Mathematics in the age of AI (Free PDF)

Artificial Intelligence is beginning to change the way mathematical research is approached. Modern AI systems are increasingly capable of assisting with mathematical reasoning, problem solving, proof development, and other research-level tasks. “Mathematics in the Age of AI”, an essay by mathematician Terence Tao, examines how the mathematical community might respond to this technological shift. The essay is based on a public lecture delivered at the 2026 International Congress of Mathematicians.

Rather than focusing primarily on whether AI will become capable of advanced mathematics, the paper considers a deeper question: What are the actual goals and values of mathematical research when AI can increasingly perform parts of the mathematical process?

Download the pdf for free: https://arxiv.org/abs/2608.16753

AI and Mathematical Research

AI has the potential to influence many stages of mathematical research. It can assist researchers with exploring ideas, investigating problems, organizing information, and developing possible approaches to difficult questions.

This creates a significant change in the traditional relationship between mathematicians and computational tools. Instead of using computers mainly for calculation, researchers may increasingly interact with systems capable of performing more sophisticated reasoning.

The Purpose of Mathematics

A central theme of the essay is that mathematics should not be defined only by the problems it solves.

Mathematical research also involves understanding, discovery, creativity, explanation, communication, and the development of new perspectives. These broader goals become particularly important when AI systems become capable of solving increasingly difficult mathematical problems.

The arrival of powerful AI therefore encourages mathematicians to reconsider what makes mathematical activity valuable.

Problem Solving and Mathematical Understanding

Problem solving provides an important case study in the discussion.

If AI systems eventually become capable of solving many research-level problems, the value of mathematics cannot depend entirely on humans being the fastest or most effective problem solvers.

Instead, mathematical understanding may become increasingly important. Knowing why a result matters, how different ideas connect, and what broader concepts emerge from a solution can remain valuable even when the computational work is assisted by AI.

Human Creativity in Mathematics

Mathematical creativity involves more than manipulating symbols or following established procedures.

Researchers choose problems, formulate questions, identify useful concepts, develop intuition, and decide which directions are worth pursuing.

AI may increasingly contribute to these activities, but the broader question is how humans should interact with such systems while maintaining meaningful intellectual participation in mathematical research.

AI as a Mathematical Collaborator

The development of advanced AI may lead to a new form of collaboration between mathematicians and intelligent computational systems.

Instead of treating AI purely as a calculator or automated solver, researchers may use it as a tool for exploration and idea generation.

This could change how mathematical research is organized, with humans focusing more heavily on high-level direction, interpretation, verification, and conceptual understanding.

Verification and Trust

As AI-generated mathematical reasoning becomes more sophisticated, verification becomes increasingly important.

Mathematical results require rigorous justification. A convincing explanation or apparently correct argument is not sufficient without appropriate validation.

This creates an important role for mathematical expertise: researchers must be able to evaluate AI-generated arguments, understand their assumptions, and determine whether the conclusions genuinely follow.

The Changing Role of Mathematicians

The increasing capabilities of AI may change what mathematicians spend their time doing.

Routine calculations and certain technical tasks may become increasingly automated. Researchers could therefore devote more attention to:

• Formulating important questions
• Developing mathematical intuition
• Understanding concepts
• Connecting different areas
• Evaluating AI-generated ideas
• Communicating mathematical insights

This does not necessarily reduce the importance of mathematicians. Instead, it may shift the emphasis toward higher-level intellectual activities.

Mathematical Education

AI also raises important questions for mathematics education.

If students have access to systems capable of solving mathematical problems instantly, traditional approaches to assignments and assessment may become less effective.

Education may need to place greater emphasis on understanding, reasoning, problem formulation, explanation, and mathematical communication, rather than simply producing final answers.

The Future of Mathematical Discovery

AI could potentially accelerate mathematical discovery by exploring large numbers of possibilities and assisting researchers with difficult problems.

However, mathematical progress is not measured only by the number of problems solved. New concepts, connections, perspectives, and forms of understanding can be equally important.

The future of mathematics may therefore involve a combination of automated problem-solving capabilities and human-driven conceptual development.

Rethinking the Values of Mathematics

The paper ultimately encourages a broader discussion about what mathematicians value.

If AI eventually becomes highly capable at solving mathematical problems, mathematics may need to place greater emphasis on the aspects of research that go beyond obtaining solutions.

Understanding why a problem is interesting, discovering useful concepts, communicating ideas, and building a shared mathematical culture may become increasingly important.

Download the pdf for free: https://arxiv.org/abs/2608.16753

Conclusion

“Mathematics in the Age of AI” explores how artificial intelligence may transform mathematical research and challenges the mathematical community to think beyond the question of what AI can accomplish. The essay instead asks what mathematics is ultimately trying to achieve and what values should guide mathematical research in an AI-driven future.

The emergence of advanced AI does not necessarily mean the end of human mathematics. Instead, it may encourage a shift from mathematics as primarily problem solving toward a broader vision centered on understanding, creativity, discovery, communication, and intellectual exploration.