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.

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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. 

Real Physics without Philosophy of Physics

There is an extensive literature on philosophy of physics developed to compensate for the fact that standard quantum mechanics does not come with an ontology of what exists, which is fundamental in classical physics:

• Real Physics without Philosophy of Physics 

Radioactive decay in RealQM

# Radioactive decay in RealQM: an honest excursion into time-dependent charge densities

Radioactive decay is the textbook poster child of quantum randomness. A nucleus sits there for  a microsecond or ten billion years and then, for no reason anyone can point to, it decays. Standard quantum mechanics says the moment is *irreducibly* random — uncaused, only its probability defined. So it is a fair question to put to RealQM, which describes matter not as probability amplitudes but as **charge densities evolving deterministically in ordinary three-dimensional space**: can a deterministic, real-space theory say anything sensible about decay?

We spent a long, disciplined excursion finding out. Here is the honest ledger — including, and especially, the parts that didn't work.

## Two decays, two verdicts

**Alpha decay is the clean case, and RealQM handles it fully.** An alpha particle (a ⁴He nucleus, charge +2) tunnels out through the daughter's *Coulomb* barrier. It is a genuine two-body decay: no weak force, no neutrino, and a sharp, *monoenergetic* alpha line whose very sharpness is the proof that no third body is emitted. Everything the process needs — extended charge, a Coulomb barrier, two-body kinematics

lives inside RealQM. And there is a genuinely RealQM-specific result underneath it: the binding of the whole alpha-cluster ladder (⁴He, ¹²C, ¹⁶O, … ⁴⁰Ca) comes out at ~107% of experiment **from Coulomb alone, with no strong force**, one scale fixed on the deuteron. Alpha decay is where RealQM is at home.

**Beta decay is where the charge-density picture ends — and we say so.** It was tempting to claim beta decay *without* a neutrino: RealQM conserves energy by construction, so maybe the continuous electron spectrum is just the conserved energy being partitioned among the electron, the recoil, and the radiated field. We tested that quantitatively. It fails. The antineutrino carries, on average, about **60% of the released energy** and the momentum imbalance; the field a charge can radiate is smaller by two orders of magnitude (the known inner-bremsstrahlung level, ~α). The recoil is negligible. So the neutrino is *not* removed — and the honest reason is deep: the neutrino is **chargeless**, and a charge-density theory simply has no object of that kind. Beta decay marks the boundary of the program, and the paper marks it plainly.

## The half-life, three ways — and no WKB

Here is the part that genuinely worked. Textbook alpha lifetimes span **twenty-five orders of magnitude**

(²³²Th at 10¹⁰ years, ²¹²Po at a fraction of a microsecond), and Gamow's 1928 WKB barrier factor famously

reproduces that Geiger–Nuttall law. But WKB is a semiclassical shortcut. Does the *full* time-dependent

RealQM give the half-life directly?

It does. Evolve a metastable charge behind a barrier in **real complex time** (the same solver as the static

relaxation, only the imaginary-time step swapped for a unitary one): the trapped charge decays

**exponentially**, and the half-life is read straight off the dynamics. Sweep the barrier and log t½ stays

linear in √(V−E) — Geiger–Nuttall, from first principles. The narrow, long-lived resonances that real-time

propagation can't reach come exactly from the **complex-energy (Siegert) width**. All three routes agree,

and none uses WKB — which is thereby *validated*, not relied upon. You can watch it happen in the browser:

the charge tunnelling through the barrier while the half-life emerges live.

## Determinism — and the mechanism that died

The most seductive idea was determinism. If RealQM is a deterministic theory, then decay isn't *really*

random — it only looks random because we don't know the exact initial state. That is the century-old

de Broglie–Bohm position, and RealQM carries it naturally: the whole history of a decaying configuration,

tunnelling included, is fixed by its **initial charge configuration**; the apparent randomness of

identical-looking nuclei decaying at different times is *epistemic*, our ignorance of that configuration.

We then reached for something sharper: coexisting charge domains, each carrying a phase clock

e^(−iEₖt/ℏ), with the escape *gated* by the coincidence of their phases — a deterministic mechanism

producing the exponential law as the statistics of a coincidence. It was a lovely picture. **It is also

wrong**, and tracing it to the end is what the excursion was really about.

The refutation is clean. In the full time-dependent RealQM, the escaping domain feels its neighbours *only*

through their **densities** |ψⱼ|², which are phase-invariant; the free boundaries carry **zero flux**. So the

neighbours' phase clocks never reach the escaping domain — the moving free boundary transmits *density, not

phase*. The decay is plain Gamow tunnelling; there is no phase gating. To manufacture gating you would have

to bolt on a **phase-permeable (Josephson) interface** — a thin overlap and a new coupling the variational

free boundary does not give — and it is *unnecessary* anyway, because the density dynamics already carry the

decay and its half-life. So we dropped it. The determinism survives (it's an interpretation); the mechanism

does not.

## So what did the excursion actually net?

No spin: **we did not find new decay physics.** The decay rate is barrier penetration, the same physics

standard quantum mechanics gives. What the full time-dependent RealQM brings, for decay, is *ontological* —

a deterministic, real-space charge-density picture in place of amplitudes and collapse — and *diagnostic*:

it was the tool that let us test and **rule out** the tempting overclaims. The science ended up being in

what we subtracted.

And that is the point worth keeping. Each attractive story — beta without a neutrino, deterministic

phase-coincidence gating — looked good until it was pushed hard, and pushing it turned it into either a

clean negative result or "it's just tunnelling." That is not a failure. It is how you end up with two papers

that claim exactly what is true and nothing more: alpha decay as deterministic Coulomb-barrier tunnelling

with the neutrino nowhere in sight; beta decay honest about the chargeless carrier it cannot supply; the

half-life captured without WKB; and the phase mechanism named, tested, and set aside.

RealQM's real power was never in single-particle escape dynamics — it is in the **static, multi-domain**

world of binding and geometry, where non-overlapping charge domains do genuine work. The one clean theory

question this excursion surfaced is the **correct time evolution of a free boundary** — advection by the

charge-fluid velocity together with a Bernoulli condition — which we identified but did not yet derive.

That, not a new decay law, is the thread worth pulling next.

 

RealQM and the Realist Quest Carried by Schrödinger

The following article has been submitted to Synthese as a journal for philosophy of science:

• Charge Densities in Real Space RealQM and the Realist Quest Carried by Schrödinger

The article compares standard quantum mechanics as probability amplitudes of configurations of N point-like particles over 3N dimensional configuration space, with RealQM as N charge densities in real 3d space interacting by Coulomb potentials.  

RealQM fulfills the quest of Schrödinger for a model of atomic physics of the same form as macroscopic continuum mechanics. The philosophical conundrums of standard quantum mechanics simply evaporate in then light of RealQM and the sense of Wittgenstein.