During the preparation of this work, essays appeared by Grant Sanderson and Lisa Valentini, which expressed similar ideas through different lenses. I highly encourage the reader to read them.
What if The Book was found tomorrow? Would we mathematicians leap for joy at the feast before us, eager to better understand even our best known theorems? Would we leave it closed, so as not to “spoil the ending” of our own quest for knowledge? Or, more melodramatically, would we just break our chalks, burn our yellow books, and go weeping into the night, bereft of new challenges to face (and papers to write)?
While not the fabled Book of Paul Erdos, AI has spawned a discourse in the mathematical community which ranges from enthusiastic to excoriating. For myself, I cannot prophesize on how AI will develop, nor exactly what changes it will bring, but as an early-career mathematician I cannot but ponder what it means for our profession going forward. Following Professor Terence Tao’s ICM 2026 talk, I think it a decent working hypothesis that we are soon to transition from an era of “proof scarcity” to “proof abundance” or even “proof indigestion,” where AI models create research-grade proofs at a scale beyond the human capacity to read. The question that we must grapple with, if this is true, is what role should humans play in this new paradigm? Professor Tao’s response is that, among other aspects of mathematical practice, humans will still perform the work of “canonicalization,” writing definitive and pedagogical presentations of results and theories to serve as references and textbooks for future learners. As far as this goes, I find this a fine benchmark, yet it gives little guidance on what most mathematicians, in our day-to-day work, should do to adapt, and even less on what we should teach to the next generation of researchers so they can have successful and meaningful careers.
In this article, I would like to propose an analogy between this potential state of affairs in mathematics and the academic study of history, and draw some lessons from the latter. A typical history graduate student, upon picking a narrow topic and plumbing the previous literature, does most of their thesis work by reading, analyzing, and synthesizing many primary sources into a narrative advancing some argument. They are not just trained in how to understand what one source is saying, or even how to establish a specific detail or fact, but rather to put them together to show the relationships between the sources, facts, actors, etc. and build a broader narrative connecting them all. Crucially, creating this narrative is not some reserve of the field’s tenured maestros, but rather the goal of each participant in historical inquiry.
Let us assume for simplicity the, to some, dire state of affairs where most mathematical propositions can be prompted to an AI for an efficient and correct answer, and that these AI’s can correctly tell us the relation between that proposition and already known mathematics (what theorems imply it, the definitions it uses, what special cases were previously proved, etc.). I argue that this would be, in effect, a well structured “archive of mathematics,” and that one response our community can take is to work with it as such.
In this world, a paper could be written like a present-day survey, except an absolute premium would be placed on motivation and “telling a story.” A thesis, too, could be like a reference-book, albeit with the same premiums and, moreover, an expectation to explicate results that have not appeared in the literature before. Here we are on well-trod ground, as mathematics has a rich store of famous pedagogical works that go beyond compiling previous results and instead present them with a new organization, new proofs, and even entirely original ideas. Most famously, though its exact origins are murky, the axiomatic method itself was popularized by the success of Euclid’s Elements, which was written as a synthesis of previous geometers’ work.1 More recently, sterling examples include inter alia Lang’s Algebra, Hörmander’s The Analysis of Linear Partial Differential Operators, Thurston’s The Geometry and Topology of Three-Manifolds, and Finite Dimensional Vector Spaces by Halmos.
Of course, it should not be expected of anyone to write at these authors’ level while in graduate school. Still, with the expanded scope that an AI-enabled mathematical archive can give us, a graduate student should be able to articulate a narrative concerning a topic and deliver a reasonable exposition. In my own field, an example might be a dissertation on Thurston’s Geometrization Theorem, tracing the main ideas up to Hamilton and Perelman,2 or a textbook describing train tracks in both the Thurstonian and Bestvinian-Handelian senses, and the flagship results they are used for. These are, of course, already established topics in the literature, but I contend that AI can be used to search for new questions where students can work a cleaner slate. Moreover, there is actual work to be done beyond understanding specific results – not every 18th-century grain contract is an historical revelation, nor is every proposition worth the wider mathematical community’s attention. Discernment, now something absorbed passively as “mathematical culture,” would in my scenario become a cultivated skill. So would the related ability of motivating a given piece of mathematics – theorems do not speak for themselves.
In any event, let us turn to the need for this form of mathematical enterprise. Concerns abound around whether humans will be able to understand the proofs AI’s write. Currently this has yet to pass, but as the length and scope of AI-based math research grows it seems more and more likely, particularly as AI begins to use its own discoveries to make yet newer ones. Indeed, it is one thing for an AI to explain its own work in terms of current knowledge, and have us understand it, but another for an AI to comprehensibly explain its work in terms of its other work, particularly after several iterations. Having a human in the exposition-loop could well be what is needed to anchor AI’s ideas in the realm of human intelligibility.
Also on this score, as I mentioned earlier, the impact that a text has depends as much on its context and readers as it does on its content or its authors, and contexts and readers change. I believe it absolutely plausible that an AI could write a good textbook on, say, Khovanov homology. I do not believe it could convince our community that it is important without our prior interest in the related questions, interest which was an intrinsically communal phenomenon. Moreover, I do not believe it could independently make a good decision on what results and details to include without access to other human-written examples on the topic. As examples of the importance of such choices in history, witness the all-too-recent effort to recover the submerged testimony of enslaved individuals from American 19th century archives, or the history of science’s “social turn” in reviewing the archives of the scientific revolution to go beyond individual scientists’ efforts to understanding the role their communities and cultures played in the scientific enterprise.
I hope I have drawn a plausible sketch for what post-AI mathematics could look like and, by analogy to history, shown that it is not unreasonable to expect from large numbers of researchers, including graduate students. There are inevitably, though, unanswered questions. First, about how and why to fund this proposed model of research, I want to remind the reader (and policy-makers!) that funding agencies have rarely funded math research projects solely to get back the solution to some problem – our grant proposals and research statements generally gesture towards understanding better some aspect or phenomenon, with conjectures serving either as very general motivation or as guideposts for avenues to investigate towards this understanding. Furthermore, with respect to applications, it is impossible to predict what mathematical ideas will be useful in the real-world or when, and such ideas become generally applied only after a large amount of digestion, understanding, and “canonicalization” have taken place. Calculus and linear algebra, as they are taught now to millions of engineering students, are paradigmatic examples of how much digestion must sometimes occur to build a useful curriculum. If my above contention on recursive AI explanations has any merit, the world will need as many mathematical researchers as ever to construct the understanding needed to apply AI’s breakthroughs.
A second concern is by what standards the novelty or merit of a researcher’s work, either past, present, or future, should be judged; if nothing else, some standard will have to be devised to determine what sort of work is sufficient for a doctoral thesis. Here the analogy with history breaks down somewhat, in that mathematics requires little of the sort of probabilistic evidence-weighing by which historians create and validate new narratives, and has none of the arguments about causality or human psychology that can be won or lost. All of this said, though, the rise of AI does nothing to dull our communal sense of aesthetic appreciation, which (subjective as it is) does influence how mathematicians are currently judged. Also, even as our field moves beyond giving plaudits to the first to prove a major theorem, the task of “digesting” the very large proofs AI creates will still require much human effort. Much like today, certain results may spawn their own bodies of literature, contributions to which would be another signal by which mathematicians can be evaluated.
AI poses many questions, most of which I have not attempted to answer here. History, though, is never a stable system, and modern ways of doing mathematics were never going to be permanent. Remember, the Pythagorean Theorem was discovered 1500 years before its first recorded proof, Isaac Newton spent much time computing logarithms by hand, and even our modern set-theoretic foundations have existed for less than two centuries. This constant change is not a flaw, and while we certainly should not submit meekly to the whims of fate, we should still recognize that the stability of professional mathematics has been relatively exceptional in the grand sweep of history. I have tried to point out one future path we might drive towards, but I truly don’t know if I’m right. Indeed, there may not be a right path for our community, only various trade-offs, hopes, and uncertainties. Nonetheless, as much as mathematics has changed, it has also proved remarkably persistent. We may do things one way now, but we are still the heirs of millennia of work by men and women pondering the same ideas and struggling over the same questions, even when separated by hundreds of years and thousands of miles. Whatever AI does to academic research, some people will succeed us, putting their own work and thought and love into this field. The question we all must decide, then, is what sort of mathematics do we want to leave them?
- I do great injustice by passing over both the Elements’ predecessors and its readers, but space permits me to go no further. Suffice it to say that it is the earliest extant example of this method in its final form, and that more information on previous mathematics can be found in R. Netz, A New History of Greek Mathematics. For a guide to its later reception, see B. Wardhaugh’s book Encounters with Euclid and the volume The History of Mathematical Proof in Ancient Traditions, edited by K. Chemla. ↩︎
- A book has already been written on this topic by L. Bessieres, G. Besson, M. Boileau, S. Maillot, and J. Porti: Geometrisation of 3-manifolds. ↩︎
Received 19 September 2026.
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