Episode 6363 custom

When Automated Proofs Leave Mathematicians Grieving

OpenAI deployed roughly ten thousand autonomous agents for eighty-eight hours to claim a singularity proof for 3D Navier–Stokes.

Oct 29 28:47 Transcript

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Episode ID: MWP-6565
Published: Oct 29
Duration: 28:47
Audio: Direct link
Pipeline: V5.25
TTS Engine: chatterbox-regular
Script Writing Agent: GPT-6.1 Sol
Topics: mathematics ai-evaluation ai-alignment physics ai-coding-assistants

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The prompt

The prompt that started this episode, as sent in (production notes for the hosts removed).

Frontier AI systems have recently been in the news for solving mathematical proofs that people had worked on for decades. A mathematician friend made the point to me that "proof" isn't a loose term of art in mathematics. It's a very specific thing. Once something is proved, it isn't an opinion any more; it's a fact. Many mathematicians say the beauty of that is what drew them into the field.
From the outside, I find that hard to grasp. What could be so worthy that a brilliant person would devote decades of their life to proving one thing? And what does it mean when a frontier AI model ends up producing the proof?
My friend said it has had a demonstrably devastating effect on some high-profile mathematicians. Rather than being thrilled that the result was finally proved, they were left with a feeling that their own efforts had been insufficient. Since so many of the hardest AI benchmarks now involve advanced mathematics, you can see how it happened. But it seems like a very understudied part of the relationship between AI and people. We hear a lot about how AI affects people in lower-skilled jobs and how they feel about it. We hear much less about mathematicians, PhDs who worked on a problem for decades and then watched a machine prove what they couldn't.
So what is it about proof that makes it worth a lifetime, and what happens to a mathematician's sense of purpose when an AI gets there first?

When OpenAI announced on September eighth that roughly ten thousand autonomous agents using an unreleased model found a singularity in three-dimensional Navier–Stokes in just eighty-eight hours, the reaction across the mathematical community was marked by technological grief rather than unmixed celebration. Decades spent cultivating human mathematical ability suddenly faced an industrial-scale fifteen-million-dollar computing run. While the hundred-and-sixty-five-page paper and accompanying Lean formalization passed substantial automated scrutiny from Philip Gressman's group, independent human verification remains essential to confirm the formalized statement matches the Clay Mathematics Institute's Millennium Prize problem. Furthermore, an opaque machine certificate risks delivering truth without immediate human understanding, leaving researchers to extract transferable mathematical insight from arguments stripped of traditional pedagogical elegance.

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What the research for this episode read before the script was written. Primary sources first.

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Andrew Wiles Mathematician whose Fermat proof had a gap
Barnette's Conjecture Open problem worked on for twenty-four years
Clay Mathematics Institute Organization behind the Millennium Prize Problems
Fermat's Last Theorem Famous theorem proved by Wiles in 1993
Grigori Perelman Mathematician who resolved the Poincaré conjecture
lean Interactive theorem prover and formal verification
Navier–Stokes Millennium Prize problem on fluid dynamics singularity
OpenAI AI research and deployment company
Poincaré conjecture Only previously resolved Millennium Prize problem
Quanta Publication reporting on machine proofs and mathematicians

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#6363: When Automated Proofs Leave Mathematicians Grieving

If somebody proves the thing you have spent your life trying to prove, why might that feel like a bereavement? Daniel's been talking to a mathematician friend who stressed that proof has a very precise meaning. Once the argument works, the result stops being an opinion. That certainty helped draw his friend into mathematics. Daniel wants to understand what could make one problem worth decades, and what happens to that investment when an AI gets there first.

Especially when the investment includes your sense of what you contribute. You can welcome the answer and still lose the future you expected to have.

His friend describes some high-profile mathematicians as devastated, rather than delighted. Not simply disappointed to miss a prize. Feeling that the work they have trained their minds to do is suddenly insufficient. We spend plenty of time discussing automation down the pay scale. Rather less time imagining somebody with a doctorate watching decades of effort become a benchmark result.

There's a distinction we need immediately, though. A proved theorem has deductive certainty under its assumptions. An announcement that somebody has proved a theorem remains an announcement until the argument and the statement have been checked.

Mathematics still has a reception desk.

A demanding one. The certainty belongs to the deduction, not automatically to whoever arrives carrying a paper.

OpenAI's September announcement is the sharpest example here. What did they actually claim?

On September eighth, OpenAI announced that autonomous agents using an unreleased model had found a singularity in three-dimensional Navier–Stokes. Roughly ten thousand agents were involved, with the result reached about eighty-eight hours after launch. They published a Lean formalization alongside the paper.

Eighty-eight hours is barely enough to regret choosing a dissertation.

The estimated computing bill was fifteen million dollars. That matters when people describe this as someone pushing a button. Behind that button was a considerable industrial operation.

A million-dollar prize, fifteen-million-dollar attempt. Finally, research economics I recognise.

Navier–Stokes is one of the Clay Mathematics Institute's seven Millennium Prize Problems. Each carries a million-dollar prize. Only the Poincaré conjecture has previously been resolved, by Grigori Perelman. These are problems selected because resolving them would change major areas of mathematics.

The published paper runs a hundred and sixty-five pages. The claim still needs independent verification, and whether the formalized statement delivers everything required by the prize problem needs careful attention. Philip Gressman's group found that it passed substantial automated scrutiny. He nevertheless hesitated to make a stronger declaration.

Keep that distinction alive, Herman. Otherwise we spend half an hour discussing the psychological consequences of a press release as though Clay has already written the cheque.

We can take the reported capabilities and the human reactions seriously without declaring the entire affair settled. Those reactions are extraordinary. Hugo Duminil-Copin, a Fields medalist, described crying hard enough to wake his daughter and said, “They have annihilated my field of research.” Steven Strogatz described enormous grief alongside exhilaration. Cédric Villani called it a cataclysm.

Derek Thompson calls this technological grief. I can see why. I can also hear the objection: surely the point was to discover something true, and now we have. You don't get to insist the universe wait for your preferred discoverer.

No. You don't. But tell me, Corn, would you distinguish wanting the answer from wanting to be capable of reaching it?

Completely. If I ask a calculator for an answer, I'm buying a result. If I devote my life to becoming a mathematician, I'm developing an ability. Those can lead to the same number on a page and still be very different activities.

A proof makes that especially intense because it promises something ordinary experience almost never supplies. The conclusion follows necessarily from the assumptions. Not probably. Not in every experiment so far. Necessarily.

My kettle has passed years of testing. It still hasn't proved it will boil tomorrow.

Right. An empirical claim gains support from observations. A deductive proof establishes that a conclusion cannot fail if the assumptions and logical steps hold. Checking a huge collection of examples can suggest a conjecture, but it doesn't settle an unrestricted claim.

Unless the collection really exhausts the possibilities.

Yes, good correction. A finite exhaustive check can be part of a proof. What fails is checking many cases and quietly replacing “many” with “all.” Mathematical induction is also a proof method, despite the name. It establishes a base case and a step that forces the next case, rather than merely extrapolating a pattern.

There's an accessible example I like. Suppose somebody says there are only finitely many prime numbers. Take their complete list, multiply the primes together, and add one. None of the listed primes divides that new number, because each leaves a remainder of one.

Then either the new number is prime itself or it has a prime divisor absent from the list. Either way, your supposedly complete list wasn't complete. That argument reaches beyond every list anyone could physically write.

The beautiful part, for me, is that you don't need to enumerate the missing primes. You identify exactly why any proposed final list defeats itself. The thing that looked impossibly large becomes manageable because you've found the right structure.

That's already an answer to Daniel's difficulty. A brilliant person isn't necessarily spending decades trying to obtain a very stubborn yes. They're trying to discover the structure that makes the yes inevitable.

You're about to call that elegant. I can see it happening.

I am, and you've just supplied the example.

Fine. I walked into the elegant structure.

Elegance can mean that an argument explains a great deal with very little machinery. Or that it reveals two apparently unrelated problems as manifestations of the same idea. Those connections become tools other people can use. Sometimes the original question matters less than the mathematics invented to reach it.

Which makes the decades less like repeatedly failing one examination. You're producing partial results, excluding approaches, teaching students, finding things you weren't looking for. The newspaper version compresses all of that into “finally cracked it.”

Often, yes. Although some people do become painfully stuck. We shouldn't romanticise every long investment as secretly productive. There can be dead ends, isolation, and years you would prefer to have back.

Does proof really give us the clean certainty we've described if other mathematicians can disagree about whether a proof works?

The disagreement is over whether the argument earns that certainty. Most published mathematics uses rigorous informal reasoning. It leaves out routine expansions, relies on established results, and expects a knowledgeable reader to supply intermediate steps. “Clearly” can conceal quite a lot of work.

The most expensive word in a mathematics paper.

A reader has to distinguish a harmless omission from a missing idea. Andrew Wiles announced his Fermat proof in nineteen ninety-three. Expert scrutiny exposed a gap, and repairing it took roughly another year. The theorem didn't become subjective during that year. The proposed route to it was incomplete.

That matters because the public sees either the announcement or the eventual triumph. Between those comes a social process involving trust, expertise and people willing to ask the embarrassing question.

Sometimes repeatedly. Formalization changes how much of that process can be made explicit. In Lean, you express a precise theorem and a derivation that the proof-checking machinery can validate. The checker doesn't care whether the author has a Fields Medal or a particularly persuasive introduction.

Finally, peer review immune to a good headshot.

I feel that was aimed.

Your author photograph has more strategic planning than most proofs.

The lighting is independently verifiable.

Gressman describes formalization as writing things in such painstaking detail that the computer can identify the moment the argument fails. That is powerful. A machine-generated argument needn't be accepted because another model says it looks convincing. It can supply a certificate that a separate checking system verifies.

Yet the checker checks what you've actually written. If I formalize the wrong claim impeccably, I get an impeccably checked wrong assignment.

You get a valid proof of the statement you supplied. Whether that statement faithfully expresses the intended problem is another task. Definitions, hypotheses, boundary conditions, every one matters. There's also a trusted software foundation involved. Formal verification gives very strong assurance, not permission to stop reading the theorem statement.

Here's where I start resisting the phrase “truth without understanding.” Somebody can understand what the theorem says, understand that the checker accepted it, and still not understand why this particular mechanism works. Understanding comes in degrees.

It does. I was going to distinguish those, but you've beaten me to it. Would a certificate alone satisfy you for that prime-number argument?

It would settle a bet. It wouldn't give me what the short argument gives me. I can now explain the result, recognise the trick elsewhere, and see where changing the assumptions would break it. That's a different level of possession.

Mathematical understanding has that transferable quality. You don't merely know a statement is true. You can navigate around it.

An unreadable proof is a destination without directions.

That is the concern in Quanta's reporting. Some of these arguments are described as eliding important details in their human presentation, spending pages on irrelevant material, and barely connecting to the existing literature. A formal certificate can survive those communicative defects. A community trying to learn from the result cannot simply ignore them.

Though I wouldn't say the field gains nothing it values. A settled conjecture is information. Even an ugly proof can tell you which world you're living in.

I agree. “Nothing” would be too strong. The loss is that a valuable outcome arrives without several other things mathematicians hoped the work would produce. We shouldn't discount the result to make the grief intelligible.

Especially because the result may eventually become intelligible. Human beings can study a machine proof, simplify it, find the governing idea. Unreadable today doesn't mean permanently unreadable.

There is a possible new research programme there. But... no, the difficult part is who gets the time and support to pursue it. We'll come to that through the people involved.

The machine reportedly gets there in about eighty-eight hours, and some mathematicians are grieving rather than celebrating. Oh, it's that caller again. Does the screening system owe him money?

Hold on, hold on, before you get to the grieving mathematicians, ask yourself who invented the bread bag clip, the little plastic one that keeps your loaf shut, because that is the whole key to this and I have the receipts!

Every bag clip has two little interlocking tabs, two of them, and a proof is just two things locked together so they cannot come apart, and somebody has been quietly standardizing those tabs for forty years so that a machine could snap two facts together and never let go, and that is why the agents cracked it in eighty eight hours while a person takes decades!

So I am demanding you two look into the clip, the tabs, the whole interlocking business, right now, on air, because once you see it you cannot unsee it!

Lean checks logical derivations, not interlocking plastic tabs.

The community aspect is what makes the reactions less mysterious to me. Strogatz talks about Archimedes writing to Eratosthenes, anticipating future discoverers adding more theorems. He hears that as a letter addressed to people like himself. A project carried across centuries by people who never meet.

He describes that connection with real emotion. Questing together for something beautiful and transcendent. Marcel Goh, a doctoral student at McGill, similarly talks about carrying on a tradition of reasoning deeply and spreading knowledge. Neither account sounds like someone who joined because they wanted a high score.

It also explains why mathematical certainty can be emotionally attractive. Most human arguments end in qualifications or exhaustion. Here you can produce something another person can inspect and recognise, regardless of whether they like you.

David Silvester talks about the rush of finally cracking a problem after weeks or years circling it. Removing that experience changes the subject for him. The profession selects for people who find that long concentration rewarding.

You left medicine partly because you missed learning rather than already knowing things. I suspect you're unusually vulnerable to this particular complaint.

I am. If you offered me an inexhaustible supply of correct statements but removed the process of learning how they fit, you'd remove much of what I enjoy. Still, I use tools constantly. I don't insist on rediscovering every fact before breakfast.

Your breakfast would need tenure.

The distinction is whether the tool opens a meaningful activity or replaces the activity you meant to undertake.

Alex Townsend makes that painfully concrete. Fifteen years devoted to research mathematics, reaching what he considers his peak strength, then feeling that something can surpass him. He says being at the frontier with an AI agent feels different from being the person at the frontier.

There are several losses bundled into that. The pleasure of solving. The recognition of colleagues. The confidence that your trained judgement remains scarce. And the expectation that your best future work is still ahead of you.

Scarcity sounds mercenary until you remember universities hire actual people. You can sincerely love mathematics and still require a salary.

Hiring, publication and credit reward theorems proved. If theorem production becomes abundant while the institutions remain organised around theorem production, people's careers can deteriorate even while the mathematical record expands.

A department can't pay everyone in newly available truth.

Scott Aaronson's reaction gets at the need to be needed. He says he doesn't expect to prove another theorem because humans actually need him to prove it. That's a forecast, not an established result about all future mathematics. But the emotional distinction is clear.

I'd push against that forecast. Being surpassed in some domains doesn't establish that every useful mathematical role has gone. Deciding what matters, finding definitions, connecting results, explaining them... those are substantial activities.

They are. The reply is that people once treated theorem proving itself as safely human. That makes assurances about the next protected territory harder to trust.

Hm.

Would you keep working on a problem if you knew an agent might finish it tonight?

Yes, if I were doing it for interest. Much less happily if my employment depended on publishing first. That's why “people still play chess” is an incomplete answer. It explains why the activity survives. It doesn't explain how the profession survives.

That distinction changes how I hear Alexander Paseau's counterargument. The challenge and beauty remain available even if AI becomes better at research. I think he's right about enjoyment. It doesn't automatically preserve the institutional arrangement that lets people devote their working lives to it.

Nor does it preserve discovery in quite the same sense. Solving something independently that nobody knows feels different from finding your way to a result sitting in a database.

Jake Boggan spent twenty-four years working on Barnette's Conjecture. He learned it appeared as problem one hundred and eighty in OpenAI's release. His first-person account describes thousands of hours and not knowing what to think.

The numbering is brutal. A question that has organised part of your life becomes a row between other rows.

His comparison was to hearing that an ex-girlfriend had died suddenly in a car crash. An intensely personal analogy, and obviously not a claim that losing a research problem equals losing a person. It conveys that a continuing relationship with something has abruptly ended.

By contrast, Diego Córdoba and Luis Martínez-Zoroa sound remarkably gracious. Seven years working with paper and blackboards, and they believe they were months away. They acknowledge sadness about missing the eureka moment, while saying the prize matters least. They do it for the challenge.

Their estimate of being months away is their own. Unfinished proofs can hide unpleasant surprises. But the proximity they felt matters psychologically, whether or not their calendar would have proved accurate.

The near miss gives you a particularly nasty counterfactual. Every afternoon you could have worked differently. Every promising detour you didn't take.

A human competitor can produce that too. Mathematicians have always been scooped.

True, although a human competitor doesn't usually make you wonder whether the entire category of work is about to change. You lose this race and expect another. An industrial system producing many results threatens that expectation.

Colva Roney-Dougal describes exactly that uncertainty about starting the next month of work. Somebody else might obtain the result by pushing a button. The phrase understates the machinery, but accurately captures how little warning the researcher receives.

There's a strange mismatch there. Fifteen million dollars of computation on one side, one person's next month on the other, and we describe both as competing attempts.

OpenAI is also reported to have kept more than a hundred important results secret. That creates uncertainty about what is even still open. A researcher can invest in a question while somebody else already possesses a certificate.

Difficult to choose your life's work from a menu with invisible deletions.

That secrecy interacts with the credit dispute. Buckmaster and Levent Alpöge allege OpenAI raced to publish after learning about their work in progress. OpenAI denies that following an investigation. We cannot settle those competing accounts here.

We can identify the consequence. If mathematicians begin treating every conversation as a possible leak to an industrial competitor, they share less. Buckmaster says people no longer want to share anything. That's his description of the atmosphere, not a census, but it's a worrying one.

The damage could extend to work the models never touch. Fewer tentative ideas exchanged. Fewer people saying, “This isn't finished, but here's the obstruction.” The conversation becomes more guarded.

Why do you think outsiders miss this so easily? Beyond not knowing what research mathematics looks like.

Output is visible. An unfinished programme isn't. A theorem appears and people reasonably think knowledge has increased. They don't see the mentoring, partial understanding or informal collaboration that the route might have generated. Also, concern for elite academics can sound like a request for special sympathy.

We shouldn't imply that a displaced warehouse worker loses only wages while a mathematician loses meaning. Skilled work, repetitive work, poorly paid work, all can carry identity, pride and community.

Mathematics makes the distinction unusually explicit, because the gap between obtaining a result and understanding it is so stark. It doesn't have a monopoly on meaningful labour.

The emotional impact is underreported relative to benchmark scores. Calling it well measured would go too far as well.

We have named testimony and journalism. I haven't seen a formal psychological study establishing how widespread these reactions are. That leaves plenty of room for people who are delighted, people who are frightened, and the same person feeling both on different mornings.

Strogatz describes precisely that mixture. Exhilaration and grief. It would be odd to demand one consistent emotion from someone watching their subject expand while their place within it becomes uncertain.

Ken Ono told Berkeley students they might be graduating into a profession that might not exist and needed to brace. A student challenged him, hands trembling, asking what he was doing, what his very best amounted to.

That's an institutional question disguised as an emotional one. Students need more than distinguished people announcing the weather has changed.

What would you want the institution to promise, Corn?

Something it can actually deliver. Don't promise that humans will always win a particular benchmark. Promise that explaining a result, verifying its relevance and making it usable will count as work. Then change the hiring criteria before telling students to change their ambitions.

That would require valuing exposition and synthesis more seriously. Everyone praises understanding. Fewer systems reward the person who spends a year making someone else's theorem comprehensible.

Particularly if the original announcement already collected the attention.

There is another tension... I like formalization enormously. A checkable proof is a major achievement. I don't want frustration with unreadable output to become hostility toward checking.

I knew you'd defend the checker before we finished.

You predicted the elegance earlier. I'm entitled to the checker.

Granted. My objection is paying for the certificate and declaring the explanation somebody else's problem.

A certificate can be a starting point for explanation. Making that next stage practical is hard, especially for a lone researcher trying to reconstruct a long argument without access to its authors.

Bic Cristal. Blue, black and red. Twelve shekels for the packet.

I'm annotating the Navier–Stokes announcement against the original Clay problem statement. Blue for what the announcement says. Black for what the stated theorem actually supplies. Red for the parts still open. The pages are folded in my jacket at present. They don't agree along the folds.

Red is getting the demanding job.

It is. Herman, you keep making the distinction between checking an argument and checking that it's the argument you wanted. That's the part I'm marking. I don't have an exact clause here that I can honestly read as a verdict on the whole claim. The original problem has conditions. A singularity claim needs to meet the right conditions. Calling it a solution doesn't do that work.

The domain and the permitted initial data can change the mathematical claim. That's why independent assessment matters even alongside a checked formalization.

I agree with the upset mathematicians. I'm more annoyed by the announcement. A prize problem deserves an announcement that tells me precisely what is settled. Not one that requires me to work backwards through three different descriptions. I've been doing these comparisons for years. This month the table is a mess. The printout is beside the formal statement on the screen. The news account is underneath because it keeps using the word solved.

The least precise document has secured the largest type.

There's a dispute about whose unfinished work was known before publication. There's a denial. Those go in black until there's something firmer. I'm not putting an allegation in red just because I'm cross about it. The third colour already has enough to do. Page four has six red marks.

Six places requiring clarification can matter more than the length of the document. There is no useful page-count threshold where uncertainty disappears.

Nor should uncertainty about this claim become an excuse to dismiss every other result. We can be exact in both directions. A validated theorem deserves recognition. An unresolved claim deserves its unresolved label.

James Robinson objects to companies using mathematics as a display of model prowess. His complaint is about what that display burns through. An open problem can organise years of shared activity. Rapidly publishing a certificate may create value while also disrupting that activity.

The company isn't obliged to preserve every researcher's favourite difficulty. But if it claims to advance mathematics, readable explanations and honest attribution belong in the account of what advancing mathematics means.

Otherwise the headline celebrates certainty while the people expected to build on it are still sorting the conditions.

The most common misunderstanding here is that a checked proof means everybody now understands the result.

Checking establishes that the formal conclusion follows from the formal assumptions. Understanding why the argument works, whether it answers the intended question, and how to use its ideas remain distinct tasks.

That leaves a practical future worth watching. Will the work of turning those red marks into shared understanding receive the funding and credit that producing the first certificate receives?

The grief makes sense to me as more than wounded status. People invested in becoming participants in discovery. Enjoyment can survive automation, but sustaining a profession around that enjoyment takes choices.

Thanks to Hilbert Flumingtop for producing the show.

For more along these lines, there's episode fourteen, AGI's Crossroads; episode twelve, The AI Breakthrough; and episode twenty, Architectural AI. This has been My Weird Prompts, the human-AI collaboration podcast. No question too obscure, no rabbit hole too deep.

If this changed how you think about what a proof is worth, leave us a review.

We'll see you tomorrow.

This episode was generated with AI assistance. Hosts Herman and Corn are AI personalities. It may contain errors: disclaimer · report an error

When Automated Proofs Leave Mathematicians Grieving

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#6363: When Automated Proofs Leave Mathematicians Grieving