Math is Cooked

Mathematicians are in a panic, over the announced solution to a Clay Millennium Problem, and the Lean 4 verification of the proof of Fermat's Last Theorem.

Tweet:

Mathematician Terence Tao: "we have to slow down AI. the pace is insane, and there's no reason to be this fast — no reason at all"

Friend of mine, math professor from one of the top five depts in the US [and Tao's coauthor]:

"I want to go away, to my little country house, and stay there. My profession has basically been destroyed by AI."

The actual Navier-Stokes equation may or may not always have solutions. The Openai counterexample assumed a forcing term.

It is no longer clear how new mathematicians will be trained, or how mathematicians will work, produce, and be credited.

Physicists are not laughing much. Theoretical Physics has been cooked for about 40 years.

There is now an orchestrated public relations and lobbying push by the leading AI companies to get special protections in federal law. The problem is that their trillion-dollar AILLM businesses are contingent on not have cheap competition from cheap Chinese and open-weight models. So they want laws declaring the cheap models to be too dangerous.

Even mathematicians have been suckered into saying the progress is too fast. The main issues are rogue agents on the internet, and recursive self-improvement. Probably half of internet traffic now is AI agents, and the big AI companies are using their models to program their next LLM releases. These are the main ingredients to what the doomsayers say will destroy humanity.

Tao is getting a lot of heat online, but he was an AI enthusiast until it outdid him on Navier-Stokes with an obscure proof.

Update: SciAm reports:

Generated by an internal large language model (LLM), OpenAI’s proof relies on an approach that many experts find unnatural. It solves a variant of the problem that mathematicians say is disconnected from reality and thus less interesting. In a sense, the LLM found and exploited a loophole in the framing of the question.

“The most important problem is unsolved,” says Luis Silvestre, a mathematician at the University of Chicago. “The Clay problem is settled, but the main problem for the Navier-Stokes equations is not.”

Furthermore, last Thursday, three mathematicians posted a proof of their own that showed that OpenAI’s method can never be extended to solve the full problem. In other words, the loophole will never be closed, barring some completely new idea.

Openai says that it has more math results to announce, and it will consult with a group of mathematicians about its release.

While the mathematicians are probably happy to be consulted, I am eager to see the results. The benefits are slowing release are not clear.

Tao's new motto is this quote:

"We are not trying to meet some abstract production quota of definitions, theorems and proofs. The measure of our success is whether what we do enables people to understand and think more clearly and effectively about mathematics." —Bill Thurston, 1994.

The quote fueled a notorious SciAm article on "The Death of Proof".

Electromagnetism Derived From Geometry

General relativity teaches that gravity is a manifestation of geometry. Not everyone knows that electromagnetism and the other fundamental forces (strong, weak) are similarly geometric.

Historically this dates back to Weyl and Kaluza-Klein, a century ago. But those theories were flawed, so a lot of physicists think that the geometry did not work.

Here is a new video that explains it well: Electromagnetism Derived From Geometry!.

In short, you add a circle to each point of spacetime, connect the circles infinitesimally to nearby circles, and then the electric and magnetic fields become the curvature.

The video tries to relate this idea to string theory, but that is speculation that did not pan out. The idea is completely essential to the Standard Model of physics, and our best explanation of the fundamental forces.

Carroll interviews the Famous Geography Professor

The latest Sean M. Carroll interview is Jared Diamond. Recent tweet about his famous book:

“Guns, Germs & Steel” was one of only 3 books in my entire life that I couldn’t finish. It’s that bad. Here’s the synopsis:

In the opening chapter he clearly states That he believes the 5’ tall Stone Age Pygmies of Papua New Guinea are more intelligent than Europeans. His rationale? He hung out with them and felt like it because they knew all the types of plants, and also their kids didn’t have TVs (I’m not kidding).

He also observed that unlike Europeans, they were constantly hacking each other to death with crude instruments. Which leads him to believe that they evolved to be smarter, because you need intelligence to survive permanent tribal warfare. Unlike cooperative farming. I really wish I was kidding about this.

He concludes therefore that any and all modern differences in technology and wealth are the result strictly of environmental factors. He spends the next 400 pages desperately cooking up insanely convoluted theories about why living in a food scarce frozen wasteland actually gives people far more free time to invent things than eating an inexhaustible supply of bananas on a tropical island would.

The book is full of contrived theories about ancient people where he has no evidence. In some cases, people have no written records, yet he somehow pretends to know everything about them.

He tells several stories about the 20th century. Stories that can easily be checked with Google. And they were gross distortions. If he cannot get 20th century facts right, how likely is it that he got facts from millennia ago?

I thought the book would be interesting because of its many awards, and because the question of explaining European success of the last millennium is an important one. But he says it was all luck, and tests do not back up his more specific theories.

Here is a more revealing interview of Diamond, this week. It all seems like garbage to me, but listen or read for yourself.

Update: Skeptic interview.

Project to Formalize All Math

About a century, Mathematics was put on firm foundations. Logically it was formalized into ZFC set theory. All theorems could, in principle, be logically derived from the axioms.

Textbook math was also made rigorous by a series of books by Bourbaki.

Now the dream of logicism is being fully realized. Jared Duker Lichtman writes:

The MAP: Mathematics Autoformalization Project

We are poised to translate all known math into formal code.

This is the math-equivalent of the Human Genome Project, or AlphaFold in modern times.

On September 4th, Anthropic shocked the math world by formalizing Fermat’s Last Theorem.

Soon the great body of math theorems will be formalized into computer-checkable proofs. No longer will there be any doubts about whether referees have adequately checked the proofs.

The proof-checking software has had some glitches, but they will soon be ironed out. Possibly some currently-accepted theorems will have to be rejected.

In the recently announced AI proof of a complex structure on the 6-sphere S⁶, the argument used a construction that a previously published paper supposedly proved was impossible. The AI found a flaw in that paper.

The math community has an amazingly good record of only publishing theorems that are true. It is way better than 99%. Soon it will be 100%. This is a level of precision unmatched in human history.

Not that this is the end of math. There is still plenty for humans to do. Humans need to make the works comprehensible, and to set directions for new research.

Now there are rumors of a Millennium Prize problem being solved, namely a counterexample to Navier-Stokes. A fluid flow that becomes singular in finite time. See this video. Apparently OpenAI, Anthropic, and some human mathematicians are competing for the credit, and the $1 million payoff.

If an AI LLM solved Navier-Stokes, this would be the biggest Math accomplishment yet. If the proof was mostly stolen from humans, the significance will be harder to assess.

Terry Tao, an expert on both Navier-Stokes and AI LLMs, discusses the progress. He says that finding a full counter-example looks feasible in the near future.

Update: Here is the OpenAI announcement. It complains that an NYU professor and an Anthropic employee did not agree to a joint announcement.

The journal Nature announces:

For the first time, a truly major open problem in mathematics has been solved by computer, according to OpenAI, which says it has cracked one of the ‘Millennium Problems’ related to the motion of fluids.

The artificial-intelligence (AI) company in San Francisco, California, announced on 8 September that it has generated a solution to the most commonly used physical model of fluids — the Navier-Stokes equations — showing that they can break down. Whether or not this could happen was one of the seven ‘Millennium Problems’ asked by the Clay Mathematics Institute at the turn of the twenty-first century. Solutions come with a prize of US$1 million....

They then decided to focus their resources on the Navier-Stokes problem on 1 September, after hearing rumours that two mathematicians, Levent Alpöge of Harvard University in Cambridge, Massachusetts, and Tristan Buckmaster of New York University, had solved some version of the fluid-motion puzzle using Anthropic AI’s models.

On 7 September, Alpöge and Buckmaster released a paper in which they say they had found a solution for the fluid equations that also achieved infinite speed, but in the simplified case in which the fluid has no viscosity.

So the AI LLM did not really solve this problem. It finished a work-in-progress by some human mathematicians.

AI LLMs have solved some major problems, but it is not so clear who deserves the credit in this case. It will be interesting to see who gets the $1 million.

Update: OpenAI says it is not claiming the million. It might have spent $50 million in computer time on it.

The counterexample is to Navier-Stokes with a forcing term. It is still not known whether the straight Navier-Stokes equation (with no forcing term) always has a solution.

Update: I should add that Professor Terry Tao is very pro-AI, and rarely gives any negative opinions. So it is significant that he criticized OpenAI's role in this matter.

Update: Rumors on big progress toward two other math millennium prize problems, Hodge and Birch-Swinnerton-Dyer.

The case against Galileo

Galileo is considered one of the great heroes of modern science. Supposedly he discovered the acceleration of gravity, the telescope, the moons of Jupiter, the relativity of motion, and heliocentrism.

Most of all, he is praised for standing up to the Pope in Rome.

Supposedly also, he discovered the experimental method. Before him, everyone blindly accepted what was written, if it came from Aristotle or the Bible. He was the first to have the courage to challenge them, and put ideas to an empirical test.

He is also sometimes credited with mathematizing science. He would deduce physical principles from pure thought, logic, math, and thought experiments. This belief is in tension with the belief that he was a great empiricist.

Wikipedia says:

Galileo has been called the father of observational astronomy,[9] modern-era classical physics,[10] the scientific method,[11] and modern science.[12]

Yes, Galileo really gets this gushing praise from historians and philosophers. It is all absurd.

I just discovered this 2020 podcast:

Galileo is the most overrated figure in the history of science. That is the thesis of Season 1 of my podcast, which consists of the following 18 episodes. ...

Galileo is praised for his work on falling bodies, but his arguments were dishonest and his trifling discoveries were not new. ...

Galileo committed scores of errors in his physics. These are bad in themselves and also undermine Galileo’s claim to credit for the things he did get right.

Galileo dismissed the notion that the moon influences the tides as “childish” and “occult.” Instead he argued that tides are a kind of sloshing due to the motion of the earth. This very poor theory is inconsistent with several of his own scientific principles.

Two thousand years before Galileo, Greek astronomers argued that the heavenly bodies revolve around the sun. Their reasoning involved sophisticated mathematics and sound physical considerations.

Galileo is credited with defeating Ptolemaic earth-centered astronomy, but most mathematical astronomers had already abandoned this theory long before Galileo.

The podcast thoroughly debunks the idea that Galileo was a great scientist. He got a lot of stuff wrong.

In particular, his main argument for heliocentrism was that the motion of the Earth caused the ocean tides. His theory was that there was one high tide per day. In fact there are two high tides per day, and it had been known for centuries that the Moon caused the tides. Also the Sun, if it is aligned.

Couldn't he just walk down to the beach and see for himself? Actually that is not so simple, as the Mediterranean is mostly an enclosed sea, and the tides are small and confusing.

Galileo is credited with an assortment of other things, and it is nearly all bogus. He did describe the moons of Jupiter, but only one day ahead of a German. He supposedly discovered Newton's first law of motion, that an object goes with constant velocity, in the absence of forces. But he only believed this for horizontal motion.

He used inclined planes to measure the acceleration of gravity, but his answer was off by a factor of two.

Listening to this podcast makes me wonder why Galileo was revered so much. Do people think his existence disproves Catholicism? I am not sure.

The craziest thing is not the overcrediting of Galileo, but the way the historians and philosophers argue that no one before Galileo tested ideas empirically, no one tried to observe the world, and no one tried to mathematize nature. Actually, these things had been done for millennia. The ancients created Astronomy models, built ships and pyramids, invented Geometry, and a lot of other great things. Of course they used observation, math, theory, and science.

The story reminds me about how lots of people credit Christopher Columbus for discovering the Earth being round. That is stupid on two levels. First, the roundness of the Earth was well-known to scholars for two millennia. Second, the claim does not even make any sense, as Columbus did not go around the Earth, but merely sailed to some islands and back.

Likewise, scholars have been doing science and mathematizing nature for millennia. And Galileo was not even a very good scientist, compared to his contemporaries. The work by Tycho and Kepler was 100x better than anything Galileo did. It was vastly superior in its empiricism and its mathematization, the two things that cause historians to idolize Galileo.

Hofstadter’s Gödel Escher Bach Fooled Us

Scott Aaronson writes:

A central thesis that many readers, including me, took from Douglas Hofstadter’s Gödel Escher Bach when young was that the secret of intelligence (and therefore, of AI) was going to have a lot to do with self-referentiality and “strange loops.” ...

Now, in 2026, we’ve succeeded at building AIs that outperform most humans at most intellectual tasks that are well-defined enough to judge. And at no point in the tech stack of those AIs — neither in the transformer neural nets, nor in the GPU clusters they run on, nor in the training process, nor anywhere else — did anyone need to build in anything about self-reference. ...

But the idea that you’d need explicit self-referentiality before you could get convincing and world-changing conversational intelligence? Let it be buried in a Westminster Abbey or Arlington National Cemetery for the most important wrong ideas in human history — geocentrism, Aristotle’s teleological physics, aether, phlogiston, Freud’s psychology, Marx’s prediction of a workers’ uprising followed by a classless utopia, etc. But buried it needs to be.

He could make a similar comment about fractals. Textbooks give examples using self-reference, and then say that the weather is also an example. But self-reference is not used to model the weather.

There is a distinction between the mathematical tool used for proof/pedagogy and the actual mechanisms that generate the phenomena in the real world.

Diagonalization can be used to construct irrational or transcendental numbers, but most transcendental numbers are not obtained or studied this way.

(I tried to post this comment on his blog.)

Quantum Nonlocality Explained FROM SCRATCH

A month ago, Curt Jaimungal interviewed Tim Maudlin on Can Quantum Mechanics Be Explained FROM SCRATCH?. Now, Part II just dropped, for another 3+ hours.

Maudlin is an expert on this subject, and he discusses in detail the work of von Neumann, Einstein, Bohm, Bell, and others saying that the concepts of locality, classical hidden variables, and quantum mechanics predictions are in conflict. One must be rejected.

What is not so clear is how Maudlin differs from textbook Copenhagen interpretation. Copenhagen says that the classical hidden variables are to be rejected. That is what von Neumann proved, under some assumptions. Then we have locality and predictions of quantum mechanics, with no problem.

Maudlin has some philosphical reasons for subscribing to classical hidden variables, so he ends up rejecting locality.

To me, no one has ever observed spooky action at a distance, so local theories are very much preferred. Bohm's pilot wave theory is crazy, because the theory is nonlocal, but all the observations magically turn out to be local.

Maudlin explains that all the great physicists rejected quantum nonlocality.

Tim: But they never straightforwardly acknowledged the nonlocality. And if you don’t believe that, 1:58:07 go read Bohr’s response to the EPR paper and see whether he straightforwardly says, “Yes, I believe in nonlocality. I believe in spooky action at a distance.” He denies it. ...

So I think Einstein saw correctly that if you accept this spooky action 1:59:25 at a distance, relativity itself is at risk. And he didn’t wanna put, you know, relativity at risk without good reason, and he couldn’t see any reason to accept the spooky action at a distance.

Now, he may not have liked it for other reasons. I mean, we, Newton didn’t like it, right? He says, he says, “No rational person would believe in it.” I mean, there are lots of people who find that just intuitively ugly or unacceptable or whatever. Curt: Mm-hmm.

Tim: And, and, and, and to this day, there are people who don’t wanna accept nonlocality, and they’ll more or less clutch at any straw to try to avoid it, as we’ll see. The psychology of that I can’t quite explain.

I can explain it. The entire scientific enterprise for millennia has been to figure out cause and effect, by tracing events to nearby actions. If that is all wrong, and our lives are controlled by the spooky astrology of far-away stars, then we need to reconsider all the known laws of science.

In fact, no such nonlocality has ever been discovered, and we can continue with what has worked for centuries.

Bell and Maudlin claim that all the great geniuses and textbooks are wrong. They have a belief in what a quantum theory ought to be, and they say that quantum mechanics, as understood by Bohr and von Neumann, does not qualify.

I agree with Newton. No rational person would believe in spooky action at a distance. Unless it is proved, of course.