to the bottom. But to go back to your point, that's why I so often add "in class society" because I do think it's dependent on social context whether it's positive or not.
John Baez (Sep 27 2026 at 22:07):Category theory submissions to the arXiv are up 33.2% this year, a lot but much less than some other subjects: the highest is combinatorics with 62%. You can see the results for different subjects here.
History and overview is up just 5.4%, the lowest of all, so the idea that humans will focus on explaining things is not a reality yet.
Joe Moeller (Sep 28 2026 at 03:56):Something I have to explain constantly to the engineers I work with is that anything you could prove with category theory could have been proven without it (except the obvious). They hate that. It also probably means that category theory gets hit less by people who just tell an LLM to find and solve an impressive open problem. This is almost the same point John was making about machismo. Eugenia Cheng’s terminology “ingressive” and “congressive” also comes to mind.
Peva Blanchard (Sep 28 2026 at 05:35):Can you explain what she means by ingressive and congressive?
David Michael Roberts (Sep 28 2026 at 07:30):ww said:
Putting together an ad-hoc group might be justifiable in a crisis, where there is not time to construct a representative democratic mechanism.
My impression is that this is not an unreasonable approximation to the situation.
John Baez (Sep 28 2026 at 08:03):As William pointed out (and you know), this business of using massive amounts of LLM time to solve famous problems before humans do is a crisis largely manufactured by OpenAI and Anthropic - and then OpenAI tried to put together a team of prestigious mathematicians to deal with the PR disaster. Luckily that team seems to have thrown off the shackles of OpenAI and reconstituted itself, and is seeking input from the rest of us.
Matteo Capucci (he/him) (Sep 28 2026 at 08:44):Joe Moeller said:
. It also probably means that category theory gets hit less by people who just tell an LLM to find and solve an impressive open problem.
Honestly I was quite surprised of how high 30% was, not how small it is. I was expecting maybe a 10%?
Matteo Capucci (he/him) (Sep 28 2026 at 08:44):I wonder how many are crackpot or cargo-culting preprints
John Baez (Sep 28 2026 at 08:53):We're getting a nice list listing of crackpot papers in category theory here, but it's unsystematic, so we may be missing a lot. It would be really great if someone could analyze this more systematically.
Matteo Capucci (he/him) (Sep 28 2026 at 09:21):I was too curious so I asked GPT-5.5 to fetch the arXiv data and run the numbers filtered as follows:
cs.AIThe first is an approximate crackpot filter, the second is a guess that the explosion in AI papers might have spilled in CT too (from the data, it doesn't seem to be case).
The results are below. It seems that the increase in math.CT submissions is dominated, but not exclusively caused by, newcomers working alone.
Matteo Capucci (he/him) (Sep 28 2026 at 09:22):I wonder if others subjects present a similar pattern
Matteo Capucci (he/him) (Sep 28 2026 at 09:27):The Mar-Aug yoy average comparison is the metric used in the post linked by John, which is a slightly odd metric. From that it looks as if math.CT submissions didn't vary at all, while the first clearly shows an upward trend even for the filtered number.
Matteo Capucci (he/him) (Sep 28 2026 at 09:31):Here's the first chart with a 5-months rolling average to smooth out the noise; doing that makes the current uptick even less pronounced
John Baez (Sep 28 2026 at 10:17):That looks pretty pronounced to me. Thanks for doing this analysis!
Jad Koleilat (Sep 28 2026 at 10:31):I am wondering whether or not it is safe for my career to talk about my research publicly. Recently, I heard the story of a researcher who shared his ideas/intuitions with a colleague. This colleague asked an LLM to generate an article based on their discussion without the consent of the other one. The LLM solved the problem and he then published it on arxiv under his own name, without mentioning his colleague who provided the ideas.
As a PhD student I fear that if I speak publicly about my research or talk about what I want to do next someone will put what I said in a prompt and generate an article in a few days. I am starting the third year of my PhD, what will happen to me If someone "solves" my thesis subject now ? I am starting to fear going to conferences or seminar before I have submitted to a journal/conference. I am wondering whether or not it is safe to include new ideas in the conclusion of the papers or presentations I am preparing. I am starting to feel that the safest thing to do is to go back to medieval mathematics where every new result and new idea is kept secret. I am starting to feel paranoia regarding all the times I talked about my research at a conference or a seminar.
The discussion that are happening here are interesting but I am sad there is very few people talking about which action to take now to make the situation better.
Nothing is done to protect us PhD student of the assault of IA. I don't know if tomorrow university will not consider it more cost effective to increase their subscription to their preferred LLM rather than open new positions. I don't know if my research will be stolen or not. If my ideas are stolen I am sure that there will be no consequences for the one who did it but, for me it will probably mean the end of my career.
Matteo Capucci (he/him) (Sep 28 2026 at 10:35):FWIW, doing maths in secret when it is (ostensibly) so easy to scoop a result might backfire even more. If you talk publicly about your research, and then a preprint appears with your idea, then the community will note that you were talking about that before, and will at least credit you alongside the scooper. You also have a timestamped proof of your work on it before in case you want to escalate.
Matteo Capucci (he/him) (Sep 28 2026 at 10:35):In other words, scooping like that is still blatant academic misconduct and we have reputational penalties against that. Those are not going anywhere.
Matteo Capucci (he/him) (Sep 28 2026 at 10:36):Still, this is another push in the direction of prizing exposition and canonicalization over mere achievement.
Robin Piedeleu (Sep 28 2026 at 10:38):Joe Moeller said:
Something I have to explain constantly to the engineers I work with is that anything you could prove with category theory could have been proven without it (except the obvious). They hate that. It also probably means that category theory gets hit less by people who just tell an LLM to find and solve an impressive open problem. This is almost the same point John was making about machismo. Eugenia Cheng’s terminology “ingressive” and “congressive” also comes to mind.
I think this point is also central to the broader debate about the value of understanding in the era of AI-generated mathematics. So far, we have had to rely extensively on finding the right abstractions to manage complexity in maths; but the process by which AI agents generate proofs is very different from that of humans, heavily skewed in favour of leveraging raw computational power. As a result, reliance on AI tools for mathematics may increase our ability to explore the search space of mathematics with existing abstractions, without necessarily needing to generate new ones.
Robin Piedeleu (Sep 28 2026 at 10:50):In fact, this suggests a thought-experiment that lets us imagine what is lost with AI mathematics. Imagine that the ancient Greeks were given DialogoGPT, an LLM trained on a huge corpus of Euclid-style geometry; they ask the oracle to solve long-standing open problems, spending millions of drachma in the process, and to everyone’s surprise it does! The proofs are very long, using extremely intricate geometric constructions that nobody can totally follow, but it comes with a Litos theorem prover certificate guaranteeing its correctness. Subsequently, whenever anyone has a new geometry question, they just ask the oracle-machine for the answer, which it diligently spits out (depending on how much money the user is able to throw at the oracle of course, which is another massive issue with AI mathematics). What would mathematical progress look like in such a world? Would we have had the analytic geometry of Descartes for example? Would high-school students thousands of years later be able to solve geometric problems that would have seemed very difficult to the ancient Greeks without the appropriate algebraic language?
John Baez (Sep 28 2026 at 11:08):They would have noticed that there's no way to trisect a general angle using compass and straight-edge moves.
John Baez (Sep 28 2026 at 11:19):Jad Koleilat said:
I am wondering whether or not it is safe for my career to talk about my research publicly. Recently, I heard the story of a researcher who shared his ideas/intuitions with a colleague. This colleague asked an LLM to generate an article based on their discussion without the consent of the other one. The LLM solved the problem and he then published it on arxiv under his own name, without mentioning his colleague who provided the ideas.
There have been stories like this long before AI: it doesn't take an LLM to "steal someone's idea". The best defense has always been to release your ideas in a documented, time-stamped way when they're ready. In the 1600s people would do this by publicly releasing anagrams:
Galileo's first cryptic report was in 1610, just a string of letters: "smaismrmilmepoetaleumibunenugttauiras." Kepler worked furiously on this, and came up with "Salve umbistineum geminatum Martia proles," usually rendered as "Hail, double-knob children of Mars." This is kind of dubious as a Latin unscrambling of Galileo's gibberish, but it fit one of Kepler's pet ideas, namely that Mars should have two moons, so that Earth (one Moon) then Mars (two) then Jupiter (Galileo had just discovered four moons there) made a geometric progression. Kepler was bit into numerology, and loved this idea.
Unfortunately, the correct solution was "Altissimum planetam tergeminum observavi" or "I have observed the highest planet to be three-bodied." The "highest planet" is Saturn, then the most distant one known, and Galileo had seen the rings, but they were unresolved by his telescope, and thus looked like two giant moons, one to either side.
Later, in 1676, Robert Hooke reported the relationship for elastic forces as "ceiiinossssttuv," which unscrambles to "ut tensio, sic vis," indicating that the force is proportional to the distance.
Skipping over more than 3 centuries, the arXiv was a huge advance because it lets you publicly release an idea where everyone can see when you did it, without waiting for peer review. If someone tries to take your idea and publish it before you, you can point to your paper and everyone - including journal editors - will know that person is a fraud.
Personally I have only once had someone release an idea I have given a talk about, out of > 100 papers. But if you're nervous you should play it safe, and you always should have: this is not new.
Morgan Rogers (he/him) (Sep 28 2026 at 11:25):There's a difference in time-frames here, though. Now you can mention an idea for future work at the end of a talk and an unscrupulous audience member can generate a paper exploring it by the next day, or by the next week if they're taking some care to read through it before putting it online. Then you cannot so easily claim to have had a piece of work stolen or scooped, even though there is no way this hypothetical audience member would have thought to generate the paper without having had this contact with your work.
Morgan Rogers (he/him) (Sep 28 2026 at 11:32):Just today I was contacted by someone who had generated a paper relating to work I did a few years ago. They were asking me to become a coauthor on a paper, stating
What remains is to understand the constructions and draw out any further lessons from them, and to communicate the results of that process.
To interpret this ungenerously, they had heard about a problem and generated an ostensible counterexample resolving it, but expected me to put in the work of understanding it on their behalf. If this was a problem I was actively working on, I would be either furious or despairing. As it is, I declined and expressed some disappointment in their choice of methodology.
Nicolai Kraus (Sep 28 2026 at 11:40):John Baez said:
History and overview is up just 5.4%, the lowest of all, so the idea that humans will focus on explaining things is not a reality yet.
Isn't this in line with the observation that LLMs are great at problem solving but poor at overview and explaining? What did you mean with the second part of your sentence?
Nicolai Kraus (Sep 28 2026 at 12:11):Matteo Capucci (he/him) said:
Here's the first chart with a 5-months rolling average to smooth out the noise; doing that makes the current uptick even less pronounced
Thank you for doing this analysis. This looks quite significant. Do we know how large the "expected" number of first-time single-authors would have been (i.e., how does this number compare to first-time single-authors that posted in, say, 2023)?
Matteo Capucci (he/him) (Sep 28 2026 at 13:21):@Nicolai Kraus I spent some more time on this question, trying to understand whether the effect in my original diagrams might not just be a systematic error: if I graph submissions from newcomers (measured as: no submission 18 months before) throughout, the effect diminishes:
Matteo Capucci (he/him) (Sep 28 2026 at 13:24):The trend line is an exponential fit for the range 2011-2022
Matteo Capucci (he/him) (Sep 28 2026 at 13:26):Here's graphs for all math.* categories on arXiv:
Matteo Capucci (he/him) (Sep 28 2026 at 13:27):These are graphs for all math.* categories showing papers by size of their set of authors:
Matteo Capucci (he/him) (Sep 28 2026 at 13:29):It's quite interesting to notice that paper writing stagnated for a few years after COVID, despite an initial uptick. In fact only the current surge in papers has recovered the distance from the expected trend (in the above it is less noticeable, but initially I computed the trend between 2010 and 2020 and the difference was really dramatic!)
Matteo Capucci (he/him) (Sep 28 2026 at 13:31):Matteo Capucci (he/him) said:
Here's the first chart with a 5-months rolling average to smooth out the noise; doing that makes the current uptick even less pronounced
I guess this is still relevant because we can see the increase is largely driven by newcomers nonetheless
John Baez (Sep 28 2026 at 14:50):Nicolai Kraus said:
John Baez said:
History and overview is up just 5.4%, the lowest of all, so the idea that humans will focus on explaining things is not a reality yet.
Isn't this in line with the observation that LLMs are great at problem solving but poor at overview and explaining? What did you mean with the second part of your sentence?
I was sort of joking, saying that we don't yet see many people moving into the job of writing histories and overviews, leaving math research to the AIs, as some have argued will eventually happen.
But yes, what we're seeing fits the idea that right now people are concentrating on using LLMs to crank out theorems.
Nicolai Kraus (Sep 28 2026 at 15:20):Interesting, thanks! @Matteo Capucci (he/him)
Joe Moeller (Sep 28 2026 at 20:06):Matteo Capucci (he/him) said:
Joe Moeller said:
. It also probably means that category theory gets hit less by people who just tell an LLM to find and solve an impressive open problem.
Honestly I was quite surprised of how high 30% was, not how small it is. I was expecting maybe a 10%?
It may be the case that category theory is quite an easy seasoning to sprinkle on things to make them sound more impressive. And we know there is a bias in the output of LLMs towards making the user feel good.
V Slicer (Sep 28 2026 at 20:34):@Matteo Capucci (he/him) Thanks for sharing your preliminary results of using AI for mathematical research on AI for mathematical research in the AI for mathematical research stream! :wink:
Matteo Capucci (he/him) (Sep 29 2026 at 07:09):Joe Moeller said:
It may be the case that category theory is quite an easy seasoning to sprinkle on things to make them sound more impressive. And we know there is a bias in the output of LLMs towards making the user feel good.
Indeed... :thinking:
Benjamin Merlin Bumpus (he/him) (Sep 29 2026 at 21:22):@Matteo Capucci (he/him) your plots are rather interesting. There are colleagues here doing similar things within combinatorics and comparing it across particular subdisciplines. They seem to get similar results to yours.
Peva Blanchard (Sep 30 2026 at 04:29):The AGMAI published a set of recommendations for responsibly releasing ai-generated maths.
Matteo Capucci (he/him) (Sep 30 2026 at 06:04):Benjamin Merlin Bumpus (he/him) said:
Matteo Capucci (he/him) your plots are rather interesting. There are colleagues here doing similar things within combinatorics and comparing it across particular subdisciplines. They seem to get similar results to yours.
Indeed, the qualitative properties if these plots are remarkably consistent across disciplines, with few exceptions. Combinatorics is notably for the massive and early increase in submissions, probably due to the Erdos problems traction?
Matteo Capucci (he/him) (Sep 30 2026 at 06:15):Peva Blanchard said:
The AGMAI published a set of recommendations for responsibly releasing ai-generated maths.
This is great! Including the fact they start by saying the AI labs just should not
Morgan Rogers (he/him) (Sep 30 2026 at 12:00):I was pleased to find the suggestions very reasonable too. No sign of them taking up John on his suggestion yet, though :sweat_smile:
V Slicer (Sep 30 2026 at 13:38):Perhaps we should be asking for the session transcripts as well before the results are considered publishable, if this is going to be part of the process and people insist on using these tools.
V Slicer (Sep 30 2026 at 13:38):And to go even further, have some requirements on the models such as the code and training data being made freely available for inquiry and modification before considering the results publishable.
I'm sure some CEOs and investors will probably be against that, but even in the case where they refuse and decide to stop meddling with mathematics it would be a positive outcome.
Oisín FC (Sep 30 2026 at 15:35):@Matteo Capucci (he/him)I'm pretty sure that the models were very good at combinatorics problems quite early primarily because combinatorics is one of the four areas tested in olympiad maths, and it overlaps a bit with research combinatorics. In 2024 and 2025, the companies were mostly using the IMO and the American olympiad pipeline as benchmarks to try improve their models at maths before they switched to Millenium problems this year :( .
Matteo Capucci (he/him) (Sep 30 2026 at 15:39):Indeed, I had a similar thought
Matteo Capucci (he/him) (Sep 30 2026 at 15:41):In my limited experience models were rubbish at CT until spring (where they could help me rewrite bits of math), and became really good only this summer (where they could come up with original arguments)
Alex Kreitzberg (Sep 30 2026 at 17:46):I would go so far as to say, there's a history of interest in AI from combinatorial thinkers. Polya would demonstrate problem solving methods on "puzzles without inherit interest" to illustrate the generality of "methods". His list of problem solving strategies roughly align with classic AI algorithms.
After the brain analogy AIs were able to solve go. Then "problem solving" became the next natural target from this perspective. Which is what the IMO, and combinatorics more broadly, are culturally about. (I say this as an occasional judge for the olympiad held at the University of Washington).
Early successes of this technology excited Knuth, Gowers, Terence Tao, etc. Gowers even had written about AI before all of this. I think this is because they are quintessential problem solvers, and that the success of AI driven approaches, validates their images of mathematics.
If category theory feels overrepresented in applications of this tech, I'd guess that's because category theory is popular in programming language theory - type theory in particular has been essential for these recent successes in AI driven mathematics.
I'm curious whether this short term emphasis on "problem solving" in AI mathematics will give way, or lead, to a broader conception of understanding. But I'm not sure how that's possible without things like "taste", "opinions", "values", etc. Thinking about this to any extent makes me uncomfortable.
John Baez (Sep 30 2026 at 20:14):Things like taste, opinions and values are incredibly important in mathematics. I suspect many mathematicians don't like talking about them because they are not easy to mathematize: some must feel embarrassed that the hard shiny rigorous structure rests on a dark, messy, shifting foundation of human values: values that determine what problems are considered important, which fields become fashionable, etc. Whether we talk about these issues or not, they are there.
Nicolai Kraus (Oct 02 2026 at 20:36):John Baez said:
Wow, and now the bastards claim they've done it. :smirk: And formalized it.
For anyone interested in this very regrettable controversy, here is Numberphile's interview with Tristan Buckmaster:
https://www.youtube.com/watch?v=Mzbtj5nkMXI&t=28s
V Slicer (Oct 08 2026 at 17:18):There's a new journal focusing on data center struggles:
https://letsgonetwork.org/reboot/
Patrick Nicodemus (Oct 09 2026 at 02:03):Nicolai Kraus said:
Matt Cuffaro (he/him) said:
Here is a screenshot of a post that circulated on LinkedIn, which is an example of my point of the changing relations of production. I don't know anything about the commenter.
I expect that many people in the general public think like this. I can't blame them. The general public is paying for our work and AI companies have spent a lot of resources on telling them that AI can replace us. (thanks for the screenshot!)
"We pay for results."
What a strange thing to say. To not be a mathematician and yet have strong ideas about what mathematics is about and why mathematical research should be socially supported. Is this person saying that they personally care whether the Riemann Hypothesis is true, but not whether a readable proof is produced?
David Michael Roberts (Oct 09 2026 at 23:08):https://www.youtube.com/watch?v=GweKOBSJvEY mathematics being discussed on mainstream news is pretty astonishing. I tried watching, but I only got through the intro before giving up. Apologies to Americans, but there's a particular accent and register of US news presenters that I find, frankly, slightly painful to listen to. I don't know why. If anyone can get any fun bits out of this, then that would be good.
Nicolai Kraus (Oct 09 2026 at 23:53):Thanks for the link. I just watched it. In terms of information and general enjoyability, the Numberphile interview is so much better, but I guess that is to be expected; this CNBC interview is for a different audience with very different constraints.
Nevertheless, it's interesting that CNBC did this interview. From what I just read, they are a niche network but specialised on financial markets. Given that this whole power demonstration by OpenAI was presumably motivated by their forthcoming IPO, they won't be happy to see this. In the end, Buckmaster questions explicitly whether it's a good idea for companies to use OpenAI products, when OpenAI has demonstrated that they're happy to use (steal) their customers' ideas; I'm not an expert, but I imagine that this could indeed hurt OpenAI.
Nicolai Kraus (Oct 10 2026 at 00:10):"We pay for results."
What a strange thing to say. To not be a mathematician and yet have strong ideas about what mathematics is about and why mathematical research should be socially supported. Is this person saying that they personally care whether the Riemann Hypothesis is true, but not whether a readable proof is produced?
I don't think this person uses the word "results" with the concrete meaning that you have in mind. For the general public, "result" does not mean "theorem" or even "paper"; for them, "result" is the thing that should come out of a mathematician's work, and they don't know what that is, but AI companies have managed to persuade them that it is what AI produces. I disagree that "we pay for results" is a strange thing to say; this is simply how most things in the world work. It's on us to explain that "results" are not simply the things that AI produces.
Zoltan A. Kocsis (Z.A.K.) (Oct 10 2026 at 04:20):@Nicolai Kraus said:
I disagree that "we pay for results" is a strange thing to say; this is simply how most things in the world work.
Maybe so. But taxpayers is a massive group that includes everyone in this thread. I think (at least in democracies) what we already pay for is a good guide for predicting what we'll pay for in the future.
And this result-orientedness is not what I see. Public funding for Film Studies does not go into producing more blockbusters. People know that, and most of them can't really explain any actual results that Film Studies produces, but they fund it anyway. Results do matter in mathematics, we should try to explain ours if we can, but I'm not convinced that results was what the public thought this funding was buying in the first place. If it was so, Hollywood would have convinced the public to redirect all^1 that Film Studies funding to them long ago. Vice versa, I don't think Film Studies could increase its funding substantially by explaining better what the results are.
You're right that we should absolutely try to understand deeply what the taxpayers (and the general public) want!
I'm used to being the most cynical person in a room, but it feels to me like general cynicism about the taxpayers, especially based on one screenshot of one tweet from one person who self-describes as a "dark enlightenment acolyte", is perhaps unwarranted.
^1 Not to mention that funding is a drop in the bucket: based on a back-of-the-envelope calculation, if all Film Studies funding was redirected to Hollywood, it would let them make one more blockbuster each year. Similarly, AI labs are not competing for math funding, even if they got all of it, it wouldn't move their bottom line much.