The fall of the theorem economy

“The product of mathematics is clarity and understanding. Not theorems, by themselves.”—Bill ThurstonHandwritten diagram by Alexander GrothendieckMy best theorem is one I never wrote down.It crystallized one bright morning in Lausanne, Switzerland, as I was preparing for my last invited conference talk. The proof felt so obvious—and the result so compelling—that I made the reckless move of editing my slides at the last minute. Time was running out and I could only include the announcement as an informal remark at the bottom of the last slide, instead of stating it as a proper theorem.1I had already quit academia and founded a machine-learning startup. I knew I would be too busy to write a clean proof and publish it. That was my excuse for being sloppy. I just wrote the remark and abandoned the slide deck as a message in a bottle.My hope was that some bright young mathematician would pick it up someday and formalize the result as part of a broader theory. If I lucked out with the intrinsic randomness of attribution, I thought, it might even be remembered as the Bessis cellular decomposition theorem.But that was stupid. By claiming the result, I had killed the incentive for anyone to write it up.If I had to pick my second best result, it would be Theorem 0.5 in my old preprint on Garside categories. I had high ambitions for this paper, yet I ended up not submitting it anywhere. The creative process had drained me, and I left active research before regaining the courage to clean up the preliminary sections.For a second best, this theorem is shockingly easy to prove. Once you get the preliminaries right, it only takes a few pages of pretty terrestrial group theory.As for the preliminaries, they are even easier. All you have to do is plagiarize a dozen or so classical papers in an arcane subfield called Garside theory, replacing the original axiom set with a slightly more general one. If you understand what you’re supposed to do, it is almost impossible to run into serious difficulties—it’s just a giant conceptual find-and-replace bulk edit. But you have to take my word for it, because I balked at producing the hundreds of pages of necessary details.

If you think that the hard part of a mathematician’s job is to prove theorems, let this serve as a counterexample—from the moment I conceived of Theorem 0.5, I knew it was true and that proving it would be straightforward.What was the hard part, then?Conjecturing the exact statement and writing it down?Not even. In this example, this part was equally straightforward.The hard part was to intuit that there should be “something like Theorem 0.5”, and to come up with a conceptual framework where it became easy to express. Once I got the definitions right, the rest followed more or less organically.Research mathematics isn’t always like that, but there are miracle days where you just put your skis on and the next thing you know is that you’re accelerating downhill. Jean-Pierre Serre famously said that writing his revolutionary paper on coherent sheaves didn’t require any thinking. Everything fell so naturally into place that his typewriter generated the 100 pages entirely by itself, as if the article had pre-existed.But I wasn’t Jean-Pierre Serre and he wouldn’t lend me his typewriter. This is why my brightest mathematical idea never made it to publication.Do I feel sad about it? Not really. My preprint remains freely available on the arXiv and has already been cited dozens of times, including by some fancy papers. The real innovation wasn’t Theorem 0.5, but the language that made it possible, especially Definitions 2.4 and 9.3—and this language found its way to a 700-page book on Garside theory that filled out much of the missing preliminaries.To be honest, I also had a selfish reason for sacrificing my most innovative preprint. It enabled me to focus on the more tedious preprint where I used Definition 9.3 as a magic ingredient in the resolution of a classical problem in my domain, the 𝐾(𝜋,1) conjecture for finite complex reflection groups, which permanently elevated my symbolic status as a mathematician.

But, in truth, the David who solved the 𝐾(𝜋,1) conjecture is a social parasite of the much better mathematician, the David who crafted Definitions 2.4 and 9.3.In the past few months, as I was grappling with the rapidly changing situation around AI and mathematics, I found myself more troubled than I ever expected to be.In theory, I should feel vindicated and happy. In practice, I am also puzzled, worried, and sad.The happy part of me sees a genuine revolution and gets excited. The vindicated part has legitimate claims to have prepared for it scientifically and epistemologically. The puzzled part is stunned by the timeline and accompanying frenzy. The sad part feels nostalgic for a lifestyle and value system that it engaged with and walked away from, and which might soon disappear. The worried part holds the synthesis. I always knew that the general public had a flawed perception of mathematics, but never expected this to become an existential threat for the discipline itself.In my book Mathematica: a Secret World of Intuition and Curiosity, I framed the misunderstanding as the tension between two versions of mathematics, official math and secret math.Official math manifests itself as a formal deduction system where you start from axioms and mechanically derive theorems. This is a nerd’s paradise, a world where truth takes binary values, reasoning is either valid or invalid, and there is technically no room for bullshit.Secret math is the human part of the story—why official math was invented, how we can successfully interact with it, its effects on our brains, and the bizarre mental techniques through which mathematicians continuously expand its territory.Secret math never made it to the curriculum, because it lacks the defining qualities of official math, and also because it feels peripheral. Official math is cold, hard, logical, objective, and it is rumored to be the language of the universe. Secret math is soft, fuzzy, subjective and, by contrast, it looks like cheap pedagogical backstory.No wonder professional mathematicians have such a dissociative view of their job.The first rule of the Intuition Club is: you don’t talk about the Intuition Club. The second rule is, if you really want to talk about intuition, make it sound casual and accessory, because we ain’t the psychology department.

The third rule is definitions are worth zero points, expository work counts negative, and the best jobs should always go to the people who proved the hardest theorems.If you think I’m exaggerating, here is what G. H. Hardy wrote in his celebrated (yet insufferable) mathematical autobiography:There is no scorn more profound, or on the whole more justifiable, than that of the men who make for the men who explain. Exposition, criticism, appreciation, is work for second-rate minds.This is peak dissociation. Behind closed doors, mathematicians are quick to complain about Hardy’s curse. They insist on the importance of teaching, even for their own comprehension of the subject matter. They lament the system’s pathological obsession with theorem-proving priority, while everyone knows the hard work often takes place outside of that loop, when trying to make sense of existing results. Yet, in public, they are bound by the honor code of mathematicians. Prove theorems and shut up!There is one exception, though. Once you get the Fields medal, you are free to say whatever you want.Bill Thurston, the 1982 Fields medallist, was a spectacular dissenter. Two years before his death, he took part in an extraordinary exchange on MathOverflow, in response to this question posted by an insecure undergrad:What can one (such as myself) contribute to mathematics?I find that mathematics is made by people like Gauss and Euler—while it may be possible to learn their work and understand it, nothing new is created by doing this. One can rewrite their books in modern language and notation or guide others to learn it too but I never believed this was the significant part of a mathematician work; which would be the creation of original mathematics. It seems entirely plausible that, with all the tremendously clever people working so hard on mathematics, there is nothing left for someone such as myself… Perhaps my value would be to act more like cannon fodder? Since just sending in enough men in will surely break through some barrier.Thurston jumped in:It’s not mathematics that you need to contribute to. It’s deeper than that: how might you contribute to humanity, and even deeper, to the well-being of the world, by pursuing mathematics? Such a question is not possible to answer in a purely intellectual way, because the effects of our actions go far beyond our understanding.

We are deeply social and deeply instinctual animals, so much that our well-being depends on many things we do that are hard to explain in an intellectual way. That is why you do well to follow your heart and your passion. Bare reason is likely to lead you astray.2 None of us are smart and wise enough to figure it out intellectually.The product of mathematics is clarity and understanding. Not theorems, by themselves. Is there, for example any real reason that even such famous results as Fermat’s Last Theorem, or the Poincaré conjecture, really matter? Their real importance is not in their specific statements, but their role in challenging our understanding, presenting challenges that led to mathematical developments that increased our understanding…Mathematics only exists in a living community of mathematicians that spreads understanding and breathes life into ideas both old and new. The real satisfaction from mathematics is in learning from others and sharing with others. All of us have clear understanding of a few things and murky concepts of many more. There is no way to run out of ideas in need of clarification…Here we need to take a short metaphysical break, because it is all too easy to brush Thurston’s words off as “feel-good” or “woke”.In my first Substack post, I (half-jokingly) declared that we had been wrong about mathematics for 2300 years, stuck in a false dilemma between formalism (“mathematics is a meaningless game of formal symbols”) and Platonism (“mathematics captures properties of actual entities living in the perfect world of ideas”).My proposed conceptualist resolution is a rephrasing of Thurston’s view: mathematics does rely on a meaningless game of formal symbols, but we only play this game because we project meaning onto it.Meaning is a cognitive phenomenon—a product of our neural architecture—and not a direct access to transcendence.When we “do math”, we manipulate formal expressions and gradually develop an intuitive feel for what they represent, as if they were pointers to objects that “existed” in a Platonic sense. Platonists take this neuroplastic side-effect at face value. Formalists view it as accessory. Conceptualists like me recognize mathematics as a critical cognitive infrastructure of the human species.A natural question is why the conceptualist resolution took so long to emerge. One reason is that it goes against the prevailing spiritualist worldview, which refuses physicalist interpretations of mathematics.