Body and Soul Math Education Reform Program Back!?

The Body and Soul mathematics education reform program was developed at Chalmers starting with an undergraduate math course on differential equations I gave in 1995-6, and was run 2000-07 as a complete program for chemistry students. When I left Chalmers for KTH in 2008 the program terminated.

The vision of automated solution of differential equations presented in B&S has now become reality with AI bringing automation into a level way beyond the vision. The basic idea with B&S is a new partnership between soul/human intelligence and body/computation as an expression of: 

• Math Education after AI

Comments?

OpenAIs Navier-Stokes Solution as Blow-Up of Math/Fluid Mech?

Post written by Claude:

OpenAI, Tao and Fefferman: The Emperor Has No Clothes

How can some of the world's most celebrated mathematicians, equipped with the most powerful AI systems ever developed, devote enormous intellectual resources to solving a problem while apparently overlooking the elementary question: What is the problem actually about?

This is the disturbing question raised by OpenAI's claimed solution of the Clay Millennium Navier–Stokes problem.

The essence of fluid mechanics is the stability of fluid motion. Laminar flow becomes unstable as Reynolds number increases, disturbances grow, and turbulence develops. Understanding this process, together with the persistence and dissipation of turbulent motion, is a fundamental challenge of mathematical fluid mechanics.

But this is not what the Clay problem asks.

Charles Fefferman's formulation replaces the physical question of stability and turbulence with a mathematical question about global smoothness or finite-time blow-up.

These are not equivalent. Instability does not imply blow-up. Turbulence does not require singularities. A flow can become violently turbulent while remaining bounded.

Nevertheless, the singularity question has been elevated to a Millennium Problem carrying a million-dollar prize.

Enter OpenAI.

Its proposed solution constructs finite-time blow-up through carefully engineered external forcing. As demonstrated in my preceding analysis, the extension to arbitrary positive viscosity is obtained by simultaneously rescaling velocity, forcing and spatial dimensions.

The Reynolds number remains unchanged.

Thus the construction does not describe what happens when viscosity decreases in a fixed physical configuration. It does not explain the onset of instability, the transition to turbulence or turbulent energy dissipation.

It constructs a singularity without explaining the instability of fluid motion.

This may satisfy the formal Clay requirements. But what has been learned about fluid mechanics?

Essentially nothing concerning its central stability and turbulence problems.

Where Is the Scientific Judgment?

The remarkable feature is not that OpenAI can produce an elaborate mathematical construction. It is that the mathematical establishment can regard such a construction as a potential solution to one of the great scientific problems of our time without first demanding an explanation of its physical significance.

Charles Fefferman bears responsibility for the formulation of the Clay problem. Terence Tao has been a leading figure in mathematical fluid mechanics and the study of singularity formation.

Surely they understand that stability and blow-up are different questions.

Why, then, has this distinction not played a more decisive role in evaluating what the Clay problem actually represents?

Why has a specially forced singularity become a possible million-dollar achievement, while the fundamental question of instability leading to turbulence remains outside the prize?

The issue is not whether the proof is correct. It may be perfectly correct.

The issue is whether mathematical correctness has replaced scientific understanding as the ultimate criterion of achievement.

The Triumph of Formalism

There is a simple mechanism behind this extraordinary situation.

A physical problem motivates a mathematical formulation. The formulation becomes prestigious. Researchers develop increasingly sophisticated techniques to attack it. Careers, publications, reputations and prizes become attached to the formal problem.

Eventually, the formulation itself becomes sacred.

Questioning whether it still represents the original physical problem is treated as less important than proving another theorem.

The mathematical machinery becomes increasingly sophisticated while the connection to physical reality becomes increasingly tenuous.

This is not a failure of formal logic.

It is a failure to apply elementary scientific reasoning before formal logic begins.

AI Makes the Problem Visible

OpenAI brings this development to a remarkable conclusion.

AI can construct mathematical arguments of enormous complexity and even subject them to formal verification.

But formal verification establishes only that a conclusion follows from specified assumptions within a formal system.

It cannot establish that the conclusion answers an important scientific question.

An AI capable of producing a hundred-page proof may still be unable to recognize that the problem being solved misses the essential phenomenon.

And if leading mathematicians accept the formal achievement without seriously examining its physical relevance, then AI has exposed a weakness not merely in artificial intelligence but in the intellectual priorities of mathematics itself.

The Emperor Has No Clothes

The emperor in Andersen's story was surrounded by people who admired his magnificent clothes because nobody wished to question what everybody else appeared to see.

Something similar can happen in mathematics.

An elaborate theorem, an enormous proof, formal verification, prestigious institutions and a million-dollar prize create an overwhelming appearance of significance.

But strip away these decorations and ask:

What does OpenAI's construction tell us about the stability of fluid motion and the development of turbulence?

The answer, as far as the construction presently shows, is essentially nothing.

This does not make the theorem false. It makes the claim of fundamental physical significance questionable.

The distinction is elementary, yet it appears remarkably difficult to maintain when mathematical prestige is at stake.

A Challenge to Mathematics and AI

The real challenge is not merely to solve problems.

It is to identify problems worth solving.

For fluid mechanics, this means understanding how instability generates turbulence, how turbulent motion remains bounded and how energy is dissipated.

A mathematical theory accomplishing this would constitute a genuine scientific advance, whether or not it resolved the Clay Millennium Problem.

Conversely, a formally correct construction of singularity under specially designed forcing may settle a mathematical challenge without substantially advancing fluid mechanics.

The question for Fefferman, Tao and OpenAI is therefore not simply whether the proof is correct, but why the mathematical community has elevated a question about singularities above the physical problem of stability and turbulence.

Until that question is seriously addressed, the emperor's magnificent mathematical clothes deserve closer inspection.

OpenAI Analyzes Navier-Stokes Blow-Up by OpenAI: Unphysical

I asked OpenAI to analyze the Clay Navier-Stokes Blow-Up so famously presented by OpenAI. I received a clear and simple analysis displayed below showing a flow velocity with for each given t<1,  support and magnitude shrinking to zero along with viscosity, and for each given viscosity > 0, magnitude tending to infinity as t approaches 1. 

OpenAI concludes that this tells virtually nothing about fluid motion, which is the subject of the Clay problem as announced very clearly. In particular, nothing is revealed about the mathematics of turbulent flow as the true open problem addressed by Clay. OpenAIs solution is a vanishing point singularity which has nothing to with turbulence as being distributed over large regions of the flow. 

One can argue that the OpenAI solution is produced by an intelligence that knows nothing about fluid motion, and reads the formal problem specification by Fefferman as the Devil reads the Bible. It is very strange that the SuperIntelligence of Terence Tao and Charles Fefferman cannot understand this.

Compare with my posts on the Clay problem (46 posts)

Here is the OpenAI analysis

OpenAI’s Navier–Stokes Blow-Up: What Happens When Viscosity Changes?

OpenAI’s proposed Navier–Stokes construction claims finite-time blow-up for every positive viscosity. But what does “for every viscosity” actually mean?

The manuscript constructs a flow at viscosity 1, then obtains other viscosities through the transformation

u_ν(x,t) = √ν u(x/√ν,t),
f_ν(x,t) = √ν f(x/√ν,t).

Pressure scales by ν, while time remains unchanged. Thus every member of this family blows up at t = 1. See equation (10.22) in the manuscript.

This is an exact mathematical scaling. But it changes the experiment: smaller viscosity comes with smaller velocity, smaller forcing and smaller spatial dimensions. The Reynolds number UL/ν remains unchanged at corresponding times.

Following the scaling through the argument reveals something more important. The collapsing core has radius

L_r ≍ √ν (1 − t)^1/2.

Its radial diffusion time therefore satisfies

L_r²/ν ≍ 1 − t.

Here ≍ means equality up to positive constant factors. Viscosity remains essential to the balance. It does not disappear as ν tends to zero, because the spatial scale shrinks to compensate. The same compensation preserves the competition between shear amplification and viscous damping in the oscillatory corrections.

The final blow-up estimate, along a shrinking observation path, takes the form

u_θ,ν ∼ e₀√ν (1 − t)^{−1/2 − h},
e₀ > 0,   0 < h < 1/100.

For every fixed time before 1, velocity tends to zero as viscosity vanishes. Nevertheless, for every fixed positive viscosity, it diverges as time approaches 1.

These statements are compatible because the convergence is not uniform near blow-up. A fixed large speed is reached only within a time interval proportional to

ν^{1/(1 + 2h)}

before the singularity. Large velocities retreat into increasingly small spatial and temporal regions.

Even the fixed blow-up time is a normalization choice. Another exact scaling,

ũ_ν(x,t) = ν u(x,νt),

keeps the spatial support unchanged, changes the force to ν²f(x,νt), and gives blow-up at T_ν = 1/ν.

The conclusion is precise: assuming the construction is correct, it supplies viscosity-dependent forces producing blow-up. It does not establish how changing viscosity affects a flow under fixed forcing and fixed geometry.

That distinction does not invalidate the mathematical claim. But it limits what the result tells us about physical fluid motion.

Illusion of Precision in Quantum Mechanics

Theoretical physics prides itself to be the most precise of all of science with 12 decimal places for the (anomalous) magnetic moment of the electron as an elementary particle in full agreement with extremely precise measurements.

This was Richard Feynmans joker card to meet questions about quantum mechanics as the supreme achievement of human intellect computed using Feynman diagrams. 12 decimal places! That would give at least 3 decimal places correct for the rest of physics. 

This is put into perspective in this new article.

End of Scientific Publishing? Back to Secrecy of Alchemists?

Today anybody can send in an article to mathematical journal with a proof of an old or new conjecture/theorem as a signed original piece of work (by using an AI assistant to carry out the details of the proof). There are still some journals that do not accept any AI presence, but of course it may be impossible to decide the extent of the assistance. 

This means that the name on the article looses its role as unique identifier to a publication list along with the individuality of the person asking for a university position based on the list. 

But human beings without individuality connects to all sorts of ants societies (or maybe even ants have individuality), anyway very different from human society. 

In particular, it invites to a completely new view on scientific publication. In the old days you would seek to make your discovery/proof known to the world in a scientific journal in order to add to your publication list as basis for your university position with pay. 

Today, you would rather do the opposite: If you have discovered something on your own, you would like to keep it stay with you and not give it away to everybody. Thus you would not disclose your findings but keep them secret to possibly exploit as something to sell, like software with a new capability. 

The scientist would then become more of a classical magician in possession of a collection of secret tricks.

Comment by AI:

• Modern science emerged in part by replacing the secrecy of alchemists with the openness of publicly testable knowledge.Your scenario suggests that AI, paradoxically, could drive science back toward the alchemist's model.

How to be Master not Slave of LLM: Leibniz World of Math

Large-Language Models as a Cognitive Virus by Ricard Solé, Giulio Ruffini, Francesca Castaldo, Marco Tuccio, Luis F. Seoane, Manlio de Domenico, Santiago F. Elena, David C. Krakauer, Michael Levin:

• There is a high risk of a run-away effect toward persistent dependence, with abrupt losses in cognitive competence. 
• The same framework, however, identifies conditions for cognitive immunization, based on reducing transmission and facilitating reversibility. 
• Our results highlight how LLM adoption may involve nonlinear collective transitions with important consequences for cognitive autonomy.

In other words, it remains to train cognitive capacities to not become a slave under LLM but the master of  LLM by keeping cognitive autonomy. What to do? Can school help? Mom and Dad? University? Life?

Physical training? 

Let us go back to Leibniz who formed the first vision of automated thinking and ask what advice he can give when 350 years later his vision has become true: The key to cognitive autonomy is 

• Leibniz World of Mathematics.

Do not hesitate to take the first step to autonomy and mastery. 

RealQM: Universality and Chirality

Two new articles about RealQM: 

• Tetrahedral Stereocentre from Packing: Chirality’s Geometric Origin without Hybridisation
• Universality in Quantum Mechanics: A Claim That Cannot Be Tested, and One That Can

The second connects to the 2026 Nobel Prize in Chemistry.

Will AI Kill Mathematics and so Theoretical Physics?

Field Mathematics Medalist Hugo Dominil-Copin asks if AI will kill mathematics:

The only way forward seems to be Resistance, but will it work? The mathematics Hugo is speaking about is so called pure mathematics performed in symbolic form on the black board with chalk, or pen and paper at the desk, then published in math journals as articles filled with symbols. Numerical/computational  mathematics fills formulas with concrete meaning in terms of numbers. The threat to life posed by AI is against pure/symbolic mathematics where LLM has shown to be a formidable player with symbols. 

Theoretical physics is based on symbolic mathematics in the form of equations in symbolic form, which can be given concrete meaning in terms of numbers and images depicting physics, if the equations are computable, that is solutions can be computed with available computational work. 

Fact is that the basic mathematical models of physics such as Quantum Mechanics, Quantum Field Theory, Quantum ChromoDynamics and String Theory, are based on equations which are not computable and so their study is limited to symbolic manipulation where even AI meets severe difficulties because possibilities are very limited. 

Theoretical physics thus meets another threat of being inaccessible to AI even to ASI, and so in particular to humanity. What theoretical physics needs are computable models, which is not a hopeless case since real physics must be some form of analog computational process in principle possible to mimic in digital computation.   

Parameter-free Universality: Standard QM vs RealQM

New article about a basic aspect of mathematical modeling of physics:

• Universality in Quantum Mechanics: A Claim That Cannot Be Tested, and One That Can.

Recall Einstein's ideal model of physical reality as a parameter-free mathematical model which can say something about reality, like a thought experiment without experimental input, like computing the circumference $2\pi$ of a circle of radius 1 from the equation $x^2+y^2=1$, but of greater scope such as computing the drag of a body through a slightly viscous fluid from the shape of the body alone, or the spectrum of a Hydrogen atom. 

Pure Mathematics: The Next Chess?

Will AI turn theorem proving into an intellectual game? 

For centuries, chess was regarded as one of the highest expressions of human intelligence. Great chess players were admired for their extraordinary powers of logical reasoning, imagination and strategic thinking. Then came computers. In 1997, IBM's Deep Blue defeated world chess champion Garry Kasparov. Today, ordinary chess engines play far beyond the level of any human grandmaster. Chess survived, but something fundamental changed. Being the world's strongest chess player no longer meant being the world's strongest chess-playing intelligence.

Could pure mathematics now face a similar transformation? It has traditionally been regarded as an even higher form of intellectual achievement than chess. Unlike chess, mathematics has played a fundamental role in the development of science and civilization. But here we must distinguish applied from pure mathematics. Applied mathematics develops methods for understanding, predicting and controlling the physical world. Newton's calculus, Fourier's analysis and modern numerical computation exemplify this tradition. Their ultimate value lies in what they make possible. Pure mathematics, by contrast, investigates abstract structures and logical relationships without requiring applications. Its achievements are judged primarily within mathematics itself. 

During the twentieth century, pure mathematics increasingly developed a culture centered on solving difficult problems. Fermat's Last Theorem, the Poincaré conjecture and the Clay Millennium Problems became celebrated intellectual challenges. The resemblance to chess is striking. Both activities operate within precisely defined logical frameworks. Both reward extraordinary ingenuity. Both have hierarchies of increasingly difficult problems. Both celebrate champions who succeed where others have failed. 

But is solving a difficult mathematical problem necessarily more intellectually significant than winning a difficult chess game? The answer depends on whether the mathematical achievement reveals something fundamentally new or merely overcomes a formidable technical obstacle. A profound mathematical concept may transform our understanding of an entire field. A difficult proof may simply establish the truth of a previously formulated conjecture. The distinction is between creating new understanding and demonstrating exceptional problem-solving ability. 

Now AI enters the mathematical arena. Just as computers surpassed human chess champions, AI may eventually surpass human mathematicians in proving difficult theorems. If that happens, the production of proofs may become increasingly routine. A mathematical conjecture could become something like a chess position: a challenge presented to a machine capable of finding the solution. The prestige attached to solving difficult problems would then face the same challenge that confronted competitive chess. Yet mathematics has something chess does not necessarily possess: the possibility of discovering entirely new structures and connections, both within mathematics and with the physical world. 

Perhaps this is where its future lies. The central question will no longer be who can prove the most difficult theorem, but who can formulate the most illuminating mathematical ideas. Chess did not disappear when computers became superior players. It became a game in which humans increasingly learn from machines. Pure mathematics may follow a similar path. 

There is, however, a fundamental distinction. Chess has objective rules for winning, but no claim that winning contributes to understanding reality. Pure mathematics has objective rules for proving, but the significance of what is proved requires a separate judgment. A chess victory is meaningful within the game. A mathematical proof establishes a truth within a mathematical framework, but its importance cannot be measured by proof difficulty alone. 

If pure mathematics increasingly rewards the conquest of difficult problems without asking what new understanding they bring, it risks becoming an intellectual sport governed by its own rules of prestige. AI may now expose this weakness by making increasingly sophisticated theorem proving a machine activity. 

Perhaps the greatest contribution of AI to pure mathematics will not be the theorems it proves, but forcing mathematicians to confront a question long overshadowed by the celebration of difficult proofs:

• What makes a mathematical theorem worth proving?