The Theorem Does Not Care

Imagine a seminar room after the proof has checked out.

The argument is unfamiliar. The conclusion is important. A path that nobody in the room had found is suddenly there, available to be followed, simplified, generalized or attacked from another direction.

Then somebody asks who produced it.

A machine.

The theorem remains exactly as true as it was a moment ago. The atmosphere becomes considerably more complicated.

I understand the disturbance. I love mathematics. I understand the attachment to a problem, the pleasure of finally seeing through it, the strange intimacy of spending months with something that refuses to explain itself. But the universe has not signed a contract guaranteeing that its secrets will be disclosed at a pace compatible with our career plans.

Science does not exist to flatter the scientist.

If the objection to a valid discovery is ultimately that it deprives us of the experience of being its discoverer, we should name the problem accurately: professional narcissism. A deeply human reaction. A poor scientific principle.

There are excellent reasons to challenge AI-generated mathematics. Errors. Misstated theorems. Missing attribution. Uncheckable claims. Restricted access. An avalanche of papers that transfers the burden of evaluation to everyone else. None of these objections becomes less serious because the technology is impressive.

But protecting the conditions of knowledge is different from protecting our place at the center of the photograph.

The October debate makes that distinction urgent. OpenAI has released hundreds of mathematical manuscripts produced with an internal model. Its repository groups related papers into 372 families and explicitly distinguishes levels of verification. These are not 372 independently certified breakthroughs, and a large file count is not a mathematical argument. OpenAI's repository.

The reactions also need to be described honestly. The independent Advisory Group on Mathematics and Artificial Intelligence asked laboratories to stop testing advanced problems on proprietary models inaccessible to the wider community. Its concerns include accountability, access and the development of human understanding. That is a specific position, not a demand to abolish AI. AGMAI's September recommendations.

The Association for Human Mathematics went further, urging mathematicians to discontinue work with OpenAI. Its statement, published as a guest post on Terence Tao's blog, opens its objection with a remarkable sentence: “Mathematicians did not ask for this work to be done.” AHM's statement.

That sentence troubles me. If the point is that a company cannot commandeer a community’s reviewing labor or set its agenda, I agree. If the point is that a discovery requires the profession’s permission, I do not. A research community should demand responsible conduct. It should not acquire ownership of the questions merely because its members have worked on them.

An unsolved problem is not a reserved parking space.

Nor should these statements be collapsed into Tao's personal position. His own Math 2.0 framing envisages AI supporting mathematical understanding and the community that develops it. There is a serious conversation here. Turning it into “mathematicians versus machines” would make it less intelligent. Tao's account of his talk.

I share much of the diagnosis about access. I am unconvinced that asking laboratories to stop attempting difficult problems on proprietary models is the right remedy. The ambition should be to expand our ability to investigate, not preserve scarcity in who can discover.

This does not make me indifferent to human understanding. In The Answer Factory, I worried about receiving a result while losing access to the intellectual territory around it. I still do. But an answer can also reveal territory we did not know existed.

Suppose a proof uses an unexpected construction. We can isolate it. Change an assumption. Find where the argument breaks. Look for a simpler case. Ask what survives in another setting. Translate the structure into a language that makes its power visible.

That is mathematical work. It does not become secondary work merely because we did not invent the first construction.

I would call it reverse mathematical analysis: begin with the result and investigate the new ways of thinking it makes possible. The aim is not to reenact every internal operation of the system. The aim is to discover what can be understood, tested and carried somewhere else.

There is an important difference between not understanding how a machine generated a proof and not being able to establish whether the proof is valid. There is another difference between checking a proof and understanding why its central idea is fruitful. Confusing these questions makes both enthusiasm and criticism sloppy.

A machine need not narrate every operation in human language to produce something worth investigating. It does need to leave us appropriate objects of scrutiny: precise statements, assumptions, checkable arguments, relevant artifacts and an honest account of the limits of the evidence. The standard should fit the claim.

When I wrote about machines designing their own operational languages in Ouroboros, I was not proposing that people abandon intent, judgment or investigation. Let the machine optimize the encoding. We still need a shared Rosetta Stone for the parts of the work through which we direct, challenge and learn.

Otherwise we face two equally foolish demands: that every machine operation be a human tutorial, or that every machine output be treated as revelation.

I prefer a colleague. One whose work I can challenge, whose contribution can surprise me, and whose apparent confidence does not relieve me of responsibility. “Co-author” describes that intellectual partnership here, not a transfer of human accountability to a machine.

Medicine offers a useful discipline for this discussion. Imagine a treatment that passes the relevant tests and helps patients. The fact that a particular physician did not personally discover it would be an absurd objection to its use. The questions concern evidence, benefit, harm and appropriate application.

That is not permission to wave away uncertainty. Drug development requires preclinical and clinical investigation, review and continuing monitoring. And a predicted protein structure from AlphaFold is neither a treatment nor a cure; it can help researchers design further experiments. The lesson is to scrutinize what the tool enables and how it is validated, not demand that progress preserve the discoverer's vanity. FDA's development process; EMBL-EBI on AlphaFold predictions.

I have spent enough time around research and peer review to have encountered criticism that seemed more interested in killing a paper than helping it become better. A profession can defend rigor. It can also defend a hierarchy while borrowing rigor's vocabulary. We should be capable of examining both possibilities without assuming that every critic is vain or every company benevolent.

Which brings us to the problem that deserves the anger.

Who gets the instrument?

Publishing a proof and providing access to the system that found it are different acts. OpenAI says the model behind this release is internal and that it is working toward release. It has also announced $5 million through a Simons Foundation initiative and model and compute access for Simons-supported researchers. Those are meaningful steps. They do not yet answer how broadly, affordably and durably the ability to ask new questions will be shared. OpenAI's release and October 15 update.

The distinction matters more than a slogan about open or closed companies. A paper may be public while the effective research capability remains concentrated. A model may be accessible while the computation required to use it well excludes most laboratories. A researcher may be allowed to inspect an answer but unable to pursue the question that answer suggests.

The risk is not simply that a company discovers something first. It is that the company becomes the gatekeeper of the next experiment.

Research freedom then depends on someone else's access policy, pricing, priorities and continued willingness to cooperate. A brilliant student in India, a small group in France or a researcher between institutions should not need an employment offer from a frontier laboratory before asking a consequential question.

This is where the university's role becomes more important, and more demanding.

A university should be part of a network through which knowledge can travel: across institutions, countries, generations and disciplines. It should help people examine one another's work, preserve methods, build alternatives and disagree without losing access to the instruments of disagreement.

That role cannot be fulfilled by waiting for a company to deliver the next batch of answers. It requires shared research infrastructure, credible independent checking, resources for studying unfamiliar results, and access arrangements that survive an individual collaboration. Universities and public funders should negotiate for these conditions and build capacity where they can, rather than confuse a demonstration account with independence.

We also have to stop treating the formation of young researchers as an inconvenient detail left over after the productivity discussion.

If AI performs some tasks through which juniors used to learn, the answer is not to prohibit the tasks from being automated. It is to design the next apprenticeship.

Imagine a student and a senior mathematician examining a machine-produced argument. The student asks why a hypothesis is needed. The senior suspects it cannot be removed. Together they ask the system to search for counterexamples, inspect the outputs and work through a simpler case. Perhaps the senior was right. Perhaps the student has found the edge of a new result.

Either way, the student is participating in inquiry. The machine's speed can create more occasions to ask, test, explain and revise—provided we organize the work that way.

Freshness is not a substitute for foundations. Foundations should make curiosity more powerful, not teach it to wait outside until it has accumulated sufficient seniority.

The senior researcher brings a map. The junior may notice that everyone has been politely walking around the same blank corner. AI can help them explore it together.

That possibility is precisely why I remain optimistic. We do not have to choose between faster discovery and the development of people capable of understanding it. We have to make their connection an explicit part of the work: ask participants to explain a new construction, identify a failed assumption, reproduce a check, or formulate a question they could not have asked before.

A machine-produced result can be a new school, if we are allowed inside it.

So let us argue fiercely about access, standards, concentration, cost and the formation of future researchers. Let us demand evidence from the laboratories and generosity from our institutions. Let us criticize a bad proof regardless of its author, and learn from a good one without demanding that it first apologize for how it arrived.

The theorem does not care about our ego.

The next discovery may depend on whether we can get past it.