Atoosa Kasirzadeh is a philosopher and mathematician. She is a research scientist at DeepMind Institute and an assistant professor at Carnegie Mellon University. Her research is focused on the social and scientific implications of AI.
In the past few months, unreleased frontier AI systems have significantly disrupted a classic conception of mathematics. They successfully established a complete formalization of Andrew Wiles’s proof of Fermat’s Last Theorem, found a solution to the Navier-Stokes problem and — just a few days ago — generated hundreds of manuscripts that each purport to prove an important mathematical problem. Most recently, on October 26, another of the Clay Millennium Prize problems was felled by an AI system.
Mathematicians reacted with a volatile mixture of optimism, anxiety and deep disorientation. In July, for example, Kirwin Hampshire, a Ph.D. student in mathematics, published an essay he titled “The Dark Night of Mathematics” in which he voiced the fear that human mathematicians might soon become obsolete next to AI systems with far greater speed and memory. “I am suffering a profound spiritual crisis due to these developments,” he wrote. “I have been screaming internally for days. It feels as though I am living inside of a nightmare.” For Hampshire, the very act of doing mathematics — the human struggle, the human aesthetic judgment, the human feeling and value of finding, understanding and sharing a new proof — could become a quaint artisanal craft, with human mathematicians no longer needed to push the frontier of mathematics.
Some mathematicians have realized that they must radically rethink their professional role and identity in light of this new technological reality. I was recently at an AI conference where a Fields Medal winner gave a talk on the frontier capabilities of AI systems for math problems. They closed their talk by saying: “We need to develop new theories of care and love.” This call was quite telling: a world-class mathematical theorist asking for theories of a very different kind. The mathematician, it seems, might become a philosopher of care and love. In a short essay, Logan Graves, a mathematical philosopher, went further, suggesting that “the era of human mathematicians is ending, if not ended”; human mathematicians should find a new purpose in life, he wrote, and “the future of the human mathematical community, and eventually all scientific communities, is the practice of priests, then that of monks. Not researchers.”
Terence Tao, a prominent Fields Medal winner and mathematics professor, acknowledged the deep complexity of this historical moment. Adopting a cautiously optimistic stance, he forecast that the mathematical “community will emerge stronger and more resilient than before.” Daniel Litt, a professor of mathematics at the University of Toronto, offered a set of core problems that should be solved for building the future of mathematics in an intriguing essay called “The End of Mathematics.” Litt wrote: “I’m broadly optimistic that mathematics will survive and flourish,” but that requires the mathematical community to address complex (not merely formal) questions such as redesigning institutions that incentivize high-quality scientific research and reorienting human capital.
Each of these reactions is entirely understandable; each represents a deeply respectable and admirable attempt to face and process a disruption to established ways of doing, being or becoming — of doing a certain type of mathematics and of being or becoming a particular type of mathematician.
Because these reactions respond to different dimensions of a complex concern, it is helpful to distinguish between four central questions that urgently need answers. First: What is mathematical progress, and how is mathematics actually advanced? Second: How is the epistemic and creative agency of practicing mathematicians being disrupted? Third: How should we educate future human mathematicians and sustain human mathematical capacities? And fourth: How should incentives, research and structures for reward and credit be reorganized?
I start by distinguishing between two models of mathematical progress. I call them the checklist model and the generative encounter model. In the checklist model, mathematical progress is primarily about incremental advancement in proving what humans, at a rather recent point in our history, have designated a checklist of “major mathematical problems.” The generative encounter model, on the other hand, conceptualizes mathematical progress as inherently open-ended and unfinished, and its development, for the time being, is subject to negotiation with definitions of human purpose, desire and experience.
“Mathematics is one of the greatest things humans have built. Like all our built things, it keeps changing as we change — and our understanding of it keeps changing too.”
The Checklist Model
Under the checklist model, mathematical progress consists of resolving well-defined, complex problems that are specified in advance. These include competition problems designed to measure the frontier of human mathematical capabilities such as those of the International Mathematical Olympiad, famous conjectures such as Fermat’s Last Theorem (or last conjecture), formalization benchmarks such as Freek Wiedijk’s list of 100 formalization challenges and prize problems such as Clay’s Millennium problems. Each of these can be considered an item on a checklist. Mathematical progress proceeds at rate at which the boxes get ticked, and reward and credit attribution goes to whoever ticks them first. This model of mathematical progress is attractive because it is neat and it makes progress easily measurable. That is why it underpins both the prestige economy of mathematics (Olympiad medals, the Fields Medal, the Millennium Prizes) and the benchmark culture of AI.
AI systems have been rapidly excelling at solving a checklist of major mathematical problems. Some mathematicians like Tao refer to a new age of “proof post-scarcity” in which checklists of mathematical conjectures, including ones to be added in the future, are quickly proved or disproved one after the other by increasingly powerful AI systems.
A subset of the mathematical community has long organized its conception of progress around this model and the straightforward systems for reward and credit that it produces. Whoever solves a major problem from a canonical checklist earns the credit, and that cements their status as a genius (often a lone genius, which is important). Entire research and academic institutions grew up around this incentive structure, and generations of students were taught that (one of) their highest aspirations should be to solve (at least) one of these problems. Now that AI systems can solve these problems, the norm that reserved such achievements exclusively for humans, and rewarded humans for them, no longer carries the social weight it once did. This means that, as Cornell mathematics professor Steven Strogatz said in a recent interview, the age of lone human mathematical heroes is over.
But the end of the lone human hero is not the end of human mathematicians.
AI is showing us that the incentives around the checklist model are not sustainable anymore. While loss and grief are valid responses to this new reality, remaining stuck there leaves the community unable to respond to radically transformative changes. We should look at this historical moment as an opportunity to redesign the structures that motivate and reward mathematical work, and we should do it soon.
“AI brings us back to an old philosophical question: Where does mathematics come from?”
The checklist model rests on a remarkably narrow conception of mathematical inquiry. Even before AI systems started solving problems, the model didn’t seem to directly serve real-world problems. Many mathematicians have long worked in “exploratory mode,” disregarding questions about whether their lines of inquiry had any practical usefulness. G.H. Hardy, an English mathematician who died in 1947, is famous for writing “I have never done anything ‘useful.’ No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world.”
If mathematical progress is bound to the values of human mathematicians who can decide what to formalize, determine which problems truly matter, co-create with software and artificial agents and invent or discover novel domains of inquiry, then AI does not signal an end for the field. Instead, it opens a vast, unprecedented space of possibility.
The checklist model of mathematical progress is not representative of the actual progress in “mathematics” when we look at the history of the field. Under its narrow view, the pessimism of some mathematicians (We are becoming obsolete; There is no need for future humans to do mathematics; No valuable rewards for doing mathematics remain for humans now) offers no guidance about what mathematicians should do next.
Under a different model of mathematical progress, which I call the generative encounter model, proving major problems is just one piece of “doing mathematics” or “being a mathematician.” Here, AI’s impact on the mathematical community, generational education and institutional design is much more open to constructive and creative adjustment.
The Generative Encounter Model
In 1939, George Dantzig arrived late to a statistics class during his doctoral studies at Berkeley and found two problems written on the blackboard. He thought they were homework assignments from the professor, Jerzy Neyman, so he solved them and handed in his work. “About six weeks later,” according to his own account, Dantzig and his wife “were awakened by someone banging on our front door. It was Neyman. He rushed in with papers in hand, all excited.” The problems weren’t homework, it turned out, but rather famous unsolved problems, and Neyman wanted to rush to publish Dantzig’s work.
After Berkeley, Dantzig became an adviser to the U.S. Air Force Comptroller. He was tasked with finding a way to mechanize the planning and budgeting of the military’s staggering logistical operations. The reality he faced was one of worldly complexity. Thousands of human clerks were tasked with drafting and creating supply-chain and personnel schedules entirely by hand, using heuristic trial-and-error.
There were no preexisting mathematical problems for Dantzig to solve here. He faced the chaotic physical reality and had to figure out how to translate it into a useful mathematical abstraction. This is an example of an “encounter” in the generative encounter model: a social and practical problem with apparently little to do with “just proving” mathematics.
As Dantzig himself later recounted in an interview, the military used the word “program” — not computer code but rather a plan or a schedule. Dantzig’s goal, therefore, became simply to more rapidly develop a program for time-staged deployment, training and logistics. He was not trying to prove a theorem or conjecture; he was not hired for that, nor did he expect an award for developing such a thing. He was trying to fix an administrative bottleneck.
To do so, he invented (or discovered, if you like) a class of problems that needed solving. He did this by structuring and interpreting the situation into an elegant formulation composed of two distinct parts: a linear objective to be achieved (maximize efficiency or minimize cost, for example) and a set of constraints expressed as a system of linear inequalities (a certain number of planes, pilots, gallons of fuel, etc.). He called this mathematical package a “linear programming problem.” The word “programming” he borrowed directly from the military’s administrative vocabulary.
By mathematically separating the physical constraints from the ultimate goal and approximating both with linear relationships, Dantzig mapped a messy real-world problem onto a clean, high-dimensional geometric object: a convex polyhedron. This abstraction transformed a question of scheduling human action and materials into a question of geometry, and in doing so, made it a part of an open-ended conception of mathematics.
Dantzig had taste. He made a crucial recognition: As a matter of mathematical necessity, the optimal solution to a programming problem would lie on one of the vertices of a polyhedron. The search for the “best” mathematical solution or the best “schedule” thus became a search for the “highest” corner of a geometric object. Dantzig then developed the now-famous simplex algorithm to systematically search for that optimal vertex by efficiently moving along the edges of the polyhedron until reaching the peak.
Before Dantzig, there was no mathematical community of “linear programmers” listing out problems in their field and seeking to solve them. He formed a community, one that grew to include operations researchers, mathematical economists and industrial engineers as additional sub-branches of the field were developed. This group defined itself by a shared interest in a set of problems that had not existed a decade earlier.
Linear programming works for continuous resources like gallons of fuel, but what if a problem demands whole units? You cannot fly half an airplane or assign a third of a pilot to a mission. That physical constraint, a new encounter, forced the invention of a branch of operations research known as integer programming. This, in turn, opened a vast new frontier of mathematics, generating entirely new branches of mathematics, theoretical models and novel algorithms, which themselves collided with new problems and produced yet more fields of inquiry.
“A revolution in automated proof discovery, on its own, guarantees nothing definite about the future of mathematics, which remains an open-ended domain of inquiry.”
Before the MIT mathematician Norbert Wiener created the field we now call cybernetics, which later formed the backbone of modern artificial intelligence, he was confronted with a wartime problem similar in some ways to the one that faced Dantzig: how to shoot a plane out of the sky using an antiaircraft gun on the ground when the enemy pilot was taking evasive action. Aiming at the plane guaranteed a miss, so where do you aim? As Wiener later put it, “It is exceedingly important to shoot the missile, not at the target, but in such a way that missile and target may come together in space at some time in the future. We must hence find some method of predicting the future position of the plane.” This problem was not on any checklist of mathematical problems. It had to be encountered.
Wiener’s move was to reframe the problem. He stopped treating a pilot as a strategic adversary to be outguessed and instead considered him a component in a system to be characterized. He saw that the pilot’s evasive maneuvers were not purely random; they were responses to the gunner’s actions and corrections against the aircraft’s physical lags. In this view, the gunner, the antiaircraft gun, the shell, the pilot and the aircraft all formed a single system with different feedback loops between them. To this loop Wiener added what he called a predictor.
In his 1948 book “Cybernetics: Or Control and Communication in the Animal and the Machine,” Wiener described the task as designing a predictor whose characteristics “depend[ed] on a long-time acquaintance with the statistics of the ensemble of time series which we desire to predict.” In this sentence, the significance of real-world encounters for mathematical progress becomes quite obvious. The physical problem of gunnery was converted into a statistical problem about time. This conversion created a cluster of new mathematical problems: What is the optimal filter to separate a signal from noise? How can one best predict the future course of a random process from its past? What, formally, does it even mean for a system to be goal-directed?
The answers gave birth to the theory of statistical filtering and prediction and ultimately the feedback formalism that organized the new mathematics and science of cybernetics. Wiener created a new branch of mathematics for describing, explaining, understanding and predicting control and communication in both machines and living organisms. It generated a new community, a new field, which brought different epistemic stances together to better understand the world. The now-famous Macy Conferences in the late 1940s and early 1950s brought together biologists, psychiatrists, anthropologists and engineers around a set of problems that previously had no disciplinary home. Cybernetics, as a new branch of mathematics, created a new home for such inquiries.
Most telling of all, the original purpose that sparked the work was soon superseded. Wiener’s antiaircraft predictor was not widely deployed; a simpler design from Bell Labs won the military contract. Had it been a problem on a checklist, Wiener’s project might be seen as a failure, seeing as how it was an inferior solution to another. But the mathematics it created survived and went on to reshape our understanding of many systems, from factory automation to human and artificial neural systems.
There Is No Distinction Between Pure And Applied Mathematics. There Is Only Mathematics.
During my doctoral studies in mathematics (in the field of “operations research”), I attended a national conference for mathematical graduate students in Canada. The atmosphere was charged with a peculiar intellectual stratification between “pure” (or “real”) mathematics and “applied” mathematics.
When some of my peers, who were working on classical analytic number theory and variations of the Riemann hypothesis, learned that others of us were researching mathematical and computational approaches for large-scale decision problems, their response was immediate and dismissive: What you do is not real mathematics. To them, real mathematical inquiry belonged exclusively to the solitary pursuit of proving major theorems within self-contained axiomatic systems. Not considered real mathematics were efforts to develop and formulate algorithms to solve large-scale decision problems, optimize resource distribution, translate frictions and decision problems into mathematical structures, and perhaps find new theories of decision-making in real-world encounters.
The culture of mathematicians has for generations harbored a hidden hierarchy between these two communities of mathematicians: Generating abstract proofs supposedly constituted “real” mathematics whereas dealing with the messy, empirical world was dismissed as an inferior craft. Historically, this divide served a practical purpose: It functioned as a tool for creating an epistemic division of labor, organizing mathematical activity and giving rise to separate institutions, journals and reward structures for tackling a variety of different cognitive engagements with mathematical problems.
By placing the generation of proofs for very complex problems at the apex of this hierarchy, the pure-applied dichotomy, perhaps unintentionally, popularized the idea that pure mathematics is (real) mathematics. In doing so, it enshrined the checklist model of mathematical progress, leading many to mistake measuring a peculiar type of epistemic activity for the entirety of mathematical progress. The conflation of the title mathematician with those who study only pure mathematics is a pervasive and uncritical cultural prejudice. It is one that relegates researchers who grapple with physical and systemic realities to second-class status. I think AI has forced us to correct that prejudice, and I do not think we can ignore it.
“A true golden age of mathematics, rooted in human purpose and new visions of education and collective reward, may have just begun.”
Dantzig too faced gatekeeping by pure mathematicians, whose insistence that their work was the only “real” mathematics generated a highly exclusionary perception of the discipline. When asked whether he thought of himself as an operations researcher or a mathematician, Dantzig answered: “I consider myself a mathematician. I don’t differentiate between pure mathematics and applied mathematics. I know other people will tell you there is a difference, but there isn’t.”
When the interviewer noted that pure mathematicians often describe operations researchers as “second-rate mathematicians playing with toy problems,” Dantzig’s response was characteristically blunt: “Mathematicians are snobs.”
“Present company excepted,” the interviewer responded. “Pure math involves clean problems, but real-world problems are messy.”
“Real-world problems are all dirty, that’s correct,” Dantzig said. “Any model of them is an approximation of reality; it’s not reality.”
Dantzig’s experience in the development of linear programming led him to become a strong advocate of encountered problems as the inspiration for interesting and challenging mathematical work. Speaking to Donald Albers in 1984, he said:
Just because my mathematics has its origin in a real problem doesn’t make it less interesting to me—just the other way around, I find it makes the puzzle I am working on all the more exciting. I get satisfaction out of knowing that I’m working on a relevant problem. I find that just as much mathematical ingenuity has to go into solving problems from a newly developing area as from some old so-called pure math area.
Operations research, Dantzig went on, is the “science of decision-making”: the art of “abstracting a problem and putting it into mathematical form.” For him, mathematical practice spans a continuous spectrum from the abstract theorist seeking formal solutions down to the practitioner needing an actionable answer to a concrete crisis. To chop this continuum in half and declare one end “noble” and the other “second-class” misses the entire dynamic essence of mathematical inquiry.
The “crisis” of AI’s ability to rapidly solve complex problems, and mathematicians’ responsibility for building infrastructure for mathematical inquiry in the near future, can be seen in a new light. A messy encounter with the world, after all, can be the birthplace of the purest mathematics, and pure mathematics can find its way back to the world.
Powerful Systems, Open Futures
We must approach this historical moment with the utmost care. The main challenge is not only to mourn a loss. It is, more importantly, to build a better future. The development and deployment of AI is forcing us to redraw the maps of human motivation, reward and research prestige in mathematics. Redrawing those maps does not have to erase human agency — but it can if approached clumsily.
If we restructure how we assign credit, set research priorities and reward work — moving away from proof discovery alone — we can cultivate “Dantzig-style” mathematical thinkers. These are people who use AI to free their creative faculties and push into deeper, more interesting or more urgent territory.
This moment has opened a vast space of possibilities. New incentives can be designed around problems that genuinely matter for better human life, for individual and social health, for planetary flourishing. And we are not short of those problems.
Perhaps the mathematical community and its future generations can move from “Who proved it?” to “Who decided this was worth proving, why, and how did we make sure it was right?” Large collaborations already exist in mathematics: the Polymath projects, the Lean community’s Mathlib, the long effort to formalize Fermat’s Last Theorem. Mathematicians can coordinate around shared goals that matter for their community, but also for the world.
They now need to scale coordination capabilities. But how?
I think the mathematical community must embrace the generative encounter model of mathematical progress. This requires that we treat the integration of AI in mathematics as a rich, independent subject of foundational analysis. Far from signaling the end of mathematics, this opens room for examining a vast frontier of urgent, encountered mathematical problems that require deep work, often in collaboration with other disciplines. Here are four critical clusters of mathematical problems where this work is urgently needed:
- Provably safe cyber infrastructure. Our global digital systems remain profoundly fragile. Human mathematicians, using AI as a high-throughput collaborative partner, can work to find the flaws and rebuild our cyber foundations from first principles, establishing provably secure and mathematically verified computing environments.
- The epistemic mechanics of deep learning. Our mathematical grasp of how neural networks learn, generalize and represent high-dimensional data is still in its infancy. Several concerns about advanced AI causing catastrophic harm originate directly from this gap in understanding. Despite significant advances, developing a rigorous “mathematical science of deep learning” remains a fundamental challenge, and solving it is a precondition for building truly trustworthy systems.
- The mathematics of multi-agent control. The dynamic alignment or misalignment of interacting, agentic AI systems remains poorly understood. We need to draw on mathematical disciplines such as mathematical security, safety engineering, robust mechanism design and game theory to analyze how individually optimized agents behave when they coordinate, compete or collide.
- The epistemology of machine-generated proofs. What should we do with AI-generated “proofs” that are too long, too complex, or too alien for human minds to understand or verify? Rather than rejecting them because they are unintelligible, we must develop new mathematical techniques for explanation, logical translation and interactive verification that make them understandable. We shouldn’t rush to rely on such proofs, but these techniques can give us reliable paths for turning raw automated output into genuine human understanding.
None of these problems appear on a checklist of mathematical problems. None of them will be easily “solved” by a single human mathematician. Each needs teams of humans equipped with technological tools to decide what and how to formalize and interpret mathematical problems that matter — and how to build institutions to apply the offered solutions.
The world around us is human-made: the buildings, the networks, the institutions, the AI models themselves. Mathematics is one of the greatest things humans have built. Like all our built things, it keeps changing as we change — and our understanding of it keeps changing too.
This brings us back to an old philosophical question: Where does mathematics come from? The cognitive scientists George Lakoff and Rafael Núñez gave one answer in their book “Where Mathematics Comes From”: embodied human cognition. Platonists give another: Mathematics is discovered, not made. Its objects exist independently of humans, and mathematicians are explorers of a terrain that has been there all along and will still be there in the absence of any human. Formalists give a third response: Mathematics is a game of symbols and rules, and it is “true” only in the sense that a legal chess move is correct. I won’t settle the debate here. But every answer locates at least part of the origin of mathematics in an encounter: between minds and the world, between us and Plato’s heaven, or between human communities and the questions they find worth asking and answering. AI systems have now become a serious participant in that encounter.
A revolution in automated proof discovery, on its own, guarantees nothing definite about the future of mathematics, which remains an open-ended domain of inquiry. That future will be shaped not only by AIs but also by the preconditions that make doing mathematics possible, or impossible: the purposes, infrastructure and institutions we build around the questions we find worth asking and answering. These include how to verify results, how to assign credit, how to train students, how to value human cognition and agency, how to support those harmed by this transition and give them new systems of recognition to rely on, how to advance human and planetary flourishing, how to make tools and infrastructure broadly accessible and who sits at the table when these questions are answered.
Answering these questions requires science and mathematics as necessary elements of the sociotechnical foundation for a better future. And that we do not yet have.
“New incentives can be designed around problems that genuinely matter for better human life, for individual and social health, for planetary flourishing. And we are not short of those problems.”
Viewed from the generative encounter model of mathematical progress, a true golden age of mathematics, rooted in human purpose and new visions of education and collective reward, may have just begun.
Suppose this situation and these questions are only temporary. What about later? What if we eventually build artificial superintelligence — systems that don’t just prove theorems but pose questions, judge which ones matter, design curricula and decide how rewards are distributed? Systems that automate the whole process of generative encounters?
The deepest version of the worry some “pure” mathematicians are articulating is not whether AI can automate mathematical proofs. It is whether AI eventually automates human purpose itself. But let’s not forget how we arrived here.
Our current moment is, among other things, a long-term consequence of the dream of formalizing mathematics, from the well-posed formulas of mathematical logic that gave rise to software engineering to the contemporary drive in the Lean community to formalize mathematics in machine-readable form so that machines might verify and, one day, prove or solve problems. In some sense, where we are today is partly the realization of a long dream held by a portion of the mathematical community. But living in a realized dream requires us to think carefully not just about the dream itself, but about its consequences. Purpose matters.
Whether the day ever comes when human purpose is fully automated none of us can know. But what is clear is that the future is shaped by what we dream and build today, and we owe it to ourselves, our children and future generations to dream and build it with care. In the end, the transition to the future is not separate from the future. And the transition is where the future is actually lived, dreamed and made.
Author’s note: Thanks to Jim Brown, Marcus Hutter and Shane Legg for comments on this essay.
