Clay Math Institute Response to Criticism of Navier Stokes Problem Formulation

Response from Martin Briden President Clay Mathematics Institute:

Dear Prof Johnson

I am familiar with the objections that you raise concerning the mathematical problem posed as the Millennium Prize Problem related to Navier-Stokes. Other applied mathematicians and engineers have expressed similar opinions. But I do not accept your conclusion that the problem “cannot be given a meaningful solution”.

I don’t think it has ever been claimed that the problem as posed dealt with turbulence in physical fluids, as you want to see addressed. It is nevertheless surely a natural and compelling question in the study of PDE. 

It may not be the problem that you would have wished to see as a Prize Problem, but from my point of view it has served its purpose well. 

Kind regards

Martin Bridson

My response: 

Dear Martin

Thank you for quick response, and acknowledgment that you are familiar with my criticism of the official formulation of the Navier-Stokes problem, which with the OpenAI solution gets new actuality. 

You say that the problem formulation has served Clay Institute well. I do not think this is so with now the AI solution presenting a potentially disastrous ground shot to mathematics as we know it: An AI proof to a problem without real meaning from physics and mathematics point of view, a 167 page proof which no mathematician can inspect in detail and so will have to be evaluated by the AI that has produced it. 

No mathematician has made any comment as if nothing has happened. But what will now be the fate of the Clay NS problem? Declared as solved but without any meaning? Debunked as nonsense AI with the problem remaining unsolved until next round of AI proof with doubled page number? How do you plan to handle the situation? Have you consulted with the experts behind the problem formulation? What do they say? How will the Clay Math Institute handle the new world of AI mathematics?

Sincerely

Claes

Letter to Clay Math Institute on Navier-Stokes Problem

I have today sent the  following letter to Clay Mathematics Institute including this analysis of the OpenAI solution.  Copies to Terence Tao, Peter Constantin and Charles Fefferman (formulated the problem).

President 

Clay Mathematics Institute

This is a follow up of my 2014 letter about the formulation of the Navier-Stokes problem motivated by the proposed OpenAI solution, which is expressed in the enclosed manuscript. I hope you will read and return with a comment after consulting with the experts. 

In short, my analysis exhibits the unfortunate elimination of turbulent fluid flow, the essence of Navier-Stokes from both physical and mathematical point of view, from the official formulation of the problem, which now has triggered an AI solution without both physical and mathematical meaning. Clay Mathematics Institute is not served by announcing prize problem missing the essentials.

Sincerely

Claes Johnson 

On Fri, 16 May 2014 at 16:50, Claes Johnson <claesjohnson@gmail.com> wrote:

President

Clay Mathematics Institute

I want to convey the information that the formulation of the Clay Navier-Stokes problem is incorrect both mathematically and physically, because the fundamental aspects of (i) wellposedness and (ii) turbulence, are not included, as exposed in detail in the following sequence of blog posts:

http://claesjohnson.blogspot.se/search/label/clay%20problem

The result is that the problem cannot be given a meaningful solution and thus does not serve well as a Prize problem. Evidence is given by the fact that no progress towards a solution has been made.

I have tried to engage Charles Fefferman, who has formulated the problem, Peter Constantin, who acts as a referee, and Terence Tao, who is working on the problem, into a discussion, but I get no response.

I hope this way to stimulate discussion, which I think would be more constructive than no discussion.

Sincerely

Claes Johnson

prof of applied mathematics 

Royal Institute of Technology, Stockholm

   

Meaningless OpenAI Solution to Clay Navier-Stokes Problem

Here is a post composed by Claude from my prompts about the Clay Navier-Stokes Millennium Problem. 

A Meaningless Solution to a Meaningless Problem

The Clay Navier–Stokes Problem asks whether solutions of the incompressible Navier–Stokes equations stay smooth, or blow up in finite time. Recently a machine-generated proof has been offered which constructs, at every positive viscosity, a forced flow whose velocity becomes unbounded.

There is a simple test any such construction has to pass, and the official formulation never asks for it: report the constants as functions of the viscosity ν. Let us apply it.

One solution, rescaled

Navier–Stokes rescales. If v solves the equations at viscosity one with forcing g, then

u(x,t) = a·v(bx, ct),    a = νb,  c = ab,  f = a²b·g

solves them at viscosity ν. The map is onto. A single unit-viscosity blow-up generates the entire family. So a theorem asserting blow-up "for every ν > 0" is not a family of results. It is one result, photographed at different magnifications, and it contains exactly what its ν = 1 member contains.

All the content is in what the rescaling does to the constants, and there are only a few ways to spend the freedom:

• Hold the solution at size one. Then the forcing amplitude is ν⁻¹: the force required diverges as the viscosity falls, and leaves any fixed admissible class — including the one Fefferman specifies, which fixes its constants once and for all.
• Hold the forcing fixed. Then the velocity amplitude is ν^1/3 and the active length ν^2/3: both vanish.
• Hold the blow-up time at t = 1 — the natural normalisation. Then everything vanishes together:

amplitude ν^1/2,   length ν^1/2,   forcing ν^1/2,   gradient 1

At ν = 10⁻⁸ the flow reaches an amplitude of 10⁻⁴ on a scale of 10⁻⁴ under a force of 10⁻⁴, and then becomes unbounded. The family converges to the zero solution driven by zero force.

The limits do not commute

Write the family out. With blow-up at t = 1,

|u_ν(t)| ~ ( ν / (1−t) )^1/2

and the two factors pull against each other. Let t → 1 first, at fixed ν: infinity. Let ν → 0 first, at fixed t < 1: the zero solution. Infinity one way, nothing the other. The joint limit has no value at all; it depends on the path taken.

The critical path is 1 − t ~ ν, where the amplitude is of order one. And there, the local Reynolds number is

Re = |u|·ℓ / ν = ν^1/2(1−t)^−1/2 · ν^1/2(1−t)^1/2 / ν = 1

identically — independent of both t and ν. The singularity sits permanently at the viscous scale. It does not pass through the inertial range on its way to blowing up; it never enters it. Turbulence is the statement that Re ≫ 1 over a wide range of scales. This object is at Re = 1 at every instant of its life.

It is a shock, with a point's extent

What is this thing? An O(1) velocity change across a thickness ν with gradient ν⁻¹ is the viscous shock profile. For Burgers, a jump U relaxes over δ = ν/U, and its dissipation per unit area is U³ — independent of ν. That is the classical anomalous dissipation, the one-dimensional model of the very mechanism Onsager's conjecture concerns.

So the construction has the profile of a shock. What it does not have is a shock's extent. A shock is a surface: thickness ν in one direction, order one in the other two, hence volume ν and dissipation of order one. This object is ν in all three directions — volume ν³, dissipation ν², carried on an energy ν³. Both vanish.

The same local profile, spread over a surface, dissipates at a rate independent of viscosity. Concentrated at a point, it dissipates nothing. The construction is made to blow up by being denied the dimensionality that lets a shock carry energy.

And it is not even a counterexample

Here is the sharpest point. On the critical path, what diverges is the gradient:

|∇u| = |u| / ℓ = 1/(1−t)  →  ∞,     while |u| ~ 1 stays bounded

— and note that the gradient does not involve ν at all: at every viscosity the same gradient is reached at the same time-to-blow-up, which is the sharpest possible statement that the family is one solution rescaled.

But Fefferman's alternatives ask for the velocity to become unbounded. It does not. At 1 − t ~ ν the family is a bounded flow with diverging gradients, which is neither horn of the dichotomy. It is the third category — non-smooth and non-singular — the one the formulation has no name for, and the one in which every flow of physical interest actually lives. The velocity becomes unbounded only afterwards, in a window of duration of order ν that shrinks to nothing.

A meaningless solution to a meaningless problem

The Clay problem is announced as a problem of turbulence — the wake behind the boat, the air behind the aircraft — and then formulated as a problem without it, asking only whether |u| stays finite. Between the description and the statement, the subject has been sorted out.

Put the turbulence back, in the only form analysis can digest — a turbulent viscosity depending on the local velocity gradients — and Fefferman's alternative (A) ceases to be open at all. Global existence and uniqueness follow by standard Sobolev-space techniques, as Lions showed in 1969, for a stress augmented by |∇u|^1/2, with a coefficient as small as one likes. And at p = 3 that added term is the Smagorinsky model. The regularisation that makes the problem tractable is a turbulence model. The formulation excludes turbulence from its alternatives, and the price of that exclusion is precisely the term whose absence makes it hard.

So we have a construction which, taken to the limit where fluids actually live, converges to the zero solution driven by zero force; which lives at Re = 1 throughout; which is a shock deprived of its surface; and which, in the regime where it has an O(1) amplitude, is not a counterexample to the stated alternative but an instance of the category the formulation declines to name.

A meaningless solution, then. But the deeper trouble is that it answers exactly what was asked. The dichotomy smooth or singular is empty because the physical solution is neither: turbulent flow has bounded velocity and gradients growing without limit — Hölder continuous with exponent 1/3, the finest structures of size ν^3/4 across which the velocity varies by ν^1/4. Not smooth, so regularity theory cannot reach it. Not singular, so no blow-up theorem describes it.

What should be asked instead is wellposedness: which outputs of this flow are stable, and to what tolerance. Drag, lift and mean pressure are stable and computable; the pointwise velocity is neither, and no theorem asserting that its trajectory is unique will make it so. Uniqueness in a function space and stability of an output are different properties, and only the second is what a computation, an experiment, or an aircraft depends on.

The prize was offered for the wrong property. It should not surprise us that what claims it is a solution to nothing in particular.

Mathematics Education vs AI vs BodyandSoul vs John Dewey

Recent posts on the OpenAI solution to the Clay Navier-Stokes Millennium Problem reflect on the monumental challenge to professional mathematicians presented by a AI agents showing to be able to prove mathematical theorems. 

The challenge also strikes mathematics education and here  (or below)is what chatGPT delivers on prompt asking about the relevance of John Dewey in the new landscape of AI. 

The BodyandSoul project also gets new actuality with AI.  

ChatGPT:

John Dewey (1859–1952) fits surprisingly well into the AI era—but AI also exposes a weakness in the way Dewey’s educational ideas have sometimes been interpreted.

Dewey’s central idea was that education should not primarily consist of transferring a fixed body of knowledge from teacher to pupil. Learning occurs through inquiry, experience, problem solving and reflection—often summarized, somewhat simplistically, as learning by doing. The teacher creates situations in which the student encounters a problem, investigates it, tests possible solutions and reconstructs his or her understanding.

AI radically changes the conditions under which this philosophy operates.

Before AI, there was a practical reason for spending enormous amounts of schooling on acquiring information and routine intellectual techniques. A student needed vocabulary, facts, algorithms, grammar, computational techniques, references, etc., because these were the tools required for subsequent intellectual work. With a powerful AI available, much of this intermediate work can be supplied almost instantly.

That pushes education toward Dewey. Consider mathematics. Instead of spending weeks learning a particular computational procedure and then being given an artificial exercise, a pupil might start with a genuine question:

Why does a suspension bridge have the shape it does?

AI can supply calculations, simulations, explanations at different levels, historical background and even computer programs. The student can formulate hypotheses, challenge the AI, modify the model and compare predictions with observations. This is recognizably Deweyan inquiry—but enormously amplified.

Yet there is a paradox. AI can also destroy Deweyan learning if it does the inquiry for the student.

Suppose the assignment is: Investigate why the Roman Republic collapsed. A student asks an AI and receives a beautifully organized essay. The product may be excellent, but virtually no education has occurred. The student has neither investigated nor struggled with competing explanations nor constructed an argument.

So the distinction in AI education becomes:

AI doing the task versus AI enlarging the student's capacity to do the task.

This makes Dewey's concept of inquiry more important than the familiar slogan learning by doing. Merely producing something—a report, presentation, program or mathematical proof—is no longer evidence that the student has learned anything. AI can produce all of these.

Consequently, I think the AI landscape points toward a modified Deweyan school with three components:

1. A substantial foundation of knowledge. A person cannot critically interrogate an AI about Roman history, mathematics or physics without already knowing something. AI therefore does not make knowledge unnecessary. In some respects it makes internal knowledge more important, because otherwise one cannot recognize plausible nonsense.
2. Authentic inquiry. Students should increasingly be given difficult questions whose answers aren't simply contained in the textbook. AI becomes an intellectual instrument—something like an extraordinarily powerful library, calculator and conversational research assistant combined.
3. Demonstrated understanding rather than submitted products. Assessment would increasingly have to involve conversation, oral examination, explanation at the blackboard, modification of an argument, criticism of an AI answer, or solving a new variation of a problem. The crucial question becomes: What can the student understand and defend?

This also changes the teacher's position. The teacher becomes somewhat more Deweyan and less like an information distributor, but not less important. The teacher chooses worthwhile problems, detects superficial understanding, asks the decisive follow-up question and possesses enough subject knowledge to know when both pupil and AI have gone astray.

There is therefore an interesting historical reversal. During the twentieth century, Dewey could sometimes be interpreted as supporting a movement away from demanding subject knowledge toward activities, projects and pupil-centered learning. AI makes that version of Dewey increasingly untenable. Activities and polished products are now almost free.

What remains scarce is understanding, judgment, curiosity and the ability to formulate a good question.

In that sense, AI may produce a more intellectually demanding version of Dewey than the schools influenced by Dewey ever did: learn through genuine inquiry—but know enough to conduct the inquiry yourself.

---

If you want, I can:

• Explore how AI can assist but not replace student inquiry in education
• Discuss challenges and solutions for assessing student understanding with AI
• Analyze the evolving role of teachers in AI-augmented Deweyan education

Two Forms of LLM for Mathematics: Words and Numbers

We have identified two forms of mathematics, (S) symbolic performed with symbols/words and (N) numerical performed by computation with numbers. The forms of AI now taking humanity with surprise are based on Large Language Models LLM exhibiting a formidable capacity to compose texts as strings of words, after training by reading many texts composed by humans over centuries. 

The OpenAI proof the Clay Millennium conjecture that solutions to Navier-Stokes equations can develop singularities in finite time, takes the form of a 167 page string of words formally verified to be logically consistent as a consequence of known already proved theorems. The proof does not present the values of the singular solution in numerical form, only that such a thing must exist as a logical consequence of know theorems taking the form a string of words thus an example of (S), exactly what LLMs are designed to do 

RealQM is a new form of quantum mechanics developed with Claude as coding agent and producing numerical solutions to concrete problems of quantum mechanics without other input than case specification thus without free parameters asking for observational input. This represents (N) as numerical computation, which is not a string of words but a string of purely computational tasks. It shows that Claude is very capable of producing efficient code performing the tasks specified by an algorithm for numerical solution of the Schrödinger equation of RealQM needed as starting point. This algorithm is not a proof but a list of tasks obeying logics. The list of basic tasks in numerical computation is limited; basically addition and gradient or fixed-point iteration, at least in the context of physics with equations such as Navier-Stokes.  

That Claude can code is not the result of reading massive text, but comes from logic combined with knowledge of numerical algorithms. The training as LLM can then help with logic.

So it is maybe not so surprising that AI can produced lengthy proofs of mathematical theorems, may better and more expedient than real top mathematicians, as an LLM. More surprising maybe that AI can code, which ultimately does not need so much of intelligence.  

There is an important difference between step-by-step formal verification of a symbolic proof of some mathematical theorem, which can overwhelming for long proofs with many steps, and assessment  of the quality of a computed solution which can be done by a posteriori evaluating its residual without checking every step.  

A numerical solution can be time-consuming to compute but quick to check. N vs NP.

 

 

Difference between Analytical and Numerical Mathematics: Capital vs Labour

The recent announcement by OpenAI of a solution to the Clay Navier-Stokes Millennium Problem exhibits a fundamental difference between analytical mathematics based on symbols/signa and numerical mathematics based on numbers. 

OpenAI thus has produced a 167 page "proof" consisting of words/signs/symbols describing a "construction" of a "solution" as a function solving Navier-Stokes equations showing development in finite time of a singularity ending existence. The "construction" consists of a 167 page string of words/signs/symbols describing the principles involved but not the numerical values of the "solution". 

The "proof" is supplemented by a formal verification of steps of the "construction" based on logic and theorems expressed in words. Full verification is impossible since the words/symbols involved are not defined in finite terms. 

No attempt is made to compute numerically the values of the solution to concretely inspect the development of the singularity. Such a thing would resort to numerical mathematics or number-crunching. This is most remarkable  and signifies a deep rift between analytical mathematics based on words/symbols and numerical mathematics base on numbers. 

A fundamental difference is that for numerical mathematics verification of correctness is possible by reducing the numerics to finite digital representation. It is thus possible to compute a numerical solution to Navier-Stokes equations and give a numerical quality measure of the constructed solution. 

AI is a form of numerical mathematics and so when AI now is brought in to help mathematicians perform analytical mathematics, as in the present case with the goal of winning a prize, the whole thing appears to collapse to numerics. 

We are thus back to Pythagoras and the basic difference appears to be finite vs non-finite. Formal verification is possible with finite but not with non-finite. 

What does this mean for mathematics and mathematics education? To start over with what?

If we want we can compare with the difference between capital (words) and labour (numbers), with now labour coming out to control capital in a revolution. Ok?

 

AI Solution of Clay Problem Sends Shock Waves into Analytical Mathematics

The Open AI solution (see previous post) to the Clay Navier-Stokes problems sends shock waves into the world of analytical (pure) mathematics. Is the proof correct? Can correctness be checked by human mathematicians or only formally by AI itself? What if AI says the proof is correct. Will AI then get the prize? 

If so the traditional split of mathematics into analytical (formulas) mathematics and numerical mathematics (number crunching), will no longer be functional. An AI proof is the result of an ultimately computational process and of course the same is true for a numerical solution. 

Traditionally, analytical mathematics has been associated with generality by offering proofs of existence of solutions (but not their values) for general data, while numerical solutions have been particular for each choice of data. Thus analytical-general and numerical-particular. 

But OpenAI offers a single counterexample to existence, not generality but extreme particularity, while numerical solution to Navier-Stokes equations for almost any data offers generality. 

We see that the distinction between analytical and numerical mathematics with computational AI gets blurred, which can be seen as a lift for numerical mathematics traditionally viewed as lower level. Mathematics is fundamentally computational.

It will be interesting to see the effects of the shock waves now sweeping over the field of mathematics, including choice of topics and education. Leibniz would have been thrilled to experience this development which he prepared 350 years ago. 

Meaningless Clay Navier-Stokes Problem Solved by AI

OpenAI announces a proof of existence of a solution to the Navier-Stokes equations (but not its numerical values), which starting from zero under smooth forcing ceases to exist in finite time: 

• We’re sharing a solution to the Navier–Stokes existence and smoothness problem, one of the Millennium Prize Problems. This proof, produced by an internal OpenAI system, shows that the dynamics of the Navier-Stokes equations for fluid motion can develop a singularity in finite time. We’re sharing both a writeup of the proof and a formalization in Lean.

Charles Fefferman, who formulated the problem in precise mathematical terms, is along with other leading mathematicians such as Terence Tao, happy that the understanding of fluid motion has now taken a big leap forward by mathematical analysis, even if the development of the singularity cannot be followed in any precise terms. Something goes wrong but what and how is hidden.

There is a further problem in this happy moment, which I have complained about over the years: Fefferman's formulation misses the essence of the physics of fluid motion, namely turbulence. The Clay problem is sold as concerned with basic aspects of fluid motion,  but does not address the most fundamental problem of all of turbulence. Fefferman's formulation directs the interest away from physics, and the unhappy result is that solution now presented by AI covering 167 pages cannot be read to learn anything, simply a mess of formulas and theorems. 

This is certainly a memento for mathematics: AI can now produce proofs of an endless number of mathematical problems without real meaning, proofs which cannot be understood by mathematicians in detail only verified formally by Lean. What will be the result?

Numerical mathematics offers a solution to the fundamental problem of turbulence, thus a different solution to a different problem formulation. See tags to this post starting with this post from 2013.

Recall that slightly viscous flow is unstable from shear and stretch and so develops into non-smooth turbulent flow which however does not break down like the Clay solution. So the solution of physical interest is non-smooth and non-singular, which is not captured in Fefferman's dichotomy of smooth or singular.  

Turbulence is an extreme form of the design of a complex world with a variety of phenomena on different scales: Develop growth from instability + curb growth to allow continued existence, not captured by Fefferman's formulation.  

PS1 When I 20 years ago complained to Fefferman that his formulation lacked true interest from physics point of view, he returned that it was enough that the problem was interesting to him.

PS2 Note that the AI solution is a proof of the existence of a very special function (unknown to details) which is a solution with a very specific particular forcing. This is not the real setting which is to study solutions under general forcing. 

PS3 Here is an interesting catch of the AI proof of existence of a singular solution. Computational solutions can be constructed for general data including turbulence and any such solution can be viewed as an AI proof of existence performed by a computer according to strict mathematical principles, including evaluation of quality. The whole process can be seen as an AI proof of existence of a solution for each given set of data. It would be strange to not consider that as a solution to the essence of the Clay problem albeit not captured in Fefferman's formulation. 

PS4 Allowing AI as computational process into the Clay problem game, we may compare the Open AI proposal as an analytical AI proof of non-existence in a very special case, with an computational AI proof of existence for any data, except one. Which proposal would you give the money to? Or 50-50? Note that the estimated cost of the OpenAI solution is several million dollars, so the Prize money will not suffice to cover, what remains is fame at price of a couple million dollars, fine for OpenAI but not for a poor pure mathematician. 

PS5 The verification by Lean in principle requires each step to be verified from logic and previous axioms/therorems/steps, which is overwhelming and cannot be done. Compare with a numerical solution produced in a number of computational steps, where a verification of solution quality can be made without verifying each step (involving round-off which propagates) because the solution produced is known. Not so with the singular solution proved to exist by AI and so only stepwise check is available (which is more impossible than possible). 

PS6 The size of the forcing appears to scale with the square root of the viscosity which means that the constructed solution is not turbulent. Another sign that the problem formulation misses the essence of Navier-Stokes. How could it go so wrong for so many mathematicians?

RealUniv vs LambdaCDM

RealUniv is a cosmological model based on a Coulomb interaction between protons and electrons on small scales according to RealQM/Nucleus, from which Newtonian gravitation on large scales emerges. All created from an initial small scale fluctuation of an electric potential. No Big Bang, no inflation, no strong/weak force, just Coulomb + Newton in a 3d Euclidean space equipped with a Laplacian differential operator. 

Check out details on GitHub Gallery with easy to read essay and and technical article. Compare with the the standard model LambdaCDM with CMB as key evidence.

The Mantra of Standard Quantum Mechanics Deconstructed to Nil

A modern physicist educated in quantum mechanics, speaks about a wave function $\Psi (x,t)$ depending on a $3N$-dimensional spatial variable $x$ for an atomic system with $N$ electrons, and a time variable $t$, evolving in time according to the Schrödinger equation 

• $i\frac{\partial\Psi}{\partial t}+H\Psi = 0$.      (S)

where $H$ is a Hamiltonian operator acting on $\Psi$. Given an initial state at $t=0$ a physicist can predict the state of any later state by time-stepping (S) from one instant of time to the next using (S).

So has time evolution of the wave function become the mantra of modern physics. In the article Unspeakable Quantum Mechanics we deconstruct this mantra and show it is empty and so misleading. Quantum mechanics is not about evolving (S) at femto/attoseconds rate of time, which is anyway impossible to compute. Read and contemplate.