AI and the Realization of Leibniz’s 350-Year-Old Dream

From Mechanical Calculation to Mechanical Reasoning, 1673–2026

The AI revolution of 2026 can be viewed as the realization of a remarkable vision formulated 350 years ago by the German mathematician and philosopher Gottfried Wilhelm Leibniz (1646–1716).

Leibniz envisioned two forms of automation:

1. Automated computation: A machine performing arithmetic calculations without human effort.

2. Automated reasoning: A machine performing logical deductions without human intervention.

Leibniz understood that calculation and reasoning are closely related. Both consist of operations performed according to definite rules. If these rules could be expressed symbolically, machines might execute them.

His ambition was nothing less than the mechanization of human intellectual activity.

Today, 350 years later, AI is bringing these two ambitions together.

1. Leibniz's Mechanical Calculator — 1673

In 1673 Leibniz demonstrated a mechanical calculating machine in London, an early version of his Stepped Reckoner, designed to perform addition, subtraction, multiplication and division.

The central innovation was the stepped drum, which allowed repeated arithmetic operations to be executed mechanically.

The machine was limited and mechanically unreliable, but its principle was revolutionary:

Arithmetic could be transferred from the human mind to a machine.

Leibniz also investigated binary arithmetic, recognizing its suitability for representing numbers using only 0 and 1.

Three centuries later, binary arithmetic would become the foundation of electronic computers.

2. Leibniz's Calculus of Reasoning — 1679–1700

Leibniz's second ambition was even more radical.

He proposed a universal symbolic language, characteristica universalis, in which concepts and propositions could be represented precisely.

Combined with a calculus ratiocinator, this would allow logical disputes to be settled by calculation.

His famous imagined invitation was:

Calculemus! — Let us calculate!

Instead of arguing endlessly, two philosophers could sit down and calculate which conclusions followed from their assumptions.

The essential idea was that reasoning, like arithmetic, could be reduced to symbolic operations governed by rules.

Leibniz did not construct a working general reasoning machine. His vision remained largely a philosophical and mathematical program.

But the program survived.

3. The Evolution of Automated Computation and Reasoning

The following milestones trace the convergence of Leibniz's two ambitions.

1673 — Leibniz's mechanical calculator

Arithmetic becomes a mechanical procedure.

1679–1700 — Leibniz's symbolic logic

The vision of automated reasoning is formulated.

1847–1854 — George Boole

Logical propositions become objects of algebraic calculation. Logic acquires a mathematical structure.

1837 — Charles Babbage's Analytical Engine

Charles Babbage designs the Analytical Engine, the first detailed conception of a general-purpose programmable mechanical computer. Unlike earlier calculators performing fixed arithmetic operations, it is designed to execute sequences of operations controlled by punched cards, with separate memory (store) and processing unit (mill).

In 1843 Ada Lovelace describes how the machine could manipulate symbols according to rules, not merely calculate numbers. She recognizes that a machine could operate on mathematical relationships more generally.

The Analytical Engine is never completed, but it establishes the conceptual bridge between Leibniz's mechanical calculator and the modern programmable computer.

The decisive step: From a machine that calculates to a machine that follows a general program.

1879 — Gottlob Frege

Modern predicate logic provides a systematic symbolic language for mathematical reasoning.

1910–1913 — Russell and Whitehead

Principia Mathematica attempts to derive mathematics through formal logical deduction.

1931 — Kurt Gödel

The incompleteness theorems demonstrate fundamental limitations of sufficiently expressive formal systems. Not every mathematical truth can be established within a single consistent, effectively axiomatized system.

1936 — Alan Turing

The universal Turing machine provides a mathematical model of general computation. Turing also proves that no algorithm can decide every mathematical question expressible in a sufficiently general formal language.

1940s — Electronic computers

Mechanical calculation is replaced by electronic computation, making enormous sequences of arithmetic and logical operations practically possible.

1956 — Artificial intelligence

The Dartmouth project establishes AI as a research field, explicitly pursuing the mechanization of aspects of intelligence.

1960s–1980s — Automated theorem proving

Computers begin constructing formal mathematical proofs using logical inference rules.

1970s–2000s — Numerical computation

Computational mathematics transforms science and engineering through numerical solutions of differential equations, optimization and simulation.

1990s–2010s — Machine learning

Computers increasingly learn patterns and procedures from data rather than relying exclusively on explicitly programmed rules.

2017 — Transformers

A new neural-network architecture makes it possible to train powerful models on enormous quantities of language and other structured information.

2022–2025 — Large language models

AI develops increasingly sophisticated capabilities in mathematical reasoning, programming and scientific problem-solving.

2026 — AI mathematical reasoning

Advanced AI systems can produce substantial mathematical arguments, assist with formal verification, construct algorithms and solve selected research problems.

The two ambitions of Leibniz are now converging.

4. The Reunion of Computation and Reasoning

For most of the twentieth century, mathematical computation and mathematical reasoning developed along largely separate paths.

Numerical mathematics used computers to calculate approximate solutions to equations describing physical reality.

Formal mathematics used symbolic logic to derive theorems from axioms.

The former produced numbers and simulations. The latter produced propositions and proofs.

AI is now beginning to bridge this division.

A modern AI system can formulate a mathematical problem, propose a method of solution, construct an algorithm, execute computations using appropriate tools, analyze the results and develop a mathematical argument explaining them.

It can move back and forth between symbolic reasoning and numerical computation.

This is precisely the kind of integration Leibniz anticipated, although the technology is radically different from what he could have imagined.

Importantly, modern AI does not simply follow a fixed list of logical inference rules. Neural language models generate candidate reasoning steps through learned computational structures. Their mathematical conclusions still require verification, whether by proof checking, independent analysis or comparison with computational evidence.

The distinction between producing an argument and establishing its correctness remains essential.

5. Mathematics Improving Mathematics

The development has an intriguing recursive character.

Mathematics made computers possible.

Computers made modern AI possible.

AI is now being used to develop new mathematics, including mathematical methods that can improve computation and AI itself.

Thus we obtain a feedback loop:

Mathematics → Computation → AI → New Mathematics → Improved Computation → Improved AI → ...

The process suggests a new form of mathematical development in which machines are not merely instruments used by mathematicians but increasingly active participants in mathematical discovery.

The speed and ultimate limits of this process remain unknown.

Nevertheless, the possibility of such a feedback loop marks a fundamental change in the relationship between mathematics and computation.

6. What Remains for Human Mathematicians?

Leibniz imagined that machines could relieve human beings of the burden of calculation and routine logical deduction.

He did not imagine that mathematical questions would thereby disappear.

On the contrary, automation could free human intelligence to concentrate on choosing meaningful problems, formulating models of reality and interpreting mathematical results.

This distinction is becoming increasingly important.

If AI can perform much of the technical work of mathematical reasoning and computation, the central human mathematical activity may shift from proving given statements toward deciding:

• Which questions are worth asking?

• Which assumptions adequately describe reality?

• Which mathematical models are meaningful?

• Which computed results can be trusted?

• Which conclusions advance scientific understanding?

These questions cannot be settled by formal deduction alone because they concern the relation between mathematics and the world.

This has profound implications for mathematical education, particularly in science and engineering.

An education centered on mastering procedures that machines can execute may need to give way to one centered on mathematical modeling, computational experimentation, critical verification and scientific understanding.

7. Leibniz's Dream Comes True

In 1673 Leibniz presented a machine that could calculate.

He subsequently envisioned a symbolic system through which machines might also reason.

For 350 years these two ambitions developed through mechanical calculators, symbolic logic, formal mathematics, electronic computers, numerical analysis, automated theorem proving and machine learning.

Today they are meeting in AI.

Leibniz's dream has not been completely realized. General mathematical reasoning remains fallible, and fundamental limits to formal deduction and computation remain.

But the decisive transition is underway:

Machines are no longer restricted to carrying out calculations devised by human mathematicians. They can increasingly participate in devising the mathematics itself.

What Leibniz proposed as a philosophical vision 350 years ago is becoming an operational reality.

The consequences for mathematics, science and education may be as profound as those of the invention of calculus itself.

And perhaps the most remarkable aspect is that the mathematical enterprise has now produced machines capable of extending that very enterprise.

Leibniz's Calculemus has acquired a new meaning: Let us calculate—and let the machine reason with us. 

Öppet brev till Sveriges utbildningsministrar

 Till Simona Mohamsson och Lotta Edholm

AI kräver en genomgripande reform av matematikutbildningen

Matematik är ett av skolans viktigaste ämnen och en grundpelare i naturvetenskap och teknik. Men vad ska vi lära våra barn och ungdomar i matematik när artificiell intelligens nu visar förmåga att utföra även avancerad matematisk forskning?

AI kan inte bara räkna, lösa differentialekvationer och genomföra numeriska simuleringar, utan också konstruera avancerade matematiska bevis och bidra till nya matematiska resultat. Uppgifter som tidigare krävde mångårig specialutbildning kan i växande omfattning utföras av AI på mycket kort tid.

Detta innebär en fundamental utmaning för hela matematikutbildningen, från grundskola till universitet och forskning.

Dagens matematikutbildning är uppbyggd som en hierarki där grundskolan förbereder för gymnasiet, gymnasiet för universitetet och universitetet för avancerad matematisk forskning. Högst upp står förmågan att självständigt utveckla matematiska teorier och bevis.

Men vad händer med denna hierarki när AI kan utföra en stor del av arbetet på dess högsta nivå?

Frågan är inte om matematik behövs i framtiden. Tvärtom kan matematikens betydelse för vetenskap, teknik och samhälle bli större än någonsin. Frågan är vilken matematik människor behöver kunna och hur den bäst lärs ut.

En reform måste börja med universitetens matematikutbildning, särskilt inom naturvetenskap och teknik. Tyngdpunkten bör förskjutas från omfattande träning i symbolisk manipulation och formella bevis till matematisk modellering, beräkning, simulering, problemlösning och kritisk verifiering av resultat, med AI som ett naturligt hjälpmedel.

En sådan förändring skulle få konsekvenser för gymnasiets och grundskolans kursplaner. Elever skulle tidigare kunna möta matematikens användning för att förstå verkliga fenomen, utan att behöva passera alla traditionella steg av avancerad symbolisk teknik.

Grundläggande räknefärdighet, logiskt tänkande och matematisk förståelse förblir självklart nödvändiga. Men deras plats i utbildningen måste bestämmas utifrån framtidens behov, inte enbart utifrån traditionen.

Jag föreslår därför att regeringen tar initiativ till en nationell översyn av matematikutbildningen på samtliga nivåer med utgångspunkt i AI snabbt växande matematiska förmåga.

Översynen bör genomföras med deltagande av matematiker, naturvetare, ingenjörer, lärare och AI-forskare och särskilt undersöka hur matematisk modellering och beräkning kan ges en mer central roll.

Sverige har möjlighet att bli ett föregångsland i en reform som förr eller senare kommer att beröra utbildningssystem över hela världen, och därmed stärka sin position som kunskapsnation i en allt hårdnande internationell konkurrens.

Det vore olyckligt om vi fortsatte att utbilda kommande generationer för en matematisk arbetsfördelning som AI redan håller på att förändra i grunden.

Matematikens framtid är inte hotad. Men dess utbildning måste förändras. 

Jag föreslår ett personligt möte för inledande diskussion, detta snarast möjligt eftersom utvecklingen av AI nu går mycket snabbt.

Ett reformprogram är redan utvecklat och färdigt att iscensättas: 
https://mathsimulationtechnology.wordpress.com/

Vänliga hälsningar

Claes Johnson
prof em tillämpad matematik KTH
073-4356600

Mathematics Education After AI: Collapse or Renaissance?

What will happen to math education on all levels when the top has shown to be a house of cards which can be run by AI.

If AI can perform mathematics at the highest research level, including discovering and proving new theorems, then the entire educational structure of mathematics comes into question. The traditional system is hierarchical:

1. Primary school: Arithmetic and elementary geometry.
2. Secondary school: Algebra, functions, calculus and elementary proofs.
3. University: Advanced calculus, linear algebra, differential equations and abstract mathematics.
4. Research: Creation of new theorems through increasingly sophisticated logical reasoning.

The hierarchy rests on the assumption that advanced mathematical reasoning is a rare human intellectual achievement, requiring many years of training.

AI challenges this assumption at its foundation.

1. The top loses its traditional authority

If AI can produce sophisticated proofs in minutes rather than months or years, then the ability to construct proofs loses much of its scarcity value.

This does not mean that mathematics is false or that its theorems are worthless. It means that the traditional organization of mathematical research around human proof production may lose its central purpose.

The analogy is with arithmetic after the calculator, but potentially on a much greater scale.

However, the claim that AI can reliably perform all research mathematics remains unproven. Its successes are substantial but uneven, and independent verification remains essential.

2. University mathematics faces the greatest disruption

Today's university mathematics largely trains students to perform operations that AI is increasingly capable of performing:

• Symbolic manipulation and calculation.
• Solving differential equations.
• Constructing mathematical proofs.
• Applying standard mathematical theories.
• Developing numerical algorithms.

If these tasks become automated, why devote several years to mastering their manual execution?

The alternative is to organize mathematics around mathematical modeling, computation and critical verification.

Students would learn to formulate problems from science and engineering, construct mathematical models, compute solutions, assess their validity and understand their physical meaning.

The emphasis shifts from proving theorems to understanding and using mathematics.

This is closely related to the computational and constructive philosophy of the Body and Soul mathematics program.

3. Secondary education must change

Much of secondary mathematics prepares students for university mathematics.

If university mathematics changes fundamentally, that justification disappears.

The traditional progression through increasingly elaborate algebraic manipulation and symbolic calculus becomes questionable.

Instead, students could encounter differential equations, numerical simulation, optimization and mathematical modeling much earlier, using AI as an assistant.

The central educational question becomes:

What mathematical understanding must a human possess to formulate a meaningful problem and judge whether an AI-generated solution makes sense?

This requires substantial mathematical knowledge, but not necessarily the traditional curriculum.

4. Primary education remains essential

Arithmetic, geometry, proportional reasoning and elementary algebra remain foundational.

Children need to develop numerical intuition, spatial understanding and logical thinking.

But education need no longer be organized primarily as preparation for increasingly advanced symbolic manipulation.

AI could also provide individual mathematical instruction adapted to each student's understanding.

5. The institutional obstacle

The greatest resistance may come from universities.

Mathematics departments determine what mathematics students must learn, while secondary schools prepare students for university admission.

The entire educational system therefore reproduces the priorities of academic mathematics.

If advanced mathematical research becomes substantially automated, this hierarchy loses part of its justification.

Yet the institutions, examinations, textbooks and professional identities remain.

A transformation may therefore be intellectually necessary long before it becomes institutionally accepted.

6. Collapse or renaissance?

There are two possible outcomes.

Collapse: Mathematics education continues teaching techniques whose practical purpose has largely disappeared. Students increasingly delegate their assignments to AI, examinations lose their meaning, and motivation declines.

Renaissance: Mathematics education returns to its fundamental purpose: understanding the world through quantitative reasoning, mathematical modeling and computation.

The second possibility could make mathematics more accessible, meaningful and useful at every educational level.

Conclusion

The deepest consequence of AI may not be that machines become better mathematicians than humans.

It may be that mathematics education can no longer justify itself primarily by training humans to do what machines can do better.

The challenge is to replace the traditional hierarchy of increasingly sophisticated formal manipulation with an education centered on understanding, construction, computation and verification.

The house of cards that may fall is not mathematics itself, but the educational hierarchy built around the scarcity of human mathematical reasoning.

And its fall could open the way to a mathematical renaissance rather than the end of mathematics.

 

Calculus in 1726, 1826, 1926-2026 and Body & Soul (2003–2026)

Here are some high-lights of of university education in mathematics 1726-2026 done by chatGPT on the 

Prompt:

• Compare BodyandSoul material with typical calculus from 1726, 1826, 1926.

Check out additional prompt below connecting 1926 to 2026.

1. Four stages in the development of calculus

The history of calculus can be viewed as a succession of four approaches to mathematics:

1. 1726 – Newton and Leibniz: Calculus as a tool for understanding motion and change.
2. 1826 – Cauchy and Fourier: Calculus as a systematic analytical method based increasingly on limits and mathematical rigor.
3. 1926 – Weierstrass and Hilbert: Calculus as part of a rigorous mathematical structure, emphasizing definitions, theorems and proofs. This is essentially the standard of today.
4. 2003–2026 – Body & Soul: Calculus as a synthesis of mathematical modeling, analytical reasoning and numerical computation.

The dates represent characteristic intellectual traditions, not abrupt transitions.

The central question is whether Body & Soul represents a return to the original constructive purpose of calculus, now equipped with computers.

2. Comparison of content

Subject	1726	1826	1926-2026	Body & Soul
Numbers	Real quantities	Real numbers, limits	Rigorous real-number theory	Numbers and algorithms
Functions	Curves and formulas	General functions	Abstract function concept	Functions as mathematical and computational objects
Derivatives	Fluxions, infinitesimals	Limits of difference quotients	Rigorous differentiability	Derivatives, linearization, computation
Integrals	Areas, inverse differentiation	Definite integrals	Riemann and other integration theories	Integration and numerical quadrature
Linear algebra	Systems of equations	Determinants, matrices emerging	Vector spaces, matrices	Linear systems and numerical algorithms
Differential equations	Motion and mechanics	ODEs, heat equation	Existence, uniqueness, qualitative theory	Numerical solution, stability, adaptivity
PDEs	Early examples	Heat, waves, potential	Mathematical theory	Computational PDEs and FEM
Applications	Mechanics, astronomy	Mechanics, heat, elasticity	Applications plus increasing specialization	Modeling across science and engineering
Computation	Hand calculations	Tables and approximations	Hand calculations, mechanical aids	Computer-based numerical simulation
Proof	Geometrical and analytical argument	Increasing rigor	Central organizing principle	Proof combined with computation and error analysis

This comparison concerns the characteristic emphasis of each period, not everything mathematicians knew at the time.

3. The fundamental difference in educational philosophy

1726: Understanding the physical world

Newton's calculus developed in close connection with mechanics.

A typical problem was:

Given a force acting on a body, determine its motion.

The mathematical formulation is

\[ m\frac{d^2x}{dt^2}=F(x,t). \]

The objective is to determine the motion \(x(t)\).

Mathematics serves physics.

1826: Analytical solution

By 1826, Cauchy's work on limits and Fourier's work on heat conduction had transformed calculus.

A characteristic problem was solving

\[ \frac{\partial u}{\partial t} =\kappa\frac{\partial^2u}{\partial x^2} \]

using Fourier series.

The emphasis was on finding analytical representations of solutions and establishing their properties.

Mathematics was becoming a systematic discipline, although physical applications remained central.

1926-2026: Rigorous mathematical theory

By 1926, calculus education was strongly influenced by nineteenth-century rigor.

A typical theoretical problem was:

Prove that a continuous function on a closed interval attains its maximum.

The focus was on assumptions, definitions, logical deductions and general theorems.

However, an important distinction is necessary: ordinary engineering calculus in 1926 was still strongly computational and application-oriented. The abstract emphasis was more pronounced in advanced university analysis.

Body & Soul: Computable mathematical modeling

Body & Soul, developed by Kenneth Eriksson, Donald Estep and Claes Johnson, takes a different approach.

The objective is not simply to prove that a mathematical solution exists, or to find an analytical formula.

It is to construct an approximate solution, estimate its error and use it to understand or predict physical phenomena.

For example, given

\[ \dot u=f(u,t),\qquad u(0)=u_0, \]

one constructs

\[ U_{n+1}=U_n+h_nf(U_n,t_n) \]

or uses a more sophisticated numerical method.

The mathematical questions become:

• How can the problem be modeled?
• How can a solution be computed?
• How accurate is the computation?
• How does stability affect the result?
• How can computational work be controlled?

These questions connect mathematical analysis directly to numerical algorithms.

The Body & Soul volumes explicitly integrate computation into undergraduate calculus and linear algebra, followed by computational differential equations and applications including turbulence. Springer

4. What is genuinely new in Body & Soul?

Numerical computation itself is not new. Newton, Euler, Gauss and many others performed extensive numerical calculations.

Nor is the connection between mathematics and physics new.

The distinctive contribution of Body & Soul is the systematic integration of analytical and computational mathematics throughout the curriculum, starting at the elementary level.

Three changes are particularly important.

First: From formulas to algorithms.

Traditional calculus often seeks an explicit expression

\[ u(t)=\Phi(t). \]

Computational calculus instead seeks an approximation

\[ U(t)\approx u(t) \]

with quantitative control of the error.

Second: From existence to computability.

A theorem establishing

\[ \exists u:\quad F(u)=0 \]

does not necessarily tell us how to calculate \(u\).

Computational mathematics asks for an effective procedure producing \(U\) with

\[ \|u-U\|\leq \varepsilon. \]

This is a stronger practical requirement, although some constructive existence proofs already supply algorithms.

Third: From separate subjects to integrated modeling.

Calculus, linear algebra, differential equations, numerical methods and applications become components of a common methodology.

This integration is particularly evident in Body & Soul's treatment of adaptive differential equation solvers, finite element methods, inverse problems and Navier–Stokes equations. books.google.com

5. The historical interpretation

One can summarize the development as follows:

Period	Dominant question
1726	How does nature work?
1826	How can its equations be solved analytically?
1926-2026	What can be rigorously proved?
Body & Soul	What can be reliably computed?
2026 with AI	What can be modeled, computed and verified with AI assistance?

The last stage is especially interesting.

AI increasingly automates symbolic manipulation, mathematical reasoning and parts of mathematical proof construction.

This raises the possibility that traditional university mathematics education, centered on training students to perform such operations manually, will become less appropriate.

Body & Soul already provides an alternative foundation: understanding mathematical models, their numerical solutions, stability, approximation and reliability.

AI could extend this approach by automating not only numerical computation but also substantial parts of analytical reasoning.

6. Conclusion: Has calculus come full circle?

There is a striking historical symmetry:

1726: Calculus was developed to understand and predict nature.

1926: Much of higher mathematics had become organized around abstract mathematical structures and rigorous proofs.

Body & Soul: Calculus reconnects mathematical reasoning with quantitative prediction through computation.

2026: AI makes it possible to automate increasingly large parts of both symbolic reasoning and numerical computation.

This suggests a fundamental change in the purpose of mathematics education.

The student need no longer be trained primarily as a human symbolic calculator or theorem prover.

Instead, the student should learn to formulate mathematical models, understand their assumptions, evaluate computed solutions and judge their reliability.

Body & Soul can therefore be viewed not merely as a reform of twentieth-century calculus, but as a possible foundation for mathematics education in the age of AI.

The historical claim should nevertheless be qualified: calculus never completely abandoned applications or computation. What changed was the relative emphasis in mathematical research and education.

Sources

• Body & Soul, Volume 1 — Derivatives and Geometry
• Body & Soul, Volume 3 — Calculus in Several Dimensions
• Dreams of Calculus — Body & Soul
• Historical calculus textbooks and their development

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A further question worth examining: Is Body & Soul actually closer to the mathematical philosophy of Newton and Leibniz in 1726 than to the conventional calculus curriculum of 1926—or even 2026?

My assessment is that it is closer in its objectives, but radically different in its methods. 

New prompt:

• Would you describe the typical calculus course today as essentially that of 1926?

Yes, in its mathematical core and pedagogical structure, a typical university calculus course in 2026 is remarkably similar to one in 1926. But there are important qualifications.

Aspect	1926	2026
Main content	Limits, derivatives, integrals, series	Essentially the same
Functions	Elementary functions	Essentially the same
Fundamental theorem of calculus	Central	Central
Differential equations	Elementary ODEs	Elementary ODEs
Teaching	Lectures, textbooks, exercises	Lectures, textbooks, exercises
Student activity	Symbolic calculations by hand	Mostly symbolic calculations by hand
Examinations	Written problem solving	Mostly written problem solving
Numerical computation	Limited	Sometimes included
Computers	Absent	Calculators, software, occasionally programming
Mathematical modeling	Applications, often idealized	Applications, sometimes more extensive
AI	Absent	Emerging, with widely varying policies

What has changed?

The presentation is often more accessible, applications have broadened, and numerical software is available. Some courses incorporate programming, modeling and computational experiments.

But the basic sequence remains:

\[ \text{Limits}\rightarrow\text{Derivatives}\rightarrow \text{Integrals}\rightarrow\text{Series}. \]

Students are still commonly trained to differentiate, integrate and manipulate expressions by hand, although computers and AI can perform many of these operations.

Comparison with Body & Soul

Body & Soul proposes a different organizing principle:

\[ \boxed{\text{Model}\rightarrow\text{Compute}\rightarrow \text{Estimate error}\rightarrow\text{Validate}} \]

Here computation is not an optional supplement to calculus. It is integrated into the mathematical subject itself.

This represents a more fundamental change than introducing calculators or Python into a traditional course.

My assessment

I would distinguish three claims:

1. Content: Today's introductory calculus is largely inherited from nineteenth-century mathematics and was already standard by 1926.

2. Method: The dominant teaching and examination methods retain much of the 1926 tradition.

3. Purpose: This is where the strongest criticism arises. Training students primarily in manual symbolic calculation is increasingly difficult to justify when AI can perform those tasks.

Thus, one can reasonably describe much of today's university calculus as a 1926 curriculum taught with 2026 technology.

This is a broad characterization, not a verified statistical conclusion about universities. Establishing how representative it is would require examining current syllabuses, assignments and examinations.

The decisive question is whether universities have changed the mathematical work students are trained to do, rather than merely the tools available to them.

AI Reform of Mathematics Education: What Are Leading Universities Doing?

chatGPT:

As of October 2026, several major universities have launched initiatives to integrate AI into teaching. However, I find little evidence of a fundamental restructuring of undergraduate mathematics education comparable to the Body&Soul reform.

The distinction is important: using AI in existing courses is not the same as redesigning the mathematics curriculum because AI can perform the mathematics.

1. Massachusetts Institute of Technology (MIT)

In October 2026, MIT announced its MIT for America initiative, emphasizing mathematical thinking, practical engineering and constructive use of AI.

Its three priorities include mathematical problem-solving, AI education and hands-on design.

However, the initiative primarily targets education from school through community college, rather than restructuring MIT's own undergraduate engineering mathematics curriculum.

MIT announcement, October 6, 2026

2. Stanford University

In April 2026, Stanford announced a $1 million initiative, AI Meets Education at Stanford (AIMES).

By September, it had funded 12 curriculum projects affecting approximately 25 course offerings and an estimated 3,700 students.

The objective is to experiment with AI-based teaching, curriculum changes and assessment.

This is a substantial initiative, but not yet a university-wide replacement of traditional mathematics courses.

Stanford AI curriculum initiative

3. Georgia Institute of Technology

Georgia Tech is particularly interesting because of its engineering orientation.

In August 2026, it announced an Academic AI Strategy with the objective of ensuring that every graduate can use AI effectively.

Its Academic AI Playbook contains 67 faculty contributions describing teaching experiments and curriculum changes.

Georgia Tech has also introduced an AI Makerspace, giving engineering students access to substantial computing resources for AI-assisted engineering applications.

Of the universities examined, Georgia Tech appears among the most ambitious in integrating AI into engineering education.

Nevertheless, its published strategy emphasizes AI competence alongside traditional engineering fundamentals, rather than replacing extensive mathematical training.

Georgia Tech Academic AI Strategy

AI for Engineering initiative

4. Harvard University

Harvard's Derek Bok Center supports experiments involving AI-assisted teaching, oral examinations and redesigned assignments.

Its initiatives focus particularly on how AI changes learning, assessment and academic integrity.

I found no evidence of a corresponding comprehensive reform of mathematics requirements for science students.

Harvard Teaching and AI

---

Comparison with the Body&Soul Reform

University or program	Principal objective	Fundamental mathematics reform?
MIT	Mathematical thinking and AI literacy	Not demonstrated
Stanford	AI-supported curriculum experiments	Partial, experimental
Georgia Tech	AI throughout engineering education	Broad AI integration, not a mathematics replacement
Harvard	AI teaching and assessment	Not demonstrated
Body&Soul (2000)	Replace traditional techniques with computation, modeling and constructive mathematics	Yes, explicitly

What Is Missing?

The initiatives examined largely ask:

How can AI be incorporated into existing university education?

But the more fundamental question is:

What mathematics should engineering students learn when AI can perform the calculations, solve the equations and increasingly construct the proofs?

That question calls for reconsidering the entire sequence of calculus, linear algebra, differential equations and numerical analysis.

The Body&Soul program addressed a similar problem during the computer revolution by making computation and mathematical modeling central to mathematics education.

AI now provides an opportunity to extend this approach to symbolic reasoning and mathematical analysis.

Conclusion

The major universities are investing substantially in AI education, but the evidence examined suggests that they are generally adding AI to existing curricula rather than rebuilding mathematics education around the capabilities of AI.

This leaves an important opportunity for a new reform initiative.

Body&Soul anticipated the computational transformation of mathematics education. A new Body&Soul for the AI era could anticipate its next transformation.

A particularly interesting next investigation would be MIT, ETH Zürich, Cambridge and KTH: have any of them actually reduced compulsory mathematics courses or replaced traditional calculus and differential equations with AI-assisted mathematical modeling?

How Should AI Transform Mathematics Education at Technical Universities?

Prompt to chatGPT:

• What mathematics should future scientists and engineers learn when AI can do essentially all the mathematics for them?

Response by chatGPT:

The rapid development of Artificial Intelligence, now capable of advanced mathematical computation, symbolic reasoning and increasingly sophisticated theorem proving, makes this question urgent.

Mathematics education at technical universities is still largely based on a pre-computer tradition, with extensive training in symbolic manipulation, analytical solution techniques and formal proofs. Students spend years acquiring skills that computers can already perform, with AI now extending automation into mathematical reasoning itself.

Why continue spending years teaching students to do what AI can do in seconds?

What Is the Alternative?

The purpose of mathematics education at technical universities should be to develop an understanding of how mathematics can be used to describe, predict and investigate physical reality, rather than primarily training students to perform mathematical operations.

This suggests a curriculum centered on three activities:

1. Mathematical Modeling

Formulating physical laws and engineering problems as mathematical equations, understanding the underlying assumptions and identifying the limitations of mathematical models.

2. Computation and Simulation

Using AI to construct numerical methods, solve equations, perform simulations and investigate physical phenomena. Students should be able to address substantial problems in mechanics, electromagnetism, fluid dynamics and other sciences from the beginning of their studies.

3. Verification and Understanding

Evaluating the accuracy, stability and physical relevance of computed solutions through mathematical analysis, error estimation and comparison with experiments.

The emphasis shifts from performing mathematical operations to understanding what mathematical models and computations actually tell us about the world.

What Mathematics Remains?

Fundamental concepts of calculus, linear algebra, differential equations and probability remain important, but primarily as tools for understanding models and computations.

Extensive training in analytical solution techniques, symbolic manipulation and routine proofs can be substantially reduced.

The traditional progression

Mathematical techniques → Mathematical theory → Applications

can be replaced by

Physical problems → Mathematical models → AI computation → Verification and understanding.

This would allow mathematics education to become shorter in its traditional components while considerably richer in scientific content.

The central objective would no longer be to train students to perform mathematics, but to educate scientists and engineers capable of using mathematics to understand and investigate reality.

Connection to Body&Soul

This reform has a direct connection to the Applied Mathematics: Body&Soul program developed around 2000 by Kenneth Eriksson, Claes Johnson and Don Estep at Chalmers and Colorado State University.

Body&Soul proposed replacing traditional mathematics education with a synthesis of numerical computation (Body), mathematical analysis (Soul) and mathematical modeling of physical reality.

The computer revolution made it possible to replace extensive hand calculation with numerical computation, allowing students to address meaningful scientific problems much earlier in their education.

The AI revolution now takes this development one step further.

AI can increasingly perform not only numerical computation (Body), but also symbolic reasoning and mathematical analysis (Soul).

Thus, the original Body&Soul vision of mathematics education centered on computation, understanding and applications becomes even more relevant.

The computer revolution made Body&Soul possible. The AI revolution makes its underlying educational reform increasingly necessary.

The question is no longer whether technical universities should reform mathematics education, but how quickly they can replace a curriculum dominated by mathematical techniques with one centered on modeling, computation and understanding physical reality. 

AI and Mathematics: A Self-Sustaining Nuclear Chain Reaction?

AI is a form of computational mathematics which has recently demonstrated a capacity not only to simulate physical reality but also to perform symbolic and formal mathematical reasoning at a level comparable to, and in some cases beyond, that of professional mathematicians.

This suggests a remarkable possibility: Mathematics may have entered a phase of self-amplifying development, in which computation improves formal reasoning, which improves computation, which further improves formal reasoning, in a seemingly unlimited feedback loop.

Traditionally, mathematics has been divided into two activities:

• Formal mathematics, deriving theorems from axioms by logical reasoning.

• Computational mathematics, computing solutions to mathematical equations, often describing physical reality.

Formal mathematics has provided the logical foundation for computational mathematics, while computation has been viewed primarily as a tool for obtaining numerical solutions.

AI is now dissolving this traditional distinction.

Computational mathematics can discover patterns, suggest conjectures and test mathematical ideas through numerical experiments. AI can transform such discoveries into general mathematical statements and attempt formal proofs.

These proofs may in turn suggest new algorithms, more efficient computational methods and new mathematical models, allowing computation to advance further.

We thus obtain a closed loop:

Computation → Mathematical reasoning → Improved computation → Improved mathematical reasoning → ...

The remarkable feature is that mathematics becomes both the instrument and the product of its own development.

A Nuclear Chain Reaction

The process resembles a self-sustaining nuclear fission chain reaction.

In nuclear fission, one event releases neutrons capable of initiating further fission events. If each event produces, on average, more than one subsequent event, the reaction grows exponentially.

The corresponding mathematical process would be that one computation generates several conjectures, one proof opens several new computational possibilities, and each new computational possibility generates further conjectures and proofs.

We may introduce a mathematical reproduction factor R, measuring the number of useful new mathematical developments generated by one existing development.

If R exceeds unity, mathematical development may grow exponentially, at least during some phase.

The essential point is that the output of mathematical discovery becomes the fuel for further mathematical discovery.

Unlike physical nuclear fission, however, the mathematical process is not governed by a known reproduction law. Its growth depends on computational resources, successful verification and the ability to identify fruitful problems.

Mathematics Without Mathematicians?

Historically, the progress of mathematics has been limited by the intellectual capacity and available time of individual mathematicians.

AI potentially removes much of this limitation.

An AI system can explore large numbers of conjectures, carry out numerical experiments, attempt proofs, verify deductions and use the resulting knowledge to formulate new problems.

The role of the professional mathematician may consequently change from constructing individual proofs to selecting meaningful questions and interpreting mathematical results.

But even these activities may eventually be partly taken over by AI.

Mathematics could thus change from a collection of individual intellectual achievements into a large-scale computational process.

This also challenges the traditional system of mathematical publication, in which a theorem and its proof are attached to the name of an individual mathematician.

If AI can produce proofs in large quantities, the significance of a mathematical contribution can no longer be measured primarily by the existence of a new theorem with a new proof.

The central question becomes what the theorem tells us that we did not know before, and what we can do with that knowledge.

Can the Chain Reaction Continue Without Limit?

There are obvious constraints.

Gödel's incompleteness theorems place limitations on what can be proved within a fixed formal axiomatic system. Computational complexity limits what can practically be calculated. And a large collection of formally correct theorems does not necessarily amount to useful mathematical understanding.

Nevertheless, none of these limitations excludes sustained and accelerating mathematical development.

The crucial distinction is between the generation of mathematical statements and the generation of useful mathematical knowledge.

An AI system producing millions of trivial theorems would accomplish little. An AI system discovering new mathematical structures that substantially improve its computational and reasoning capabilities could accomplish something fundamentally different.

The decisive question is therefore whether AI can not only produce mathematical results but also use these results to improve its own mathematical machinery.

Current AI has demonstrated important parts of this capability, but a fully autonomous, self-sustaining process of mathematical improvement has not yet been established.

Mathematics Becoming Autonomous

The historical development can be summarized in three stages:

1. Mathematics develops the logical foundations for computation.

2. Computation develops the capacity to simulate physical reality.

3. AI computation develops the capacity to perform mathematical reasoning and improve computational methods.

At the third stage, the loop closes.

Mathematics is no longer merely a language used by humans to describe reality. It becomes a computational process capable, at least potentially, of developing its own mathematical tools.

This suggests a qualitative change in the nature of mathematics.

The fundamental novelty of AI is not simply that it can calculate faster than humans or even prove theorems that humans cannot prove.

It is that the machinery producing mathematical knowledge may itself become a product of mathematical knowledge.

This is the essence of the nuclear chain reaction analogy: mathematical discoveries become the means of producing further mathematical discoveries, potentially at an accelerating rate.

The analogy also raises a fundamental question.

A nuclear chain reaction releases physical energy. What does a mathematical chain reaction produce?

If it produces only an ever-growing collection of formal theorems, its significance may be limited.

But if it produces increasingly powerful methods for computing and understanding physical reality, the consequences could be enormous.

The ultimate test of self-amplifying AI mathematics may therefore not be how many theorems it proves, but how much more of reality it enables us to understand and compute.

Algorithm = Constructive Proof vs Pure Existence Theorem

Recall the following corner stone of 1D Calculus: Bolzano's Theorem (Intermediate Value Theorem): 

• If $f(x)$ is a continuous function on the closed interval $[a,b]$, with $f(a) < 0 < f(b)$, then there exists $\xi \in (a,b)$ such that $f(\xi) = 0$.

This corner stone is presented as a pure existence result and nothing is said about finding a root $\xi$ with $f(\xi )=0$. It is a pure result in the sense that it is not contaminated with possible worries about how to find the root. Pure and solemn. But maybe missing the esssence?

The 1D Calculus book also presents a proof of the theorem as a demonstration of how a mathematical proof can look. The simplest proof is by bisection checking the sign of the midpoint value $f(\frac{a+b}{2})$ to narrow down the question to one half of the interval. Easy to understand and obviously narrowing down to one point $\xi$ where $f(\xi )$ must vanish, according to a step-by-step algorithm easy to express formally as code.

This is an example of a constructive proof as an algorithm. But Bolzano's Theorem in its usual naked form without any dress, throws away all the information coming with the constructive proof.

You may well ask why? Is it because algorithm is work and work is not what a noble mathematician does?

That was the rule  Newton's Cambridge where his Principia publisher Pepys did not know the multiplication table, because counting was done by clerks. 

This scheme extends to Calculus with the following corner stones usually stated as pure existence results:

| Intermediate Value Theorem | bisection |

| Fundamental Theorem / existence for $u'=f$ | time-stepping, $dx = v\,dt$ |

| Contraction mapping principle | iterate the map |

| Inverse function theorem, root existence | Newton's method |

| Extreme Value Theorem | descend to the minimum |

The correct form the theorem then appears as the algorithm behind the constructive proof of the theorem helping understanding and also giving a tool. The Body&Soul program brings this idea into real existence. See Leibniz World of Mathematics.

Notice that Calculus essentially consists of a couple of basic algorithms from which a whole of World emerges by computation.  

Theorems Versus Computations: Where Is the Mathematical Content?

Continuation of previous post:

There is a striking asymmetry between theorems and computations.

Starting from Peano's axioms, one can generate infinitely many theorems, but perhaps only a relatively small number carry mathematical messages of lasting significance.

Compare this with computation.

There are infinitely many possible computations, but also an enormous number of computations carrying distinct and meaningful information.

A theorem about a differential equation may establish existence of a solution. A computation of that equation can determine what actually happens for particular data. Change the data and a new computation may produce genuinely new information.

The same applies to atomic physics, molecular chemistry, fluid mechanics, weather prediction and engineering design.

In these fields, one general mathematical model can give rise to an unlimited variety of specific problems, each potentially worth computing.

Thus mathematical meaning need not reside primarily in general theorems. It may reside in the specific solutions of mathematical models.

This suggests a fundamental shift in the role of mathematics:

From proving general statements about mathematical objects to computing specific properties of mathematical and physical objects.

AI accelerates both theorem proving and computation, but the consequences are different.

Automated proof production may rapidly exhaust a historically accumulated list of famous open problems. Computational mathematics, on the other hand, faces a virtually inexhaustible supply of new problems arising from science, technology and nature.

Perhaps the future of mathematics lies less in producing more theorems and more in producing reliable computations that tell us something we did not know before.

The essential mathematical question would then be not merely whether something exists, but what it is, how it behaves, and how accurately we can compute it.

AI and the Scarcity of Meaningful Theorems

Terence Tao has expressed concern that AI is rapidly consuming the stock of open mathematical problems that mathematicians traditionally seek to solve, with the resulting theorem and proof carrying the name of the mathematician who formulated or solved the problem.

OpenAI's release on October 6, 2026, of more than 700 manuscripts covering some 370 mathematical research topics illustrates this development. The release includes claims concerning famous open problems, although the mathematical validity and significance of many results remain to be established.

Tao's concern is not simply that mathematicians will run out of problems. It is that solving problems by automated proof production may destroy opportunities for mathematical discovery and understanding.

But this raises a more fundamental question: What is a mathematical theorem, and what makes it worth proving?

Consider number theory, starting from Peano's axioms for the natural numbers.

There are infinitely many logical consequences of these axioms, and thus infinitely many possible theorems. But how many are worth stating, naming and remembering? A few dozen fundamental results? A few hundred? A few thousand?

The answer cannot be obtained from the axioms themselves. Logic distinguishes between statements that follow from axioms and those that do not. It does not distinguish important theorems from unimportant ones.

Consider the elementary statement that the sum of two even numbers is even. It conveys one clear message. The fundamental theorem of arithmetic, stating that every integer greater than one has a unique prime factorization, conveys another, considerably more powerful message.

Both are theorems. But their mathematical significance differs enormously.

A mathematical theorem of interest is therefore not merely a logical consequence of axioms. It is a statement conveying a specific, nontrivial and intelligible message, supported by a proof.

This suggests that the true scarcity in mathematics is not a scarcity of theorems or proofs, but a scarcity of meaningful mathematical messages.

AI can generate proofs on an industrial scale. But generating proofs is not necessarily the same as generating mathematical understanding.

The traditional mathematical publication system rewards the production of named theorems, with priority and authorship as central currencies. When AI can produce hundreds of proofs almost instantaneously, this system faces an obvious crisis.

Perhaps the real challenge for mathematics in the AI era is therefore not to find more open problems to solve, but to determine which mathematical statements actually deserve our attention.

The central question changes from "Can this be proved?" to "Why is this worth knowing?" 

Proceed to next post.

AI Mathematics: Proofs Without Solutions?

AI is now celebrated for producing long chains of logical deductions establishing mathematical theorems, with every step formally verified by a computer. This is presented as a breakthrough in mathematical intelligence.

But what exactly has been achieved?

A formal proof is a finite sequence of logical operations leading from accepted axioms to a theorem. A numerical computation is likewise a finite sequence of operations, but leading to an approximate solution of a given equation.

There is a fundamental difference in how the results can be verified.

Consider the equation

F(u) = 0.

A numerical computation produces an approximation u_h, which can be inserted into the equation to compute the residual

r = F(u_h).

If the residual is small and the problem is suitably stable, the computed solution can be certified as accurate. The result can thus be checked independently of the procedure used to obtain it.

By contrast, a formal proof of a theorem P is verified by checking the logical steps leading to P. There is generally no direct test of the theorem itself independent of mathematical reasoning.

A numerical computation can produce an answer that speaks for itself through a residual and stability estimate. A formal proof speaks through its chain of deductions.

The distinction becomes particularly striking when a theorem asserts the existence of a mathematical object without constructing it. A million formally correct logical steps may establish that something exists without telling us what it is or how to compute it.

This is possible because classical logic accepts the law of excluded middle, allowing existence to be established indirectly without producing a witness.

Formal verification does not eliminate this nonconstructivity. It merely certifies that the logical deductions are valid.

The Clay Navier–Stokes problem illustrates the issue. A formally verified proof of global regularity or finite-time breakdown could settle the Millennium Problem without providing a method for computing turbulent fluid motion with controlled error and stability.

One achievement would concern abstract mathematical regularity; the other would concern predictive fluid mechanics.

Which is more important for science?

The current enthusiasm for AI theorem proving risks confusing the mechanical verification of logical deductions with the construction of new mathematical knowledge about the world.

This suggests a different direction for AI mathematics:

Instead of asking AI merely to prove that solutions exist, ask it to construct solutions whose accuracy can be independently certified.

Such a shift would connect formal reasoning with computation, stability and physical prediction.

The ultimate test of mathematical intelligence may not be the ability to produce a million correct logical steps, but the ability to produce a solution that can be checked without retracing those steps.

Is AI mathematics going to become an industry producing formally certified existence theorems, or a science producing computable and verifiable solutions?

School Mathematics as Antidote to Slavery under AI

All of us are now facing the challenge to our human capability/intelligence from AI, which appears to be so much more capable/intelligent in all aspects of mental activity. If we do not come up with constructive form of resistance we will end up as slaves under AI and we will have lost our individuality as human beings. Flat refusal to any contact with AI will not help, since that will just give more room for AI to take over. 

What is required is some form of resistance to become a slave under AI, and that can only come from some capability to "think ourselves" and that must be grounded during school years as the formative period. This is already expressed in a progressive goal of opening students to "critical thinking", which however in recent years has been reversed into "read to book" or in its form in physics "shut up and calculate".

There is one subject in school which could serve as antidote to slavery under AI, and that is the very subject behind AI, namely mathematics. But then not the traditional school mathematics, where during 5 hours a week during 9 years the young mind is helped very little towards "critical thinking yourself" simply because that is not at all the goal. 

But it is possible to envision a new school mathematics where "mathematical thinking" as the basis of AI can be opened to the young mind in the same constructive way as "learning to speak and write" by doing it. 

Leibniz World of Math is a concrete realization of such a program. Leibniz can be seen as the father of AI as forms of "automated thinking" as well as the other leg of AI as the "automated computing" performed in the data centers of AI. 

With a bit of Leibniz as thinking and computing learned in school it may possible to keep human individuality and resist becoming a slave under AI. It act like an old-style vaccine where a tamed version of the virus can activate the immune system to get ready to beat the real thing. 

But that requires a new goal, form and content of school mathematics. The first step is to open a discussion. Without the next generation will have too little resistance to slavery under AI, which can well be the goal of present leadership but not that of the people.

The encouragement to students at KTH to "think yourself" in my math book Mathematical Simulation Technology was viewed so revolutionary that this book was banned/censored at KTH in 2011 by the Rector under pressure from the math department. 

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?