At a recent conference on mathematics in the age of automated proofs, mathematician and Fields Medalist Akshay Venkatesh presented "How do we talk to our students about AI?'' He quoted an email he'd received from a young student who asked, "Do you believe that mathematics is worth being studied in a world in which a machine can answer everything for you? What do you believe would be the 'job' of a mathematician in this world?" Venkatesh framed AI as an opportunity to correct what he called an "essential gap that has opened between the practice of mathematics and our values." Mathematician William Thurston has explained these values by writing, "mathematics is not about numbers, equations, computations, or algorithms: it is about understanding." But Venkatesh argued that the record on this is terrible, lamenting that "for a typical paper or talk, very few of us understand it." He is not alone in thinking that something is wrong with the current state of mathematics research.
The challenges of communicating mathematical ideas to the general public are well-known. But what may be surprising to people outside the field is how little mathematicians themselves understand each other. Over the past decades, the field has been blessed with stunning advances, some of which can be easily described, such as the discovery of an "einstein" shape that generates an infinitely irregular tiling of the plane and the solution to the 400-year-old problem of finding the densest packing of 3-dimensional balls. But other advances are abstruse to all but a handful of insiders. Recent progress on the Langlands Program, a challenging vision suggesting how to connect some seemingly distant mathematical fields, has led to a great expansion in the global corpus of mathematical knowledge. But the objectives of this program, laid out in 1967 by mathematician Robert Langlands, are known to be "fiendishly difficult to describe." Last summer's resolution of one of the goals, known as the geometric Langlands conjecture, consisted of a series of five papers totaling almost a 1000 pages. But the celebration of this milestone was tempered by the realization of how few people can credibly claim to understand any of it at all.
Mathematics is supposed to be a "universal language" and indeed there is a remarkable degree of consensus across the field. When students enter the mathematical community at a college or university, they are taught to communicate mathematical ideas efficiently and precisely. Over a year-long abstract algebra course, students develop fluency with a strange new vocabulary in which common words like groups, rings, and fields take on specialized new meanings.
Because mathematical terms have a clear meaning and, with a few qualifications, mathematicians share the same beliefs about logical axioms and systems of deduction, disagreements in mathematics tend to be resolved quickly and definitively, even in the case of a particularly difficult conundrum called the abc conjecture. Yet, as the mathematical universe has expanded beyond the familiar numbers and geometric figures that one learns during pre-college years, there is no single chart that can be used to navigate the expanse. Research mathematics has been fragmented into an increasing list of subfields. The Mathematics Subject Classification taxonomy divides the field into 63 primary classifications partitioned further into 529 subfields, each of which has developed its own specialized language used to state and prove technical theorems and that requires years to learn.
The sophistication of modern mathematical proofs is paralleled by the increasing complexity of many scientific theories, from the standard model of particle physics to gene regulatory networks to biogeochemical cycles. Every scientific or technical field has its jargon, which hinders communication with the uninitiated. Researchers have found that the readability of scientific texts is decreasing over time. But there are communication challenges that are specific to mathematics when the abstract concepts being talked about don't necessarily correspond to anything in the everyday physical world. The lack of relevant personal experience contributes to the difficulty in understanding something like the Langlands Program, where expert mathematicians in different fields find it difficult not only to understand the solutions but to even grasp what questions are being asked.
This can be illustrated with an example from my own research area called category theory, which has contributed a substantial new vocabulary of technical terms that even those who know sometimes describe as "abstract nonsense." The category theorist and mathematical physicist John Baez jokes that concepts in the field are so abstract that even their examples need examples before getting to something that a mathematician outside the fold might find familiar. A paradigmatic example is the notion of a "natural transformation," which has a technical meaning first articulated in the 1940s that I could explain, only with some difficulty, to another mathematician in about half an hour. Examples of natural transformations can be constructed using a technical result called the Yoneda lemma. Although those in a small circle are steeped in this foundational theorem, it is unfamiliar to almost everyone outside of it. With sufficient time and attention, mathematicians can communicate technical concepts with each other, but this process is frustratingly slow. It is challenging to explain what my colleagues and I mean by naturality in the middle of a lecture that aims to describe some recent research.
When a mathematical proof is presented in a research paper, the convention is to use the first-person plural pronoun "we," a discursive method that was popularized much earlier in mathematical texts (with frequent examples in the 18th and 19th centuries of the French "nous" or German "wir"). As stilted as that might sound, Paul Halmos described the reason for this usage in his 1970 essay "How to write mathematics." The "we" refers collectively to the paper's authors and its readers, because to really understand a mathematical argument, readers must recreate it for themselves in their own heads. In other words, mathematical communication proceeds at the pace of mathematical understanding.
Where do mathematicians look for solutions to the challenge of communicating mathematics, to each other and to the general public as well? Perhaps too much energy has been devoted to new discoveries, no matter how obscure, with not enough effort reserved for improving ways to make sense of what is already known. As Thurston wrote in "On proof and progress in mathematics," "We need to focus far more energy on understanding and explaining the basic mental infrastructure of mathematics -- with consequently less energy on the most recent results." Thurston, who like Venkatesh focuses on the human experience, suggests that technical mathematical jargon must be supplemented by an alternate effort to develop "mathematical language that is effective for the radical purpose of conveying ideas to people who don't already know them."
One strategy is to provide more opportunities for communications training and practice. The upcoming Joint Mathematics Meetings, to be held in Washington, DC, in January 2026, will feature a reprise of a 2022 conference on communicating mathematics, with workshops and panels that will offer practical suggestions for communicating with other mathematicians, policy-makers, the general public, and even children.
As Venkatesh concludes in his lecture about the future of mathematics in a world of increasingly capable AI, "We have to ask why are we proving things at all?" Thurston puts it like this: there will be a "continuing desire for human understanding of a proof, in addition to knowledge that the theorem is true." In this process of advancing human understanding of mathematics, the proof and theorem is a means and not an end. For what ultimately matters is how mathematical ideals enrich and transform the human relationship with, and perspective on, the world. Even if an AI can tell people what is true, there will always be more to understand about why.