For a very long time, my favorite activity was to sit and think about mathematical questions. Staring at a blank sheet of paper, trying to grasp in my mind abstract concepts, bouncing ideas off colleagues and students. There is nothing quite like the feeling that you are exploring the unknown by only using your mind.
Mathematicians have different styles and motivations for doing math. For some, it is about the challenge of solving hard problems. Others might have some application in mind, or a question they feel compelled to find the answer to. But I believe that none would stick with this profession if they did not find the process of doing mathematics satisfying. We cannot deny that this process is going through a profound change with AI. Some mathematicians have been meeting this change with excitement, while others with grief. I share both sentiments.
When considering the future of math (and science at large), it is worth reflecting on its past. For centuries, mathematicians and scientists typically worked under the patronage of nobles and royals: Archimedes advised King Hiero II, al-Khwarizmi was supported by the Abbasid caliphs, and Galileo by the Medicis, while medieval universities remained largely teaching institutions. It was only after the emergence of the royal academies and research universities from the seventeenth century onwards that scientists were paid by the state for pure research. Beyond patrons and academies, scientists have supported themselves in varied ways: military engineering (Archimedes), day jobs in law (Fermat), personal wealth (Darwin), the Royal Mint (Newton), tax farming (Lavoisier), and of course, examining patents.
But if there is one constant that has held true from Archimedes to our time, it is the importance of a scientific community. Scientific communication evolved from personal correspondence and networks of letters, through academies, to modern journals. But throughout this evolution, scientists valued the community of their peers.
What these communities focused on has changed over time. These days proving theorems is considered the prized activity in mathematics, but throughout much of history, the focus was on calculations or solving problems, rather than rigor. In the sixteenth century, Italian mathematicians earned jobs through equation-solving duels, which caused them to keep formulas such as the solution to the cubic equation secret. Euclid and al-Khwarizmi are known to this day not because of their new discoveries as much as for organizing known results.
Today, there are many self-selected and self-organizing scientific communities, which set up their own publication venues, norms, and processes. They are by and large self-governing, and each scientist chooses in which of these communities to participate. The university, which typically pays the scientist’s salary, largely defers to the judgment of their community as to the value of their work. Together with the mechanism of tenure, this leads to a remarkable lack of direct employer oversight of scientists’ output.
Many discussions of science’s culture, including peer review, the publication process, and tenure, focus on their various defects and problematic cases. Yet science has been immensely successful. To use AI terminology, the last “pretrain” humanity got was hundreds of thousands of years ago with Homo sapiens (internal codename: homo-erectus-v5-pro-max). And yet we have managed to advance so much on that basis.
I am also surprised by how we managed to keep science legible. Theories like general relativity and quantum mechanics are the results of hundreds, if not thousands, of years of work by humanity’s most brilliant scientists and mathematicians. And yet we are able to routinely teach them to undergraduate students.
Impact of AI on Math
At this point, it is undeniable that AI can make significant contributions to solving mathematical problems. Solving open problems has been a prized activity in mathematics for a long time. One reason is that it is the easiest way to verify that one has done something that is both novel and interesting. And if we’re lucky and the problem was chosen well, the solution will not be a highly specific trick, but a more general insight or technique that can be used broadly. However, this does not mean that solving open problems is the only or even the most important contribution, and thankfully the “inefficiency” of the mechanisms for evaluating researchers allows many other types of contributions to still flourish.
For the same reason as above, solving open problems with AI is a straightforward way to dispel the skepticism of people who believe it cannot do research-level mathematics. But once we go beyond such skepticism, it is not clear that solving open problems should be the only or even the main use of AI. If you hope or believe AI is inherently incapable of posing problems, writing exposition, or building theories, then you are bound to be disappointed.
What does this mean for mathematics and the role of human mathematicians? There have been radically different responses to this. On one extreme, Weinreich called for a “total opposition to artificial mathematics,” and in particular for mathematics departments to “[establish] anti-AI policies for student work, [reserve] hire lines for mathematicians who eschew AI, [and value] AI-free papers more highly in tenure promotion.” (It is unclear if, under his proposal, “AI-free” papers would be allowed to cite results that did use AI.) On the other hand, Tao said that AI will change how we do math, and that “we do have to somehow let go of conventional assumptions of what intellect is.”
Weinreich’s point of view resonates with the view of mathematics as an art, which is about human expression. On the other hand, many of the strongest mathematicians in history, including Archimedes, Newton, Euler, Gauss, and von Neumann, had a deep interest in its applications. If you care about mathematics’ applications, eschewing AI-enabled discoveries is not an option. Hence, I do not believe that an AI-free vision of mathematics, of the type promoted by Weinreich, is a viable future for it as an academic field. Recreational and competitive mathematics will have different standards, but the lessons of chess (which is actually thriving!) suggest that a complete rejection of AI is unwise even in these domains.
This does not mean that we will have no need for human mathematicians. As mentioned above, the modes of scholarship and funding models for mathematicians have changed over the years, and can change again. In particular, education has long been one of the primary missions and occupations of mathematicians, and it will be as important as ever. If we want (as I do) humans to keep control of their destiny, an educated society will be only more important as AI systems become more powerful.
I believe that human curiosity and legibility will also continue to play a crucial role in science and mathematics. One lesson from history is that it’s extremely hard to predict which directions will have practical applications, and the answers to questions pursued out of pure intellectual curiosity can have great practical impact. In 1940, the mathematician G. H. Hardy wrote that “Real mathematics has no effects on war. No one has yet discovered any warlike purpose to be served by the theory of numbers or relativity, and it seems very unlikely that anyone will do so for many years.” Needless to say, since then people have found many useful, and even warlike applications for these fields, and in particular GPS and public-key cryptography.
Given the track record of curiosity-based science, it would not be wise to eliminate humans from this process and replace them with “artificial scholarly communities” any time soon. Even if it were possible to get the same value, given the unpredictability and time lag of basic science applications, we will not be able to verify this in the near future. Also—and here I am biased—curiosity-based science is good in itself. Number theory is beautiful and would have been beautiful even without cryptography.
Could humans even keep up with understanding AI advances? I believe the answer is yes. Humans have always developed increasingly powerful abstractions to handle complexity. This is the only way we can use brains much like those of our cave-dwelling ancestors to grasp quantum mechanics, write complex software, manage companies, and organize societies many orders of magnitude larger than they did. Abstraction is what allowed us to compress thousands of years of scientific progress into an undergraduate program, and will allow AI to compress its findings and make them legible to us. It may well be that such levels of abstraction will mean that we do not always follow all steps of a proof, just like we do not verify today that a program correctly multiplied two large numbers.
AI will impact much more than science and math. Mathematicians are people too, and they face many greater risks (as well as potential benefits) from AI than those related to its impact on their profession. If AI leads (as I very much hope) to a flourishing human society, then it would be one that values education, curiosity and creativity. We might not explore math using a blank sheet of paper in the same way as I did as a graduate student, but we would still be making and sharing new discoveries. Emma Goldman is often (mis)quoted as saying “If I can’t dance, I don’t want to be part of your revolution.” Similarly, I don’t want to be part of an AI revolution that has no room for human scientists, mathematicians, or artists.
Windows On Theory 