AI Calls for the Reeducation of Mathematics Teachers — But Is It Possible?

AI is fundamentally changing mathematics education. When AI can explain mathematical concepts, solve equations, prove theorems and construct computational models, the traditional role of the mathematics teacher as a provider of mathematical knowledge and techniques is rapidly losing its justification.

What remains for the mathematics teacher?

A new role is emerging: to guide students in mathematical investigations, identify meaningful problems, connect mathematics with the sciences, critically examine results and distinguish mathematical understanding from mere mathematical production.

The teacher of yesterday teaches students mathematics. The teacher of tomorrow does mathematics together with students.

Can teachers educated for the old role be reeducated?

But there is a fundamental difficulty: Can mathematics teachers educated for the old role be reeducated for the new one?

A university mathematics professor is the product of perhaps twenty or thirty years of education and research within a system emphasizing specialized knowledge, analytical techniques and the proof of theorems. Professional identity, academic prestige and career advancement are built upon these achievements.

AI now challenges the very foundations of this system.

A professor who has spent a lifetime mastering a particular mathematical technique may discover that AI can perform it in seconds. A first-year student equipped with AI may explore mathematics far beyond the professor's specialization.

The necessary reeducation is thus not simply a matter of learning to use AI. It requires a fundamentally different conception of mathematics and mathematics education.

The teacher must learn to move across mathematical disciplines, integrate computation with analytical reasoning, engage with scientific problems and participate in investigations without knowing the answers in advance.

Consider a professor of differential equations who has spent a career proving existence theorems but has never computed a solution of the Navier–Stokes equations. Can this professor be transformed into a teacher who guides students in constructing their own fluid solvers with AI and investigating the emergence of turbulence?

Possibly. But it requires more than acquiring new skills. It requires questioning the values and priorities that have shaped an entire professional life.

Who will organize the reeducation?

And who will organize this reeducation? The very mathematics departments whose culture and traditions are being challenged by AI?

There is a real risk that the response will be to defend the old system: restrict AI, insist on traditional examinations and continue teaching mathematical techniques whose practical importance is rapidly diminishing.

The alternative is to embrace AI as an opportunity to reunite mathematics, computation and the sciences, and to transform the classroom into a laboratory of mathematical discovery.

The reeducation of teachers must itself take the form of the new education: teachers must undertake AI-assisted mathematical investigations, explore unfamiliar territory and experience the intellectual freedom now available to their students.

Some teachers will welcome this transformation. Others may find it difficult or even impossible. Universities may have to educate and recruit an entirely new kind of mathematics teacher.

The decisive question

The decisive question for mathematics education after AI may not be whether students can learn mathematics in a new way. It may be whether mathematics teachers can.

And if they cannot, will mathematics education have to wait for a new generation of teachers — or will students simply move ahead without them? 

Yang–Mills vs RealNucleus: Strong Force vs Coulomb

The gauge theory of the Standard Model meant to explain why nuclei hold together has not yet been shown to exist as mathematics, and it does not compute a nuclear binding energy; a Coulomb model with no strong force exists because it runs, and it computes one. That number can be checked, and it comes out at about 87%.

Where Yang–Mills came from

In 1954 Chen Ning Yang and Robert Mills wrote down a generalisation of Maxwell's electrodynamics in which the field carries its own charge. The aim was the strong interaction between protons and neutrons: isospin made into a local gauge symmetry, with the nuclear force carried by the new field. Pauli raised the objection at once. The quanta of such a field are massless, so the force should reach to infinity, but the nuclear force dies off within a few femtometres.

Twenty years later the theory came back as Quantum Chromodynamics (1973), Yang–Mills with gauge group SU(3), acting on quarks. In this version the force between nucleons is no longer fundamental. It is a residue of the colour force, much as the van der Waals force is a residue of electrostatics. Asymptotic freedom earned a Nobel prize, and at collider energies the theory is superb.

The Millennium Problem: does the theory exist?

In 2000 the Clay Institute listed Yang–Mills existence and mass gap among its seven Millennium Problems. Jaffe and Witten's statement asks for a proof that, for any compact simple gauge group, a quantum Yang–Mills theory exists on four-dimensional spacetime and has a mass gap: a strictly positive energy for the lightest excitation.

Take the two halves separately.

• Existence. Nobody has constructed the quantum theory in four dimensions as a mathematical object satisfying the standard axioms. The theory used every day is a set of perturbative rules plus a lattice regularisation whose continuum limit is assumed, not proved.
• Mass gap. The gap is what Pauli asked for in 1954. A gap means a short range, and a short range is what makes the nuclear force nuclear. Seventy years on it is still a conjecture, supported by lattice numerics and not proved.

So the property that would make Yang–Mills a theory of short-range nuclear binding is the property that is open.

What it delivers on nuclei

Grant the theory anyway and ask for the number it was introduced to explain, the binding energy of a nucleus.

• Lattice QCD has reached light nuclei only at unphysical quark masses, with pions several times heavier than real ones. The extrapolation back to the physical point is not controlled, and different groups have disputed even whether two-nucleon systems bind at those masses.
• Chiral effective field theory, the working theory of nuclear forces since Weinberg (1990), is fitted. Its low-energy constants are read off nuclear data, and then nuclear data are reproduced.

Seventy years after Yang–Mills and fifty after QCD, there is no parameter-free computation of the deuteron or the alpha from the theory.

RealNucleus

RealNucleus asks whether the binding can come from the force everyone agrees is there. It has protons and electrons as charge clouds on non-overlapping domains, the same free-boundary construction RealQM uses for atoms, interacting by Coulomb's law only. There is no strong force and no weak force. A neutron is a proton–electron pair, so a deuteron is two protons held by one electron. Binding comes from dual confinement: the electron clouds hold the protons together against their repulsion, and the protons cage the electrons against theirs.

The model is computable. It runs on a GPU in a web browser, so its numbers can be checked against measurement:

• One scale, fixed on the deuteron (2.22 MeV). Nothing else is fitted.
• The alpha comes out at about 87% of its measured binding, with an alpha/deuteron ratio of about 11 against 12.7.
• The model neutron (1p + 1e) is computed unbound, so free-neutron decay into a proton and an electron follows from the same Coulomb energy, with no weak force.
• Binding does not run away under compression. Scaling the configuration down passes through an interior minimum, so these nuclei have a preferred size chosen by the variational principle rather than imposed.

A correction, made openly. Earlier posts and papers in this series gave the ratio as 13.1 and the alpha at 103%. Those were single-mesh numbers. On a four-mesh ladder the deuteron settles about 27% deeper, and the ratio levels off near 11. The heavier rows of the ladder (⁸Be, ¹⁶O) are still one arrangement selected from a scan, not mesh-converged, and are provisional. That is the cost of a model that computes: when its numbers are wrong, the error can be found and has to be reported.

The comparison

	Yang–Mills / QCD	RealNucleus
Exists as mathematics	open (Clay problem)	yes: it is an algorithm, and it runs
Short range	needs the mass gap, which is unproven	electron glue localised into alpha-sized packets (2e + 4p)
Parameter-free nuclear binding	none	alpha ~87%, ratio ~11 vs 12.7, one scale
Inputs for nuclei	quark masses, plus fitted constants in practice	proton and electron charge; deuteron fixes the unit
What sets nuclear size	the gap (unproven)	open: the model's own unsolved problem

The last row is where the two meet. Both put the hard question in the same place: what sets the nuclear scale. Yang–Mills puts it in a mass gap that has not been shown to exist. RealNucleus finds an interior size minimum but cannot yet predict the size law R ∝ A^1/3. One of the two can be tested against measurement as it stands.

The point

This is the distinction drawn in an earlier post, The Algorithm Is the Proof. A Millennium Problem that asks whether a theory exists is asking for an existence theorem. An algorithm that runs needs no such theorem, because each run exhibits a solution.

Running is not the same as being right. RealNucleus is at 87%, not 100%, and its heavier nuclei are not yet derived. But its errors can be measured, and the measurement has already corrected it once. Seventy years after Yang and Mills, the theory chosen to explain the nucleus still owes both a proof that it exists and a computed nuclear binding energy. That justifies asking how much of the nucleus Coulomb's law already explains.

The model, the mesh ladder and the simulations are in RealNucleus v4 and at claes542.github.io/RealMolecule. 

Fluid Mechanics Education After AI: From Equations to Discovery

A fundamental change in how fluid mechanics can be taught 

(or any technical subject)

A traditional university course in fluid mechanics introduces the Navier–Stokes (NS) equations as the mathematical foundation of the subject, but rarely allows students to solve them computationally themselves.

Instead, students learn simplified analytical solutions, boundary-layer theory, dimensional analysis, empirical correlations and turbulence models. Computational fluid dynamics (CFD), if included, often means using software developed by others.

The student learns about fluid mechanics without independently computing the fluid flows described by its fundamental equations.

Artificial intelligence now makes a radically different approach possible.

Start with Navier–Stokes

A modern introductory course can begin directly with the incompressible Navier–Stokes equations:

• $\frac{\partial u}{\partial t}+u\cdot\nabla u+\nabla p =\nu\Delta u+f$
• $\nabla\cdot u=0$.

Instead of spending weeks preparing students for equations they may never solve themselves, the course immediately asks:

Can you construct a computer program that solves these equations?

With the help of AI, students can develop their own numerical solvers, including discretization, time stepping, pressure computation, boundary conditions and visualization.

What previously required substantial programming experience and specialized numerical expertise can now become an introductory exercise.

Students no longer need years of preparation in numerical analysis and programming before attempting their first NS solver. AI can provide much of the technical assistance, allowing students to concentrate on the mathematical and physical problems.

The important point is that students do not merely operate software. They construct it, inspect it, modify it and test it.

From computation to understanding

Once equipped with their own NS solvers, students can explore fundamental phenomena:

• Laminar flow in pipes and channels.

• Flow separation around obstacles.

• Vortex formation and vortex shedding.

• Transition from laminar to turbulent flow.

• Energy transfer and dissipation.

• The influence of viscosity and Reynolds number.

They can change geometry, initial data, forcing and viscosity, and observe the resulting flows.

They can investigate which computed features are robust under mesh refinement and which may be numerical artifacts.

The equations are no longer abstract formulas in a textbook. They become instruments of exploration.

Students write their own fluid mechanics textbook

The next step is even more radical.

Instead of reproducing explanations from a textbook, students use their own simulations to develop their own descriptions of fluid phenomena.

Each student becomes an investigator who asks questions, performs computational experiments, interprets results and formulates conclusions.

AI assists with programming, debugging, mathematical analysis and presentation, but the student remains responsible for the scientific questions and the credibility of the results.

The final examination can take the form of a personal computational fluid mechanics report, supported by reproducible simulations.

The student does not merely learn established explanations but constructs and tests explanations.

A fundamental difference

Traditional education typically follows the sequence:

Theory → Simplification → Textbook Examples → Examination

AI-supported computational education can instead follow:

Fundamental Equations → AI-Assisted Solver → Computational Experiments → Understanding → Explanation

This is not simply a more efficient way of teaching the same material.

It changes the student's relation to knowledge.

In the traditional course, knowledge is primarily something transmitted by the teacher and textbook.

In the new course, knowledge is something students actively produce and critically examine.

Commercial CFD software offers powerful computational capabilities, but its internal algorithms may remain inaccessible to students. An AI-assisted solver developed by the student can instead be completely open to inspection and modification.

There is, of course, an essential distinction between generating code and obtaining trustworthy numerical solutions. Stability, convergence, conservation and validation must be central parts of the course. AI cannot replace these requirements, but it can help students investigate them.

What remains for the teacher?

The teacher's role becomes more demanding rather than less important.

Instead of primarily explaining established solutions, the teacher guides students in asking meaningful questions, designing numerical experiments and distinguishing physical phenomena from computational artifacts.

The teacher brings mathematical and physical judgment that cannot be replaced by simply producing executable code.

Students learn not only how to obtain computational results, but also when and why those results deserve to be believed.

A new beginning for fluid mechanics

For more than a century, the Navier–Stokes equations have been presented as the foundation of fluid mechanics, while their direct computational exploration has largely remained outside introductory education.

AI changes this situation.

A student can now begin with the fundamental equations, construct a working solver and use it to discover the subject through computation.

The ambition is not necessarily to reproduce the capabilities of industrial CFD or to resolve fully developed turbulence at arbitrary Reynolds numbers. It is to make the mathematical foundation of fluid mechanics computationally accessible from the beginning.

This may be the beginning of a broader transformation of scientific education, in which AI makes constructive computational mathematics accessible to every student.

The textbook is no longer the starting point and final authority.

The equations are the starting point. Computation is the laboratory. Understanding is the goal.

The revolution is not that AI can solve the equations for students, but that AI enables students to solve the equations themselves—and thereby construct their own understanding of fluid mechanics.

Unresolved Problems of Quantum Mechanics Waiting for AI

I asked chatGPT to write a post on unresolved problems in QM and this is what came out:

Artificial Intelligence is now attacking major unsolved problems in mathematics. But what about quantum mechanics (QM), where fundamental problems have remained unresolved for a century?

QM is celebrated as the most successful theory of physics. Yet its mathematical formulation is notoriously difficult to compute, its physical interpretation remains disputed, and many complex phenomena cannot be predicted directly from its fundamental equations without substantial additional modeling.

Is quantum mechanics waiting for an AI breakthrough?

Consider ten challenges:

1. Many-electron systems: Can the exponential complexity of conventional many-body wavefunctions be avoided?

2. The periodic table: Can the sequence 2, 8, 8, 18, 18, 32, ... emerge constructively from electron–nucleus interactions?

3. Chemical bonding: Can formation and breaking of chemical bonds be computed dynamically in three-dimensional space?

4. Electrical conduction: Can conductivity and resistance be computed directly from microscopic electron dynamics?

5. Superconductivity: Can unconventional superconductivity be explained and predicted from first principles?

6. Measurement: How does a quantum wavefunction produce a definite observed outcome?

7. Classical mechanics: How does classical physics emerge from quantum dynamics?

8. Dissipation: How does irreversible macroscopic behavior arise from reversible microscopic equations?

9. Protein folding: Can folding dynamics be computed directly from atomic interactions?

10. Electromagnetic coupling: Can electron dynamics and electromagnetic fields be computed together efficiently?

These are not all completely unexplained phenomena. Standard QM offers powerful partial answers and successful approximations. What remains missing is a generally applicable, computationally feasible description connecting microscopic equations directly to complex physical observations.

AI Mathematics Versus AI Physics

AI mathematics starts from given equations and seeks proofs of mathematical statements.

AI physics faces a more fundamental task: starting from observed phenomena, discover equations and computational algorithms capable of explaining and predicting them.

AI mathematics asks: Can we prove a theorem within an existing theory?

AI physics asks: Can we construct a better theory of physical reality?

The second question may be more consequential than the first.

RealQM: A Challenge for AI

Real Quantum Mechanics (RealQM) proposes an alternative computational foundation based on electron charge densities evolving in ordinary three-dimensional space according to time-dependent partial differential equations with moving free boundaries.

Its objective is to replace the exponentially complex configuration-space description of standard many-electron QM by a constructive model with computational work scaling linearly with the number of electrons. 

To see what RealQM delivers on challenges 1-10 go to The World as Computation. Discover that all 1-10 are addressed. Give your evaluation in comments below.