A Beginning for Mathematics

Three years ago, artificial intelligence systems lacked the ability to reliably add two numbers. A year ago, internal models at OpenAI and DeepMind received a perfect score on the International Mathematical Olympiad. Now, these systems are autonomously resolving major open questions in mathematics. It seems improbable that this trend will continue for another year, but it is expected to persist.

It is evident that this will necessitate a fundamental restructuring of our profession. A few weeks ago, I delivered a talk with the title "The End of Mathematics." While some might assume that the talk was about AI "solving" mathematics, that is not the case. Instead, the talk presented a pessimistic view of the future, in which, despite the potential for AI systems to be robustly superior at mathematics, the structure of our institutions could lead to a halt in human understanding of mathematics, and potentially even a stagnation in mathematical progress itself.

Despite my general enthusiasm for utilizing AI in mathematics, I share this concern with many critics. I aim to present a positive perspective on the future of mathematics and the human practice of it. I argue that we can enhance human understanding while the creation of intriguing mathematical results becomes less reliant on human involvement.

This essay assumes that robustly superhuman AI systems for most or all aspects of mathematics will soon exist. However, the specific changes to our institutions that I propose can be implemented even if we only accept the less stringent premise that the production of mathematical text is becoming increasingly detached from mathematical understanding.

It is clear that there is no consensus within the mathematical community regarding our objectives. Some individuals seek to solve problems; others view mathematics as a form of play or poetry. Some believe we are uncovering the secrets of a platonic realm. Some see the goal as embodying a love and comprehension of mathematics, and sharing that love and comprehension with the next generation.

My personal, albeit self-referential, answers are: these goals should be broad in scope. The definition of what constitutes high-quality mathematics has evolved significantly over time; we define it as a community. We are not merely training PhD students to conduct research in mathematics. A significant portion of our responsibility, albeit perhaps underappreciated, is to educate the general public about high-quality mathematics and mathematical thinking.

My personal goals include broadening the general public's understanding of mathematics and fostering the development of mathematical thinking skills. While our goals may vary, most of our objectives have been realized primarily through proving theorems. Almost all papers or doctoral theses revolve around a central theorem, with a corresponding proof.

However, it is evident that the goal of mathematics is not merely to prove theorems; if it were, it would be trivial to automate. A computer or monkey could begin with the axioms of ZFC and iteratively apply deduction rules, without considering their meaning. Proving theorems is not the primary objective of mathematics; our computer or monkey could easily generate all mathematical propositions in alphabetical order, conjecture all valid proofs, and discover countless mathematical truths.

The importance of theorem proving has been particularly evident when a theorem resolves an open question, particularly one that has persisted despite extensive research. Automating such theorem proving is straightforward; a computer or monkey can simply generate all mathematical propositions in alphabetical order. The prevailing attitude within our community towards a technology capable of proving theorems and solving open problems suggests that our operational definitions of values are at best incomplete.

The prospect of automating mathematics by systematically enumerating all conjectures and their proofs within ZFC may not be troubling for you. However, let us assume that the computer or monkey possesses a high level of intelligence; perhaps it comprehends the results it generates, writes elegant explanations, and primarily focuses on questions deemed interesting by the mathematical community.

If the machine successfully answers many of our pressing open questions and poses even more fundamental inquiries, is there still a need for human mathematicians? I believe there is. This machine may produce valuable answers, but it would not inherently provide the human understanding necessary to fully grasp those answers. In fact, I contend that we are at the onset of a remarkable, beneficial surge in mathematics, and if we value human understanding, the demand for human mathematicians will increase.

However, the profession must adapt. In this transition, we must determine what to retain and what to discard. Some aspects I wish to preserve include learning seminars, serendipitous conversations that spark new ideas, and students approaching professors to discuss mathematics. A thriving community engaged in the study of emerging exciting mathematics is something I would like to maintain.

Thousands of individuals collaborating to gradually resolve their uncertainties. I worry that much of the recent discourse on this subject focuses more on preserving the exact structure of academic mathematics institutions rather than our values. How can we preserve the journal and peer review system? How can we safeguard the arXiv?

How can we continue to act as gatekeepers? If you have internalized the fact that existing AI systems can generate relatively high-quality results at a minimal cost, the notion that any semblance of the current equilibrium can endure the impending changes appears implausible. As we seek a new equilibrium, we should consider the future landscape.

If the models remain inadequate in certain domains, we will need human mathematicians more than ever before.