On September 8, 2026, OpenAI announced a resolution of the Navier-Stokes millennium prize problem, claiming a construction of a finite-time blow-up from smooth forcing. As a young student I had been inspired by the Navier-Stokes problem—I could understand the statement and was surprised at the contrast between the ease of solving linear systems of ODEs and the challenge posed by completely reasonable looking PDEs (the naïveté of youth). This curiosity had a significant impact on my course selection as an undergraduate, and I even joined a computational fluids research group as a student assistant, before my own mathematical whims took my journey elsewhere. I had expected to feel excitement if a solution to the Navier-Stokes problem was announced, a drop everything and try to read the paper kind of day. Instead, my reaction on September 8 was one of grief and revulsion.
Grief is borne of loss, and to lose something it must be valued. Revulsion, similarly, is a reaction to a contradiction of values. Many mathematicians have called for an examination of value in light of the Navier-Stokes announcement and other AI developments this summer.1 In response, a flurry of exposition of the aspects of mathematical practice the community values has followed, with some common themes receiving broad endorsement.2 I will use the Navier-Stokes announcement to explore the universal human desires that lead me to value the aspects of mathematics other than answers, I hope in a fashion that will reach beyond only mathematicians.
On values
The need for mathematicians to reflect on and declare a system of values, particularly one relatable to the broader public, is not new. This is fortunate: we can draw on a rich tradition, particularly from the parts of our community focused on teaching and on building an equitable mathematics, as well as the philosophy of mathematical practice. These voices have not held center stage3 but they will be essential to meet the moment.4
My first encounter with the teaching community’s practice of values and reflection was in January of 2017. I was on the eve of finishing my doctorate and filled with agita about job prospects. I knew that I cared about teaching well, about opening doors for others, that I desired to dismantle the “calculus machine” in favor of something better—but I could not say clearly why I wanted to do these things or why I believed they were good to do. A ball of nerves and underdefined conviction I settled into Francis Su’s outgoing MAA Presidential Address, entitled “Mathematics for Human Flourishing”. I was awestruck. I wept. Since then Su’s central message has been close to mind, you can find its impact across my career and in how I conduct my day to day work. Having a clear statement of values is immensely powerful.
Allow me to summarize Su’s message. “What is mathematics for?” seems like the pressing question of the moment, but just as pressing is the question “Who is mathematics for?”. Su answers: “mathematics is for every human, and it is for human flourishing”. The word flourish has appeared repeatedly in the recent discourse on AI and mathematics,5 but what do we mean by it? For Su, flourishing is in the Aristotelian sense: the exercise of virtue that leads to the realization of one’s potential, the potential and dignity of others, and the fulfilment of basic human desires.
What desires are fulfilled by the practice of mathematics? Su identifies “exploration, meaning, play, beauty, permanence, truth, struggle, power, justice, freedom, community, and love”, found implicitly or explicitly in mathematicians’ writing about the discipline’s values spanning centuries. Meeting these desires through the practice of mathematics leads to flourishing in a unique way. This is why we value mathematics.
For a full accounting, read the book by the same title that grew out of the talk. Better yet, read it with a circle of colleagues or students and meet to discuss the questions for further reflection Su provides.6 A clear statement of values was illuminating as a finishing Ph. D. student, and it remains so in understanding my reaction to the moment.
On what was lost: Grief
OpenAI’s announcement included a description of token and agent usage for the successful run that produced the proof. Not mentioned were negative runs. Nor do we know the margins on external API pricing. So it is an educated guess that the computation that produced the paper and lean certificate, plus negative runs, cost $10 million internally. This figure is murky: how should one also count ammortized infrastructure spend, salary, etc. but it suffices for my illustration. If you’ve got a different number in mind feel free to re-scale this analysis.
To put $10 million into perspective, U.S. National Science Foundation standard grants in mathematics are around $250,000.7 A standard grant provides 3 years of funding that supports all aspects of the grantee’s research: hiring and mentoring students and postdocs, travel, and their own salary or time to pursue the research.
Consider a counterfactual: the $10 million spent on compute was instead distributed as NSF funding in the area, and the activities so funded also lead to a solution of the Navier-Stokes problem. For concreteness, suppose 20 grantees received 2 awards each over a 6 year period total, each ultimately supervising 4 graduate students and 1 postdoc supported by their awards.8 In total, the $10 million would have supported 120 people, let’s call them the Argonauts.9 What was lost, if not in actuality then in feeling, is at a minimum all of the byproducts of the funded activity.
Human mathematics is a messy product, full of false starts, scenic dead ends, surprising tangents, and self-reinforcing crossroads of alternate proof techniques and analysis. Along the way to a resolution of the Navier-Stokes problem the Argonauts would have explored the mathematical landscape. Over the course of their exploration, the Argonauts would have made meaning for all of us, relating their partial progress through intermediate dispatches from the frontier (papers, yes, but also seminars, conferences, surveys, and public outreach), stories of struggle and beauty.
The resolution found by the Argonauts would be the conclusion of this journey. An answer, the truth of the matter, but not only. A wonderful source of beauty in mathematics is the self-reinforcing nature of mathematical truth. Roads of inquiry that diverged long ago return from a thick forest of confusion to cross, inviting a change of perspective that inspires play. These self-reinforcing roads to a proof are also a sign of deep knowledge and are a result of deep investigation, virtues that lead to a deeper sense of truth than simply knowing an answer. The meanders of the Argonauts would produce a rich web, not a single path.
Our Argonauts would certainly have found community within their niche of mathematics, many conferences and special sessions are smaller than 120. This niche would be a single patch in the fabric of mathematics: one poised to offer hospitality to the rest of us in the new country they discovered. The community of Argonauts would erect a monument to their success: surveys and expositions, a call to the new frontier, and the start of a new long road of human flourishing. Mathematical knowledge, unlike mathematical truth, is not so ironclad: while a proof charts a path to truth, without readers it provides no knowledge. Our access to mathematical knowledge is social, grounded in the living tradition of practitioners.10 Only by nurturing this tradition, as the Argonauts would, do we achieve a human kind of permanence.
I want to be explicit: I am not evaluating this alternate scenario on utilitarian grounds, but from virtue ethics. While the resource calculation factors into the opportunity cost to illustrate this thought experiment, it is to contrast the underlying motives and their attendant virtues, or lack thereof.
Not all of this is lost for good. The Clay problems were selected for the expectation the adventures in their pursuit would be fruitful far beyond an answer. In the near term, mathematicians driven by the desires above will still generate some of the richness of the counterfactual Argonauts.11 But I fear it will be a diminished effort. In part, this is our own fault. The incentives of the mathematics profession, both explicit and in the social contract, are against it—we have not changed much since Thurston wrote “On Proof and Progress” or Perelman refused his prizes. These incentives are bound up in institutions with significant inertia, so will not change overnight. Even if they do, an oracular insight can have diminishing effects.12
On what was gained: revulsion
With the announcement, mathematicians received a 166 page manuscript and lean certificate, poorly sourced and cited, written without the authors taking the responsibility for its content—contrary to the emerging community consensus.1314 The paper has none of the qualities we would expect from the Argonauts’ resolution. To me, this is ugly, but alone it is not sufficient to account for my revulsion.
Mathematics is not the only thing that gained on September 8. OpenAI gained a marketing coup—the prestige associated to answering a big hard math problem in the public’s eyes, and the ensuing media circus. As demonstrated by the quality and responsibility taken for their paper, this gain, and not a contribution to mathematics, was their primary goal.15 This exploitation runs contrary to the basic desire for justice, which adds to the distaste.
The affront to justice gets worse. Beyond the low quality of scholarship there is a spectre of dishonesty about the OpenAI paper. There is the alleged possibility that the OpenAI result was achieved by incorporating the work in progress of others without credit, and possibly without consent.16 OpenAI’s initial statment did not dispel this concern, “[w]hile unlikely, we cannot rule out that de-identified data derived from their usage of our products helped improve our models.” That statement was removed in an update to the press release, replaced with the follow-up ‘categorical’ denial to the New York Times, “it was “categorically” impossible for its A.I. system to have been influenced by anything Dr. Buckmaster had done in the past two months,” ignoring that Alpöge and Buckmaster had been using a variety of models, including OpenAI products, for the past year in their work. I do not have the means to determine if something untoward happened, but it does not matter. As a species we have long understood that even the appearance of dishonesty is counter to our desire for justice.
Damaged trust is also contrary to our desire for community. Scientific community depends on the free exchange of ideas, grounded in shared values that incubate the trust that the community will trace these ideas and uplift all contributors. Moreover, collating open problems, publishing partial results, and discussing work in progress are the essential fabric of a communal sense of exploration and shared struggle. The shadow cast by this breach of trust is not isolated and the situation is degrading rapidly.17 Even if, in the end, transparency is provided by OpenAI or other AI companies and we are satisfied that in fact nothing unseemly happened, trust has been damaged. The loss is instant, rebuilding trust is a slow and deliberate process.18
The injury to trust is magnified by the broader ethical orientation demonstrated by OpenAI. Land and labor exploitation, as documented by Karen Hao and many others,19 do not paint a picture of a trustworthy partner. Nor does the glib disregard for the climate arson implied by the plans for scaling of infrastructure to support their mission. The values reflected by OpenAI’s leadership and senior employees, such as Sam Altman’s remark that “I think that AI will probably, most likely, sort of lead to the end of the world. But in the meantime, there will be great companies created”20 or the belief that working for OpenAI in order to acquire sufficient wealth to avoid the coming “permanent underclass,”21 are antithetical to any form of human flourishing—mathematical or otherwise. I find these positions abhorrent and ethically indefensible.
This is the heart of my revulsion: that the thing that I love, the community and ideals carried by a flotilla of mathematical Arguses22, has been unceremoniously dashed against the rocks in service of a project and organization that is repugnant to its core.
To the future, guided by values
I have focused on a case study of what has happened and how the actors in the story have behaved for good reason: this summer has demonstrated the challenge of prognostication, particularly in asserting upper bounds. Also, it is easy to hypothesize a future AI system to fit any view on the utopian—dystopian spectrum, and this hypothesizing obscures the current human values of the organizations who might create such a system. I anticipate many responses of the form “but in the future it won’t be like that.” Says you.
So I will attempt to paint a portrait of what is next, less as prognostication and more as a continuation of how our desire for human flourishing can be fulfilled, grounded in the specifics of what has been done. Necessarily, this sketch will be incomplete.
The market incentives driving the large private AI companies, and their ethical frameworks, are not likely to change in the near term. Change will almost surely require governmental action, which does not appear forthcoming in the United States. Even if it were, human institutions have significant inertia. This poses an additional obstacle, beyond the ethical failures of the AI companies, to rebuilding trust and the incentives for open science.
Difficult is not impossible. As a community we are not powerless. The wave of signatories to the opposition to the Mathathon led to its revision into a more productive form. The Leiden declaration, and the Mathematics and AI Statement, have changed the local behavior of the large AI companies. The market incentives remain the same, but satisfying these incentives requires the prestige that comes only from our collective endorsement. This is a delicate social lever: “a proof so complicated no mathematician can follow” plays into public perceptions in favor of marketing hype and against our desires for truth, beauty, and meaning. Used with care, our currency of prestige does offer one lever to align market interests with our own.
Aligning the market interests with our own would not sate our desires for justice and freedom. The ethical vacuum at the helm of the large AI companies would remain. Mathematics would also risk becoming indentured to these companies for model access and compute time. In the epilogue to Empires of AI, Hao tells the story of te hao Maori: the endangered language of the indigenous Maori of New Zealand. Large companies automatic translation tools under-represented and under-prioritized the language. Working with community members, in a framework grounded in consent and community stewardship, a local te hao radio station built a curated high quality data set that would be owned in trust by the station in service to the community. On top of that data set the station built a translation model for te hao that was trained and runs locally on two GPUs. The resulting model has played a significant role in the preservation and teaching of te hao.
Could mathematics follow the te hao route? It is possible—we have a long tradition of community stewarded software projects, as well as community infrastructure investment.2324 A successful effort, by a consortium, professional society, or public or private foundation, to provide a local model (call it Delphi) that places an emphasis on attribution and scholarship would free us from private AI companies and the injustice of their noxious ethical framework. Creating such a model would not be the end of the story. As with other open-source models, there is the possibility to diminish any market incentive for private companies with misaligned values to pursue mathematics in a way disruptive to human flourishing, though it would not eliminate the misalignment.
The hypothetical Delphi model reveals the ambiguity in the question “Should mathematics use AI?”: what do you mean, AI? The question of whether to use a large private AI company’s models is one of: is the mathematical understanding worth the moral cost of collaborating with those who would at best subjugate you into a permanent underclass, and at worst kill us all for a few cool quarterly earnings calls? My rhetorical question reveals where I stand. The question of whether to use a community stewarded model such as Delphi is a question of values and the philosophy of the practice: emerging from the many essays that open by setting aside the odious nature of the current private AI providers is a spectrum of answers depending on the particular use case, guided by a framework grounded in mathematical values.
Much has been made of the potential existential risks posed by AI development, particularly by the AI companies themselves. I remain a skeptic. But, supposing the threat is real, the threat is in the form of a failure of ethics, and the behavior and public statements of the private AI companies to date does not give them much credibility when they claim to be qualified to prevent this failure. Indeed, in informal discussions many AI boosters assert “we have no idea how to solve the problem but a more powerful AI surely will”, an uncanny echo of early advocates of nuclear proliferation “we have no idea how to solve the threat of annihilation but a larger bomb delivered faster surely will”. The distinction between these two is the assertion of a future entity without evidence for its possibility to justify their already held conclusion. I do not think such claims hold much water. Across disciplines, the achievements of current models are interesting, but, particularly in mathematics, largely identifiably a clever recombination of previous, human, work in the area. If we really have no idea at all, then remixing our previous work, even in unexpected in novel ways, is not going to produce a solution.
My earlier mention of government regulation twins with the reference to nuclear proliferation. In the realm of existential threats, we are also not powerless. Here, physicists efforts in shifting the conversation on nuclear proliferation point a way forward. In 1986 there were over 60,000 nuclear warheads in arsenals around the world; due to cooperation founded in a trust in the shared desire for a future of human flourishing, about the only trust available in the moment, treaty negotiation reduced that number to less than 10,000 as of 2023. This progress was hard won, and the international scientific community’s advocacy was instrumental in achieving it.25 Mathematicians now have a similar role to play. In the flurry of writing following the Navier-Stokes announcement I am confident that as a community we will rise to the challenge and not give in to despair and cynicism.
Postscript
Between when I prepared my first paper manuscript (September 11) and submitted the draft to Proofs & Prompts, Francis Su posted his own reaction. The explicit declaration of the core value of human flourishing also appeared in Po-Shen Loh’s guest post on Tao’s blog. The IAS and OpenAI announced an independent advisory panel based at IAS to guide private AI companies in responsibly engaging with the discipline. The creation of this panel demonstrates the social capital the mathematics community holds, as well as its limitations. On October 6, OpenAI posted the infamous dump, claiming in the press release to follow the recommendations of AGMAI, but following only the substring “release” and not any of the subsequent recommendations, beyond a thin promise to fund some workshops. From where I sit, AGMAI’s response was immediate capitulation, and I am disappointed. The judgement that their (sensible) recommendations were not followed is not a hard one to make, and did not need to be “left to the mathematical community to assess“.
Acknowledgements
I’d like to thank Rae Mondo for a lovely afternoon by Lake Merritt where I began formulating this essay, and for her encouragement to take my thoughts from conversation into writing. My colleagues Noah Friedman-Biglin (SJSU Philosophy) and Curtis Asplund (SJSU Physics) read an initial draft and provided thoughtful commentary, including lines of discussion that go beyond the scope of what I aimed to set out here: one thing the three of us agree on is that to meet the future the training of new mathematicians and scientists will require significantly more explicit instruction in ethics, values, and the impact of our work in broader society. Finally, I appreciate Proofs & Prompts willingness to publish this at the pace I finished polishing it, knowing that I’ve somewhat missed the boat on a timely reaction.
- See for example, Tao’s 2026 ICM address, or the many posts on this blog & elsewhere across the mathematical web. ↩︎
- The “Severe Misalignment” statement, the founding of the Association for Human Mathematics, for example. ↩︎
- Indeed these voices have met strident opposition in the community, with some mathematicians going so far as to found a new professional society in opposition. ↩︎
- Tao has a post collating some of these voices via crowdsourcing. They have often found a home in the Mathematical Association of America. ↩︎
- See for example, the posts on this blog using the word. ↩︎
- Practitioners of mathematics surely recognize the importance of doing the exercises in a community. ↩︎
- I arrived at this figure initially based on my gut sense of the funding landscape. Examining the NSF active award database shows an average of $230,000 for all active DMS awards, but this includes conferences and special programs. As a rough filter for these, the average of all grants without a colon in their title is $243,000. An attempt at a more curated list of grants not funded through special programs produced an average of $239,000. So it is an overestimate! ↩︎
- The exact trainee numbers are of course a ballpark, but reasonable based on my own experiences in geometric group theory. Navier-Stokes and its surrounds are not my area, but the timeline is also plausible: see for example Buckmaster’s remark in the New York Times “maybe we solve the Millennium Prize problem in 10 years without AI”. The exact timeline does not matter much for the counterfactual, only the order of magnitude. ↩︎
- Su’s Aristotelean framing put me in a Hellenistic mood and I found the essay read weird if I kept using “the 120” for a collective noun. ↩︎
- This is where a better facility with the philosophy of mathematics and its practice than standard mathematical training provides is important. I was exposed to some excerpts of Penelope Maddy’s writing on the subject as an undergraduate, but the philosophers of mathematics have written much more. ↩︎
- Indeed, the effort began shortly after OpenAI’s announcement. ↩︎
- Tao’s reflections on the impact and process of the Equational Theories Project on mathstodon provide an example of this phenomenon. ↩︎
- OpenAI “We hope the mathematical community will engage deeply with these results, place them in context, and bring the ideas behind them to life through new research and discovery.” in the press release for Ten Advances. ↩︎
- The above press release, and further releases, despite citing the Leiden declaration and later AGMAI recommendations, are dubious when it comes to item 4, completely ignore item 5, and do poorly on item 6 of the Leiden declaration. ↩︎
- I am more interested in understanding people and organizations by what they do, than what they say. ↩︎
- See Buckmaster’s statement and post on Mathstodon. Since that statement this possibility has appeared in regard to other recent advances of the summer, see Andreas Thom, and Borisov, Gabber, and Vasiu’s remarks in their paper on the Jacobian Conjecture. ↩︎
- The October 6 “dump” has led to a host of further questions. While https://citedbyagi.com illustrates the knowledge drawn on, there remains in my mind (based on personal interactions with mathematicians working on some of these problems) the question of whether strategies, partial progress, etc., entered into the training data. Again, I do not have the means to know, but in light of the above and the general character of OpenAI the spectre only grows. ↩︎
- OpenAI is both capable of knowing better, with many Ph. D. mathematicians on staff, and of doing better: many have remarked that their models are capable of taking an uncited manuscript and locating possible relevant citations. That they did not for Navier-Stokes, and continued to not take such action after the broader backlash and AGAMI recommendations, further tarnishes their character. This is a footnote because the subsequent paragraph makes clear there is no character to tarnish. ↩︎
- Hao, Karen. Empire of AI. Penguin Press, United States (2025). See also the bibliography of Empire of AI. ↩︎
- As reported here, amongst many other places. ↩︎
- This Op-Ed discusses the use of the phrase and my own personal interactions, living in the San Francisco Bay Area, have included meetings with people who sincerely believe this is what they are up to. ↩︎
- Jason’s vessel the Argus is the ship from which the Argonauts derive their name in myth. ↩︎
- For example, Sage, GAP, Pari, and many other open source systems. Or, the Simons Magma license (not without its own controversy) ↩︎
- Since preparing the draft of this essay SAIR announced exactly such an effort. ↩︎
- The progress is also fragile: the number of weapons is once again trending upwards. ↩︎
Received 9 October 2026.
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