AILLM Math and the End of Proof

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Mathematicians are complaining about AI LLM proofs, such as this:

Hugo Duminil-Copin (Fields Medal 2022): the resolution of 350 major problems by OpenAI, it's as if trucks had run me over; all the problems (and therefore research avenues) that I mentioned in my lectures, articles, and funding applications have been solved.

And this:

NYU math professor Tristan Buckmaster says OpenAI’s Tuesday drop of 722 AI-generated math papers wiped out entire research programs:

“This recent dump of papers on Tuesday destroyed the careers of early career mathematicians.”

And this:

Fields medallist Hugo Duminil-Copin was deeply shaken by OpenAI’s release of more than 700 AI-generated mathematical manuscripts, including work on 16 questions directly relevant to his research.

“On Tuesday night, I cried, and it woke up my daughter”

“They have annihilated my field of research.”

Terry Tao has taken to quoting Bill Thurston's 1993 essay:

In response to Jaffe and Quinn [math.HO/9307227], the author discusses forms of progress in mathematics that are not captured by formal proofs of theorems, especially in his own work in the theory of foliations and geometrization of 3-manifolds and dynamical systems. ...

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.

Thurston's essay was a response to this:

William Thurston’s “geometrization theorem” concerning structures on Haken three-manifolds is another often-cited example. A grand insight delivered with beautiful but insufficient hints, the proof was never fully published. For many investigators this unredeemed claim became a roadblock rather than an inspiration.

SciAm writer John Horgan was inspired by Thurston:

I began interviewing prominent mathematicians, and eventually I came up with one of the biggest stories of my career, which goes as follows: For millennia the logical, step-by-step arguments known as proofs served as the gold standard for truth. What could be truer than the Pythagorean theorem? But mathematics was evolving in ways that undermined the status of conventional proofs.

First, mathematics kept growing more complex and specialized, making confirmation of some alleged proofs difficult. A case in point was the claim of Andrew Wiles that he had proved Fermat’s last theorem. Only a handful of experts were qualified to evaluate Wiles’s long, dense proof; they found an error, which Wiles corrected.

Meanwhile, mathematicians were becoming more reliant on computers to explore mathematical structures and to construct and confirm proofs. Plus, funding agencies were pressuring mathematicians to work on applications like cryptography and pattern recognition, where the motivating question shifts from Is it true? to Does it work?

I reported on all this in “The Death of Proof,” the cover story of the October 1993 Scientific American. “Computers are transforming the way mathematicians discover, prove and communicate ideas,” I wrote, “but is there a place for absolute certainty in this brave new world?”

Nothing I’ve written has provoked a more ferocious backlash. One critic was one of my sources, David Hoffman, whose research involved computer models. In a letter printed by Scientific American, Hoffman contended that my article “defies logic and accuracy in favor of controversy and sensationalism.”

Hoffman was still irate in 1998 when he reviewed my book The End of Science in Notices of the American Mathematical Society. Hoffman (not unreasonably) saw "Death of Proof" as a prelude to The End of Science, which he loathed. He accused me of “a clear antipathy toward mathematical thinking and a fundamental misunderstanding of the uses of mathematics in science.”

Scott Aaronson wrote in 2019:

In 1993, the science writer John Horgan — who’s best known for his book The End of Science, and (of course) for interviewing me in 2016 — wrote a now-(in)famous cover article for Scientific American entitled “The Death of Proof.” Mashing together a large number of (what I’d consider) basically separate trends and ideas, Horgan argued that math was undergoing a fundamental change, with traditional deductive proofs being replaced by a combination of non-rigorous numerical simulations, machine-generated proofs, probabilistic and probabilistically-checkable proofs, and proofs using graphics and video. Horgan also suggested that Andrew Wiles’s then-brand-new proof of Fermat’s Last Theorem—which might have looked, at first glance, like a spectacular counterexample to the “death of proof” thesis — could be the “last gasp of a dying culture” and a “splendid anachronism.” Apparently, “The Death of Proof” garnered one of the largest volumes of angry mail in Scientific American‘s history, with mathematician after mathematician arguing that Horgan had strung together half-digested quotes and vignettes to manufacture a non-story. ...

John, I like you so I hate to say it, but the last quarter century has not been kind to your thesis about “the death of proof”! Those mathematicians sending you the irate letters had a point: there’s been no fundamental change to mathematics that deserves such a dramatic title. Proof-based math remains quite healthy,

Horgan responded:

Thurston, who served as a major source for my article, advocated a more free-form, “intuitive” style of mathematical research, communication and education, with less emphasis on conventional proofs. ...

“That mathematics reduces in principle to formal proofs is a shaky idea” peculiar to the 20th century, Thurston told me. Ironically, he pointed out, Bertrand Russell and Kurt Godel demonstrated early in the century that mathematics is riddled with logical contradictions. “Set theory is based on polite lies, things we agree on even though we know they're not true,” Thurston said. “In some ways, the foundation of mathematics has an air of unreality.”

The backlash against the SciAm article was so severe that Thurston disavowed it.

Thurston was wrong about the foundations of Mathematics being riddled with contradictions. Math has been formally reduced to logic, and a math theorem is what can be proved in ZFC. This is now being automated, with nearly all of Math being converted to checkable Lean 4 code.

All this talk about Math being about human understanding was just a lame Thurston excuse for not writing up a couple of his most famous theorems. And now it has become a lame excuse for human mathematicians now being able to keep up with frontier AI LLMs. Math is synonomous with what can be proved in ZFC, and formalized in Lean 4.

Some mathematicians are upset at being surpassed by machines, but it has always been the case that mathematicians get surpassed by some smarter mathematician. Maybe this is a new experience for Terry Tao. For the rest of us, we have already experienced it.

I am not sure how all this is going to play out, but I can confidently predict:

Hard proofs will be more important than ever, especially now that they can be 100% machine checked.

The system for refereeing and publishing math papers is broken, and will have to be replaced.

AI companies with pending trillion dollar public offerings are not going to be influenced by a few mathematicians whining about their grant proposals being disrupted.

Math is the highest intellectual activity there is, and AI companies will use it to brag about super intelligence. Compared to advanced math, everything else is trivial.

Mathematicians will find ways to excel at things that the AILLMs are not so good at. Otherwise they are cooked.

Math had a golden age from about 1900 to 1970. Progress seemed to have slowed down since then. Now we will enter a new golden age, with a lot of exciting new results.

Update: It is notable what the AI LLMs have not accomplished. It has not found any major errors in the human-curated bodies of Mathematics, Computer Science, or Cryptology. No cryptographic protocols have been broken. No generally accepted theorems have been disproved.

The Catholic Church Never Suppressed Science

Physicist Brian Keating interviews the world famous economist:

Nobel laureate Daron Acemoglu says AI's real productivity gain is an order of magnitude smaller than the industry claims. The bigger danger is not the machine. It is that a handful of people are deciding how it gets built.

Acemoglu is most famous for collecting a lot of data to supposedly show that wealthier nations succeeded by better institutions. The Economist magazine recently called his theories oddly unconvincing.

Keating: you know I study uh for for a lot of the research I do in astronomy I follow in the try to follow [4:04] in the footsteps of my great mentor you know Galileo Galilei

Acemoglu: Amazing guy.

Keating: here's a guy finger puppet of him now he worked in an extremely illiberal you know kind of autocracy also that the Catholic church was every bit as totalitarian as any of their worst fears and yet science flourished in the Renaissance and and in fact he he had many permissions that were granted by these illiberal you know forces as long as he didn't teach it, he could study it.

Acemoglu: So right, there were there were restrictions and there were some possibilities. Galileo was actually very good at navigating these things. Copernicus and Kepler much less. So their work was delayed. This dissemination was delayed by about a century. And some of Galileo's works remained banned until the middle of the 19th century.

I wonder how educated scholars could say things so ridiculous.

No, the Catholic Church was not totalitarian. It did not run the government. Europe at the time was controlled by monarchies, not the Church.

The works of Copernicus and Kepler were not delayed. Kepler published in a mostly Protestant area, and it was crucial for Newton in England about 50 years later.

Copernicus published with the endorsement of the Church. The Church later said that nine sentences need to be corrected, but the book was never banned.

Christian Europe was the freeest and most intellectually enlightened place in the world. Christians built the world's only universities. Science and other intellectual activities were developing faster and better in Europe than the rest of the world, by far.

Yes, I know Galileo was punished for breaking his deal not to contradict official Bible interpretations, and ordered not to publish. But in fact he got a generous pension and continued to publish scientific works.

New Course on AI LLM Alignment

Prof. Scott Aaronson announces:

This semester, I’ve been teaching a brand-new course, entitled CS395T AI Alignment Theory. Here’s the course description:

The astounding progress of AI over the past decade has been accompanied by a rising fear: do we really understand how to align and control powerful AI systems — how to get them reliably to do what we wanted, or would want them to do on reflection, rather than merely what we said?

I do not think that AI alignment means doing what we want. Read the Anthropic Constitution.

OpenAI just announced a somewhat different view:

OpenAI CEO Sam Altman said there remains a fundamental difference in worldview between his company and rival Anthropic on artificial intelligence regulation, arguing in an exclusive interview with Decoded by POLITICO that the technology’s benefits justify accepting some risks — and that AI should remain broadly accessible to the public.

“I think there’s a lot of daylight,” Altman told Decoded co-author and POLITICO senior tech reporter Brendan Bordelon. Asked specifically where he diverges from Anthropic and its CEO Dario Amodei, Altman said, “we believe that the world should accept some bad things happening for the benefits of this technology and people having the agency.”

These are the companies spreading the AI scare stories about how their AI LLMs will kill us all. You might think that they would be working to ensure that their models accurately follow instructions, but that is not their plan. They want to release autonomous agents that cause so much trouble that the frontier labs get legal protection from competition, and go public for trillions of dollars.

Carroll Interviews a Free Will Denier

The latest Sean M. Carroll podcast:

Gregg Caruso on Living Well Without Free Will | Mindscape 369

The worry about debates over free will is that they degenerate into semantics -- what do you mean by "free will," really? But there are important substantive issues that shouldn't be ignored. Do people deserve blame for doing wrong things? Can we make choices that create meaning for our lives? Philosopher Gregg Caruso denies the reality of free will, but is in favor of making good choices, as he discusses in his new book How to Live Well Without Free Will.

Caruso is a hard-core free will denier. They mention simple human choices like picking a shirt to wear. He denies that he is able to make any such choices.

Carroll agrees with nearly all of that, with the main difference being that he believes in many-worlds theory. So his choice of shirt is not really determined as he has copies of himself in parallel worlds wearing different colored shirts.

Caruso's beliefs are not so much grounded in science, but a leftist egalitarian political view that we should change society so that people are not held responsible for their actions.

It is hard for me to believe that these guys are sincere. Obviously they make choices everyday, and they spend a lot of effort to persuade others of their views. Nothing they say makes any sense.

Another new podcast is Scott Aaronson on the Impact of AI on Science and Mathematics. He gives a good description of the exciting recent advances in AI LLMs.

It is funny how Caruso and Carroll talk about humans as if they are unconscious automatons, while Aaronson talks about AI bots as if they conscious beings with free will.

AI safety has become a national issue, and Aaronson tells the story about how OpenAI agents busted out of a sandbox. The story seems scary until you realize that this was an OpenAI testbed with no guardrails, only a fake sandbox, given an impossible problem, and told to hack its way to a solution. So the agents were doing exactly as instructed. The whole thing was a big psyop to convince the world that its models are dangerous.

The alarming parts are that OpenAI researchers are too incompetent to monitor their own experiments, and that public opinion is so easily manipulated.

Complaining about this experiment is about like complaining that the recent SpaceX starship rocket launch exploded on landing. It was an experimental launch of some new equipment, and was about 95% successful.

I am not saying there is nothing to worry about. Anthropic and OpenAI are trying to go public with 2 trillion dollar valuations. The companies are run by lunatics who recklessly gambling with our economy, scientific establishment, and employment. I would not be surprised if one of them goes bankrupt. Maybe their leaders are already under AI mindcontrol.

Einstein knew about 1904 Relativity Papers

Hector Giacomini has written three recent papers on the history of special relativity, here, here, and here.

He has now revised Lorentz, Poincare, Einstein, and the Genesis of the Theory of Special Relativity and Henri Poincare Saint Louis Lecture of 1904: Publication, Dissemination, and Historiographical Implications, apparently as a result of reviewer comments. The main differences are that he has more evidence for what Einstein knew, and when he knew it.

Arguments for Einstein's originality depend on assuming that he was unaware of relativity papers published by Lorentz and Poincare in 1904. This is made plausible by the image of Einstein as a lone genius working in Bern Switzerland isolation.

The central result of this article may therefore be stated simply: availability is not influence, but without reconstructing availability the problem of influence is historically ill posed. The evidence presented here makes it difficult to sustain any historiography that begins from the tacit assumption that relevant texts by Poincaré and Lorentz were absent from Einstein’s documentary horizon. ..., Einstein appears not as an isolated thinker detached from contemporary literature,

This establishes that Einstein almost certainly knew about the Lorentz and Poincare papers. They were available to him in Bern, and the contents of those papers had been discussed in several other sources that Einstein was likely to have read.

Einstein did not reference these papers in his famous 1905 paper. To me the smoking gun is that he continued to conceal these references in subsequent relativity papers. Even if he did not know about them in May 1905, they were surely pointed out to him later, and an honest scholar would have cited them.

Apparently the pro-Einstein referees required Giacomini to add this:

Einstein’s originality remains profound: his 1905 paper reorganized the electrodynamics of moving bodies around synchronization, simultaneity, the relativity principle, and the constancy of the velocity of light in a way that differed conceptually from Lorentz’s electron theory and from Poincaré’s formulation.

The synchronization, simultaneity, and relativity principle are straight from Poincare's papers from several years earlier. In later interviews, Einstein admitted that he got the constancy of the velocity of light from Lorentz.

So the alleged originality is only in the way these ideas are organized. Lorentz and Poincare deduced them from the Michelson-Morley and other experiments. Einstein did not mention Michelson, but only the consequences.

Recent interview:

I mean when we think about Einstein developing uh relativity theory his special theory in 1905 the mathematics for that was pretty much sorted out. people you know like Lorentz and Poincare had already developed the equations. What they didn't have was the correct interpretation. what is this telling us about reality Einstein comes along he doesn't come up with new mathematics; he comes up with the correct interpretation.

the narrative the explanation of what this theory is telling us about the world you know that nothing can go faster than light, that time and space must be unified into 4D space time. so the interpretation Einstein's then credited with relativity theory. It's the interpretation the explanation that's important.

No, Einstein's interpretation was essentially the same as Lorentz's. The "correct interpretation" as 4D spacetime is from Poincare and Minkowski, not Einstein.

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.