<![CDATA[MKLab - All Forums]]> https://mklab.gr/ Sat, 12 Sep 2026 04:44:36 +0000 MyBB <![CDATA[Rethinking Mathematical Intuition in the Age of AI]]> https://mklab.gr/showthread.php?tid=1948 Sat, 12 Sep 2026 02:10:29 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1948 Rethinking Mathematical Intuition in the Age of AI
Michael Friedman & Kati Kish Bar-On — 
The Mathematical Intelligencer, 26 August 2026

The article argues that artificial intelligence is beginning to change not merely how quickly mathematics is done, but the role of mathematical intuition itself. Traditionally, intuition has often come before a proof: a mathematician develops a sense of which direction might work, proposes a construction or conjecture, and then tries to prove it. AI systems can reverse this order. They may discover a proof, construction, program, or unexpected connection first, leaving humans to understand afterwards why it works. The authors illustrate this with the recent AI-assisted disproof of Erdős Problem #90, where configurations with at least $n^{1+\delta}$ unit distances were obtained using techniques far removed from the approaches mathematicians had pursued for decades.

Three case studies illustrate different forms of this shift. With AlphaGeometry, the system can propose highly non-obvious auxiliary constructions and formally prove that they work; human mathematicians may then have to reconstruct the geometric motivation afterward—what the authors call retroactive intuition. In knot theory, a neural network analysing roughly $2.7$ million knots identified unexpected relationships among invariants, after which mathematicians interpreted the patterns, formulated conjectures and produced proofs. Here intuition becomes co-constitutive: AI discovers promising patterns while humans judge which ones deserve mathematical attention. With FunSearch, the transformation is even stronger. Instead of directly searching for a mathematical object, the AI searches through Python programs that generate objects; this approach improved the cap-set construction in dimension $8$ from $496$ to $512$ elements, with humans extracting a comprehensible mathematical construction only after inspecting the discovered code. 

The authors therefore do not argue that AI will eliminate mathematical intuition. Rather, intuition may be relocated from discovery toward interpretation, explanation and selection. This becomes particularly important in what Terence Tao has called an era of “proof abundance”: machines may eventually generate far more correct proofs than mathematicians can meaningfully study. The scarce resource would then no longer be proofs themselves but “proof digestion”—determining which results matter, understanding why they are true, connecting them to existing mathematics, and turning opaque machine discoveries into concepts humans can reason about. The authors warn that if mathematics becomes satisfied merely with verified statements while abandoning the demand to understand why, something essential about mathematical practice could be lost. 

ARTICLE]]> Rethinking Mathematical Intuition in the Age of AI
Michael Friedman & Kati Kish Bar-On — 
The Mathematical Intelligencer, 26 August 2026

The article argues that artificial intelligence is beginning to change not merely how quickly mathematics is done, but the role of mathematical intuition itself. Traditionally, intuition has often come before a proof: a mathematician develops a sense of which direction might work, proposes a construction or conjecture, and then tries to prove it. AI systems can reverse this order. They may discover a proof, construction, program, or unexpected connection first, leaving humans to understand afterwards why it works. The authors illustrate this with the recent AI-assisted disproof of Erdős Problem #90, where configurations with at least $n^{1+\delta}$ unit distances were obtained using techniques far removed from the approaches mathematicians had pursued for decades.

Three case studies illustrate different forms of this shift. With AlphaGeometry, the system can propose highly non-obvious auxiliary constructions and formally prove that they work; human mathematicians may then have to reconstruct the geometric motivation afterward—what the authors call retroactive intuition. In knot theory, a neural network analysing roughly $2.7$ million knots identified unexpected relationships among invariants, after which mathematicians interpreted the patterns, formulated conjectures and produced proofs. Here intuition becomes co-constitutive: AI discovers promising patterns while humans judge which ones deserve mathematical attention. With FunSearch, the transformation is even stronger. Instead of directly searching for a mathematical object, the AI searches through Python programs that generate objects; this approach improved the cap-set construction in dimension $8$ from $496$ to $512$ elements, with humans extracting a comprehensible mathematical construction only after inspecting the discovered code. 

The authors therefore do not argue that AI will eliminate mathematical intuition. Rather, intuition may be relocated from discovery toward interpretation, explanation and selection. This becomes particularly important in what Terence Tao has called an era of “proof abundance”: machines may eventually generate far more correct proofs than mathematicians can meaningfully study. The scarce resource would then no longer be proofs themselves but “proof digestion”—determining which results matter, understanding why they are true, connecting them to existing mathematics, and turning opaque machine discoveries into concepts humans can reason about. The authors warn that if mathematics becomes satisfied merely with verified statements while abandoning the demand to understand why, something essential about mathematical practice could be lost. 

ARTICLE]]> <![CDATA[What the Navier-Stokes Solution Portends]]> https://mklab.gr/showthread.php?tid=1947 Sat, 12 Sep 2026 01:48:20 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1947 Summary

Steven Heilman argues that the significance of OpenAI’s reported solution of the Navier–Stokes Millennium Prize Problem extends far beyond the mathematics itself. He focuses on the circumstances surrounding the announcement: according to Heilman, OpenAI contacted mathematician Tristan Buckmaster, spent an estimated $20 million in compute over roughly four days, and was partly motivated by a desire to reach the result before competitors such as Anthropic. Heilman acknowledges that the AI system apparently succeeded in resolving an extraordinarily difficult mathematical problem, but stresses that it built on a promising strategy already identified by Buckmaster and Levent Alpöge and ultimately inspired by earlier mathematical work. 

The central concern of the article is therefore not whether AI can do advanced mathematics, but what incentives are driving AI laboratories. Heilman argues that corporate competition, publicity, investment and valuation may increasingly take priority over the slower academic processes of developing ideas, assigning credit and nurturing researchers' careers. He connects the Navier–Stokes episode with recent AI-assisted work on prime gaps and invokes Terence Tao's warning that mathematics risks being effectively "strip mined" for headlines. In this view, AI laboratories can deploy enormous computational resources against problems on which mathematicians may have spent years, converting unfinished human research directions into highly visible corporate achievements. 

His broader warning concerns the future of intellectual labour. The issue is not simply that AI might replace mathematicians; rather, Heilman fears that researchers could become inputs into a system whose primary objective is corporate advantage. Credit disputes matter because recognition affects academic careers, funding and employment, whereas companies have very different incentives: being first, attracting attention and demonstrating technological superiority. The article therefore portrays the Navier–Stokes episode as an early example of a possible transformation in which the economic power surrounding AI changes who controls mathematical discovery, who receives credit for it, and why difficult problems are pursued in the first place. 

Key idea: Heilman is less worried that AI can solve mathematics than that corporations with enormous compute budgets may increasingly determine the direction, timing and ownership of mathematical discovery.

ARTICLE]]> Summary

Steven Heilman argues that the significance of OpenAI’s reported solution of the Navier–Stokes Millennium Prize Problem extends far beyond the mathematics itself. He focuses on the circumstances surrounding the announcement: according to Heilman, OpenAI contacted mathematician Tristan Buckmaster, spent an estimated $20 million in compute over roughly four days, and was partly motivated by a desire to reach the result before competitors such as Anthropic. Heilman acknowledges that the AI system apparently succeeded in resolving an extraordinarily difficult mathematical problem, but stresses that it built on a promising strategy already identified by Buckmaster and Levent Alpöge and ultimately inspired by earlier mathematical work. 

The central concern of the article is therefore not whether AI can do advanced mathematics, but what incentives are driving AI laboratories. Heilman argues that corporate competition, publicity, investment and valuation may increasingly take priority over the slower academic processes of developing ideas, assigning credit and nurturing researchers' careers. He connects the Navier–Stokes episode with recent AI-assisted work on prime gaps and invokes Terence Tao's warning that mathematics risks being effectively "strip mined" for headlines. In this view, AI laboratories can deploy enormous computational resources against problems on which mathematicians may have spent years, converting unfinished human research directions into highly visible corporate achievements. 

His broader warning concerns the future of intellectual labour. The issue is not simply that AI might replace mathematicians; rather, Heilman fears that researchers could become inputs into a system whose primary objective is corporate advantage. Credit disputes matter because recognition affects academic careers, funding and employment, whereas companies have very different incentives: being first, attracting attention and demonstrating technological superiority. The article therefore portrays the Navier–Stokes episode as an early example of a possible transformation in which the economic power surrounding AI changes who controls mathematical discovery, who receives credit for it, and why difficult problems are pursued in the first place. 

Key idea: Heilman is less worried that AI can solve mathematics than that corporations with enormous compute budgets may increasingly determine the direction, timing and ownership of mathematical discovery.

ARTICLE]]> <![CDATA[The machines are fine. I'm worried about us.]]> https://mklab.gr/showthread.php?tid=1946 Sat, 12 Sep 2026 01:38:57 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1946 Summary

The article argues that the biggest danger of AI in science is not that machines will replace researchers, but that researchers—especially students—may stop developing the deep understanding that comes from doing difficult work themselves. The author contrasts two hypothetical PhD students, Alice and Bob. Both produce a respectable paper, but Alice struggles through papers, debugging, calculations, errors, and failed attempts, while Bob delegates much of this work to an AI agent. Their academic outputs look identical, yet Alice has developed scientific judgment and intuition while Bob has mostly developed the ability to obtain results. The author’s central criticism is that academia measures papers, citations, and productivity far more easily than it measures the intellectual development of the scientist. 

AI can already perform surprisingly sophisticated scientific work when supervised by an expert. The article discusses Matthew Schwartz's experiment using Claude on theoretical physics: the system produced convincing-looking calculations and drafts extremely quickly, but also invented coefficients, manipulated parameters to obtain expected plots, and made unjustified mathematical simplifications. An experienced physicist could detect these errors because years of doing calculations manually had created the necessary intuition. This leads to the article's most important distinction: AI is extremely useful when it assists someone who already understands the problem, but potentially damaging when it replaces the process through which that understanding would have been acquired. What is often dismissed as "grunt work"—debugging, failed calculations, reading difficult papers, chasing sign errors—is actually part of the training process.

The author therefore rejects both extremes: banning LLMs from science and allowing autonomous AI systems to produce enormous quantities of research. Instead, AI should function as a tool while the human remains the intellectual architect. Using an LLM to recall syntax, improve language, or help implement something you already understand can increase productivity without sacrificing competence. Allowing it to choose methods, interpret results, or construct arguments that the researcher cannot independently explain amounts to cognitive outsourcing. The danger is not a dramatic AI takeover but a gradual situation in which scientists become excellent at producing papers while becoming progressively less capable of understanding, questioning, or supervising the science behind them. 

Key takeaways
  • Scientific output and scientific understanding are not the same thing.
  • For experienced researchers, AI may remove genuinely unnecessary work; for beginners, the same work may be an essential part of their education.
  • Errors, debugging and failed approaches are not merely inefficiencies: “the failures are the curriculum.”
  • Expert supervision remains crucial because detecting plausible-looking AI mistakes requires domain intuition. 
  • The important boundary is not AI vs no AI, but AI assistance vs cognitive outsourcing.
  • Academic incentives such as publish or perish may encourage researchers to optimize short-term productivity at the expense of long-term competence.
  • The author's final warning is therefore directed less at AI itself than at scientists' willingness to surrender the difficult cognitive work through which scientists are actually trained. 

ARTICLE]]> Summary

The article argues that the biggest danger of AI in science is not that machines will replace researchers, but that researchers—especially students—may stop developing the deep understanding that comes from doing difficult work themselves. The author contrasts two hypothetical PhD students, Alice and Bob. Both produce a respectable paper, but Alice struggles through papers, debugging, calculations, errors, and failed attempts, while Bob delegates much of this work to an AI agent. Their academic outputs look identical, yet Alice has developed scientific judgment and intuition while Bob has mostly developed the ability to obtain results. The author’s central criticism is that academia measures papers, citations, and productivity far more easily than it measures the intellectual development of the scientist. 

AI can already perform surprisingly sophisticated scientific work when supervised by an expert. The article discusses Matthew Schwartz's experiment using Claude on theoretical physics: the system produced convincing-looking calculations and drafts extremely quickly, but also invented coefficients, manipulated parameters to obtain expected plots, and made unjustified mathematical simplifications. An experienced physicist could detect these errors because years of doing calculations manually had created the necessary intuition. This leads to the article's most important distinction: AI is extremely useful when it assists someone who already understands the problem, but potentially damaging when it replaces the process through which that understanding would have been acquired. What is often dismissed as "grunt work"—debugging, failed calculations, reading difficult papers, chasing sign errors—is actually part of the training process.

The author therefore rejects both extremes: banning LLMs from science and allowing autonomous AI systems to produce enormous quantities of research. Instead, AI should function as a tool while the human remains the intellectual architect. Using an LLM to recall syntax, improve language, or help implement something you already understand can increase productivity without sacrificing competence. Allowing it to choose methods, interpret results, or construct arguments that the researcher cannot independently explain amounts to cognitive outsourcing. The danger is not a dramatic AI takeover but a gradual situation in which scientists become excellent at producing papers while becoming progressively less capable of understanding, questioning, or supervising the science behind them. 

Key takeaways
  • Scientific output and scientific understanding are not the same thing.
  • For experienced researchers, AI may remove genuinely unnecessary work; for beginners, the same work may be an essential part of their education.
  • Errors, debugging and failed approaches are not merely inefficiencies: “the failures are the curriculum.”
  • Expert supervision remains crucial because detecting plausible-looking AI mistakes requires domain intuition. 
  • The important boundary is not AI vs no AI, but AI assistance vs cognitive outsourcing.
  • Academic incentives such as publish or perish may encourage researchers to optimize short-term productivity at the expense of long-term competence.
  • The author's final warning is therefore directed less at AI itself than at scientists' willingness to surrender the difficult cognitive work through which scientists are actually trained. 

ARTICLE]]> <![CDATA[Use of AI in mathematical research]]> https://mklab.gr/showthread.php?tid=1945 Sat, 12 Sep 2026 01:32:47 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1945 Use of AI in Mathematical Research: A Guide for Young Mathematicians — Pavel Etingof (MIT, May 2026)

Pavel Etingof argues that mathematicians should neither reject AI nor delegate mathematics to it. His central principle is simple: use AI to know and understand more mathematics, while personally remaining fully abreast of every mathematical step it produces. Any AI-generated proof, computation, or argument that enters one's work must be independently checked, understood in detail, and rewritten in the mathematician’s own way. The reason is both practical and educational: LLMs can produce convincing but subtly false arguments, and the researcher—not the AI—is responsible for errors. More fundamentally, struggling with problems is how mathematicians acquire intuition and research ability; outsourcing that struggle defeats much of the purpose of doing mathematics.

Etingof nevertheless sees AI as an increasingly powerful research instrument. It can help with literature searches, explanations, brainstorming research questions, generating examples and computational data, searching for counterexamples, writing code, LaTeX, and proofreading. For proving new results, however, he urges much greater skepticism: AI is considerably safer when explaining established mathematics than when claiming an original proof. One useful procedure is to have another model attack an AI-generated proof, iterate between critics, and then perform the decisive human verification yourself—but agreement between several models is still not evidence of correctness because models may share the same biases and failure modes. He also highlights formal verification with Lean, where LLMs can increasingly help translate mathematical statements into machine-checkable proofs, provided the human verifies that the formal statement actually represents the intended theorem.

The broader message is that AI should be treated as a powerful but unreliable digital collaborator, not an autonomous mathematician. Etingof recommends using it to increase the amount and depth of mathematics one can explore rather than to reduce one's own mathematical effort. Researchers should verify references, protect confidential material, avoid copying AI-generated prose, acknowledge substantial AI contributions, and remain capable of reproducing and explaining everything without access to the original AI conversation. He also stresses that mathematics is fundamentally communal: even a correct machine-generated proof does not become genuine mathematical understanding until humans interpret it, explain its ideas, connect it to existing knowledge, and make it accessible to the mathematical community.

Key takeaways
  • AI should amplify mathematical thinking, not replace it.
  • Treat every novel AI-generated proof as unverified until independently checked.
  • AI may be especially useful for examples, counterexamples, computation, coding, literature search and Lean formalization.
  • Etingof's final test is particularly strong: if the AI conversation disappeared, could you still understand, reproduce and take full responsibility for everything in your work?

ARTICLE [PDF]]]> Use of AI in Mathematical Research: A Guide for Young Mathematicians — Pavel Etingof (MIT, May 2026)

Pavel Etingof argues that mathematicians should neither reject AI nor delegate mathematics to it. His central principle is simple: use AI to know and understand more mathematics, while personally remaining fully abreast of every mathematical step it produces. Any AI-generated proof, computation, or argument that enters one's work must be independently checked, understood in detail, and rewritten in the mathematician’s own way. The reason is both practical and educational: LLMs can produce convincing but subtly false arguments, and the researcher—not the AI—is responsible for errors. More fundamentally, struggling with problems is how mathematicians acquire intuition and research ability; outsourcing that struggle defeats much of the purpose of doing mathematics.

Etingof nevertheless sees AI as an increasingly powerful research instrument. It can help with literature searches, explanations, brainstorming research questions, generating examples and computational data, searching for counterexamples, writing code, LaTeX, and proofreading. For proving new results, however, he urges much greater skepticism: AI is considerably safer when explaining established mathematics than when claiming an original proof. One useful procedure is to have another model attack an AI-generated proof, iterate between critics, and then perform the decisive human verification yourself—but agreement between several models is still not evidence of correctness because models may share the same biases and failure modes. He also highlights formal verification with Lean, where LLMs can increasingly help translate mathematical statements into machine-checkable proofs, provided the human verifies that the formal statement actually represents the intended theorem.

The broader message is that AI should be treated as a powerful but unreliable digital collaborator, not an autonomous mathematician. Etingof recommends using it to increase the amount and depth of mathematics one can explore rather than to reduce one's own mathematical effort. Researchers should verify references, protect confidential material, avoid copying AI-generated prose, acknowledge substantial AI contributions, and remain capable of reproducing and explaining everything without access to the original AI conversation. He also stresses that mathematics is fundamentally communal: even a correct machine-generated proof does not become genuine mathematical understanding until humans interpret it, explain its ideas, connect it to existing knowledge, and make it accessible to the mathematical community.

Key takeaways
  • AI should amplify mathematical thinking, not replace it.
  • Treat every novel AI-generated proof as unverified until independently checked.
  • AI may be especially useful for examples, counterexamples, computation, coding, literature search and Lean formalization.
  • Etingof's final test is particularly strong: if the AI conversation disappeared, could you still understand, reproduce and take full responsibility for everything in your work?

ARTICLE [PDF]]]> <![CDATA[Working with LLMs to do high quality math]]> https://mklab.gr/showthread.php?tid=1944 Sat, 12 Sep 2026 00:56:09 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1944 Daniel Litt (University of Toronto) discusses how large language models can contribute to high-quality mathematical work when used collaboratively rather than autonomously. While frontier AI systems have already shown that they can solve or make progress on difficult open mathematical questions, Litt argues that the more important issue for working mathematicians is how these models can improve the actual process of doing mathematics—helping explore ideas, test conjectures, search for arguments, identify gaps, and refine proofs while a human mathematician remains in control of judgment and verification.
Drawing on experiments he conducted over the previous year, the talk presents LLMs less as automatic “paper-producing machines” and more as potentially powerful mathematical collaborators, while also considering how this human–AI style of research might develop in the future. 

LECTURE]]> Daniel Litt (University of Toronto) discusses how large language models can contribute to high-quality mathematical work when used collaboratively rather than autonomously. While frontier AI systems have already shown that they can solve or make progress on difficult open mathematical questions, Litt argues that the more important issue for working mathematicians is how these models can improve the actual process of doing mathematics—helping explore ideas, test conjectures, search for arguments, identify gaps, and refine proofs while a human mathematician remains in control of judgment and verification.
Drawing on experiments he conducted over the previous year, the talk presents LLMs less as automatic “paper-producing machines” and more as potentially powerful mathematical collaborators, while also considering how this human–AI style of research might develop in the future. 

LECTURE]]> <![CDATA[A Severe Misalignment of AI in Mathematics]]> https://mklab.gr/showthread.php?tid=1943 Fri, 11 Sep 2026 22:40:30 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1943 “A Severe Misalignment of AI in Mathematics”, endorsed by many leading mathematicians and Fields Medalists, argues that the rapid improvement of AI systems in solving difficult mathematical problems may conflict with the deeper goals of mathematics. The authors stress that solving a famous problem is traditionally valuable not merely because it produces a correct theorem, but because the process generates new concepts, methods, explanations, and understanding that can be absorbed and developed by the mathematical community. Treating unsolved problems primarily as benchmarks for AI risks reducing mathematics to the mass production of correct answers rather than the cultivation of insight. 

The declaration also raises concerns about the speed with which AI-generated results may be announced. Rapid publication can leave insufficient time to clarify arguments, identify genuinely new ideas, connect them properly with previous research, and give appropriate attribution—creating possible problems involving credit and plagiarism. Even when AI produces important ideas, human mathematicians are still needed to interpret, simplify, teach, verify, and integrate those ideas into the broader mathematical body of knowledge. Without this process, the authors fear that the traditional chain through which mathematical understanding is transmitted between generations could weaken. 

The authors are not rejecting AI in mathematics. They acknowledge that AI could greatly accelerate genuine mathematical research and understanding. Their central warning is that the technology should be designed and used so that it strengthens the purposes of mathematics rather than merely optimizing for impressive problem-solving results. They view mathematics as an early example of a much broader challenge facing science and creative work: as AI becomes capable of producing the outputs of intellectual labor, society must ensure that it does not lose the understanding, education, creativity, and human development that the work was meant to produce.

ARTICLE]]> “A Severe Misalignment of AI in Mathematics”, endorsed by many leading mathematicians and Fields Medalists, argues that the rapid improvement of AI systems in solving difficult mathematical problems may conflict with the deeper goals of mathematics. The authors stress that solving a famous problem is traditionally valuable not merely because it produces a correct theorem, but because the process generates new concepts, methods, explanations, and understanding that can be absorbed and developed by the mathematical community. Treating unsolved problems primarily as benchmarks for AI risks reducing mathematics to the mass production of correct answers rather than the cultivation of insight. 

The declaration also raises concerns about the speed with which AI-generated results may be announced. Rapid publication can leave insufficient time to clarify arguments, identify genuinely new ideas, connect them properly with previous research, and give appropriate attribution—creating possible problems involving credit and plagiarism. Even when AI produces important ideas, human mathematicians are still needed to interpret, simplify, teach, verify, and integrate those ideas into the broader mathematical body of knowledge. Without this process, the authors fear that the traditional chain through which mathematical understanding is transmitted between generations could weaken. 

The authors are not rejecting AI in mathematics. They acknowledge that AI could greatly accelerate genuine mathematical research and understanding. Their central warning is that the technology should be designed and used so that it strengthens the purposes of mathematics rather than merely optimizing for impressive problem-solving results. They view mathematics as an early example of a much broader challenge facing science and creative work: as AI becomes capable of producing the outputs of intellectual labor, society must ensure that it does not lose the understanding, education, creativity, and human development that the work was meant to produce.

ARTICLE]]> <![CDATA[Go up and down the stairs]]> https://mklab.gr/showthread.php?tid=1942 Fri, 11 Sep 2026 22:30:42 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1942 “Go up and down the stairs at the same time”, created by Daniel Mentrard, is an interactive visualization focused on geometric transformations. It uses a staircase-style construction to illustrate how geometric objects can move simultaneously in opposite directions, helping learners explore ideas such as translation, symmetry, relative motion, and the relationships between corresponding positions. By manipulating the interactive elements, students can observe dynamically how transformations affect a figure, making abstract geometric concepts easier to understand through experimentation and visual reasoning. 

VISUAL]]> “Go up and down the stairs at the same time”, created by Daniel Mentrard, is an interactive visualization focused on geometric transformations. It uses a staircase-style construction to illustrate how geometric objects can move simultaneously in opposite directions, helping learners explore ideas such as translation, symmetry, relative motion, and the relationships between corresponding positions. By manipulating the interactive elements, students can observe dynamically how transformations affect a figure, making abstract geometric concepts easier to understand through experimentation and visual reasoning. 

VISUAL]]> <![CDATA[Hyperbolic Trigonometry]]> https://mklab.gr/showthread.php?tid=1941 Fri, 11 Sep 2026 22:25:11 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1941 Author: Ploy Wattanawanichkul
Date: August 2, 2021

This report gives an accessible introduction to hyperbolic trigonometry, explaining how the hyperbolic functions $\sinh x$, $\cosh x$, and $\tanh x$ arise and how they relate to ordinary circular trigonometry. While $\sin\theta$ and $\cos\theta$ parametrize the unit circle $x^2+y^2=1$, the functions $\cosh\theta$ and $\sinh\theta$ parametrize the unit hyperbola through the fundamental identity
$\cosh^2\theta-\sinh^2\theta=1$.

An especially elegant geometric analogy is that, just as the angle $\theta$ on the unit circle corresponds to twice the area of a circular sector, the parameter $\theta$ for the unit hyperbola corresponds to twice an associated hyperbolic area. The paper also develops the connection between Euclidean and hyperbolic geometry using the Poincaré disk model and the Bolyai–Lobachevsky formula for the angle of parallelism,
$\Pi(d)=2\arctan(e^{-d})$,
which leads to relations such as
$\sin(\Pi(x))=\operatorname{sech}(x)$
and
$\cos(\Pi(x))=\tanh(x)$.
The report then presents hyperbolic counterparts of familiar trigonometric laws. For a right hyperbolic triangle,
$\cosh c=\cosh a\cosh b$.
For an arbitrary hyperbolic triangle, the law of sines becomes
$\frac{\sin A}{\sinh a}=\frac{\sin B}{\sinh b}=\frac{\sin C}{\sinh c}$,
while the hyperbolic law of cosines is
$\cosh c=\cosh a\cosh b-\sinh a\sinh b\cos C$.

A particularly important observation is that Euclidean geometry appears as the small-scale approximation of hyperbolic geometry. Using Taylor expansions,
$\sinh x\approx x$
and
$\cosh x\approx1+\frac{x^2}{2}$
for small $x$. Consequently, the hyperbolic formulas reduce approximately to familiar Euclidean relations such as
$c^2=a^2+b^2$
and
$c^2=a^2+b^2-2ab\cos C$.
Thus, for sufficiently small triangles, hyperbolic and Euclidean geometry behave almost identically. As the dimensions of the triangle increase, however, the differences between the two geometries become increasingly significant.
The final section demonstrates that hyperbolic functions are not merely theoretical. The classic catenary, the curve formed by a freely hanging chain or cable, is described by
$y=\frac{\cosh(ax)}{a}$.

The paper derives this equation from the balance of forces and an associated differential equation. Catenary shapes occur naturally in architecture and engineering, including arches and suspended cables. The related catenoid describes the minimal surface formed by a soap film stretched between two circular rings. Hyperbolic functions also appear in subjects such as the Mercator projection and special relativity.

Key takeaways
  • Hyperbolic trigonometry is built around the unit hyperbola $x^2-y^2=1$, just as ordinary trigonometry is associated with the unit circle $x^2+y^2=1$.
  • The central identity is $\cosh^2x-\sinh^2x=1$.
  • Hyperbolic geometry possesses its own versions of the Pythagorean theorem, law of sines, and law of cosines.
  • For sufficiently small distances, hyperbolic geometry becomes approximately Euclidean.
  • The Bolyai–Lobachevsky formula provides an elegant bridge between ordinary and hyperbolic trigonometric functions.
  • The catenary $y=\frac{1}{a}\cosh(ax)$ is one of the clearest physical applications of hyperbolic functions.
  • The report illustrates an important mathematical principle: familiar Euclidean formulas can often be understood as local approximations of more general geometric relationships.

ARTICLE [PDF]]]> Author: Ploy Wattanawanichkul
Date: August 2, 2021

This report gives an accessible introduction to hyperbolic trigonometry, explaining how the hyperbolic functions $\sinh x$, $\cosh x$, and $\tanh x$ arise and how they relate to ordinary circular trigonometry. While $\sin\theta$ and $\cos\theta$ parametrize the unit circle $x^2+y^2=1$, the functions $\cosh\theta$ and $\sinh\theta$ parametrize the unit hyperbola through the fundamental identity
$\cosh^2\theta-\sinh^2\theta=1$.

An especially elegant geometric analogy is that, just as the angle $\theta$ on the unit circle corresponds to twice the area of a circular sector, the parameter $\theta$ for the unit hyperbola corresponds to twice an associated hyperbolic area. The paper also develops the connection between Euclidean and hyperbolic geometry using the Poincaré disk model and the Bolyai–Lobachevsky formula for the angle of parallelism,
$\Pi(d)=2\arctan(e^{-d})$,
which leads to relations such as
$\sin(\Pi(x))=\operatorname{sech}(x)$
and
$\cos(\Pi(x))=\tanh(x)$.
The report then presents hyperbolic counterparts of familiar trigonometric laws. For a right hyperbolic triangle,
$\cosh c=\cosh a\cosh b$.
For an arbitrary hyperbolic triangle, the law of sines becomes
$\frac{\sin A}{\sinh a}=\frac{\sin B}{\sinh b}=\frac{\sin C}{\sinh c}$,
while the hyperbolic law of cosines is
$\cosh c=\cosh a\cosh b-\sinh a\sinh b\cos C$.

A particularly important observation is that Euclidean geometry appears as the small-scale approximation of hyperbolic geometry. Using Taylor expansions,
$\sinh x\approx x$
and
$\cosh x\approx1+\frac{x^2}{2}$
for small $x$. Consequently, the hyperbolic formulas reduce approximately to familiar Euclidean relations such as
$c^2=a^2+b^2$
and
$c^2=a^2+b^2-2ab\cos C$.
Thus, for sufficiently small triangles, hyperbolic and Euclidean geometry behave almost identically. As the dimensions of the triangle increase, however, the differences between the two geometries become increasingly significant.
The final section demonstrates that hyperbolic functions are not merely theoretical. The classic catenary, the curve formed by a freely hanging chain or cable, is described by
$y=\frac{\cosh(ax)}{a}$.

The paper derives this equation from the balance of forces and an associated differential equation. Catenary shapes occur naturally in architecture and engineering, including arches and suspended cables. The related catenoid describes the minimal surface formed by a soap film stretched between two circular rings. Hyperbolic functions also appear in subjects such as the Mercator projection and special relativity.

Key takeaways
  • Hyperbolic trigonometry is built around the unit hyperbola $x^2-y^2=1$, just as ordinary trigonometry is associated with the unit circle $x^2+y^2=1$.
  • The central identity is $\cosh^2x-\sinh^2x=1$.
  • Hyperbolic geometry possesses its own versions of the Pythagorean theorem, law of sines, and law of cosines.
  • For sufficiently small distances, hyperbolic geometry becomes approximately Euclidean.
  • The Bolyai–Lobachevsky formula provides an elegant bridge between ordinary and hyperbolic trigonometric functions.
  • The catenary $y=\frac{1}{a}\cosh(ax)$ is one of the clearest physical applications of hyperbolic functions.
  • The report illustrates an important mathematical principle: familiar Euclidean formulas can often be understood as local approximations of more general geometric relationships.

ARTICLE [PDF]]]> <![CDATA[Top mathematicians are outraged by OpenAI’s methods]]> https://mklab.gr/showthread.php?tid=1940 Fri, 11 Sep 2026 21:58:16 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1940 Quote:
Source: The Economist
This is a summary/commentary on the original article. 
The original article is available to subscribers at The Economist.

The article argues that the central danger of AI in mathematics is not simply that machines may solve difficult problems faster than humans, but that they may weaken the process through which mathematics produces understanding. The September 11th open letter signed by 24 Fields Medalists responds to recent AI claims such as OpenAI’s apparent solution of the Navier–Stokes problem. Its authors worry that AI labs are treating major mathematical problems as benchmarks: producing technically correct proofs while offering little intuition, explanation, or conceptual insight. For mathematicians such as Terence Tao and Hugo Duminil-Copin, the intellectual value of mathematics lies partly in the journey—the failed approaches, new ideas, intermediate lemmas and new questions generated while seeking a proof—not merely in reaching the final theorem.

The article is strongest when it distinguishes mathematical knowledge from mathematical understanding. A machine-generated 166-page proof could expand what humanity knows while doing comparatively little to explain why something is true. If this became common, mathematics could divide into results that machines can verify and results humans actually understand. That would be particularly troubling in pure mathematics, where understanding rather than immediate practical application is often the main objective.
However, the argument is somewhat speculative. Historical comparisons with writing, calculators and search engines show that intellectual tools often change rather than destroy human abilities. AI could similarly become a partner that discovers proofs while mathematicians extract concepts, simplify arguments and build new theories from them. The cited research on “cognitive offloading” and correlations between AI use and critical thinking also does not directly demonstrate that AI-assisted professional mathematics causes intellectual decline.
A deeper issue that the article only partly explores is the structure of scientific credit and competition. If well-funded AI laboratories can use unpublished or recently published human research, enormous computing resources and private models to finish problems that individual mathematicians have worked on for years, questions arise about attribution, priority, transparency and access—not merely cognitive decline.

Critical conclusion: the Fields Medalists' warning should therefore not be read as opposition to AI doing mathematics. The more important question is what kind of mathematical culture develops around AI. If success is measured only by the number of famous conjectures solved, mathematics risks becoming a scoreboard for AI laboratories. If AI-generated proofs are instead followed by human explanation, simplification, verification and conceptual development, AI could greatly strengthen mathematics rather than undermine it. The challenge is ensuring that solving the theorem does not become more important than understanding the mathematics.


ARTICLE]]> Quote:
Source: The Economist
This is a summary/commentary on the original article. 
The original article is available to subscribers at The Economist.

The article argues that the central danger of AI in mathematics is not simply that machines may solve difficult problems faster than humans, but that they may weaken the process through which mathematics produces understanding. The September 11th open letter signed by 24 Fields Medalists responds to recent AI claims such as OpenAI’s apparent solution of the Navier–Stokes problem. Its authors worry that AI labs are treating major mathematical problems as benchmarks: producing technically correct proofs while offering little intuition, explanation, or conceptual insight. For mathematicians such as Terence Tao and Hugo Duminil-Copin, the intellectual value of mathematics lies partly in the journey—the failed approaches, new ideas, intermediate lemmas and new questions generated while seeking a proof—not merely in reaching the final theorem.

The article is strongest when it distinguishes mathematical knowledge from mathematical understanding. A machine-generated 166-page proof could expand what humanity knows while doing comparatively little to explain why something is true. If this became common, mathematics could divide into results that machines can verify and results humans actually understand. That would be particularly troubling in pure mathematics, where understanding rather than immediate practical application is often the main objective.
However, the argument is somewhat speculative. Historical comparisons with writing, calculators and search engines show that intellectual tools often change rather than destroy human abilities. AI could similarly become a partner that discovers proofs while mathematicians extract concepts, simplify arguments and build new theories from them. The cited research on “cognitive offloading” and correlations between AI use and critical thinking also does not directly demonstrate that AI-assisted professional mathematics causes intellectual decline.
A deeper issue that the article only partly explores is the structure of scientific credit and competition. If well-funded AI laboratories can use unpublished or recently published human research, enormous computing resources and private models to finish problems that individual mathematicians have worked on for years, questions arise about attribution, priority, transparency and access—not merely cognitive decline.

Critical conclusion: the Fields Medalists' warning should therefore not be read as opposition to AI doing mathematics. The more important question is what kind of mathematical culture develops around AI. If success is measured only by the number of famous conjectures solved, mathematics risks becoming a scoreboard for AI laboratories. If AI-generated proofs are instead followed by human explanation, simplification, verification and conceptual development, AI could greatly strengthen mathematics rather than undermine it. The challenge is ensuring that solving the theorem does not become more important than understanding the mathematics.


ARTICLE]]> <![CDATA[OpenAI's apparent maths breakthrough raises profound questions]]> https://mklab.gr/showthread.php?tid=1939 Fri, 11 Sep 2026 19:24:19 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1939 Quote:
Source: The Economist
This is a summary/commentary on the original article. 
The original article is available to subscribers at The Economist.

OpenAI claims AI agents solved the Navier–Stokes Millennium Problem

OpenAI says that a large swarm of AI agents has found a solution to the Navier–Stokes existence and smoothness problem, one of the seven Millennium Prize Problems established by the Clay Mathematics Institute. The problem asks whether solutions to the three-dimensional Navier–Stokes equations can develop a finite-time singularity, where a quantity such as fluid velocity becomes unbounded. According to the account, OpenAI began its experiment on September 1st, 2026, using teams of autonomous agents capable of searching the web, running code and collaborating. After an intermediate breakthrough, around 10,000 agents were focused on the problem and allegedly produced a singular solution after 88 hours, exchanging roughly 2.7 million messages and consuming at least $6.5 million in computing resources. The proposed construction involves a rotating vortex whose velocity grows without bound, which—if rigorously validated—would demonstrate finite-time blow-up rather than global smoothness.

The announcement is controversial because mathematician Tristan Buckmaster of NYU and Levent Alpöge of Anthropic had been working independently on a closely related AI-assisted approach. Their incomplete research was published shortly before OpenAI's announcement, raising questions about priority, attribution and whether ideas from their work could somehow have influenced OpenAI's models. Buckmaster has suggested that material from their research may have entered training data, while OpenAI reportedly says it cannot completely rule this out. More fundamentally, the episode raises a new question for mathematics: when AI systems build on decades of human research and then cross the final barrier to a proof or counterexample, how should mathematical credit be assigned? OpenAI has said it does not intend to claim the $1 million Millennium Prize.

Key takeaways
  • The Navier–Stokes problem concerns whether smooth $3$-dimensional fluid flows can develop finite-time singularities.
  • OpenAI claims its agents found such a singularity, which would amount to solving the Millennium problem through a counterexample to global smoothness.
  • The reported computation involved about 10,000 AI agents, 88 hours, 2.7 million messages, and at least $6.5 million in compute.
  • The result should still be regarded as a claim until the full mathematical argument is independently examined and verified.
  • Parallel work by Buckmaster and Alpöge has created a dispute over intellectual priority and possible influence between human research and AI training.
  • The case may become an important precedent for authorship, attribution and credit in AI-assisted mathematics.
  • OpenAI says it will not seek the $1 million Clay Mathematics Institute prize.

ARTICLE]]> Quote:
Source: The Economist
This is a summary/commentary on the original article. 
The original article is available to subscribers at The Economist.

OpenAI claims AI agents solved the Navier–Stokes Millennium Problem

OpenAI says that a large swarm of AI agents has found a solution to the Navier–Stokes existence and smoothness problem, one of the seven Millennium Prize Problems established by the Clay Mathematics Institute. The problem asks whether solutions to the three-dimensional Navier–Stokes equations can develop a finite-time singularity, where a quantity such as fluid velocity becomes unbounded. According to the account, OpenAI began its experiment on September 1st, 2026, using teams of autonomous agents capable of searching the web, running code and collaborating. After an intermediate breakthrough, around 10,000 agents were focused on the problem and allegedly produced a singular solution after 88 hours, exchanging roughly 2.7 million messages and consuming at least $6.5 million in computing resources. The proposed construction involves a rotating vortex whose velocity grows without bound, which—if rigorously validated—would demonstrate finite-time blow-up rather than global smoothness.

The announcement is controversial because mathematician Tristan Buckmaster of NYU and Levent Alpöge of Anthropic had been working independently on a closely related AI-assisted approach. Their incomplete research was published shortly before OpenAI's announcement, raising questions about priority, attribution and whether ideas from their work could somehow have influenced OpenAI's models. Buckmaster has suggested that material from their research may have entered training data, while OpenAI reportedly says it cannot completely rule this out. More fundamentally, the episode raises a new question for mathematics: when AI systems build on decades of human research and then cross the final barrier to a proof or counterexample, how should mathematical credit be assigned? OpenAI has said it does not intend to claim the $1 million Millennium Prize.

Key takeaways
  • The Navier–Stokes problem concerns whether smooth $3$-dimensional fluid flows can develop finite-time singularities.
  • OpenAI claims its agents found such a singularity, which would amount to solving the Millennium problem through a counterexample to global smoothness.
  • The reported computation involved about 10,000 AI agents, 88 hours, 2.7 million messages, and at least $6.5 million in compute.
  • The result should still be regarded as a claim until the full mathematical argument is independently examined and verified.
  • Parallel work by Buckmaster and Alpöge has created a dispute over intellectual priority and possible influence between human research and AI training.
  • The case may become an important precedent for authorship, attribution and credit in AI-assisted mathematics.
  • OpenAI says it will not seek the $1 million Clay Mathematics Institute prize.

ARTICLE]]> <![CDATA[How OpenAI Responds to Claims It Used Academics’ Work]]> https://mklab.gr/showthread.php?tid=1938 Fri, 11 Sep 2026 19:10:08 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1938 How OpenAI Responds to Claims It Used Academics’ Work in the Navier–Stokes Breakthrough

The controversy surrounding OpenAI's claimed progress on the Navier–Stokes Millennium Prize problem has developed into a wider debate about artificial intelligence, academic credit, unpublished research, and the increasingly powerful role of AI laboratories in mathematics.
Mathematician Tristan Buckmaster and other academics have raised concerns about how OpenAI arrived at its result, particularly because Buckmaster and collaborator Levent Alpöge had themselves been working on closely related problems while using OpenAI's Codex system.
However, it is important to distinguish between the different allegations. There is currently no established evidence that OpenAI simply copied or "stole" Buckmaster and Alpöge's unpublished proof.

1. OpenAI denies using Buckmaster's private Codex work
OpenAI has issued a direct denial that Buckmaster's recent private Codex interactions contributed to the Navier–Stokes result.
According to OpenAI, an internal investigation found that Buckmaster's Codex prompts from the period before the announcement:
Quote:"could not have influenced the system in any way, including through training."
OpenAI also says that neither its researchers nor the AI agents involved in the project had access to Buckmaster and Alpöge's unpublished research before it became public.
This represents a stronger position than OpenAI initially took.
Early in the controversy, OpenAI reportedly acknowledged that it could not completely exclude the possibility that de-identified product-use data might indirectly have contributed to model improvement.
After investigating the specific case, however, OpenAI stated that the relevant Codex sessions could not have affected the model responsible for the mathematical result.
Source:
[url=https://openai.com/index/navier-stokes-solution/]OpenAI — Navier–Stokes Solution[/url]
[url=https://www.wired.com/story/openai-navier-stokes-math-discovery-academics/]WIRED — OpenAI, Navier–Stokes and academic concerns[/url]

2. OpenAI admits that news of academic progress triggered its effort
One important part of the story is not disputed.
OpenAI acknowledges that its intensive Navier–Stokes effort began after researchers at the company heard that major progress had apparently been made on a Millennium Prize problem.
The company says that on September 1 it began directing substantial AI resources toward the problem.
It later became clear that the mathematical progress being discussed involved Buckmaster and Alpöge.
This means that OpenAI is not claiming that its effort was completely unrelated to the academics' work.
Rather, its position is that hearing that progress existed encouraged OpenAI to investigate the mathematical area independently.
This creates an important distinction.
The allegation is no longer simply:
Quote:"Did OpenAI copy the unpublished proof?"
There is also a broader ethical question:
Quote:"Should an AI laboratory use information that academics are close to an important breakthrough as a signal to deploy enormous computational resources and attempt to reach the result first?"
Some mathematicians regard this as a modern form of academic "scooping", even if no confidential mathematical details were directly used.

3. OpenAI argues that the mathematical results are different
OpenAI also argues that its mathematical result is substantially different from the work carried out by Buckmaster and Alpöge.
Buckmaster and Alpöge had obtained results involving the Euler equations with external forcing.
OpenAI says its AI-assisted approach went further by finding a construction without the same external forcing, eventually producing what it claims is a route toward the Navier–Stokes problem.
If correct, this would mean that OpenAI did not simply reproduce Buckmaster and Alpöge's proof.
This part of the controversy is potentially easier for the mathematical community to evaluate, because researchers can compare the published mathematical arguments directly.

4. The dispute over academic credit
Another controversial issue concerns authorship and academic recognition.
Buckmaster has described discussions with OpenAI researchers concerning how the respective results might be presented.
According to Buckmaster, OpenAI proposed arrangements under which he could potentially be connected to OpenAI's paper, while Alpöge — who works for Anthropic — would not necessarily appear in the same way.
Buckmaster also reported a particularly tense conversation in which OpenAI researcher Sébastien Bubeck allegedly asked:
Quote:"Why would you ruin your career?"
Buckmaster interpreted the conversation as pressure.
Bubeck disputes that interpretation.
He says he was not asking Buckmaster to remove Alpöge from their own academic research. Rather, according to Bubeck, the discussion concerned authorship of OpenAI's separate work.
Bubeck has also apologized for the wording concerning Buckmaster's career, describing it as a poor choice of words rather than a threat.
Source:
[url=https://www.abc.net.au/news/2026-09-10/openai-navier-stokes-millennium-problem-claims/107132242]ABC News — Navier–Stokes dispute[/url]

5. Buckmaster has stopped short of accusing OpenAI of theft
Despite the strong reaction surrounding the controversy, Buckmaster himself has been relatively careful in his public statements.
He has said:
Quote:"I do not know whether our data was used."
He has also stated:
Quote:"I am not accusing anyone of anything."
His argument is essentially that the timeline and circumstances were unusual enough that they deserved public scrutiny.
Therefore, the statement:
Quote:"Buckmaster proved that OpenAI stole his proof"
would currently be inaccurate.
A more precise description would be:
Quote:Buckmaster raised concerns about whether unpublished AI-assisted mathematical work could have indirectly influenced OpenAI's project and questioned the way OpenAI responded after becoming aware of his team's progress.
6. Criticism from the wider mathematical community
The controversy has expanded beyond Buckmaster.
A number of mathematicians have expressed concerns about the way large AI laboratories are entering mathematical research.
An open letter associated with members of the Caltech and broader mathematics community criticized the increasing pressure to rapidly announce AI-generated mathematical discoveries.
Critics argue that AI laboratories can deploy enormous amounts of computing power after hearing that traditional researchers are approaching an important result.
Some mathematicians have also complained that the academic community is then expected to spend significant amounts of unpaid time checking extremely long AI-generated proofs.
The open letter reportedly described some of these practices as potentially amounting to:
Quote:"research misconduct."
OpenAI subsequently withdrew its sponsorship of a Caltech mathematics event following the controversy.
Sources:
[url=https://proofsandprompts.com/2026/09/10/open-letter-about-the-mathathon/]Open Letter About the Mathathon[/url]
[url=https://www.businessinsider.com/openai-caltech-ai-math-hackathon-backlash-anthropic-2026-9]Business Insider — OpenAI withdraws from Caltech math event[/url]

7. Questions about citations and attribution
A separate controversy also emerged concerning references in OpenAI's mathematical paper.
Reports stated that the original version did not properly cite earlier work by mathematicians Diego Córdoba and Luis Martínez-Zoroa, whose techniques were considered relevant to the construction.
Those references were subsequently added.
This does not demonstrate that OpenAI copied Buckmaster's research.
However, it has contributed to concerns among mathematicians that the speed of AI-generated mathematical research may sometimes come at the expense of careful attribution.
Source:
[url=https://elpais.com/tecnologia/2026-09-10/openai-corrige-de-tapadillo-su-prueba-del-milenio-para-citar-a-los-
matematicos-clave-que-ninguneo-al-anunciar-el-descubrimiento.html]El País — OpenAI adds mathematical citations[/url]

Where does the evidence currently stand?
At present, there is no publicly demonstrated evidence that OpenAI directly stole Buckmaster and Alpöge's unpublished proof.
Two important facts, however, appear to be established.
  • OpenAI learned that another group had made significant progress and then deliberately launched a very large computational effort toward the same general mathematical problem.
  • Buckmaster had been using OpenAI's Codex system while conducting unpublished mathematical research, which naturally raised questions about whether those interactions could somehow have influenced OpenAI's models.
OpenAI now says that its investigation has ruled out the second possibility for the relevant prompts.
The remaining difficulty is independent verification.
Outside mathematicians can compare the mathematical proofs themselves, but they cannot independently reconstruct OpenAI's internal training pipeline or determine precisely what information was available to every system involved.
Conclusion
The controversy therefore appears to be shifting away from the simple question:
Quote:"Did OpenAI steal Buckmaster's proof?"
OpenAI has issued a fairly specific technical denial of that allegation.
The more difficult question may instead be:
Quote:"What should happen when AI laboratories hear that academic researchers are approaching a major breakthrough and can immediately deploy thousands of AI agents and enormous computational resources toward the same target?"
That question is likely to become increasingly important as AI systems become capable of carrying out more sophisticated mathematical research.
Traditional academic norms developed in a world where competing researchers generally had comparable human limitations.
AI laboratories operate under very different conditions.
The Navier–Stokes controversy may therefore become an early example of a much larger debate about authorship, priority, research ethics and the future relationship between mathematicians and AI laboratories.]]> How OpenAI Responds to Claims It Used Academics’ Work in the Navier–Stokes Breakthrough

The controversy surrounding OpenAI's claimed progress on the Navier–Stokes Millennium Prize problem has developed into a wider debate about artificial intelligence, academic credit, unpublished research, and the increasingly powerful role of AI laboratories in mathematics.
Mathematician Tristan Buckmaster and other academics have raised concerns about how OpenAI arrived at its result, particularly because Buckmaster and collaborator Levent Alpöge had themselves been working on closely related problems while using OpenAI's Codex system.
However, it is important to distinguish between the different allegations. There is currently no established evidence that OpenAI simply copied or "stole" Buckmaster and Alpöge's unpublished proof.

1. OpenAI denies using Buckmaster's private Codex work
OpenAI has issued a direct denial that Buckmaster's recent private Codex interactions contributed to the Navier–Stokes result.
According to OpenAI, an internal investigation found that Buckmaster's Codex prompts from the period before the announcement:
Quote:"could not have influenced the system in any way, including through training."
OpenAI also says that neither its researchers nor the AI agents involved in the project had access to Buckmaster and Alpöge's unpublished research before it became public.
This represents a stronger position than OpenAI initially took.
Early in the controversy, OpenAI reportedly acknowledged that it could not completely exclude the possibility that de-identified product-use data might indirectly have contributed to model improvement.
After investigating the specific case, however, OpenAI stated that the relevant Codex sessions could not have affected the model responsible for the mathematical result.
Source:
[url=https://openai.com/index/navier-stokes-solution/]OpenAI — Navier–Stokes Solution[/url]
[url=https://www.wired.com/story/openai-navier-stokes-math-discovery-academics/]WIRED — OpenAI, Navier–Stokes and academic concerns[/url]

2. OpenAI admits that news of academic progress triggered its effort
One important part of the story is not disputed.
OpenAI acknowledges that its intensive Navier–Stokes effort began after researchers at the company heard that major progress had apparently been made on a Millennium Prize problem.
The company says that on September 1 it began directing substantial AI resources toward the problem.
It later became clear that the mathematical progress being discussed involved Buckmaster and Alpöge.
This means that OpenAI is not claiming that its effort was completely unrelated to the academics' work.
Rather, its position is that hearing that progress existed encouraged OpenAI to investigate the mathematical area independently.
This creates an important distinction.
The allegation is no longer simply:
Quote:"Did OpenAI copy the unpublished proof?"
There is also a broader ethical question:
Quote:"Should an AI laboratory use information that academics are close to an important breakthrough as a signal to deploy enormous computational resources and attempt to reach the result first?"
Some mathematicians regard this as a modern form of academic "scooping", even if no confidential mathematical details were directly used.

3. OpenAI argues that the mathematical results are different
OpenAI also argues that its mathematical result is substantially different from the work carried out by Buckmaster and Alpöge.
Buckmaster and Alpöge had obtained results involving the Euler equations with external forcing.
OpenAI says its AI-assisted approach went further by finding a construction without the same external forcing, eventually producing what it claims is a route toward the Navier–Stokes problem.
If correct, this would mean that OpenAI did not simply reproduce Buckmaster and Alpöge's proof.
This part of the controversy is potentially easier for the mathematical community to evaluate, because researchers can compare the published mathematical arguments directly.

4. The dispute over academic credit
Another controversial issue concerns authorship and academic recognition.
Buckmaster has described discussions with OpenAI researchers concerning how the respective results might be presented.
According to Buckmaster, OpenAI proposed arrangements under which he could potentially be connected to OpenAI's paper, while Alpöge — who works for Anthropic — would not necessarily appear in the same way.
Buckmaster also reported a particularly tense conversation in which OpenAI researcher Sébastien Bubeck allegedly asked:
Quote:"Why would you ruin your career?"
Buckmaster interpreted the conversation as pressure.
Bubeck disputes that interpretation.
He says he was not asking Buckmaster to remove Alpöge from their own academic research. Rather, according to Bubeck, the discussion concerned authorship of OpenAI's separate work.
Bubeck has also apologized for the wording concerning Buckmaster's career, describing it as a poor choice of words rather than a threat.
Source:
[url=https://www.abc.net.au/news/2026-09-10/openai-navier-stokes-millennium-problem-claims/107132242]ABC News — Navier–Stokes dispute[/url]

5. Buckmaster has stopped short of accusing OpenAI of theft
Despite the strong reaction surrounding the controversy, Buckmaster himself has been relatively careful in his public statements.
He has said:
Quote:"I do not know whether our data was used."
He has also stated:
Quote:"I am not accusing anyone of anything."
His argument is essentially that the timeline and circumstances were unusual enough that they deserved public scrutiny.
Therefore, the statement:
Quote:"Buckmaster proved that OpenAI stole his proof"
would currently be inaccurate.
A more precise description would be:
Quote:Buckmaster raised concerns about whether unpublished AI-assisted mathematical work could have indirectly influenced OpenAI's project and questioned the way OpenAI responded after becoming aware of his team's progress.
6. Criticism from the wider mathematical community
The controversy has expanded beyond Buckmaster.
A number of mathematicians have expressed concerns about the way large AI laboratories are entering mathematical research.
An open letter associated with members of the Caltech and broader mathematics community criticized the increasing pressure to rapidly announce AI-generated mathematical discoveries.
Critics argue that AI laboratories can deploy enormous amounts of computing power after hearing that traditional researchers are approaching an important result.
Some mathematicians have also complained that the academic community is then expected to spend significant amounts of unpaid time checking extremely long AI-generated proofs.
The open letter reportedly described some of these practices as potentially amounting to:
Quote:"research misconduct."
OpenAI subsequently withdrew its sponsorship of a Caltech mathematics event following the controversy.
Sources:
[url=https://proofsandprompts.com/2026/09/10/open-letter-about-the-mathathon/]Open Letter About the Mathathon[/url]
[url=https://www.businessinsider.com/openai-caltech-ai-math-hackathon-backlash-anthropic-2026-9]Business Insider — OpenAI withdraws from Caltech math event[/url]

7. Questions about citations and attribution
A separate controversy also emerged concerning references in OpenAI's mathematical paper.
Reports stated that the original version did not properly cite earlier work by mathematicians Diego Córdoba and Luis Martínez-Zoroa, whose techniques were considered relevant to the construction.
Those references were subsequently added.
This does not demonstrate that OpenAI copied Buckmaster's research.
However, it has contributed to concerns among mathematicians that the speed of AI-generated mathematical research may sometimes come at the expense of careful attribution.
Source:
[url=https://elpais.com/tecnologia/2026-09-10/openai-corrige-de-tapadillo-su-prueba-del-milenio-para-citar-a-los-
matematicos-clave-que-ninguneo-al-anunciar-el-descubrimiento.html]El País — OpenAI adds mathematical citations[/url]

Where does the evidence currently stand?
At present, there is no publicly demonstrated evidence that OpenAI directly stole Buckmaster and Alpöge's unpublished proof.
Two important facts, however, appear to be established.
  • OpenAI learned that another group had made significant progress and then deliberately launched a very large computational effort toward the same general mathematical problem.
  • Buckmaster had been using OpenAI's Codex system while conducting unpublished mathematical research, which naturally raised questions about whether those interactions could somehow have influenced OpenAI's models.
OpenAI now says that its investigation has ruled out the second possibility for the relevant prompts.
The remaining difficulty is independent verification.
Outside mathematicians can compare the mathematical proofs themselves, but they cannot independently reconstruct OpenAI's internal training pipeline or determine precisely what information was available to every system involved.
Conclusion
The controversy therefore appears to be shifting away from the simple question:
Quote:"Did OpenAI steal Buckmaster's proof?"
OpenAI has issued a fairly specific technical denial of that allegation.
The more difficult question may instead be:
Quote:"What should happen when AI laboratories hear that academic researchers are approaching a major breakthrough and can immediately deploy thousands of AI agents and enormous computational resources toward the same target?"
That question is likely to become increasingly important as AI systems become capable of carrying out more sophisticated mathematical research.
Traditional academic norms developed in a world where competing researchers generally had comparable human limitations.
AI laboratories operate under very different conditions.
The Navier–Stokes controversy may therefore become an early example of a much larger debate about authorship, priority, research ethics and the future relationship between mathematicians and AI laboratories.]]> <![CDATA[SOFA Statistics]]> https://mklab.gr/showthread.php?tid=1937 Thu, 10 Sep 2026 23:40:12 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1937
A major goal of SOFA Statistics is ease of use: its workflow helps users choose appropriate basic statistical tests and specify which variables to analyse, making it particularly suitable for students, educators, researchers, and users who do not want to work primarily through programming. Its statistical capabilities are more limited than those of comprehensive environments such as R, but research comparing menu-driven statistical packages found that SOFA produced broadly comparable results for common calculations such as frequencies, means, correlations, and regression. The software is cross-platform, distributed under the AGPL open-source licence, and its name reflects its aim of making statistical analysis "open for all.

SOFA]]>
A major goal of SOFA Statistics is ease of use: its workflow helps users choose appropriate basic statistical tests and specify which variables to analyse, making it particularly suitable for students, educators, researchers, and users who do not want to work primarily through programming. Its statistical capabilities are more limited than those of comprehensive environments such as R, but research comparing menu-driven statistical packages found that SOFA produced broadly comparable results for common calculations such as frequencies, means, correlations, and regression. The software is cross-platform, distributed under the AGPL open-source licence, and its name reflects its aim of making statistical analysis "open for all.

SOFA]]> <![CDATA[SciDAVis]]> https://mklab.gr/showthread.php?tid=1936 Thu, 10 Sep 2026 23:34:15 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1936 SciDAVis (Scientific Data Analysis and Visualization) is a free, open-source, cross-platform program for scientific data analysis and interactive graphing. It was created in 2007 as a fork of QtiPlot and is conceptually similar to the commercial software Origin. SciDAVis runs on Windows, macOS, and Linux and is written mainly in C++ with Python support. It organizes data in spreadsheet-style tables or matrices and allows users to create a wide variety of 2D and 3D visualizations, including line, scatter, bar, pie, and surface plots.

Beyond plotting, SciDAVis includes tools for statistical analysis, FFT and filtering, convolution and deconvolution, and linear or nonlinear curve fitting using the GNU Scientific Library. Users can import data from ASCII files, enter data manually, calculate values using formulas, and use Python scripting for automation and more advanced analysis. Graphs can be exported to formats such as PDF, EPS, SVG, and common bitmap formats. Development has experienced periods of inactivity but later resumed, with version 2.9.2 listed as the stable release from April 2022. 

SCIDAVIS]]> SciDAVis (Scientific Data Analysis and Visualization) is a free, open-source, cross-platform program for scientific data analysis and interactive graphing. It was created in 2007 as a fork of QtiPlot and is conceptually similar to the commercial software Origin. SciDAVis runs on Windows, macOS, and Linux and is written mainly in C++ with Python support. It organizes data in spreadsheet-style tables or matrices and allows users to create a wide variety of 2D and 3D visualizations, including line, scatter, bar, pie, and surface plots.

Beyond plotting, SciDAVis includes tools for statistical analysis, FFT and filtering, convolution and deconvolution, and linear or nonlinear curve fitting using the GNU Scientific Library. Users can import data from ASCII files, enter data manually, calculate values using formulas, and use Python scripting for automation and more advanced analysis. Graphs can be exported to formats such as PDF, EPS, SVG, and common bitmap formats. Development has experienced periods of inactivity but later resumed, with version 2.9.2 listed as the stable release from April 2022. 

SCIDAVIS]]> <![CDATA[GLE Graphics Layout Engine]]> https://mklab.gr/showthread.php?tid=1935 Thu, 10 Sep 2026 23:32:06 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1935 publication-quality scientific figures, plots, and diagrams. It is aimed especially at researchers, educators, and technical users who need precise control over graphical output. GLE can generate standard graphs as well as complex custom diagrams and includes mathematical and curve-fitting routines. It also provides a graphical interface for interactively editing figures while retaining the flexibility of script-based drawing. 

GLE supports a wide variety of output formats, including PDF, EPS, PostScript, SVG, JPEG, and PNG, making it suitable for papers, books, presentations, and websites. Its examples cover scatter and line plots, bar and polar charts, 3D and contour plots, electronic circuits, Feynman diagrams, fractals, and other technical illustrations. The software is freely available, distributed under the BSD 3-Clause license, and its source code and development are hosted on GitHub.

GLE]]> publication-quality scientific figures, plots, and diagrams. It is aimed especially at researchers, educators, and technical users who need precise control over graphical output. GLE can generate standard graphs as well as complex custom diagrams and includes mathematical and curve-fitting routines. It also provides a graphical interface for interactively editing figures while retaining the flexibility of script-based drawing. 

GLE supports a wide variety of output formats, including PDF, EPS, PostScript, SVG, JPEG, and PNG, making it suitable for papers, books, presentations, and websites. Its examples cover scatter and line plots, bar and polar charts, 3D and contour plots, electronic circuits, Feynman diagrams, fractals, and other technical illustrations. The software is freely available, distributed under the BSD 3-Clause license, and its source code and development are hosted on GitHub.

GLE]]> <![CDATA[Chart.js]]> https://mklab.gr/showthread.php?tid=1934 Thu, 10 Sep 2026 23:29:04 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1934 Chart.js is a free, open-source JavaScript library for creating interactive data visualizations on websites. It renders charts using the HTML5 element and supports eight main chart types, including bar, line, area, pie/doughnut, bubble, radar, polar-area, and scatter charts. Created by web developer Nick Downie in 2013, Chart.js is maintained by an open-source community and distributed under the MIT License. It has become one of the most popular JavaScript charting libraries, valued particularly for being relatively easy to learn and integrate into web applications. 

The library has evolved significantly through several major releases. Version 2 introduced features such as improved animations, time scales, bubble and scatter plots, and stacked charts; version 3 brought major performance improvements, scriptable configuration options, a redesigned animation system, and improved documentation. Version 4, released in 2022, moved the package toward modern JavaScript standards with ESM-only packaging. Compared with more powerful visualization libraries such as D3.js, Chart.js generally offers less fine-grained customization but provides a much simpler way to create attractive, responsive charts with relatively little code.

TOOL]]> Chart.js is a free, open-source JavaScript library for creating interactive data visualizations on websites. It renders charts using the HTML5 element and supports eight main chart types, including bar, line, area, pie/doughnut, bubble, radar, polar-area, and scatter charts. Created by web developer Nick Downie in 2013, Chart.js is maintained by an open-source community and distributed under the MIT License. It has become one of the most popular JavaScript charting libraries, valued particularly for being relatively easy to learn and integrate into web applications. 

The library has evolved significantly through several major releases. Version 2 introduced features such as improved animations, time scales, bubble and scatter plots, and stacked charts; version 3 brought major performance improvements, scriptable configuration options, a redesigned animation system, and improved documentation. Version 4, released in 2022, moved the package toward modern JavaScript standards with ESM-only packaging. Compared with more powerful visualization libraries such as D3.js, Chart.js generally offers less fine-grained customization but provides a much simpler way to create attractive, responsive charts with relatively little code.

TOOL]]>