A Beginning for Mathematics

A few years ago, artificial intelligence systems were not able to reliably add two numbers together. Just a year ago, the internal models at companies OpenAI and DeepMind achieved the equivalent of winning a gold medal in the International Mathematical Olympiad. Now, these systems are autonomously solving major open questions in mathematics.

It is difficult to imagine this trend continuing for another year, yet I believe it will. This necessitates a significant shift in our discipline. I recently delivered a presentation titled "The End of Mathematics." One might assume this talk was about AI potentially "solving" math, but it instead presented a pessimistic view of the future.

Despite the possibility of AI systems being robustly superior to humans in mathematics, our current institution design could still stifle human understanding and possibly even mathematical progress itself. I believe we can avoid this grim future, but I also think it is plausible if academic mathematics does not adapt. Although I am generally enthusiastic about the use of AI in mathematics, I share the concerns of many detractors.

My aim is to present a positive vision for the future of mathematics and the human practice within it. I believe we can deepen human understanding even as the production of interesting mathematical results becomes less dependent on it. This essay assumes that AI systems capable of consistently outperforming humans in most or all aspects of mathematics will soon exist.

However, the proposed changes to our institutions only require acknowledging the weaker premise that the production of mathematical text is becoming increasingly detached from mathematical understanding. It is now evident that there is no consensus within the mathematical community regarding our goals. Some of us strive to solve problems, while others consider mathematics as play or poetry.

Some seek to unravel the mysteries of the platonic realm, while others aim to embody love of and understanding of mathematics, and to transmit that to future generations. Personally, my objectives are to produce high-quality mathematics and to cultivate high-quality mathematicians. These goals should be understood broadly. What constitutes high-quality mathematics has evolved over time, and we define it as a community.

The job of mathematicians is not just to train students for research in mathematics; it also involves educating the general public about high-quality mathematics and mathematical thinking. Regardless of our goals, we have primarily measured success through theorem proving. Most papers or PhD theses have a main theorem, accompanied by a proof.

However, it is clear that proving theorems is not the ultimate goal of mathematics; automating this task would be trivial. A computer or monkey could easily start from the axioms of Zermelo-Fraenkel set theory and apply deduction rules, without paying any attention to their meaning. While it is easy to see how this would resolve open problems, our computer or monkey would need to pay attention to the meaning of the results it produces.

This highlights that the general attitude within our community towards a technology that can prove theorems and solve open problems may be incomplete. The prospect of automating mathematics by enumerating all conjectures and proofs of ZFC might not disturb you, but imagine a scenario where this computer or monkey is incredibly intelligent.

It understands the results it proves, writes elegant explanations, and possesses a good sense of what we consider interesting. It answers many pressing open questions and raises many more fundamental ones. Is there still a need for human mathematicians? I believe there is. This machine may yield valuable answers, but it would not generate human understanding of those answers.

In fact, I contend that we are at the beginning of an extraordinary mathematical explosion, and if we value human understanding, there will be an even greater demand for human mathematicians than ever before. However, our profession must change. As we transition to a new equilibrium, we must decide what to retain and what to discard.

I would like to preserve: engaging in learning seminars, serendipitous conversations that spark new ideas, and students approaching professors to discuss mathematics. A thriving community that collectively uncovers new insights together will also be preserved. I worry that much of the recent discourse on this topic, including some of my previous writings, focuses too much on preserving the exact shape of academic institutions, instead of our values.

How can we maintain the journal and peer review system? How can we safeguard the arXiv? How can we retain our role as gatekeepers? If you have internalized the fact that existing AI systems can produce relatively high-quality results at a low marginal cost, the notion that any semblance of the current equilibrium can survive what's coming seems absurd.

As we search for a new equilibrium, we could strive to harness the edge of model capabilities. Currently, AI systems arguably underperform humans in theory-building, posing questions, and exposition. Therefore, we could prioritize and reward these skills. I believe this approach is unwise; consider the rapid pace at which academic institutions adapt compared to the speed at which model capabilities advance.