Artificial Intelligence and Mathematical Research
An editorial by Christophe Besse, Director of CNRS Mathematics (Insmi).
In recent months, mathematical research has been transformed by the impressive advances in artificial intelligence.
The progress and changes of recent months have been considerable. Several proofs of remarkable conjectures, achieved with the help of artificial intelligence, have caused quite a stir. This obviously does not mean that artificial intelligence can now solve any open problem. But the frontier has shifted, and it is shifting rapidly.
The interesting question is therefore no longer whether AI can ‘do mathematics’. It is what sort of mathematics we want to do with it.
Mathematics is a human endeavour to understand the abstract world. Doing mathematics means exploring, formulating the right questions, inventing concepts, establishing unexpected connections, and seeking the right perspective on a problem. It also involves establishing proofs. However, a proof is not merely a sequence of arguments leading to a correct result: it seeks to make it clear why that result is true, to bring out ideas and to convey them. The challenge, therefore, is not simply to produce more mathematics, or to produce it more quickly, but to understand how these new tools can enrich this collective endeavour.
Our discipline also has an extraordinary characteristic: until now, it has required relatively little costly infrastructure, compared with many other sciences. Sometimes all it takes is a quick sketch, a sheet of paper, a pencil, a blackboard, some reference material, and colleagues with whom to exchange ideas: this relative lightness of resources has profoundly shaped mathematical research.
AI could shift this balance. Mathematics may be entering a period in which, more than ever before, access to computational infrastructure, models and data will become a key factor in the ability to remain at the forefront of research.
It is absolutely essential that we maintain a public and academic capacity to develop, study and use these tools. Our scientific independence depends on it. We must also defend open knowledge: a mathematical advance only realises its full value when it can be circulated, understood, verified, taught and reused. Our model for exchanging knowledge, sharing results, conjectures and questions must not evolve into a state of individual isolation.
In light of the current debates and concerns, I would suggest three words: observe, support, structure.
Observe, first of all. Look very closely at what researchers are doing, monitor emerging practices and determine what new science is actually being produced thanks to these tools.
Support, next. Give those who wish to do so the means to experiment with and access these new tools, without imposing a single way of conducting research.
Structure, finally. If the transformations currently underway are confirmed, we will need to reflect on infrastructure, evaluation, recruitment, best practice and doctoral training.
In my view, there is a fundamental paradox: the more powerful these tools become, the more important foundational training will be. The more AI advances, the more proficient we will need to be in mathematics to know how to query these systems, understand their outputs and limitations, and draw new ideas from them, whilst exploring other avenues.
We must not underestimate what is happening. The scientific community must engage with these developments with curiosity and rigour, without confusing speed with haste.
To build on the work already underway within various groups, we are facilitating a discussion on how we conduct research and train young scientists, as well as on the scientific, environmental and ethical issues at stake. This discussion must involve all stakeholders: the mathematical community, research organisations, universities, partners, learned societies and companies developing AI solutions.
Change is already here. It is up to us, as a community, to decide what we want to make of it.
Christophe Besse, Director of CNRS Mathematics (Insmi)