Proofs and Prompts

Notes from the common room

Open letter to the European Mathematical Society

Snorre H. Christiansen, Professor at the University of Oslo

The EMS should establish a structure to shepherd the use of artificial intelligence in mathematics research. First version sent to the EMS on 20 September 2026.

A European institute for AI enhanced research in mathematics

This open letter is a response to the latest developments in AI (including contributions to the Millennium problem on Navier-Stokes) as they impact the mathematical community.

My impression is that many research mathematicians would feel a loss of meaning if the most advanced research were to consist mainly in efficient prompting and agent management, especially if these tools are outside of our control. This sense of loss compounds with scepticism about where the broader society is driven, by the interests of a handful privately owned outsized AI companies. What’s more, in most cases these companies are subject to politics and jurisdictions that we can hardly count on swaying.

I believe that the mathematical community is well placed to develop its own AI models, to regain control over its destiny. The community has competent and motivated manpower, especially if research as we knew it is upended, as seems to be already the case. I am sure that mathematicians can develop the AIs that are best at doing maths and that they are best placed to help regulate the use of this tool, notably in regards to the education and recruitment of future generations of mathematicians. Mathematical thinking is as needed now as it has ever been.

This communal AI could take the form of an institute owned by EU/EEA through the European Research Council (ERC), governed by the EMS and partnered with promising European AI companies for mutual benefit (I have noted that Mistral has developed Leanstral).

It would be subject to European law, contribute to European sovereignty and would give European mathematicians a hand on the steering wheel in matters that are important to them. Communal values have been expressed in particular in the Leiden declaration. The declaration by Fields medalists relates our concerns to the ones expressed in other arts and sciences. The initiative should be mindful of the solidarities felt by mathematicians across continents.

Mathematicians have always had a broad spectrum of views on matters of ethics, especially regarding military technology. I suppose this will also be the case regarding AI. I understand that some will prefer to stay away from AI all together, and that’s fine. Protecting human agency in mathematics can take many forms.

This initiative would provide a structure that gives willing mathematicians a possibility to defend their values and regain a sense of control over their destiny. If prompting is the future, I for one would like to prompt inside a structure I can support. This way we can at least try to align future AIs with the interests and sense of dignity we defend as mathematicians and as human beings united by the Universal Declaration of Human Rights.

Traditional ways of doing maths would coexist with AI enhanced mathematical research developed in house and whose findings are shared on a need to know basis and published according to consensual protocols as collective endeavors (high energy physicists working at CERN have more experience with this).

I would also suggest that the EMS establish a Millennium type prize related to AI. It could be for instance to explain mathematically why deep neural networks work so well on some problems otherwise considered intractable (due for instance to the curse of dimensionality). This would give the EMS an occasion to communicate to the greater public about the role of mathematics in modern technologies.

Yours sincerely,

Snorre H. Christiansen

Received on 22 September 2026.

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15 responses to “Open letter to the European Mathematical Society”

  1. Nilima Nigam Avatar
    Nilima Nigam

    Thank you! A publicly-supported AI infrastructure, developed by mathematicians, would be an excellent resource. It would address many-a-problem I see with the present situation.

    Reply
  2. Galyna Livshyts Avatar
    Galyna Livshyts

    Thank you! I keep saying this all the time! https://m.facebook.com/story.php?story_fbid=pfbid0uUL6dDimDGs4hNSiEiPZgyEnHkWLEwAzWaLjfY2SKuVhhbAC3ayhUeQqq3XLP3Zbl&id=100001436827244

    I think that math does not even require a language model, one could develop an a lot more efficient neural network specifically for math, operating in formal logic, requiring far less computational power and servers. The academic infrastructure is plenty sufficient, both in financial capabilities, and even more so in manpower, to build this tool, and reshape the field for the better. This would mean — access to knowledge available to all, not depending on consumer interests of the AI companies. This is a far better approach then saying “let’s shut AI down” — one cannot uninvent it 🙂 Best to take control over it!

    Reply
    1. Anonymous Avatar
      Anonymous

      “I think that math does not even require a language model, one could develop an a lot more efficient neural network specifically for math, operating in formal logic, requiring far less computational power and servers.”

      This strategy has failed countless times by now. Every specialized model has been weaker in capabilities than a general LLM.

      “Best to take control over it!”

      If it was this trivial, no one would be panicking over it.

      Even this post does not address how Europe would compete when it is much harder to build a data centre in EU, and even if the permitting was easier, Europe cannot match a trillion dollars per year spent by the US to build and serve AI models.

      Reply
  3. Mister Avatar
    Mister

    The power of the current generation of models that are currently available commercially will be after not too long also be available (from thieves of thieves) in open-weight models, which are cheaper to run. That means that the capital cost of developing some EMS coordinated system from scratch (or near scratch), if they are based on similar technology, will be a hard to sell to any potential backers. It also means that soon hobbyists will, at lower expense, be able to prove very strong theorems, like they can now by paying 100 or 200 a month. Unless this EMS coordinated system will also be running giant swarms of agents, I don’t foresee how this could help (even if it were adopted) in terms of the most “advanced” research, whatever that means anymore. And there are many initiatives already that are developing/running truly GDPR-compliant inference, which makes the privacy benefits of such an initiative also less appealing (even if perhaps still appealing).

    Reply
    1. Snorre Harald Christiansen Avatar
      Snorre Harald Christiansen

      I don’t think it needs to be developed from scratch. I see the maths part as an add on to some system that is already good at language processing and programming and that is not primarily the responsability of the maths community.

      Reply
    2. Stefan Dawydiak Avatar
      Stefan Dawydiak

      Maybe I misunderstood the proposal, but was buying an open-weight model and renting some EU-located data centre capacity not precisely the proposal?

      I want also to point out that history since 2016 has made it clear that abdicating all science to the US is not really an option for the rest of the world. While delegating all thinking the two US companies might be an option for Americans, it’s a really silly one for anyone outside the US to consider. Indeed, we could already have done this long ago: just divert all ERC funding to US and Chinese institutions only, fire all public-centre scientists in the EU, and “just read the papers” that result. Obviously this is not a wise move. The fact that there is now a new variant of this blunder available doesn’t change that.

      I also don’t think “zero skill, one-shot” hobbyist prompters are something we need to “worry” about: by definition they have no advantage versus a large institutional effort running 24/7. It’s also not clear why anyone would try do this, forever. Like, for the moment it’s exciting to them to “cause science to be done” (it is really inaccurate to say they are proving anything, just as we don’t say we are cooking when we order food via an app) or do “disrupt the ivory tower,” but how many people will think this is fun three months from now? Hobbyists who want to actually help can perhaps donate computing power via something like BOINC, if that’s not happening at small scale already.

      Reply
    3. Morten Krogh Avatar
      Morten Krogh

      The proofs that can be done for $100 will be totally uninteresting. Math research will accelerate and the interesting results will cost millions of dollars in compute. Models, hardware and energy needed will increase and increase. Cheap proofs will be uninteresting and will be considered self evident by the powerful data centers.

      Reply
  4. Snorre Harald Christiansen Avatar
    Snorre Harald Christiansen

    Some additional thoughts:

    – Funding. Very soon project proposals to Horizon Europe will include budget items for AI capabilities. The total cost of this will be determined by the real cost of providing such services, the perceived productivity gain of users, and market strategies of companies offering these services. It’s hard to predict, but if this becomes 10% of the total budget, we are already above a billion euros annually.

    – Intellectual property (IP). Many AI giants have demonstrated questionable practices regarding IP, as witnessed by the many current lawsuits. On the other hand their expansive view of what constitutes “fair use” is now protected in the US by a Department of Justice “statement of interest” (dated September 1st 2026) invoking national security. In this situation Europe might want to take measures to protect its IP and research resources against aggressive ambitions of “AI dominance”. What is “fair use” of our prompts?

    Reply
  5. wp1651893662 Avatar
    wp1651893662

    This is the first institutional proposal in this discussion, and I am largely convinced by it. I want to add one structural caution and one suggestion.

    The caution: the risk is not that the commons model might be unclean. The risk is that there is only one of it.

    A shared model does not forbid mathematicians from asking different questions. It changes the cost of asking them. Once the tool exists, the path it is good at becomes cheap — promptable, reviewable, fundable — and the path it is bad at becomes expensive: no tooling, no reviewers fluent in it, no students trained for it. Over a decade that is not censorship; it is a directional filter installed at the entrance of research. Mathematicians will still disagree, but increasingly they will disagree inside one basin.

    This discussion has already produced a counter-example, from a different direction entirely: Mathathon’s new theme, “Old Problems, New Proofs,” makes the search for alternative proofs of solved problems an official goal — four-colour, ABC, Navier–Stokes. That is a community saying, formally, that alternative routes are themselves a contribution. The difference between mathematicians is not friction to be optimised away; it is where new routes come from. A single shared model optimises exactly that away.

    The suggestion: alongside the institute’s commons model, explicitly support private models — small models trained on, and adapted to, a single mathematician’s own corpus: their papers, notes, referee reports, failed attempts, correspondence. This is not a convenience feature. It differs structurally in three ways.

    First, opacity is tolerable when the loop is local. Much of the alarm about large models concerns scale: one model’s output simultaneously shapes millions of people, and no one can audit what it is doing to them. A private model’s loop is physically bounded by one person. It can be strange, idiosyncratic, even wrong, without that wrongness propagating.

    Second, it keeps the mathematician’s own difference in the loop. A shared model converges toward the mean of the corpus it was trained on. A private model is anchored to one person’s history, including their dead ends — the parts a shared model has no reason to keep.

    Third, it makes provenance shown rather than declared. “We did not distil” is unverifiable, whether said by a company or by an institute. A model trained on one person’s traceable corpus has provenance that can be exhibited, not asserted.

    On engineering: I am not arguing that every mathematician should pretrain from scratch. Fine-tuning a strong open base is the realistic route, and that is fine for the commons model. But fine-tuning produces branches, not lineages: everything derived from one ancestor still carries that ancestor’s priors. So if the institute wants genuine diversity, it should also maintain, deliberately and at a loss, a small number of models trained independently from the start — a seed bank rather than a product. They need not be good. They need to be different.

    Two honest limits. Private models do not solve alignment; they only shrink the blast radius. And nothing here guarantees “clean” — provenance will have to be shown, not claimed.

    One addition to the proposed prize: alongside explaining why deep networks work, I would welcome a prize for measuring and maintaining cognitive difference between models — a way to tell whether two systems are genuinely independent, or whether we are counting one perspective several times. If we cannot measure how much difference remains, we will not know when we have lost it.

    Reply
    1. breadpurple02138716ec Avatar
      breadpurple02138716ec

      It is sort of ironic that this posts argues against the creation of a new model serving different needs from the already available models, using an argument that more models are better than a single one.

      It is not like people suddenly cannot use other models anymore when this new one is created.

      Reply
      1. wp1651893662 Avatar
        wp1651893662

        Fair catch — I should be clearer.

        I’m not arguing against it; I think it should be built. My point is narrower: more models and more independent lineages are not the same quantity. A model fine-tuned from an existing base adds a model, not a lineage — and models sharing a base tend to fail in the same direction.

        On “people can still use other models”: true in principle. What I’m less sure about is what happens once the models start talking to each other.

        If that worry is real, you won’t need me to argue for it. You’ll notice it yourself — the first time several different models give you the same answer, for the same reason.

        That’s the only thing I’d ask you to keep. Not to accept it now. Just to remember where to look.

        Reply
  6. S Avatar
    S

    I fully back this response to the ongoing situation. It seems more sensible to provide better options and incentivise people to use them, rather than count on making new “cannot”s: new laws, which require other agencies to enact and enforce.

    The recognition that AI is a public good has been announced by some private AI-companies, but has less of a place in the zero-sum landscape of competing for profit, than in publicly funded research where positive-sum is often the goal. It also seems a natural opportunity for the EU, together with the UK perhaps, to take the lead on showing how it is done; such a narrative-backing means an increase in expected funding for the endeavour.

    Full Disclosure: I’ve used Snorre’s work on Diff. Forms on Cellular Complexes. This has no bearing on my attitude toward the content of his post, although it did encourage me to read it.

    Reply
  7. b Avatar
    b

    I love the spirit of this post, but the author seems oblivious to multiple realities at play.

    There is a ton of institutional knowledge that goes into training even a moderately-sized LLM. (“In theory, theory and practice are the same. But in practice, they’re different.”) Any such effort that does not incorporate existing expertise would be reinventing many wheels, and would probably be extremely cost-ineffective.

    Such an academic effort certainly couldn’t compete with the big AI labs in terms of funding, and probably couldn’t complete in terms of expertise either (precisely _because_ it couldn’t compete in terms of funding).

    On the other hand, there already exist multiple startups that are aiming to apply AI to mathematics (including Harmonic, Axiom Math, Theorem, Math Inc, Thinking Machines, and perhaps others). These efforts are probably the most well-positioned to compete against the big labs in this realm, although I for one would certainly not bet on them. But they strike me as the best shot the mathematics community might have at some version of this proposal. They are much more likely to be amenable to real dialogue and engagement, and would probably jump at the opportunity to serve the mathematical community in a way that feels respectful and mutually beneficial.

    Reply
    1. Snorre H Christiansen Avatar
      Snorre H Christiansen

      Thank you. Yes I am probably naive. On the other hand the proposal is not to create an ultracompetitive llm from scratch but to collaborate with (European) industry to design some system that works for mathematicians. So instead of aiming for some system that can do cutting edge research on its own, it’s to create a system that would help mathematicians in their daily work: digest the whole literature, prove lemmas, reliably find references, formalize proofs, all in a way that respects intellectual property (in Norway the Employee Invention Act of 1970 probably is in contradiction with many current practices) and provides a clear path towards publication (for instance by making the process a posteriori transparent). The ERC could create a dedicated institute even if the prospect of a fully functional AI is a couple of years into the future. The cost of that is small compared with the stakes.

      Reply
  8. Snorre H Christiansen Avatar
    Snorre H Christiansen

    A “CERN for AI-assisted science” was also advocated for here.

    Reply

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