OpenAI's 722 Mathematics Papers and the Closed Laboratory of the General Intellect
Who Audits the Proof? A Class Reading of a Science in Which the Result Is Public and the Machine Is Private
Who Audits the Proof? OpenAI's 722 Mathematics Papers and the Closed Laboratory of the General Intellect
Dear Young Comrades, What Does This Piece Say?
This week the world of mathematics woke to a noise it is not used to. On the night of 6–7 October, OpenAI uploaded 722 mathematics papers to GitHub and announced that they were the product of an internal model that has not been opened to the public. This is the news that went through the press as "more than 300 open problems solved." Mathematicians are now going through the pile one by one; some are excited, some are angry, and most are both at once.
This piece takes up the real question behind that news: is knowledge produced by a monopoly's machine behind closed doors scientific knowledge? Who audits, who verifies, who pays, who gains? These questions concern not only mathematicians, but everyone who finds the product of their labour, every day, in a model's training data.
The core of the piece is this: OpenAI's model was run on about 4,000 open problems, and the 372 result families the company found "meaningful" were published. Great names such as Riemann, Hodge and Kakeya appear, yet most of the results are narrower than their titles; none passed peer review, about a third have no formal verification, and the model and the prompts were not shared. This publication came after a month of crisis: the priority fight over Navier–Stokes, the warning declaration of 25 Fields medalists, and the proposals of the independent advisory group mathematicians formed, which were treated as "not binding."
I read the class meaning of these facts as follows: what is happening here is the expropriation of the general intellect. Mathematics is a knowledge commons, the product of thousands of years of common labour, and the model was trained on this commons. The order that has emerged works in four ways:
- The result is public, the means of production private: The norms of reproducibility and organized skepticism in science cease to function in the face of a closed machine.
- The cost is social, the profit private: The burden of verification is left to mathematicians free of charge; the reputation and the market value go to the company preparing for a public offering.
- Priority turns into a market: Capital that can run ten thousand agents for days and the researcher who gives a year to one problem are not competing on equal terms.
- Mental Taylorism: The metric shifts from insight to the number of problems solved; the brightest minds are drawn from the public university into private laboratories.
But this is not a hostility to technology. That the machine can produce new mathematics is a real increase in productive force; the problem is the property relations into which this force is set. At the end of the piece I have listed, as concrete tasks, what scientific workers and young researchers can do to change these relations.
Let us now go into the detail.
What Happened? The Facts
Let us first get the numbers straight, because figures that do not agree with one another are circulating in the news.
- In the catalogue OpenAI published there are 722 papers; they are distributed across 17 fields of mathematics, under 372 "result families" (ThePrint, OfficeChai). The figure 377 appeared in a New York Times headline; the difference may come from the catalogue's numbering, but OpenAI did not explain it (Tech Insider).
- According to the company's account, the model was tested on about 4,000 open problems. The published catalogue is the subset OpenAI found "meaningful enough." For each result, an average of three hours of ChatGPT Pro "thinking" computation was spent (OfficeChai).
- The model's name was not disclosed. The model is not public. The full prompts were not published; only the average computation data was shared (Yahoo Tech / Gadget Review).
- None of the papers passed peer review. OpenAI itself writes that there "may be problems" in results that have not been formally verified. In 235 of the 372 families, that is in about two thirds, there is a Lean formal-verification file (BigGo Finance). These Lean files too are reported to have been written by artificial intelligence, and their review status marked as "not checked" (ThePrint). Shortly after publication, three papers were retracted because of errors (Crypto Briefing).
The size of the claims, and their real measure
There are titles in the catalogue that sound very large: the "quasi-Riemann hypothesis," Khot's Unique Games Conjecture, the Hodge conjecture, the Kakeya conjecture, that the plane cannot be coloured with 5 colours, the form of Hilbert's tenth problem over the rational numbers, and the refutation of dozens of old conjectures (OfficeChai).
But most of the titles are wider than their contents. The "quasi-Riemann" result is not the Riemann hypothesis itself; it is a weaker statement, saying there are no zeros in the region where the real part is greater than 7/8. The Hodge result does not cover the whole conjecture, but a particular family of shapes. Kakeya is taken up only in 3 and 4 dimensions (OfficeChai). None of these results is trivial; if they are correct, some of them are genuinely large. But the word "if correct" is, at this moment, the heaviest word in the sentence.
Background: three crises in one month
This publication did not appear in a vacuum. The preceding weeks went as follows:
- 8 September — the Navier–Stokes fight. OpenAI announced a 100-page proof on Navier–Stokes, a Millennium Prize problem, and said that about 10,000 agents had worked in parallel for 88 hours. The same day Tristan Buckmaster of NYU published a similar study he had been carrying on for nearly a year with Levent Alpöge, who works at Anthropic, and claimed that Sébastien Bubeck of OpenAI had said to him "Why would you ruin your career?" and that pressure had been applied over the order of publication. Bubeck and Sam Altman rejected this account. OpenAI said it had not seen the pair's work; it did not, however, entirely rule out that de-identified data obtained from product use may have improved the company's models (Decrypt).
- 11 September — the declaration of 25 Fields medalists. Twenty-five Fields medalists, among them Terence Tao, Peter Scholze, Maryna Viazovska, Manjul Bhargava, June Huh and Martin Hairer, published a text titled "The Serious Misalignment of Artificial Intelligence in Mathematics." The central sentence was this: "Problem solving is merely a means to, and a proxy for, reaching the real aim, such as conceptual understanding and insight." (ExplainX)
- 21 September — the Independent Advisory Group. OpenAI proposed to some mathematicians an advisory board inside the company. The mathematicians instead formed a group that is independent, unpaid, and without decision-making authority over any company: the "Mathematics and Artificial Intelligence Advisory Group" under the roof of the Institute for Advanced Study (IAS) at Princeton (Charles, De Lellis, Gowers, Hairer, Srivastava, Tillmann, Vakil, Witten, Wood) (Let's Data Science). The group's proposal to OpenAI was plain: share the model, the full prompts, and the computation time for each result (Yahoo Tech). OpenAI said it had followed the guidelines but was not bound by them (BigGo Finance).
What are the mathematicians saying?
The responses are not of one voice, and that is a good thing:
- Bryna Kra of Northwestern describes the closed meeting in August as a moment "in which extreme excitement and extreme fear were lived together"; she writes that rigorous papers were asked for, and that "these views were openly ignored." Her most important sentence is this: announcing mathematics on Twitter and in press releases is not a treatment that befits the field that produces the training data (BigGo Finance).
- Nestor Guillen of NYU says that mathematicians see the companies as structures that behave "like a mafia," and that he feels "deep concern" at power concentrating in a few companies.
- Andrew Sutherland of MIT stresses that the results should be counted as unverified until they are audited.
- Daniel Litt of Toronto says something important from the other direction: publishing is better than hiding; there is no reason for companies to hide the answers from mathematicians.
- Terence Tao finds the pace at which the frontier laboratories produce results "wild" (BigGo Finance); he had warned before that artificial-intelligence proofs can look flawless and hide subtle errors (OfficeChai).
One more detail: Fields medalist Jacob Tsimerman is taking leave from the University of Toronto and joining OpenAI (OfficeChai). I will come back to this detail below.
Mathematics: The Purest Form of the General Intellect
In the famous "Fragment on Machines" in the Grundrisse, Marx writes that social knowledge accumulates and turns directly into a productive force, and he calls this the general intellect. Knowledge is the product not of individual brains but of social labour accumulated across generations; the machine is the "objectified" form of this accumulation.
No field fits this definition as well as mathematics does. Since Euclid, every theorem has been built on the one before it. Mathematicians share their proofs largely in the open; the preprints on arXiv, the discussions on MathOverflow, lecture notes, theses, Lean's open-source library Mathlib… All of these are a knowledge commons that no one possesses alone.
OpenAI's model was trained on this commons. Bryna Kra's phrase "the field that produces the training data" says exactly this. The model digested centuries of common labour; it now draws new results with the methods this labour produced, and presents these results under a company's name and in its shop window. The process I earlier called "the expropriation of the general intellect," in the piece The Revolt of Crystallized Labor, stands before us here in its most naked form: common knowledge is taken free of charge as input, and the output is marketed as the success of a machine in private property.
We should also note a contradiction here. As we saw in the distillation debate, the monopolies count learning from everyone's labour as "fair use," while they brand learning from their own outputs as "theft." The same logic works in mathematics: Mathlib is open, arXiv is open, humanity's mathematics is open; but the machine that processes this accumulation is closed.
The Heart of Science: Organized Skepticism and Reproducibility
In 1942 the sociologist of science Robert K. Merton gathered the norms of the scientific community under four headings. The first he named — interestingly — "communism": scientific findings are common property, and the scientist yields a finding to the community only in return for recognition (citation, reputation). The others are universalism, disinterestedness, and organized skepticism: every claim is submitted to the community's open audit.
These norms were never fully applied; science too is inside class society. But in today's event each of these norms is directly violated:
- Common property: The results are shared, but the instrument that produces these results (the model, the weights, the prompts) is not shared.
- Organized skepticism: For an experiment to be reproducible, the apparatus must be known. While the model is closed, a researcher outside cannot run the process again, and can examine only the finished product (ThePrint).
- Disinterestedness: The timing of the publication falls in the same period as the company's preparation for a public offering; the mathematicians interviewed say openly that displays of talent matter to investors (BigGo Finance).
At this point an epistemological distinction has to be made. Mathematics differs from the experimental sciences: if a proof is correct, it does not matter who or what produced it. Formal verification tools such as Lean are very valuable for this reason. Why, then, is the closed model a problem?
For three reasons:
- Lean audits only the logic. Human experts have to check separately what has been proved, that is, whether the formal statement really expresses the mathematical problem that is claimed (ThePrint). Besides, for a third of the families there is no formal verification at all.
- Selection bias is invisible. 4,000 problems were tried, 372 families were published. What happened in the rest? How many failed attempts, how many wrong proofs were produced? To speak about the model's reliability without knowing this is like deciding that the lottery is profitable by seeing only the winning tickets.
- Knowledge is not only the result; it is the method. As the Fields medalists emphasize, the aim of mathematics is insight. Understanding why a proof works also includes how it was found. Shortened reasoning summaries were shared for only ten results.
In short: the result is public, the means of production private. This is the summary of capitalist knowledge production.
The Bill for Verification: The Socialization of Cost, the Privatization of Profit
The least discussed side of this event is the labour of verification.
722 papers were published in a single night. Who will read them? Peer review of a mathematics paper takes months, sometimes years, and is almost always done free of charge. As the news openly writes, the work of sorting out what is correct, original and valuable "was loaded largely onto the shoulders of the mathematics community" (Yahoo Tech).
Now look at this picture:
- The company met the cost of computation with its own capital, and wrote the reputation of the results to its own brand.
- Verification, criticism and debugging were left to mathematicians who work in universities and whose salaries come largely from the public budget.
- The nine mathematicians in the advisory group do this work free of charge (Let's Data Science).
- And this community's proposals are "not binding."
This is a familiar pattern: profit is privatized, cost is socialized. Imagine a factory that closes its quality-control department and tells the customer "if you see an error, report it"; here the customer is humanity's common scientific mind. Mathematicians are being signed up, without noticing, as unpaid workers in a company's product-verification department.
This is akin to the situation we called, in the swarm dossier, "the hurricane's accounting department": the machine produces quickly, but the burden of digesting, sorting and making sense of the results of production falls again on human labour.
The Priority Race, and the Crushing of Scientific Norms by Capital
The Navier–Stokes event shows another side of the matter. In science, priority (who found a result first) matters, because the scientist's only "wage" is often recognition. In Merton's words, the scientist who yields a finding to the community receives citation and reputation in return.
If what Buckmaster recounts is true, a company executive pressed an academic on the order of publication and threatened him with his career. OpenAI and Bubeck reject this; so, rather than deliver a final verdict, the two sides' opposing accounts have to be noted side by side (Decrypt). But independently of the accounts, the structural fact is plain:
- On one side there is a capital that can run 10,000 agents for 88 hours.
- On the other side there are two researchers who gave a year to one problem.
- And it is not even known for certain whether data obtained from one of the researchers' product use entered the rival company's model.
This is a priority market, in which competition runs on unequal terms. The mathematicians' warning is exactly at this point: if this pace continues, researchers will begin to hide their work instead of publishing it, and scientific openness will be damaged (Morning Brew). The monopoly's speed itself erodes the culture of openness that feeds the knowledge commons. Enclosure advances not only through the closedness of the machine, but also through people closing their own work out of fear.
Mental Taylorism: The Separation of Insight from Production
Taylorism was a way of organising work that took the worker's knowledge out of their hands and carried it into management, splitting the job into "those who think" and "those who execute." What we call mental Taylorism is the application of this split to mental labour.
Let us read again, with this eye, the sentence in the Fields medalists' declaration: "Problem solving is merely a means of insight." The company's metric, however, is the number of problems solved; because a number can be marketed, and insight cannot. Mathematics is being reduced to an economy of metrics: how many problems, how many hours, how many papers. A year ago the measure of success was high-school olympiad questions; now the measure is the Millennium problems. Each time a metric is saturated, a new one is found (OfficeChai).
In this process the mathematician's role changes too. They are under pressure to turn from a creative researcher into a quality controller who audits the machine's output. This is a reorganisation of the labour process, and, as with every reorganisation in history, in whose favour it will work is not a technical question but a class question.
Tsimerman's move to OpenAI is part of this picture too. The brightest minds trained in the public university are drawn into private laboratories by wages and computing resources that public institutions cannot provide. This is not something to be judged as an individual preference; it is a structural enclosure of brains. The precarisation of the academy, the lack of posts for young researchers, and the narrowing of public research budgets are the objective conditions that increase this power of attraction.
Are We Against Technology? A Dialectical Note
No. Care is needed here.
Daniel Litt's objection is well placed: publishing is better than hiding. If a machine can really produce new mathematical results, this is a great increase in productive force for humanity. The development of formal verification tools such as Lean and Mathlib can make mathematics more reliable and more accessible. The Fields medalists are not against mathematics assisted by artificial intelligence either; what they are against is the relations surrounding the results (ExplainX).
As Marx showed, the problem is not in the productive forces but in the relations of production. If the same machine worked in public property, with open weights, and with a research agenda directed according to social needs, a large part of today's debate would disappear. The labour of verification would not go uncompensated; the priority race would not be tied to a monopoly's market value; the question "which problems should be solved" would be answered according to scientific and social need, not according to an investor presentation.
Let us also note an irony: Lean and Mathlib, which make a portion of these papers verifiable, are an open-source commons that volunteer mathematicians accumulated over years. Even the monopoly's shop window of "correctness" leans, in the end, on the community's common labour.
Two Columns: The Scientific Norm and Monopoly Practice
| The scientific community's norm | OpenAI's practice in this publication |
|---|---|
| The method and the apparatus are explained, the experiment is reproducible | The model is closed, the prompts were not shared, there is only average computation data |
| Peer review comes before publication | 722 papers, without peer review, uploaded to GitHub in a single night |
| Failed attempts are knowledge too | Of 4,000 problems, only the subset found "meaningful" was published |
| Previous labour is cited | In August, criticisms of missing citation of previous work; in September, the Navier–Stokes priority fight |
| Verification is a labour whose return is recognition | The burden of verification was left to the community free of charge |
| The aim is insight | The metric is the number of problems solved |
| The community's proposals are a guide | The advisory group's proposals are "not binding" |
| Knowledge is common property | The result is common, the means of production private |
Concrete Tasks
This debate may look to us like a distant Princeton debate. It is not. Every mathematics department in Turkey, every doctoral student, every scientific worker is inside this transformation. What can be done?
- Make the demand for openness concrete. For every scientific result claimed to have been produced with artificial intelligence, the model, the prompts, the computation data, and the rate of failed attempts should be disclosed. This is the IAS advisory group's demand too. Science organisations (such as the Turkish Mathematical Society) and journals can make this a condition of publication.
- Public computing infrastructure. That universities and public research institutions have a common, open and public computing infrastructure is a precondition of scientific independence. This should be asked about in the budget priorities of TÜBİTAK and the universities. We also discussed this demand in the Socialist AI Manifesto.
- Grow the open verification commons. Contributing to projects such as Lean/Mathlib is, for young mathematicians, a practice both scientific and political. As long as the instrument of verification stays open, the monopoly's monopoly on correctness is broken too.
- Recognition of the labour of verification. Refereeing and verification should cease to be invisible labour in academic evaluation; they should find a return in the criteria for posts and promotion.
- The right of citation and of data. The right of scientists to know how their work and their usage data are used in model training, and to object, should enter the agenda of scientific workers' unions.
- The struggle against the precarisation of the academy. The antidote to the enclosure of brains is secure employment and a sufficient research budget in public scientific institutions. This is a direct part of the trade-union struggle of university workers.
- Scientific literacy. Learning, and teaching, to look for the word "quasi" when reading headlines that say "Riemann has been solved." Separating the language of marketing from the scientific claim is the defence of the public mind.
To Whom Does the Proof Belong?
Dear young comrades,
A proof, by its nature, belongs to everyone. No one can buy the Pythagorean theorem; once it is understood, it becomes everyone's. This is also what has made mathematics a commons of humanity for thousands of years.
Are OpenAI's 722 papers a contribution to this commons, or a new form of its enclosure? The answer lies not in the correctness of the papers, but in the relations that produce them. If the results are open but the machine is closed; if the burden of verification goes to the community, and the reputation and the market value go to the company; if the community's proposals are counted as "not binding," then what stands before us is not the emancipation of science, but the expropriation of the general intellect once again.
The organized skepticism of science is today turning into a class demand: who audits the proof? The answer to this question should not be "a board of directors behind closed doors." The answer should be humanity's common mind, that is, those who produced it together across centuries.
Knowledge belongs to everyone.
Sources
News and analyses
- ThePrint — Weeks after claiming AI cracked 90-yr-old maths puzzle, OpenAI drops the mother lode on mathematicians
- OfficeChai — OpenAI Releases Over 300 Mathematical Results Produced By An Internal Model
- The Washington Post — OpenAI releases progress on more than 300 math research problems
- Morning Brew — OpenAI said it solved 300+ complicated math problems, mathematicians are livid
- Crypto Briefing — Mathematicians are combing through OpenAI's AI-generated results on over 300 problems
- Yahoo Tech / Gadget Review — OpenAI Drops 722 AI Math Manuscripts on GitHub. Experts Must Now Verify Them.
- BigGo Finance — OpenAI Drops 722 Math Manuscripts Overnight… Sparking Fury in the Math Community
- Tech Insider — OpenAI 377 Math Problems: 372 Families, One Prompt
- Decrypt — OpenAI Says It Solved a $1M Math Problem. A Rival Mathematician Says He Did It First
- Let's Data Science — Mathematicians Announce Independent AI Advisory Group
- ExplainX — Fields Medalists vs OpenAI: The Math AI Declaration
Theoretical background
- Karl Marx, Grundrisse, "Fragment on Machines" (1857–58)
- Robert K. Merton, "The Normative Structure of Science" (1942)
Pieces on Knowledge Commons on which this piece is built
