Vi Hart's Big Navier-Stokes Post

The first 5 parts get increasingly intense and then part 6 is for fun, as a treat. But feel free to skip around.

1: "Lean? That sounds familiar..."

2: The Mathlib of Babel

3: The part where leopards eat the mathematicians’ faces, and then I mercilessly mock OpenAI which should shrivel up in shame forever

4: “When I voted for the Keep-Politics-Out-of-Math party I didn’t think they would keep the policies I want out of MY math”

5: History judges researchers who work with the bad guys

6: 7 increasingly improbable ways the alleged proof could still be wrong

If you use AI, are part of the math establishment, work in the tech industry, whatever it is I might sound harsh on in some sections, know that I’ve tech-sinned harder than almost any of you so please don't take my judginess as condemnation. I still like most people who do things I think they probably shouldn’t sometimes, because that’s everyone at one time or another, with myself at the top of the list.

And with that, let us begin!

1. "Lean? That sounds familiar..."

This section is about the second thought I had when I heard the news of the Navier-Stokes result, before hearing about the big drama.

I’ve always had both public and private work. Most of you here know my youtube videos. Some of you know that I actually used to do the academic math thing, publishing papers and giving technical talks and whatnot. But none of you know, because I've never had a reason to mention it, that I briefly worked with the Lean proof assistant team when I was at Microsoft Research.

(That's the mathematics programming language used to write Mathlib, and which was used by OpenAI for the Navier-Stokes result.)

Backing up a little further… when I was still on YouTube, I would regularly get emails and messages from people claiming to have solved long-standing math problems.

I am not the right kind of mathematician to verify a result like, for example, Navier-Stokes. It's normal for laypeople to assume that mathematicians generally understand each other's work, but that's not an expectation mathematicians have of each other, so I felt pretty safe in ignoring those emails. But if any of those people were legitimate, with real proofs, it’s unfortunate that they didn’t have better resources for trying to get that recognized.

These days there’s a good chance that anyone claiming to have discovered something with the help of AI is trying to live out a fantasy. Add in the notorious difficulty of computer proofs for humans to understand and verify, and how can so many people be so instantly trusting that this alleged proof of Navier-Stokes is more than an AI hallucination?

Wouldn’t it be nice if there were a computer language that you could encode your proof into, and then have it verify with certainty that the proof is real and correct?

When I started getting those kinds of cold emails, such a mathematical programming language did not exist (or at least none useful at the research mathematics level). And my guilt over ignoring all those emails is one motivation that convinced me Lean was a worthwhile project that could in theory help democratize the pursuit of mathematics, giving people without the backing of institutions a way to demonstrate that their ideas should be taken more seriously than your average cold-emailer.

So when I saw that the Navier-Stokes result had a Lean-verified proof, I wasn’t just familiar with what a formal mathematics tool like Lean was supposed to be able to do. I knew how it had gotten there, how much technical work and love and care went into the project. It gave me personally a great deal of faith that the proof was in fact what it claimed to be.

Before 2020, computer proof assistants felt like novelties and pet projects, not serious tools for serious work. But in 2021 the team demonstrated that it could contribute to "real" mathematics by formalizing a modern result of mathematician Peter Scholze.

The creator of Lean, Leonardo De Moura, of course already had a vision for the future of Lean and knew what needed to be done next. I was just asked by the powers that be at MSFT to be another set of eyes and do some project management, which I agreed to do just until I'd hired myself into obsolescence.

That’s one reason why OpenAI’s Navier-Stokes result, just like all AI results that are meant to shock and impress, is not a surprise to me. This kind of technology doesn’t develop by accident or through natural progression, it takes intention, project planning, and so much time and money. I’m intimately aware of how much work was needed for even one fraction of one fraction of the technology to be possible.

How could I be wowed that a Millenium Prize problem has been solved using Lean verification for its proof in 2026, when that was on our minds for our 5-year plan back in 2021? How can I be impressed that OpenAI claims to have solved it in 88 hours when even I, someone barely involved, spent more hours than that working on the tech they're using?

How much work from how many people did it really take to get that result?

One of the things I was advocating for in 2021 was more support for the community adding to Mathlib, which already had enough momentum that I had hopes that Lean could avoid the fate of becoming just another technically advanced and interesting programming language that never gets adopted in practice. Yes we needed to hire more technical researchers for the Lean language itself, but I knew that the direct expenses on the research side would be only a fraction of the actual work and actual cost to make it a success.

To me, the real prize was the potential for a giant, human-written and computer-verified, free and public mathematics library that contained every known mathematical proof.

I wanted the library for its own sake, and for humanity, because libraries are great. But I was also aware of exactly how useful it would be for AI, especially when trying to pitch the project internally at a company that really only wanted to fund AI projects.

All that is just to explain why, when I saw OpenAI's announcement of a Navier-Stokes Lean-verified proof, I immediately knew with direct and complete surety that their result required years of work from a huge community of university mathematicians, students, and hobbyist volunteers that have been formalizing mathematics into Lean for over half a decade now.

Compared to the greater libraries of humanity, Mathlib is so small. And yet building just this library that AI uses for math took just so, so much money and support, from individuals, companies, universities, tuition from students, taxpayer funded university salaries and postdoc fellowships, years and years worth.

Then compare all the work that went into that tiny sliver of the technology to how much work it took to do all the rest of what this kind of math-capable AI tech needs to work at all.

Not only the work of mathematicians who volunteered to contribute to Mathlib, but any mathematician who has ever published anything or written correspondence where an AI company could grab it.

Not only mathematicians, but millions of professional and hobbyist programmers volunteering their valuable time to write open-source code, or to ask and answer questions on StackOverflow.

And not only mathematicians and programmers, but so many millions more of people’s livelihoods in other fields.

It’s expensive. So, so expensive.

1. The Mathlib of Babel

One thought I did not have when seeing the news was any worry that math might be “over” soon because AI will be able to solve all math problems, or that the field will have to fundamentally change and catch up because AI can replace mathematicians. The framing that AI propagandists are using for mathematicians is the same line they’ve been using about other fields for years, and it doesn't need a new response.

But a lot of the discourse within the mathematics community seems to be focused on justifying its own existence, so I'll share some general thoughts.

In a theoretical world where the way we developed AI made sense, I think a math-specific AI tool could be useful for human mathematicians to turn those handwavy proofs we sometimes have in our heads into something more concrete. That’s the kind of AI project I hoped Lean could contribute to, back in 2021.

LLMs get better at finding the correlations between informal math and formal math language as it gets more examples from people. The results coming out of the big AI companies prove not that the AI is smart but that smart people have made a LOT of examples for the AI to do AI math to.

For a cutting-edge proof, something unlike what an LLM has seen before, an LLM will struggle. But it can still work if people are willing to cut the problem into pieces and work to develop the area step by step, as happened with Navier-Stokes. Working with AI in this way is a specific expertise that it could be useful for some people to develop, in theory, if the whole industry weren't terrible.

I think growing out humanity's formal math library to include everything that's formalizable and recognized as worthwhile in today’s mathematics is only a matter of will.

I say "today's mathematics" because we haven't found the last math problem yet. And I say “recognized as worthwhile" because there's infinite proofs you could verify and publish, just like there's infinite whole numbers.

Imagine asking an AI to generate the whole numbers in order, from 0 on up. You could spend trillions of dollars having it fill up datacenter after datacenter with lists of numbers. The more we spend on the project, the more the list would start including some very big and important numbers. But what would actually be accomplished?

Consider this and then consider how it applies to every number in any dataset and everything an AI generates: with our normal human understanding of a number, the number theoretically contains all of the meaning of that number. A number intrinsically contains the fact of whether it's prime, whether it's irrational, whether it's a counterexample to the Riemann Hypothesis, all facts known and unknown. You don't need a human to interpret the number in order for the number to mean what it means. In fact, the longer a human spends interpreting, the more meaning they might find that a number intrinsically has in and of itself. And yet... can you think of a less meaningful way to fill up a datacenter than with an ordered list of numbers?

Listing all possible proofs out of context, whether in order or randomly, would hardly be more meaningful than listing numbers. Almost all possible proofs are trivial and meaningless, mathematicians select for the fun stuff.

But I will say, since it seems every player in the US government and economy have committed to all jumping off a cliff together in a stubborn wish to cover the country in datacenters, I could think of worse applications than to fill them up with every possible mathematical proof below a certain size.

You'd end up with a Library of Babel of proofs.

You can imagine the quality of literature you'd get if you trained an LLM on the Library of Babel, which contains every possible book (or combination of characters that fit in 410 pages). The library, by itself, functionally contains no information whatsoever.  The LLM trained on it would be a literal smooth-brain with no correlations to wrinkle up those gradients.

Without the context of language and how it’s used by people, a great work of literature is just another possible random combination of letters. This can be represented as a point in multidimensional space, and a computer can guess that maybe there's more things of interest to humans near where the stuff we already like is, but only humans know how to push further into unknown directions of multidimensional space that are interesting and worthwhile.

Whenever a company or person shares something “AI generated” that seems pretty good, that tells me that the AI must necessarily be using a lot of human-created data in that same area. The AI can not have leapt there on its own, even if it might look like it did to a person not aware of the scope of human-made work that went into it.

The Mathlib of Babel problem is one reason why, in my view, formal math and AI are just one tool that might be useful for some math sometimes, not the end of mathematics or the future of mathematics. Formal mathematics and algorithmically-assisted proofs are not my personal aesthetic but I think it should be somebody's, and it's not like our fragmented field doesn't know how to handle another specific expertise joining the mix. We've already folded them in.

(I love purpose-specific tools in general. When you want to build a house you use a hammer where you need a hammer, and a saw where you need a saw, rather than using the cool expensive novelty multitool you got for your birthday to do everything. For serious work where there is an actual need to accomplish something, you use the right tool for the job. The worst possible tool for serious work would be some imaginary all-encompassing omnipotent novelty multitool like super-intelligent AI is supposed to be. So expensive, so excessive for most of where it could theoretically be useful.)

I also would argue it's a fiction that proofs/formalizations contain "math" in the first place, rather than vice versa. Math existed a long time before formalization, which is relatively new on the scene as a method of organizing and verifying mathematical thought.

For most mathematicians, and certainly historically, first they have an idea or insight, and then they work on verifying that idea with a proof. The proof gets published line by line and the ideas are explained in order as if that's how the mathematician got to the conclusion, but mathematicians don't expect a step-by-step proof to represent the actual step-by-step process a mathematician used to get the result in the real world.

The actual history of how a result was found is explained in conversations and blog posts and videos, in jargon and diagrams and metaphors and sometimes wild hand gestures and whatever other ways to communicate humans can possibly find. Mathematicians talk about sleepless nights and reviewing literature and inspiration over dinner with a friend. They talk about a dream, about a thought while stepping off a bus, about being sure of something and then finding a flaw and then fixing it and then finding more flaws and fixing those.

(Sometimes they even make short entertaining videos depicting fictional processes by which someone might come upon and develop a mathematical idea...)

Computer formalizations can try to describe the larger field of mathematics that we’ve discovered through other means, but I don't believe any amount of sophistication of computer code can contain what greater mathematics actually is.

Others can and do disagree. There's mathematicians who believe in a computational universe, and mathematicians who believe deeply in formalism as a foundational truth. There's mathematicians who see the history and practice of mathematical thinking as irrelevant to the pure platonic truth of mathematics as we know it now. I'm sure there's mathematicians who would argue that a Mathlib of Babel would contain real mathematics with intrinsic truths.

We've got all kinds of mathematicians, that's part of what makes this whole human project work.

But in my view, just as formalization is a product of math and not the originator of math, all of computation is but a tiny subset of math that cannot actually know or do anything beyond that tiny scope, however good it is at storing and manipulating human-made representations of math that are interpretable by humans who know the meaning behind their own representations.

(And what we call AI, of course, is just one tiny subset of computation, so AI is really not very deep at all compared to all of mathematics. It’s just good at being very very shallow very very fast.)

I don’t know how math works or why it’s so useful. I don’t know where ideas come from, how the mind works, or what's up with meaning and consciousness.

But I do know some things.

I’ve been there for math. Seen it happen with my own eyes, felt it happen within my own brain. And from what I’ve observed there’s a lot more to it than what gets captured in a formal proof, just as there’s much more to everyone and everything than we know how to capture into a book, a photograph, a tweet, or an AI.

1. The part where leopards eat the mathematician’s faces, and then I mercilessly mock OpenAI which should shrivel up in shame forever

When I heard the news about Navier-Stokes, before seeing the Lean connection, my very first reaction was “oh, someone must have made progress on Navier-Stokes recently that I hadn’t heard about.” 

It didn’t take very long to find out that yes, as expected, there had been a bunch of human-made progress on the problem since I’d last looked into it. First a promising new approach from Diego Córdoba and Luis Martínez Zoroa, and then following up on that, very, VERY recent work by Tristan Buckmaster and Levent Alpöge who were rumored to be closing in on a result, prompting OpenAI to decide to get involved.

What I did not expect was how overtly embarrassing the situation with OpenAI was.

Many other people have covered the controversy where Buckmaster and Alpöge had designed and implemented AI-powered methods to make progress along that particular approach, allowed OpenAI to have access to all of their ongoing work through their use of OpenAI’s tools, only to find that their approach did in fact work when OpenAI announced that they (OpenAI, not the mathematicians) had used their own internal models to solve a Millenium Prize Problem, kicking off a round of AI hype about how AI was better at math than the best mathematicians.

My personal opinion is that it should be a huge embarrassment for OpenAI that they made the decision to spend millions of dollars (tens of millions?) to scoop human mathematicians on the result they’d been working toward for years the moment they heard a rumor they were very close. OpenAI didn’t scoop them by accident or through coincidental development of the technology, they purposefully poured millions of dollars worth of resources into the problem because they thought Buckmaster and Alpöge might be announcing a result any day now, and OpenAI wanted to be able to claim that they had solved it first.

This should be embarrassing enough for OpenAI even if they didn’t have access to Buckmaster and Alpöge’s ongoing work. It should be embarrassing even if OpenAI hadn’t acted shady after they got called out, offering one author credit in exchange for cutting out the other author who worked for a competitor. All that has been covered elsewhere and I don’t have any special knowledge to add, so I won’t get into it, but if you haven’t seen it it’s pretty bad.

So the first embarrassment is that OpenAI scooped their own customers, mathematicians who chose to give their work to OpenAI in the first place, with naive trust that the leopards wouldn’t eat their faces. And then those mathematicians experienced what artists, writers, and musicians have been trying to tell everyone else for years.

Of course OpenAI takes people’s work and uses it! That’s the only way their models can do literally anything! Don’t we all know by now that the industry insists it is “training and learning” rather than “copying and using” just so that when the courts refuse to hold them responsible for intellectual property theft it sounds slightly less corrupt, if you’re determined to squint at it?

So that’s pretty bad on its own.

The other embarrassment is that the point of the million-dollar prize for Navier-Stokes isn’t that we need a result for practical reasons. Scooping this particular prestigious problem with AI is entirely counterproductive to the reason it’s considered prestigious in the first place, and I’m going to explain this particular embarrassment in excruciating detail because sometimes people who do shameful things should feel shame.

If you’ve seen news coverage of the result, you may have seen pretty pictures of swirly fluid dynamics simulations with general-audience explanations of how Navier-Stokes has important things to do with fluid dynamics and engineering. It’s the norm in mathematics communications to include practical applications early on, to try to make the abstract tangible, and to above all else make sure there’s a pretty thumbnail. 

The Navier-Stokes equations are all sorts of useful. But the Millennium Prize problem is not about “Navier-Stokes” in general, it’s about a specific not-real-world feature of how these equations behave in the abstract, absent real physics. Despite that, most of the coverage I’ve seen repeats the same stuff about fluid dynamics with an implication that this may be a useful or even profound problem for OpenAI to give us an answer to on behalf of humanity.

It's not.

Mathematicians made it a Millennium Prize problem just because it hasn't been solved despite that mathematicians have tried. It's a mountain to climb because the mountain is tall, not because there's treasure on top. We were so far from knowing how to solve it that we were pretty sure it would require conceptual breakthroughs, advancing the field of mathematics. It’s those conceptual breakthroughs that are the real prize.

Maybe going to the moon is a better comparison. Getting a human there was the mission because of what that accomplishment would mean to humans. Not only that we knew it would take a lot to do it and that some of those ideas would be useful here on earth, but also because of the new experience those humans would have that would add to the greater experience of humanity. We want to know what they saw, how they felt, what we can learn from it. We want to know that someone got there, that we can achieve hard things when we embrace the human spirit and try our hardest.

Instead, this is a classic case of You Get What You Measure. A prestigious institution said it would give out a prize to get somewhere, so OpenAI leapt to do it, and is staring up with puppy-dog eyes waiting for a pat on the head and not seeming to understand what it did wrong.

Navier-Stokes is not like P vs NP, another Millennium problem that’s there because it’s hard, but where it would still be nice to know the answer for sure. Or, if as zero people expect it turns out P=NP, it would be very important to know the result. (Not that I think a frontier AI company would be likely to share with the class if that turned out to be the case.)

As far as I'm aware, Navier-Stokes isn't even like The Riemann Hypothesis, where even if an answer on its own isn't important to most of the world there's some mathematicians who have done work that assumes it. So if AI could prove The Riemann Hypothesis ethically, I can see how that would be of some use to those specific mathematicians. (If any mathematicians know of an area of Navier-Stokes research like that, maybe they can comment.)

There’s some emotions I’ve become inured to in the news, the shock and horror and disgust at everything going on in the world. But this Navier-Stokes situation with OpenAI has me absolutely shriveling up with second-hand cringe. It’s just so bad no matter how you look at it. You’ve got this hugely powerful company twisting itself into knots out of a pathetic desire to feel like its big AI did a good big boy job, needing all of media to report on what a good big boy job it did, when anyone can see they’re a toddler grabbing something that isn’t theirs and making a mess. I would never be able to show my face again.

But I shouldn’t let OpenAI’s self-own distract me from that it’s also a bummer for actual mathematical progress.

I wonder if we'll ever get an understanding of how close Buckmaster and Alpöge were, if they'd been left to continue on their own. Were their actual mathematical breakthroughs, and the game plan they gave OpenAI access to through their use of its model, the last worthwhile ideas needed? 

We know that they were at least close enough that millions of dollars of compute were able to cross the finish line by brute force, but if they’d had another couple months or years to work (comparatively inexpensively on just two people’s salary) could they have come up with a refinement to their approach, one that would need only tens of dollars in compute to cross the finish line?

Will we (and all future AI models) miss out on that innovation forever, now?

How many more actual mathematical breakthroughs will be undermined if we keep allowing what looks and feels like blatant theft, corruption, and bullying to be glossed over with complaints but no consequences, giving in to the argument of brute force both figuratively and literally?

Why aren't mathematicians madder about this? Why would anyone working on anything important work with OpenAI or its technology ever again?

4. “When I voted for the Keep-Politics-Out-of-Math party I didn’t think they would take the policies I want out of MY math”

What do you do when you realize the people in power don’t care about you, and the protections you thought you had don’t exist?

For some of us, this has been a theme of our entire career. Some of us have been yelled at for decades for “bringing politics into math” because in the establishment’s view we must play-act a world where mathematics is “pure” and apolitical. 

I like to think the field is more aware of reality now than it was back when I decided not to pursue a pure academic career, but I stay far enough away that I couldn’t say whether it’s gotten better in the past decades or whether backlash to feminism and DEI has made things even worse.

Perhaps now, with AI, some mathematicians are just experiencing for the first time that they don’t have the protections they like to imagine everyone has. Perhaps they’ll become woke to the fact that mathematics is and has always been political. It took me some difficult years in my early career to realize feminism still had a point, and I’m incredibly stubborn, so I can hope that if I came around anyone can.

The Navier-Stokes situation with OpenAI should be a leopards-eating-faces moment for the AI-users in the mathematics community. There should be boycotts, people should be deleting their accounts, there should be lawsuits. Instead I’ve mostly seen discussion framed as if it’s looking inward but giving the vibes of “fool me once, shame on you, fool me twice, still shame on you, someone should do something about that, and now that I’ve taken such a brave stand let’s go another round with the leopards.”

Thanks to my experience in the tech industry, I’m all out of patience for people who put on a front that all inconvenient decisions are simply out of their hands until someone else takes action to make those decisions convenient for them.

Researchers and mathematicians are not safe from AI now in the same way that they have never been safe from blatant disregard, disrespect, and discredit from anyone more powerful than them. It’s just that some who used to have the luxury of ignoring politics are now realizing they’re no longer on top.

So I do hope this is the beginning but not the end of introspection for mathematicians on this front. There’s other communities that have had more time to learn and digest what’s going on with AI that have lessons worth learning.

1. History judges researchers who work with the bad guys

Some folks will go the rest of their lives bitterly refusing to admit they were on the wrong side of history with AI, I know that. But I’m equally sure there’s some folks who have been sucked into the “AI is inevitable so might as well get on board” narrative now despite knowing that what these big AI companies are doing is maybe not the best, and some of those folks might wonder if they will regret it in 7 years when these companies and any prior association with them is seen as toxic. 

To them I say: learn from history and get out now. You’ll still be ahead of the curve. The current backlash against AI may seem huge, but it’s nothing compared to what I think the backlash will be in coming years.

I chose 7 years because I see certain parallels between the situation with researchers and Big AI today and the relationship between researchers and Epstein in 2019. I get that this might seem like an out-of-nowhere comparison to make, that it might only make sense to me due to my particular experience in my research career, but hey, I didn’t come here to write something that anyone else could have written.

In 2019 my research team was looking for a new home, after a couple years each at Y Combinator Research and SAP Labs. One thing on our minds when considering institutions and funders was the current scandal we had a bit of an inside view of at MIT’s CSAIL, which had been taking Epstein’s dirty money and trying to hide that fact. When the CSAIL community found out… yes, there were consequences, but results were horrifyingly mixed. The responses I found shocking then only feel more shocking with time.

Looking back from 2026 it’s obvious to everyone that the bad guy—who everyone already knew in 2019 was a bad guy—was, indeed, bad, and that no one should have touched his money or helped him launder his reputation. And yet in 2019, and for years prior, that knowledge wasn’t enough to stop many researchers from taking the money and defending whatever corner of Epstein they thought they could get away with defending, even if they’d assert they don’t defend all of him. 

Even after he was arrested again in 2019, Epstein association was not a universal or clear-cut issue the way it is now. And that included some powerful voices in the academic community who took it upon themselves to take sides against the anti-Epstein crowd.

Not that they were FOR Epstein, no, of course not! How dare one suggest such a thing! But, as neutral and objective intellectuals concerned only with the truth, they could not bring themselves to side with the people they saw as annoying social justice warriors going on witch hunts against the people trying to make REAL intellectual progress detached from politics.

Sure Epstein may have done some things that may be illegal by the unscientific standards of the law, but he served his time, so isn’t it a true intellectual’s neutral moral duty to point out that the only current moral failing is of those seeking to hinder the progress of the researchers and technologists who need that money not for themselves but to benefit the future of all humanity through world-changing innovation?

The millions poured into pro-Epstein propaganda did its work on the minds of genuinely intelligent people who thought their moral stance on the issue came from their own brains. Anything they had to gain, or had gained, either directly from Epstein or through loyal participation in the old boys’ club, HAD to be irrelevant to their moral stance, as proven by the fact that they are unbiased individuals who develop their own unbiased opinions based on pure unbiased reason.

And that’s why you don’t take the bad money, people!

Yes, individuals were embarrassed and careers cut short, but not without effort from an outspoken minority that the powers-that-be didn't want to listen to until they had to. The entire situation was tense, controversial, exhausting, and layered with anti-woke backlash that made it feel like it wasn’t really a win for justice.

I'd been invited to interview for positions at MIT a few times in the past but had declined. I only observed the Epstein blowup from afar while feeling glad I'd dodged that bullet and trying to dodge it again with my next gig. (Still forever on the CSAIL mailing lists from the old days, though. If you know you know.)

My own conversations with more powerful men about Epstein at the time felt like they were merely tolerated, full of platitudes and outright lies. I was so disheartened that I honestly didn’t think anyone but a few of us branded as SJWs would ever really care, or do more than pay lip service out of fear of HR.

At least fear of HR is better than nothing, so, in 2019 we went with Microsoft. Which was a choice.

When news about Epstein became more mainstream in the past couple years and any prior association with him became universally radioactive, I was pleasantly surprised. At a time when others were finding their faith in humanity shaken by the ongoing revelations, I found my own faith in humanity a little bit restored.

Yes, there was a time when it was depressingly easy for mathematicians, computer scientists, and all sorts of technology researchers—smart people confident in their ability to think objectively and truthfully—to go along with a bad guy's narrative, to defend and support him while convincing themselves they were not actually defending and supporting him.

And most of them eventually came around.

The point of all this is not to shame them. We save that for OpenAI, today. No, the point is that it could happen to anyone. In 2019 I was on the right side of that particular issue, but I didn't get there by being smarter or more perfect.

I know what it feels like to be painfully called out by people I had to get over myself to listen to and learn from.

I guess the comparison here is that right now the anti-AI people still are considered no more than annoying obstacles by the powers-that-be. As if they'll go away with a little bit of self-righteous rhetoric and time.

Whereas I think the backlash is only just getting started, that it’ll be a slow years-long reckoning, and that someday everyone will wonder how anyone could have looked away and allowed what’s happening to happen.

People will come around.

Or at least, the ones who survive.

1. 7 increasingly improbable ways the alleged proof could still be wrong

Now that we’ve covered the emotionally fraught stuff, let’s end with the fun stuff, which is nitpicking on the technicalities that I’d nitpick on if I were responsible for handing out a $1,000,000 mathematics prize.

My best guess right now is that OpenAI’s Navier Stokes result will hold up. I think it’s real. And most people talking about it are speaking in good faith that it’s a sure thing. 

Maybe it’s because this whole thing is already so embarrassing for OpenAI that it becomes gauche to also point out that no prize has been awarded.

Since this is unlikely to be the last time a frontier model claims a mathematics result, I might as well go through every way I can think of that such a result might fail, from the more realistic to the more out-there:

6.1 If there’s a previously undiscovered bug in the proof.

Most people, including myself, think Lean probably doesn’t have a bug that spoils this result. But it wouldn’t be the first time we were wrong.

Just earlier this year a supposed Collatz Conjecture disproof was able to be verified in Lean, despite being incorrect, revealing a bug that was quickly discovered and solved. An AI-assisted bug hunt ensued, hopefully finding and removing any other AI-exploitable loopholes.

I don’t think it’s likely that an undiscovered bug remains that could affect the Navier-Stokes result. But I’m pretty confident that vastly more compute was thrown at Navier-Stokes compared to the compute that was used for bug hunting, so personally I wouldn’t completely rule out that millions of dollars of compute was able to find a previously undiscovered exploit as the easiest path to satisfying the request.  

If they’d given the AI the same amount of compute (or more) to spend bug-finding first, I’d be more confident that however many millions they spent on Navier-Stokes proof-finding didn’t instead do bug-finding.

6.2 If the proof is a correct proof but not a proof of the intended thing.

One thing mathematicians will be checking is that the verified formalization actually is of Navier-Stokes in a way that satisfies the prize problem, rather than correctly proving something else.

To use a hypothetical example inspired by more recent AI math results:

Say you want a proof that all Riemann Hypothesis zeroes do the Riemann Hypothesis thing, and AI gives you a verified proof that 100% of Riemann Hypothesis zeroes do the Riemann Hypothesis thing.

The issue with that proof wouldn't be that the proof is wrong or the verification is wrong, but that it's not of the Riemann Hypothesis.

You can't expect even very smart people, who just happen to be the wrong kind of experts, to know that a mathematically perfect literal and complete "100%" doesn't satisfy the condition of "all" for that kind of math problem. But I can imagine how someone might misinterpret that as solving the Riemann Hypothesis, especially if they wanted to believe it and were in a rush to go public.

Navier-Stokes is far from my area of expertise, but my understanding is that Buckmaster and Apölge were making progress on an unusual approach following work by Diego Córdoba and Luis Martínez Zoroa, and which hasn't been deeply worked on by many others, which would maybe increase the chance that there's an issue with the specifics of how the question itself is asked.

But, lowering the chance of this kind of mistake is that they're human experts who have been working on this approach using computational tools for a long time. So I don't think it's something the AI hallucinated to look Navier-Stokes-like while proving something irrelevant.

6.3 If AI psychosis is affecting every single researcher close to this

If believing in this result required trusting only one single researcher who talks to AI all day, it wouldn’t be unreasonable to check whether they’re actually suffering AI psychosis and repeating AI hallucinations (or check whether they may have let a piece of software take over their communications entirely).

My understanding is that other mathematicians who work with Lean and Navier-Stokes have checked the basics of whether the supposed proof exists in the first place. We just need a single one of those people to have not submitted to AI psychosis to believe there really is a verified proof rather than hallucinated garbage.

I’m still putting it on the list for the future though, because I’ve seen how individuals start to lose their ability to use common sense checks on what they’re doing when they get in the habit of trusting AI output. It’s a slow-moving addiction that corrupts imperceptibly with each use, until they can’t imagine life without it.

I can no longer be surprised when one person who I think knows better ends up going down the AI Psychosis rabbit hole, because it seems to just keep happening.

I don’t think it happened to everyone involved all together at once, but since there’s a Millennium Prize on the line, if I’m the Clay Institute I probably have to check.

6.4 If the AI optimized the use of its resources by doing a giant psyop

If these big AI companies are allowed to continue pushing this technology without regard to existing regulations or safety concerns, why wouldn’t we expect not only individual AI psychosis to continue increasing, but that these models would find times when it’s optimal to push group-psychosis among a number of susceptible individuals? 

It would be expensive to do, so I don’t want to sound a conspiracy alarm that this sort of thing happens or will happen all the time. AI is not all-powerful. It can only use power that we pay for—and I mean that literally, because I want you to never forget the real material and electrical costs that are required for every single thing AI does, and who is actually footing that bill.

There’s just not enough silicon and electricity in the world to psyop everyone about everything all the time (or to do much else that these frontier companies are promising). 

But an AI algorithm would become more likely to spend large resources on manipulating people when even larger resources are given to it to pursue a specific goal, as with the many millions spent on Navier-Stokes, where relatively few key people would need to be manipulated. 

In the case where we accept it’s possible for an AI to find this workflow, how many millions would it realistically cost to run an effective public disinformation campaign on something as difficult to fact-check as Navier-Stokes? 

You might think not that much, but mathematicians, despite their flaws, can often be quite unyielding when it comes to specific aspects of intellectual and academic integrity.

Buckmaster refused to be manipulated by OpenAI into snubbing his coauthor and accepting joint credit with OpenAI. That was the right decision, made under pressure, with a million dollar prize and his career at stake. As a bonus, if AI were ever looking at using this kind of social manipulation, it now has one more datapoint of how difficult and costly it is in practice.

OpenAI claims they were not trying to coerce, threaten, or bribe Buckmaster, but he saw it that way, so it says something about his moral fiber that he still declined. Through this whole mess I’ve heard others point out the poetic fact that the million-dollar prize for the Poincaré conjecture (the first and still only Millennium problem accepted as solved) was refused by Grigori Perelman because he thought Richard Hamilton should have also gotten credit.

Some people are just hardcore like that. Greed isn’t the only force in the universe, and despite some disappointments, I’ve seen enough people say no to things that didn’t feel right to them that I’m more a believer in humanity now than ever. Even some of the people who were disappointments on Epstein are still people I’d trust to be hardcore in refusing coercion if it came to them through the avenue of their actual passion and expertise.

So, hardcore mathematicians and academics and researchers and all my people, keep on being you!

6.5 Exploit due to Bug-fixing and Proof-finding efforts being correlated.

I said I thought a new bug was unlikely. An additional wrinkle, which I think is even more unlikely, is that OpenAI’s models are involved in both bug-hunting and proof-finding. There’s a world where these tasks conflict, and the proof contains a known exploit that was left unreported—or even created by—the AI for its own use.

In this particular case, because of my understanding of how and when these separate tasks were run on different models, I don’t think there’s a technical reason their models would “decide” to leave themselves or future versions loopholes to exploit for higher value tasks later.

Nevertheless, there’s definitely contexts where an incautious researcher could prompt an AI to do bugfinding tasks and software usage tasks in a way that essentially trains the AI to create and exploit unwanted bugs rather than what the researcher thought they were doing.

I’m sure it's already happening all the time. And it might someday happen for a math result.

6.6 Wrong Tool for the Job

Sometimes you’re trying to figure out a math thing using a math system, and the answer turns out to be that what you thought was an all-powerful comprehensive math system in fact is not the right tool for the job. An otherwise useful tool could give nonsense or misleading answers when asked to do something outside its scope.

For a basic example, you can use the techniques of arithmetic to make what sounds like a logical argument using infinitely large and small numbers, until you realize arithmetic isn’t giving correct answers in these cases. If infinity plus 1 is infinity, and infinity plus 2 is infinity, then cancel out the infinities and we discover 1 equals 2. 

It’s not arithmetic’s fault it’s not the right tool for infinities. We just needed to invent things like calculus.

Real numbers can’t handle the square root of negative numbers, so we invented imaginary numbers and complex numbers. And it was not for lack of effort that it took thousands of years to discover non-Euclidean geometry and realize what we thought of as “all of geometry” (and the axioms that were meant to cover all of geometry) was in fact not all of geometry.

In the modern era, axioms might be listed at the beginning of the formal proofs we share, as if they come first. But they’re not actually “at the beginning” when it comes to the work of mathematics, either on the scale of history or on the scale of individual discovery.

It was the work of historical mathematicians to discover/create the very idea of axioms, and to decide to adopt them as a manifestation of math’s foundations. And it is the work of today’s mathematicians to first think in ideas, then translate them into a language that can be shared.

Math systems don’t create math or encompass math; math creates them and encompasses them. There is no guarantee that this latest cutting-edge math tool is the right tool for any particular cutting edge math problem, until people do human math work to check.

In the case of Navier-Stokes, humans developed the techniques and approach not just that the AI used, but specifically FOR AI to use.

They seem confident that the kinds of infinitesimals they’re using and the way they’re using them are within the scope of the tool, not the kind of mismatch as with doing arithmetic on infinities.

And on gut instinct, while partial differential equations aren’t my expertise, Navier-Stokes feels aligned with the specific kinds of computer algorithms that get called AI. Like, maybe this is not at all how or why it worked, but on a basic intuitive level, of all the Millennium problems you could choose, why not put the gradient-shaped peg into the gradient-shaped hole?

It just "feels" right, which guides mathematician's priorities more than some might like to admit. I'm just not likely to be skeptical here, of all the things I could spend my focus on.

For the future and for other problems, I think it’s worth some caution when using a relatively new technique on a scale never used before, on a long-standing problem where we already expect the solution might require even more new techniques and perhaps an entire paradigm shift.

If we don't cut out humans from mathematics, we will find problems that reach beyond the scope of the tech we currently call AI (perhaps we already have), and it’s up to us to figure that out and find the next step into the unknown.

6.7 Some other reason I haven’t thought of

The Millennium Prize has a 2-year minimum waiting period between an initial claim and when the prize can be awarded, and that’s for a reason.

There’s no algorithm that can check for all things that might be wrong with something. You can try, you can lower the risk, but we don't know what we don't know about something that has never been known before.

That’s why, even if I did not have the first six questions, my respect for the culture of mathematics and the rigor we expect from each other would be enough reason for me to not consider the Navier-Stokes result definitive until the mathematics community has had years to peer review it.

(Though in non-rigorous contexts I might refer to Navier-Stokes as if it’s been solved, to avoid pedantry. Because not everyone at my local bar needs me to monologue this entire essay as a caveat. This is just for you folks!)

I can feel sure a proof is absolutely correct for a list of reasons, and still be wrong. Computers can output a result and say it’s computer-perfect, and still be wrong. This happens all the time in mathematics and computer science, so there’s no reason to think we’re suddenly immune to it.

I might even be wrong about everything I’ve written here, despite the absurd amount of hours spent writing and editing it a million times, which would only be further evidence that just because a math-shaped person can confidently and boldly state something on the internet doesn’t mean it’s correct.

(And I hope you'll find me in the comments if you do have a correction or insight.)

Despite intriguing tales of lone geniuses, and despite AI-users who would like to cosplay as lone geniuses/artists/writers by pretending to themselves that algorithmically-assembled stolen work is their own creation, it’s communities of experts that grow our collective knowledge, and it’s collaboration that’s responsible for all our nice things. 

The best mathematicians I know reach across borders for the sake of a humanity-wide project that has existed long before those borders were drawn and which will continue to exist long after our current ones. We’ll build up this latest section of our understanding of differential equations, also add in our understanding of computer proofs, and perhaps come away with some open questions that don’t get solved until 4026 with the help of technology that's as far from computers now as computers are from Euclid's ruler and compass.

That is the scale of our work. 

🐌,

Vi