AI Changes Mathematics: From Solving Equations to Asking Questions

 What remains for the human mathematician?

Mathematics as a tool for understanding the world can be divided into three fundamental steps:

1. Formulate a question: Identify a phenomenon or problem that calls for understanding or explanation.

2. Formulate a mathematical model: Translate the question into mathematical language, typically equations.

3. Solve the equations: Analyze the model, compute solutions, derive consequences, and compare predictions with observations.

Traditionally, mathematics education and research have concentrated on Step 3, with considerable attention also given to Step 2, while Step 1 has largely been taken for granted.

Artificial Intelligence is now fundamentally changing this division of labor.

A New Division of Labor

AI is rapidly developing the ability to perform Step 3: solving equations, constructing proofs, writing numerical algorithms, and carrying out simulations.

What previously required years of mathematical training can increasingly be accomplished with AI assistance in minutes or hours.

AI can also contribute substantially to Step 2 by suggesting mathematical models, identifying assumptions, and translating physical principles into equations.

But Step 1 remains fundamentally a human activity.

The new division of labor is thus:

• Human: Formulate the question.

• Human + AI: Formulate the mathematical model.

• AI: Solve and explore the equations, with human verification and interpretation.

This represents a fundamental shift in the nature of mathematical work.

The Importance of Asking the Right Question

Consider Newton's mechanics.

The fundamental achievement was to ask how the motion of physical bodies could be described and predicted, and then formulate equations relating force, mass, and acceleration.

Once the equations were available, their mathematical consequences could be explored.

Similarly, Maxwell formulated a mathematical description of electricity and magnetism, while the Navier–Stokes equations describe the motion of fluids.

In each case, the decisive scientific contribution involved identifying a meaningful question and finding a mathematical model capable of answering it.

Solving the resulting equations was a further task.

Today AI is increasingly capable of carrying out that task.

Mathematics Education Must Be Reversed

Traditional mathematics education typically begins with Step 3.

Students are given equations and trained to solve them. They spend thousands of hours learning techniques for finding answers to questions already formulated by others.

Step 2 receives less attention, and Step 1 is often almost entirely absent.

But if AI can perform Step 3, this educational order loses much of its justification.

A new mathematics education should instead begin with Step 1:

What do we want to understand?

Why does an aircraft fly? Why does a molecule form? What causes turbulence? How does an electric current flow?

Students should then proceed to Step 2, developing mathematical models with AI assistance.

Finally, AI can perform Step 3, allowing students to investigate the consequences of their models and compare them with reality.

The educational emphasis shifts from learning how to solve equations to understanding which equations should be formulated, why they are meaningful, and what their solutions tell us.

Can AI Take Over All Three Steps?

AI may eventually become increasingly capable of proposing scientific questions as well as formulating and solving mathematical models.

But there is an essential difference between generating possible questions and deciding which questions are worth asking.

Scientific questions arise from human curiosity, experience, needs, and the desire to understand the world.

A mathematical model must be judged not only by its internal consistency but also by its relevance to the question and its agreement with observations.

AI can assist in these judgments, but human responsibility remains essential.

A New Mathematics

The emerging mathematical process can be summarized as:

Human Question → Human/AI Model → AI Solution → Human Evaluation

The evaluation may lead to a revised question or an improved model, initiating a new cycle.

This changes the role of the mathematician from primarily being a solver of equations to becoming a formulator of questions and models.

It also changes the role of the mathematics teacher.

The teacher of the future must teach students to ask meaningful questions, formulate mathematical models, and critically evaluate AI-generated solutions.

This is a radically different task from teaching techniques for solving given equations.

Conclusion

The traditional mathematician was primarily trained to solve mathematical problems.

The mathematician of the AI era will increasingly be concerned with deciding which problems should be posed and how they should be expressed mathematically.

The three steps of mathematics are not equally affected by AI:

Step 1 belongs primarily to humans. Step 2 becomes a collaboration between humans and AI. Step 3 is increasingly performed by AI.

This suggests a fundamental transformation of mathematics education and mathematical research.

The future of mathematics is not primarily about finding answers to given questions. It is about asking questions worth answering.