To resolve a central tension in the practice of mathematics by academics, I suggest the parallel with fast food and its attendant slow food movement.

Three hearts

Mathematical work comprises many activities and experiences. Among my favorites are:

  1. Reading a single paper over the course of several months.
  2. The thrill of the chase, when the scent of a theorem, discovery, or the right definition is in the air.
  3. The moment a PhD student shows me their first theorem or paper.

The first and second constitute gaining and creating knowledge. The third is about transmitting meta-knowledge: the ability to perform 1 and 2.

Mathematics for me has always been about the joy of personal understanding and the engagement in bringing others around to that understanding, usually slowly over the course of many months or years.

In recent months, a deep unease has been growing within the mathematical community around the role of language models in our research. The fear is fundamentally economic. LLMs can produce mathematical statements and proofs, some certainly interesting.

Are we ourselves, mathematicians, obsolete? Will our students fail to get jobs? Will our funding agencies, or even our universities, abandon us? Is mathematics in fact solved?

No. Mathematics is not the collection of true statements implying, by complement, the collection of false statements as well. It cannot be solved any more than human experience can be.

Mathematics and society

Most academic mathematicians are slow math practitioners at heart, even if the pressures of paper-writing, grant-proposing, teaching, and administration seem rather far from our idealized contemplative life.

Yet, math also forms the bedrock of the sciences and engineering and as such is central to how human societies solve technical problems. Mathematicians cannot expect our esteemed place within our culture to survive unchanged if we do not use every tool possible to help it face its grand challenges.

Alexander Nazaryan wrote in the New York Times on 19 June 2025: “Problems in mathematics take decades or centuries, sometimes, to solve,” [Patrick Shafto] said in a recent presentation at DARPA’s headquarters on the Exponentiating Mathematics project, which is accepting applications through mid-July. He then shared a slide showing that, in terms of the number of papers published, math had stagnated during the last century while life and technical sciences had exploded. In case the point wasn’t clear, the slide’s heading drove it home: “Math is sloooowwww. …”

We exist, professionally, in part to educate engineers and scientists, to provide researchers to national laboratories, and to provide highly-trained workers to companies. Occasionally, theoretical results open up new areas of specific application, for example the use of lattice-based methods in post-quantum cryptography standards. If we turn our back on this part of our role, we can certainly say goodbye to our grants and teaching reductions. We can go the way of the Department of —, permitted to exist out of nostalgia, but decreasingly relevant.

To address the central contradiction between our personal practice and our external purpose, I propose fast math and slow math as ways of thinking about different types of mathematics. Just as we all prefer slow food, sometimes fast is what we need, or crave.

These different ways are not meant to separate human mathematicians into “slow math” and “fast math” camps. Indeed, I strongly urge against the temptation to split into us and them. I don’t want to sit on hiring committees fractured by debates over AI use or virtue testing instead of honest evaluation of candidates.

The goal of slow math and fast math is the same: to increase human understanding of what is true and false in those regions of “proposition-space” of interest to us. These take their places as different modalities of doing mathematics, alongside computer algebra, exposition, the creation of visualizations, editorial work, and so on. Not every mathematician does, or should do, all of these.

Dream bigger dreams

Thanks in part to technological advances, but largely driven by an unending thirst to know, physicists have gone in the span of a couple of centuries from tabletop experiments to running some of the largest collective endeavors of humankind.

LLMs represent a tremendous technological advance and some might choose to dream bigger dreams. We should deliberately and thoughtfully use these models and other AI tools to have those dreams. Some of these might be to answer new, pressing questions created by the very existence of the tools themselves, like the problems of interpretability in LLM systems. These tools were built with mathematics and our discipline provides one of the principal means by which we understand them and use them safely.

Other dreams should be to undertake audacious programs, which would not have been practical in the past. I have in mind research directions at the scale of the Langlands program or Grothendieck’s dream of motives.

One example could be higher-dimensional algebraic geometry. Many counterexamples that have been established with LLM assistance have to do with the fact that we do not know the zoo of algebraic varieties in higher dimension the way we do curves and surfaces. (Although, of course, there have been many, many amazing results, like Beauville–Bogomolov.)

Such programs should be pursued rigorously with a back-and-forth between humans and LLMs. These programs should have directors, postdocs, students, standards. The goal would be to leverage LLMs to understand high-dimensional phenomena – for humans to understand. It is not to produce a slurry of results no human has read or understood.

These programs are fast math: far-reaching, ambitious, almost reckless. They would aim at transformative new discoveries and organizations of mathematical knowledge, perhaps particularly motivated by grand challenges at the boundary of mathematics and applied mathematics. But they would not end with a given discovery. They would also be responsible for producing lecture notes, textbooks, databases, and other tools to aid human apprehension of their discoveries. They would exist to truly harness AI capabilities and prepare the outputs for human understanding.

Collective standards

I suggest the following standards for the mathematical community. Please consider these as a first proposal and feel free to send me feedback. The overall goal is that we should use LLMs to aid in human mathematicians’ understanding, not merely to increase our production.

  1. Disclose all collaborations, including with LLMs. It is my hope that eventually we can use open source models for much of the day-to-day work that people might turn to LLMs right now. We already cite computer algebra systems as soon as the work goes beyond the totally routine. We should do so here.

  2. Do not judge mathematicians for their LLM use. The standards of our community are still in flux. We can fight over what those standards should be, but individuals have to act in the meantime and we should not litigate their behavior.

  3. Posting a paper to the arXiv or publishing it means three criteria have been met by the authors:

    • you could reproduce the main ideas if on a desert island (excepting large computer computations);

    • you have written it for human understanding, to the best of your ability;

    • you take intellectual and reputational responsibility and credit for the results.

    In the second point, using LLMs for polishing writing, spell-checking, finding inaccuracies, or creating diagrams is completely fine. Using an LLM to generate a first draft of a paper is not acceptable.

  4. LLMs should not be used to compete with other mathematicians, to scoop them or otherwise embarrass them. It goes without saying that we are somewhat competitive, but there is nothing to be gained from this behavior and a lot to lose in terms of collegiality. We are extremely lucky to be in a field where we talk openly at conferences not only about what we’ve done but what we are working on, sometimes years before those results see the light of day.

  5. Refrain from asking every random question to LLMs, or at least from making the answers public. Simple results are important for training students. Many have been excited to ask questions long-abandoned in their research programs, including myself. Of course, we cannot say that any given question is off-limits. But I expect that this fervor will die down in the coming weeks or months. And we do not need to repeat it with every new “release” from the big labs.

  6. Focus, if you choose to use LLMs, on developing your own research, intuition, and results. No one owns an area of mathematics or a certain problem. But it is not acceptable for me to wake up one day and say, you know I know famous mathematician — is working on problem — and I’m just going to solve it. They will be impressed with me, or I will get that next job I need. The easiest way to guard against this behavior in the long run is to just not reward it.

  7. Do not work faster than you can understand or effectively communicate. This golden rule informs all of the others.

  8. Do not recruit, hire, promote, or praise based on the number of papers. In the past, the production of high-quality papers has provided a proxy for evaluating understanding and brilliance. Of course, you want to hire the individuals who’ve written these papers to access their insight for your own research or for the sake of the students in your department’s charge. Each department will have to find new ways of judging.

  9. Do not delegate the previous task entirely to LLMs.

Before LLMs, I typically had 5-10 projects on the burners or in the fridge at any one time, probably working actively on 2-3 at any one time. This reflects my interests and personal capability to pay attention to them. Having used language models, specifically agentic coding systems (basically nonstop for coding for 7 months), I can attest that there is a tendency to do more and more and more, but that the pleasure in doing so drops, the quality is hard to verify, and my understanding of the output goes down as well. As such, I strongly disagree with the premise of Shafto’s slide that stagnation in mathematics is measurable by slower growth in publication rates than in other sciences. Papers are only a proxy for understanding.

There are also areas where I believe it is explicitly permitted to use LLMs. They can provide translation services, whether to improve writing in a non-native tongue, or to translate into one’s own language. They can proofread. Let’s face it, no one is doing this for us anymore at most journals. They can act as tutors.

Experiments will inevitably produce writing largely generated by an LLM and possibly understood by no human. These texts deserve a home too, because they might later be useful for human work, but that home is not the arXiv.

Pascal

We have known since von Neumann, Turing, and Gödel in the 1930s and 40s that mathematical theorems are nothing but provable propositions in a formal language. These could be produced automatically by any computer.

Humans care about individual statements and theorems. Perhaps in part because of our intuition from the natural world, these propositions are not equal in our eyes. Mathematics is what human mathematicians understand. Beyond is darkness: true absence, or inscrutable trophies of alien intelligence.

In Pascal’s Sphere, Borges invokes Blaise Pascal: “nature is a fearful sphere whose center is everywhere and circumference nowhere” to propose that “universal history is the history of the different intonations given a handful of metaphors.” Indeed, mathematics is an infinite sphere whose center is everywhere and circumference nowhere.

Like Pascal, overwhelmed by humanity’s simultaneous insignificance and ability to understand our insignificance, mathematicians work in a habitable zone in which the theorems, questions, and conjectures are configured so that some structure is visible.

The fact that LLMs can seemingly be organized into their own societies is interesting. The mathematics that they produce there is not interesting to me. Language models have not solved mathematics for humans. They are practicing mathematics for a nascent civilization.

Closing

I believe civilizations beyond ours exist, have existed, and will exist in the future. Our theorems have already been proved. They will be forgotten, and proved again and again until the last ember goes out. The exercise of mathematics is the work of civilization, and we will continue.