Mathematics in the Age of AI

If AI proves theorems, what is left for people?

This is not just a question of dividing tasks between people and machines. It is about the meaning of the profession: why someone chooses to do mathematics and what counts as their contribution.

Our answer starts with the purposes of mathematical work, not with assumptions about the limits of AI.

1. The change affects a way of life, not just productivity

For many mathematicians, working on a proof has value in itself. Spending a long time with one problem, trying different approaches, and gradually understanding its structure is not just time spent on the way to a result. It is a large part of why people choose mathematics.

So automation cannot be described only as getting rid of routine work. It also affects the work people find most meaningful and enjoy most. New opportunities do not remove the difficulty of this change.

Adapting the profession does not mean declaring long, independent work unnecessary. We need to understand its place under the new conditions and how to assess its value without reducing everything to a race to produce results.

2. Producing results is not the only goal of mathematics

AI can take on more and more research work. But the existence of a system that builds proofs does not, by itself, mean that people understand those proofs, see why they matter, or know how to use the results.

We still need people who can ask their own questions, understand theories, discuss ideas, connect them to other problems, and pass that understanding on.

This is not an argument against mathematics developed independently by machines. It is a reason to maintain and develop mathematics as a human activity and culture, not just a collection of established statements. Training new researchers is part of this task.

3. A growing stream of proofs needs careful assessment

It is not enough to know that another proof has appeared. We need to ask:

• What has been established, and how has its correctness been checked?
• What is the main idea, why does it matter, and what is it connected to?
• What does the result help us understand or do, and how can we explain it to others?

These questions matter regardless of who produced the proof. As the number of results grows, they cannot be treated as secondary.

4. We need a broader view of research contributions

New theorems remain an important part of research. But a mathematician’s contribution cannot be measured only by the number of statements they have proved themselves.

Asking questions, making ideas simpler, finding connections, organising reliable research, explaining, and training new people also matter.

This includes looking for possible applications and talking with people outside the field. Not every piece of work needs to be applied, but making mathematics understandable and connecting it to other fields should not be left to a few enthusiasts.

Five roles of a mathematician

1. The philosopher of a field — depth

A philosopher of a field understands its main ideas, methods, and structure. They stay informed about what is happening across the field. They see what is missing, which questions should be asked differently, and which definitions and proofs could be made simpler. They do more than understand the field: they actively develop and change it themselves.

While reading the field’s “newspaper” every day, they think about future directions. Their work is to ask not only what has been proved, but what we really understand and where we should go next.

How we try this role ↗

2. The connector — breadth and possible applications

A connector finds meaningful links between areas of mathematics, other sciences, and practical problems. They distinguish a shared structure from a superficial analogy.

They bring mathematical ideas into new settings and bring new questions into mathematics from problems in other fields. They understand both the possibilities and the limits of this transfer. They do more than notice these links: they actively develop them and, in doing so, change the fields involved.

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3. The research architect — organisation and responsibility

A research architect plans the research process: choosing directions and tools, dividing tasks between people and AI, and combining partial results.

They organise verification and understand what makes the final claims trustworthy. Responsibility cannot be passed to a tool along with the task.

How we try this role ↗

4. The explainer — understanding and openness

An explainer identifies the main idea, the real difficulty, and a useful example. They can discuss the same mathematics with specialists, students, other scientists, and people without specialist training.

Explaining helps test one’s own understanding and makes a result accessible to others. Clarity does not mean giving up rigour: it helps people see the ideas behind the technical language.

How we try this role ↗

5. The mentor and culture-builder — people

A mentor helps others become independent researchers. They help others develop mathematical taste, encourage curiosity, and create trust and space for serious discussion.

They pass on not only knowledge, but also an approach to work: honesty, persistence, and responsibility.

How we try this role ↗

How to understand these five roles

This is not a list of abilities that AI cannot have. AI can take part in each of these roles. It is a description of the skills mathematicians themselves need.

Not everyone needs to be equally strong in all five roles. Not every researcher needs to become a laboratory leader. This model should broaden our view of mathematical work, not push out people whose strength is deep, focused research.

Education: what we need to teach now

Strong foundations still matter

Having access to ready-made solutions does not remove the need to learn to reason. Students must still try, make mistakes, check their work, and rethink their ideas. Otherwise, knowing how to use a tool can take the place of understanding the subject.

Education must therefore protect time and space for independent work, including work that is slow and does not lead to an immediate result.

Five areas of training

1. Understand the structure of a theory. Explain the meaning of definitions and the assumptions in theorems, choose meaningful examples, and formulate your own questions.
2. Find connections and possible applications. Study an idea in different settings, translate a problem from outside mathematics into mathematical language, and understand what the answer means in the original situation.
3. Organise research. Plan small projects, choose tools, work with people and AI, check results, and explain the reasons for your decisions.
4. Explain to different audiences. Present the same idea to your own group, students in another subject, and non-specialists. Answer questions and notice what remains unclear.
5. Help others learn. Give useful feedback, discuss mistakes, and support independent thinking rather than replace it with ready-made answers.

Practical formats

This training needs small, stable groups with a mentor, classes on how mathematical fields develop and connect, and regular spoken explanations and discussions. These should be part of the curriculum, not depend only on individual teachers’ initiative.

The aim is not to replace learning mathematics with learning to use AI. It is to teach students to use new tools while continuing to think for themselves.

PhD training: preparing independent researchers

Why do we need new mathematicians?

Less demand for people to carry out individual tasks does not mean that we no longer need new researchers. PhD students are future colleagues and independent partners in discussion. They bring different questions, experience, and ways of thinking.

PhD training should therefore not be built mainly around carrying out a supervisor’s assignments. It should prepare students for independent work in all five roles.

1. Taking part in choosing research questions

PhD students need to learn how to choose questions, understand their place in a field, compare approaches, and organise verification.

They need to explain not only a proof, but why the question deserves attention and what difference the answer makes.

2. Assessing understanding and research decisions

Using AI does not, by itself, tell us whether a student is working independently. What matters is whether they can defend an argument, check a key step, recognise a limitation, answer a new question, and clearly identify their own contribution.

New results remain a necessary part of a thesis. But readiness for independent research cannot be judged only by how many results someone produces or how quickly they produce them.

3. Working beyond a narrow topic

Training should make room for neighbouring fields, joint projects, discussion of possible applications, and talks for different audiences.

Not every thesis has to be applied. But PhD students need to learn to see their work in a broader context and explain it to people outside their immediate circle of specialists.

4. Mentoring and time to think

Supervisors should not only check results. They should discuss research directions, help develop mathematical taste, and gradually hand over responsibility. PhD students, in turn, need experience helping younger students and taking part in their group’s research life.

At the same time, new requirements should not become an endless list of duties that crowds out research itself. PhD training must preserve the chance to think deeply and at length, rather than demand constant displays of productivity.

Our position and an invitation

What we need to do

AI is already part of the conditions in which mathematical research takes place. This requires us to rethink professional roles, education, and the way we assess research contributions.

Our aim is to preserve independent human thinking and a meaningful life in research. At the same time, we need to learn how to work with a growing body of results, find connections and possible applications, and make mathematics easier for others to understand.

We will have to adapt. What to change and what to preserve is for the mathematical community itself to work out.

Join this work

We invite researchers, teachers, and students to discuss and try new ways of doing mathematics and teaching it.

We are interested in approaches to research and education that develop all five roles, take the real difficulties of this change seriously, and preserve space for deep, independent thought.

Ways to join ↗[email protected]