My 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.1

I 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.

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Cracking the honor code

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.

It also goes against the honor code of mathematicians. Hardy’s curse is so powerful that even Thurston found it hard to overcome. When multiple MathOverflow users thanked him for his take, he noted in reply:

Thanks for the comments. I try to write what seems real. By now, I have no cause to fear how I will be judged, which makes it much easier for me. It’s gratifying when my reality means something to others.

But then, how could such a toxic honor code survive for so long?

The answer is simple. The honor code was useful to mathematics as an academic discipline. It helped it stay exceptionally healthy and meritocratic, as noted in the epilogue of my book:

This system has its merits. It reduces arbitrariness and helps mathematicians guard against complacency and nepotism. When a discipline deals with eternal truths, it offers a neat way to evaluate careers.

The honor code also served as a guide to researchers themselves, when evaluating new ideas and new directions of research. Concept-building and problem-solving, the two facets of mathematics, are in a symbiotic relationship, as remarked by 2018 Fields medallist Peter Scholze:

What I care most about are definitions. For one thing, humans describe mathematics through language, and, as always, we need sharp words in order to articulate our ideas clearly… Unfortunately, it is impossible to find the right definitions by pure thought; one needs to detect the correct problems where progress will require the isolation of a new key concept.3

This is how the system worked for millennia. Mathematicians created value by introducing new concepts, but the rule was that only theorems could put bread on the table. The deal was fine because the two aspects almost always walked hand in hand. David, the social parasite who claimed credit for the 𝐾⁡(𝜋,1) conjecture, was the same person as the David who crafted Definitions 2.4 and 9.3.

Solving a big conjecture was a cryptographic proof that you had come up with a genuine conceptual innovation.

I am using the past tense because this is no longer the case. There is a structural vulnerability in the honor code of mathematicians and AI has started exploiting it in a systematic manner.

The way of Go and Chess

The trigger for this post was a speech by Geoff Hinton, which caught me off guard:

I agree with Demis Hassabis, the leader of DeepMind, who for many years has said AI is going to be very important for making scientific progress…

There’s one area in which that’s particularly easy, which is mathematics, because mathematics is a closed system…

I think AI will get much better at mathematics than people, maybe in the next 10 years or so. And within mathematics, it’s much like things like Go or Chess that are closed systems with rules...

I was used to the general public being profoundly wrong about the nature of mathematics. But I wasn’t prepared for a Turing awardee and Nobel prize winner comparing it with Go and Chess.

I wrote a short response on X and tried to move on, but it kept troubling me.

Then it all clicked into place. About a year ago, I had been approached by a young mathematician friend who had done his PhD in my domain. He was thinking about launching an “AI for pure math” business and I mentored him for a while.

Like him—and like Hinton and Hassabis—I was fully convinced that AI was about to transform mathematics and science in general. But I was unsure about the business model and minimum viable product.

Mathematicians may look like Luddites, but they rarely are. They love pen and paper, blackboard and chalk, but they jumped on Donald Knuth’s typesetting revolution. A century ago, they chose to rebuild their entire knowledge stack on a new operating system, set theory, that promised massive gains in reliability and scalability. A few decades later, they recognized that there was no real difference between a mathematical proof and a calculation, and set out to build the first computers. Deep learning, with its heavy use of linear algebra and stochastic gradient descent, is a brainchild of mathematics.

In the 1970s, when Kenneth Appel and Wolfgang Haken built a computer-assisted proof of the four-color theorem, this opened an intense debate on the epistemic nature of such proofs and their admissibility in peer-reviewed journals. Although, to be honest, there never was much suspense—the barbarians won, because there were barbarians on both sides.

Computers had always been part of my mathematical life4 and the promise of AI and autoformalization had long felt irresistible to me. This is what got me excited when my friend reached out and asked for my advice.

I started looking at the “AI for math” space and couldn’t understand what was going on. Why were these startups raising so much money? Pure mathematics is such a tiny market. The investments felt disproportionate.

My preferred strategy, the one I would have pursued, was to create the Wolfram Research of the AI age. Mathematics-enabled science and technology is a much larger market than pure mathematics and, as Wolfram demonstrated, there is room for simplifying and productizing the experience of interacting with mathematical objects. The users love the product and it is sticky.

But my friend insisted he wanted to do something specifically about pure math.

I didn’t know what to say, because I was stuck. The only useful products I could think of were literature spotters and interactive proof assistants—hard to package, hard to price, and even harder to sell. I could see a long term business strategy, but it was one I wouldn’t touch with a ten-foot pole—becoming the Elsevier of the AI age, the most hated brand in science, an arm-twisting extractive monopoly that repackages the mathematical commons into a mandatory experience.

There was a third strategy, though. But it was risky and, like the previous one, it did require a certain degree of cynicism. I’d call that plan the luxury acquihire: 1/ build a useless product that is striking enough, 2/ give the impression that you have solved a major scientific problem, 3/ pray for a quick M&A by a tech giant or a major AI lab.

Still, the numbers didn’t add up. The “AI for math” startups were rumored to be raising hundreds of millions. There must have been a smarter investment thesis, which I was failing to comprehend.5

Then I heard that Google was leading a massive effort to solve the existence and smoothness of Navier–Stokes equations. I thought OK, I get it, that’s a Millennium Prize problem. But, wait, that still doesn’t make sense—the payout is one million dollars, peanuts. As one great mathematician remarked to me, Google likely mobilized more brainpower on this single effort than the entire community ever did.

It only started to make sense after I heard Hinton’s speech.

If mathematics really was a closed system—or if this is what all the stakeholders around the table are willing to believe—then the investment pitch becomes trivial: “DeepMind solved Go and Chess, we’re going to solve mathematics!”

At a time when the leading AI labs are betting trillions that humans are soon to become obsolete, the promise of “solving mathematics”, the crown jewel, the pride of the human race, is simply irresistible.

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Bringing a caliper to a gunfight

On February 5th, a team of eleven high-profile mathematicians (including Martin Hairer, the 2014 Fields medallist) announced the First Proof project and released a first batch of ten “research-level math questions”:

This manuscript represents our preliminary efforts to come up with an objective and realistic methodology for assessing the capabilities of AI systems to autonomously solve research-level math questions. After letting these ideas ferment in the community, we hope to be able to produce a more structured benchmark in a few months.

One of our primary goals is to develop a sophisticated understanding of the role that AI tools could play in the workflow of professional mathematicians. While commercial AI systems are undoubtedly already at a level where they are useful tools for mathematicians, it is not yet clear where AI systems stand at solving research level math questions on their own, without an expert in the loop.

From a purely scientific perspective, there is nothing to complain about. These are incredibly smart people, engaging a real-world controversy with an open-minded attitude and a creative approach.

The First Proof team was doing everything right—and this is what terrified me.

But before I explain, I must reiterate that I have a very high opinion of this project. The team represents the mathematical community at its best, people driven by curiosity and integrity, willing to experiment outside of their comfort zone, and they did come up with genuinely good ideas.

Daniel Litt wrote an excellent essay, Mathematics in the library of Babel, on the First Proof project and his own first-hand assessment of the AI-for-math situation. His perspective is that of a radical non-Luddite, a pure mathematician who has engaged with LLMs for years and is genuinely impressed by the recent progress.

He was surprised by how many of the First Proof open problems ended up being solved by the teams at Google, OpenAI, and others. By his own count:

It seems likely that somewhere between 6 and 8 [of the 10] problems were solved correctly if one combines all attempts.

There are serious caveats, though:

The models (and the humans supervising them) generated an enormous amount of garbage, including some incorrect solutions claiming to be formalized in Lean. Even the best models/scaffolds seem not to be able to reliably detect when they are producing nonsense.

Even when AI-for-math systems operate autonomously, which few in the current crop actually do, they still require humans to intervene upstream and downstream, if only to assess the results and filter out the junk. This isn’t anecdotal. The AI labs are investing so much human intelligence into these projects, much more than any real-life mathematician can ever mobilize, that the delineation is never entirely clear. The most damning illustration is this:

It was not clear to OpenAI which of their solutions to First Proof are correct.

In other words: without the pro bono effort of the good old academic community, they might have never known. (Litt was himself a contributor.)

In truth, this also applies to human-generated proofs. But there is a fundamental nuance. In the human way of doing math, theorem-proving and concept-building walk hand in hand, which forces proofs to be intelligible (if only to their authors).

This is where the metaphysics comes back to bite. If mathematics was just a formal game of meaningless symbols, intelligibility would be a vacuous notion. The reality of mathematical practice forcefully points in the opposite direction. As it happens, published research is full of bugs, but these bugs tend to be contained and fixable, precisely because human-generated proofs are meaningful and (almost always) directionally correct—two notions that are impossible to reconcile with the formalist worldview.

This isn’t a cosmetic nuance. This is a mandatory condition for mathematics to exist as a sustainable endeavor.

As noted by Litt, the essential characteristic of being “truth-seeking” was missing from the AI solutions to the First Proof questions:

Many correct solutions are very poorly written, and their correctness is exceedingly difficult to check because of this... In [human solutions] the main ideas, obstructions to be overcome, etc., are usually identifiable; in [AI solutions] they are often completely unclear. And in the course of writing their solutions, the human authors often develop useful new objects, terminology, etc. to capture what they’re doing, while the models usually just plow ahead.

This is where the second leg of the math-for-AI revolution comes into play: autoformalization, the capability to transform the mildly informal style of human-produced proofs—and the outputs of LLMs trained on them—into bulletproof, machine-verifiable logical derivations, expressed in specialized languages such as Lean.

On paper, this can address the issue of correctness and remove the need for human validation. But, as noted by Litt, “there was only one credible formalized solution to any problem, impressively and manually orchestrated by Tom de Groot.”

This is of course an evolving situation. The stakes are high and the investment massive. LLMs, scaffolding, and autoformalization, are all making steady progress.

What would happen if, a year from now, the First Proof team released another set of 10 problems of equivalent difficulty? Litt doesn’t answer this specific question, but he expects AI to autonomously produce results “at a level comparable to that of the best few papers” within the next few years.

I share his sentiment. In my view, the likely outcome would be that the leading AI labs would score a perfect 10, coming up with fully automated & fully correct solutions to all problems.

Does that mean that AI will have “solved mathematics” by early 2027?

Of course not. There are three additional caveats that I am sure all insiders will have already spotted, but non-specialist readers may have missed.

Caveat #1: Oceans

The first one is straightforward—the First Proof “research-level questions” were neither profound nor difficult. They were closer to technical lemmas, well-calibrated intermediate subproblems that occur in the course of proving a theorem and are typically handled in a few paragraphs or pages. There is an ocean between a technical lemma and a serious paper, another ocean between a serious paper and a breakthrough, another ocean between a breakthrough and a Fields-medal-level contribution, and yet again several oceans above that.

First Proof is now working on a second batch, which is likely to include harder problems.

From a technical perspective, they did pick the right level for their first batch. This only appeared in hindsight: the final score was above 0/10 and below 10/10, in the sweet spot for their stated goal of finding an “objective and realistic methodology for assessing the capabilities of [current] AI systems”.

Yet I think that releasing a benchmark that didn’t include serious problems was extremely dangerous. The general public doesn’t read the fine print, and may not even know what a lemma is. If Google or OpenAI had scored a perfect 10, the headline would have read: “GAME OVER: The world’s top mathematicians challenged an AI with 10 research-level problems; the AI nuked each and every one of them.”

Similar distortions are being made every single day on social media, feeding the very confusion that First Proof was trying to dispel.

Caveat #2: Accretiveness

The second caveat is much more profound. It is also extremely subtle and hard to communicate to the general public, and this combination is creating a tricky situation for the mathematical community.

The problem with unintelligible proofs goes way beyond correctness, and cannot be resolved by autoformalization alone: even if correct, unintelligible proofs aren’t accretive to the mathematical corpus.

I know this sounds barbaric. Let me explain with an example. A few weeks ago, Math Inc, one of the best-funded AI-for-math startups, produced a Lean formalization of Maryna Viazovska’s spectacular work on the sphere packing problem in dimensions 8 and 24, results which earned her Fields medal in 2022. That was impressive in its own right: never before had theorems of this caliber been autoformalized. Yet this clear success was met by a massive pushback from the “formal mathematics” community, the very people who lead the effort to port “human mathematics” into machine-verified code.

Luddites?

Well, it’s more complicated, as it emerges from their actual conversation. The thing is that autoformalization isn’t a full solution to the problem of formalizing mathematics, just like Tesla’s Full Self-Driving isn’t a full solution to the problem of driving cars.

Yes, I know, that sounds counterintuitive. This is why outsiders are likely to miss the nuance and label the pushback as Luddism.

A clear explanation can be found in Alex Kontorovich’s account of his own learning curve with formalized mathematics. In a nutshell: Mathlib, the dominant Lean library, is a human-curated formalization of an ever-growing fraction of existing human mathematics. It exposes clean APIs and abstractions, without which no autoformalization could take place. By contrast, Math Inc’s autoformalized proof of Viazovska’s results exposes no intelligible interface. Who in their right mind would merge a 200,000-line unaudited vibe-coded blob into the master branch of global human science?

Kontorovich has a great expression for what is missing—canonization:

By canonization, I mean the process of taking a local, one-off formalization and turning it into library mathematics: general, reusable, coherent, efficient, and compatible with the rest… Canonization often changes the picture itself: the definitions, the abstractions, the API, and sometimes even the statement… This is extremely difficult and time-consuming.

But this, again, might be mistaken for Luddism. “General”, “reusable”, “coherent”, “efficient”… Aren’t these reactionary arguments made up by dumb humans with narrow context-windows? Why would an AGI care about that? Isn’t autoformalization, by design, the definitive solution of all problems with vibe-coding?

Doesn’t proven code meet the highest conceivable standards for software? Why are you constantly shifting the goalposts?

The software analogy helps understand the core issue. For normal software, quality is first and foremost a pragmatic notion—good code should compile, run smoothly in production and satisfy user needs. But Lean code never runs anywhere. It is just a library that sits in a repository and might be imported in the future, who knows when, in the process of formalizing another theorem.

You cannot “ship and let the users do the QA for you” because your only user is the future of mathematics. This is why canonization, from Euclid to Zermelo-Fraenkel to Bourbaki, has always been a core concern of the formalist school.6

We’re back to the metaphysics. The problem with unintelligible mathematics isn’t that it might be false. It is that it is literally meaningless, in the sense that it doesn’t compile on the only hardware that is currently able to make sense of it and appreciate its value—the human brain.

This, of course, could change in the not-so-distant future. There is nothing magical about the human brain. Artificial sense-making and truth-seeking architectures will certainly emerge, and at some point they will probably surpass humans on all aspects of mathematical creativity, including the canonization of formal proofs (which, from an algorithmic complexity perspective, is a mandatory requirement for all forms of mathematics, whether human or superhuman).

Yet this is absolutely not our present—nor a straightforward continuation of current trends—and I don’t think the conditions are met for a fruitful debate. In any case, if humans are still around, they will still feel the need to understand the world, and this will continue to drive them to engage with mathematics.

No one knows the timeline to AGI. Meanwhile, the autocanonization capabilities of frontier models barely exceed zero and Math Inc’s autoformalized proof of Viazovska’s theorems sits as an orphaned 200,000-line blob.

This is what made the Mathlib community so angry. They had been working on a multiyear project to formalize Viazovska’s work. Math Inc jumped in on this collective effort, leveraged prior insights, then abruptly went silent, until they made their spectacular announcement.

Is this necessarily bad news? Now that the brute-force autoformalization is done, why can’t the Mathlib community refocus on the value-creating canonization?

Because of Hardy’s curse and the honor code of mathematicians. Math Inc captured the prize (“first formalization of a Fields-medal-level theorem”) and there is no social reward left for cleaning up after them. Hence this comment by Patrick Massot, a non-Luddite expert in formalized mathematics:

I think the situation is pretty clear: AI companies, and especially Math Inc, will indeed thoroughly bomb this area to turn it into a giant radioactive wasteland that will never be able to sustain life again, so we will never get the benefits expected from formalization (improved understanding and accessibility). I strongly advise young people to contribute to less shiny projects that are less likely to be destroyed.

What makes the situation really tricky is that unintelligible formal proofs may hold significant residual value, even if they aren’t accretive to the canonized corpus.

And, to be honest, the issue existed well before AI, with the four-color theorem, with the classification of finite simple groups, or with Tom Hales’s monumental work on the Kepler conjecture (which led him to seek a formalized proof).

The likely outcome is that formalized mathematics will now develop in two separate layers, an intelligible layer embodied by Mathlib, and an unintelligible layer we might call Mathslop, a library of results that are known to be correct via proofs that no human has ever understood.

Caveat #3: The Overhang

When the First Proof problems were released, my greatest fear was that the big AI labs were going to one-shot every single one of them, for the wrong reasons—because the answers might have been available somewhere in one shape or another.

Litt notes that 2 of the 10 problems had solutions7 in the existing literature (which, no surprise, most LLMs were able to leverage), while a “sketch of proof” was available for a third problem (which, nevertheless, no LLM was able to solve).

But what about the remaining 7 problems? Were they really open? In what sense?

My view is that it is impossible to say for sure, due to a structural feature of the mathematical corpus which, no doubt, is going to play a central role in the AI-for-math debate. In fact, I suspect that our legacy notions of “creativity” and “innovation” are ill-founded, and that AI is about to teach us brutal lessons about them.

Most mathematicians are intuitively aware of this structural feature, although its exact shape and size are impossible to chart. I haven’t seen any serious attempt to theorize it and the feature has no agreed-upon name—let me call it the Overhang.

I expect the Overhang to be absolutely gigantic.

I met it on several occasions in my career. Most encounters were fairly casual, except for one mystifying experience that took place as I was doing my best work, right after I came up with the core idea in my Definition 9.3. That was an entirely new concept, built on a stack of new ideas that I had just come up with, unlocking the solution to a classical conjecture—subjectively, that felt like a stroke of genius.

As I was “canonizing” my most creative idea ever, I made a shocking discovery. My proud invention, divided Garside categories, was essentially equivalent to a seemingly esoteric construction in an entirely different branch of mathematics, namely the Bökstedt-Hsiang-Madsen subdivision of Connes’s cyclic category in algebraic K-theory.

In other words, my one genius idea was déjà vu. Or was it? Two decades later, I continue to view this moment as the high point of my career. I don’t really care if my idea was or wasn’t “original”, as I doubt this is the most salient notion. What struck me was the sudden flash of meaning, which was otherworldly. (A clear cognitive sign of long-term value.)

In any case, a good LLM might have spotted the syntactic analogy in the preliminary phases of my work, when I still had no clear idea where it was leading. Then it could have scooped me by “front running.”

The existence of profound correspondences between seemingly unrelated topics is one of the most celebrated aspects of mathematics, a source of joy and marvel. It might be the purest expression of mathematical beauty. Descartes changed the world by noticing that algebra and geometry were essentially the same, and by building a bridge between them. Once you “invent” cartesian coordinates, many legacy problems become trivial—and interesting new problems instantly emerge.

But modern mathematics is so fabulously complex that most correspondences go unnoticed.

Sometimes connecting the dots leads to initial mystery rather than discovery. A striking example is John McKay’s famous remark that 196 883 + 1 = 196 884. The number 196 883 appeared in the study of the Monster, while the number 196 884 appeared in the study of modular forms, two areas of mathematics that were mutually alien. The conjectural vision that they should be related seemed so ludicrous that it was labelled the Moonshine theory, or the Monstrous Moonshine. Richard Borcherds received the 1998 Fields medal for proving it correct.

But sometimes connecting the dots is enough to solve a major problem. The last mile of proving a conjecture is often about realizing that the missing bit was already present somewhere in the literature.

The Overhang consists of the unrealized capital gains of past mathematical creativity, the latent value from connecting the dots in the existing corpus. It is a dividend of canonization. Mathematician X states problem A, mathematician Y crafts concept B, then mathematician Z notices that B trivially solves A and “captures” the social reward. But in the process of capturing the reward, Z usually introduces new concepts and new open problems, reinjecting latent value into the Overhang.

LLMs can be trained on the entirety of the mathematical corpus. Thanks to their phenomenal memorization and pattern-matching abilities (without always being able to map out their associative logic and attribute due credits), they are in a unique position to harvest the Overhang. By contrast, professional mathematicians have typically read a few hundred articles in their career, out of millions of existing references, less than 0.1% of the total.

This will lead to great discoveries, which is unambiguously exciting. But it could also lead to a sad new deal, where human slaves painfully curate the Overhang while AIs systematically beat them at the finish line.

We are very far from it, though, which in and of itself is disorienting. Litt adds this sharp remark:

The mystery is this: a human with these capabilities would, almost certainly, be proving amazing theorems constantly. Why haven’t we seen this from the models yet? What are they missing?

The answer seems quite obvious—current AI systems and humans process mathematics in entirely different ways. The best models are insanely stronger on certain aspects, which necessarily implies that humans are still insanely stronger on others.

AI is at the same time superhuman and subhuman, depending on how you look at it.

This is a compliment to both AIs and humans. This plurality of aspects makes the whole “benchmarking” enterprise extremely fragile, if not ludicrous. There is a real possibility that AI will achieve problem-solving supremacy long before it achieves concept-building adequacy. What are we going to call that? “Superintelligence”? Seriously?

This is my core issue with the First Proof approach. It is a meaningful benchmark for working mathematicians wondering what AI can do for them in the present, from within the constraints of an obsolete honor code, at a moment when they urgently need to break away from it.

Why don’t we construct benchmarks that are fairer to humans? Because we can’t. Because the true value of mathematics, the collective and individual elevation of our worldviews, is ill-defined and intangible. This intangibility was the raison d’être of the honor code.

For millennia, we had agreed to only benchmark human intelligence on its problem-solving facet, as we had found that it was the best objective proxy. Theorem proving is so inordinately difficult for our cognition that the only progress path was through patient concept building and neuroplastic internalization of these new concepts.

Yet this was only ever a proxy. The thing we really care about is different in kind. Industrial robots are far stronger than humans, yet we still go to the gym. Blenders have been outchewing us for over a century, yet we still don’t eat exclusively through straws.

I know the optics are horrible. Declaring that math is first and foremost about comprehension, an unbenchmarkable aspect, sounds like an all-too-convenient excuse. Yet this isn’t something I am pulling out of my hat at the last minute. Here is what Thurston wrote in 2011:

Mathematics is commonly thought to be the pursuit of universal truths, of patterns that are not anchored to any single fixed context. But on a deeper level the goal of mathematics is to develop enhanced ways for humans to see and think about the world. Mathematics is a transforming journey, and progress in it can be better measured by changes in how we think than by the external truths we discover.

I cited this passage in my book, written five years ago, before the AI storm started. I also cited a 1628 rant by Descartes against the Ancient Greek mathematicians who, “with a kind of pernicious cunning”, had “suppressed” the “true mathematics”(which he identified with the inner cognitive methodology) and only published “childish and pointless” stuff, “the fruits of their method, some barren truths proved by clever arguments, instead of teaching us the method itself.”8

Terry Tao made a similar comment in his recent interview with Dwarkesh Patel:

In math, the process is often more important than the problem itself. The problem is kind of a proxy for measuring progress.

He also made this remarkable prediction:

I think within a decade, a lot of things that mathematicians currently do—what we spend the bulk of our time doing and a lot of stuff we put in our papers today—can be done by AI. But we will find that that actually wasn’t the most important part of what we do.

The feedback on social media was overwhelmingly positive, as is often the case with Tao’s public interventions. Yet I did notice some unusual dissent, specifically on the notion that problem solving in itself isn’t all that valuable.

To the AGI prophets, this passed as a refusal to see the writing on the wall. There were some actual sneers.

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This is not a drill

On March 3rd, Jeremy Avigad, the first director of the newly-created Institute for Computer-Aided Reasoning in Mathematics, posted a five-page essay on Mathematicians in the Age of AI.

It contains a detailed account of the Math Inc vs Mathlib incident, of which Avigad was a prime witness. He too is a radical non-Luddite, ideally positioned to observe the rapidly changing situation:

We have to face up to the fact that AI will soon be able to prove theorems better than we do…

We need to remember our strengths: mathematicians are problem solvers and theory builders extraordinaire. Rather than fight the use of AI in mathematics, we should own it. It is not enough to keep up with current events and design benchmarks for AI researchers; we need to play an active role in deploying the technology and molding it to our purposes.

Regarding the First Proof initiative, Avigad admits having been “surprised” that Google could solve “six out of the ten problems.”

As written above, I would have put it differently—I felt more “relieved” than “surprised.” But I was “surprised” too, in the sense that I had no intuition of how difficult the problems really were and no clear anticipation of how many would be solved.

What I did find disconcerting was Avigad’s congratulatory comment on the interest sparked by First Proof:

A few AI developers took the bait

I am not sure if this is irony or naivety. To people familiar with the tech ecosystem, it was pretty clear that the First Proof problem set would trigger a five-alarm scramble at all the leading AI labs. Who could miss out on an opportunity like that?

Here is how Greg Brockman, the president of OpenAI, commented on it:

“First Proof” (http://firstproof.org) is a very cool concept and approach. congrats and thank you to the team who put it together!

The timeline is remarkable. Brockman’s comment came on February 14th, as OpenAI was on a roadshow to secure the largest private technology fundraise in history. Even more remarkable, Brockman wasn’t the only member of OpenAI’s leadership team to get “distracted” by geeky side-projects in the midst of these incredibly tense weeks. A few days before, Sam Altman had found the time to do a bit of coding on his own and express his sadness at realizing that Codex had rendered him “useless.”

But was this ever a distraction? You could see this as a massive fish taking the bait, or a fox thanking the hens for handing the keys to the henhouse.

To be clear, I don’t think OpenAI—or any of the other AI labs—was doing anything evil. The First Proof team hadn’t left them any choice. Once the problems had been released, they had to compete, and they had to go all in, because they couldn’t afford to be outclassed.

In fact, all this is normal. The AI labs have genuinely exciting tech, and theorem-proving is a genuinely exciting use case.

The only problem is what we are all tempted to read into it.

This is what made Hinton’s speech so troubling. He presented mathematics as a low-hanging fruit, a “particularly easy” terrain for AI, because it is “a closed system with rules,” “like Go and Chess.” Yet Go and Chess aren’t ordinary closed systems. They are actual games with winners and losers. Even more so, they are finite two-person games with perfect information and, by a 1913 theorem of Zermelo, they admit optimal strategies.9

But if AI ever “wins” at mathematics, then someone else will have to be the “loser,” and the only other player in the room is human thought.

Who cares if mathematics isn’t an adversarial game—and we are still figuring out the rules. When the news is this sensational, it becomes fit to print.

Here is what the nuclear scenario looks like. A year from now, an AI lab assembles a secret A-team, throws a few billion dollars of compute at the Riemann hypothesis, and comes up with a 2,000,000-line Lean proof. What are mathematicians going to say?

That the proof is unintelligible? Right, but who cares? Isn’t the whole of math already unintelligible to most people?

That the theorem was still proven by the humans in the loop? Yes and no—and in that specific case I would probably vote no.

That a few billion dollars represents decades of the entire mathematical sciences budget at the National Science Foundation? Correct, so what? And what if the conjecture happens to be much easier, accessible with only nominal resources?

That the conjecture itself was still produced by a human brain, along with the entire mathematical stack underneath, starting from the very language in which the conjecture was written, the concepts of infinite sums, complex numbers, meromorphic functions and analytic continuation, the interpretation of the conjecture as a profound statement about the distribution of prime numbers, and the prophecy that it should be true? And that, at this point, humans are the only beings who care about it?

This objection is profound and valid. But when you look like a sore loser, nobody cares what you say.

I don’t expect this exact scenario to unfold. But I do expect to see, in the near future, one or more claims that AI has “solved math”, or “achieved mathematical supremacy.” And, given the current state of the conversation, I expect the actual evidence supporting these claims to be framed and/or understood as a defeat for human intelligence.

If Geoff Hinton doesn’t get it, then 99.9999% of people won’t get it either.

Whatever they will feel the urge to say when the disaster happens, mathematicians should start saying now. It is probably too late to entirely dodge the bullet, but at least they have a little time to anticipate and coordinate.

The clearest sign of urgency is the growing defiance of younger mathematicians, many of whom are joining the AI labs. Full disclosure: I understand their choice, and would likely do the same if I were in their shoes.

Meanwhile, non-tenured academics are showing signs of despair. Ashvin Swaminathan, a postdoctoral fellow at Harvard, commented that Litt’s essay, which he agrees with, left him “with a lot of questions about the longevity of research mathematics as a profession outside of the most elite institutions.”

Swaminathan’s thread is a great read that touches on an essential aspect of mathematics—teaching:

One might ask whether research mathematicians can support their work through teaching. But that ignores the dramatic impact that AI is having on teaching. A university degree may very soon be worthless, given how many students use AI to do their work for them. There’s a reason why the number of tutoring requests I’ve received, and the number of students who attend office hours, has decreased dramatically. There’s a reason why students do incredibly well on homeworks and far worse on exams. Many kids are graduating with no real skills.

Let me emphasize that this is the same exact issue as in research. Problem solving is just a means to an end. The core payout of pure mathematics is the neuroplastic elevation of our worldview, and we can’t get there by copy-pasting the result.

The false notion that mathematics is only about proving stuff (or finding the right results) is toxic at all levels, starting in primary school.

In particular, the frequently heard claim that “AI can already perform PhD-level research”—if and when it will be correct—isn’t as compelling as it may seem. PhD students are students. Reasonable advisors direct them to problems with accessible strategies. In fact, most published research sits comfortably within the Overhang, where it serves a dual purpose, part research in the true sense, and part cognitive maintenance of the research community. That is not a dig at my former colleagues, but a testament to the genuine difficulty of frontier mathematics, which only exists as the spearpoint of a collective endeavor focused on intelligibility.

If theorem-proving remains their only official currency, all research mathematicians run the risk of becoming demonetized in the course of the next few years.

Coming clean

Is there anything the mathematical community can do?

I don’t see any miracle solution, but there are obvious approaches to limiting the damage. The first one is to finally come clean about the nature of mathematics.

All the active mathematicians I have cited, from Tao to Avigad, from Litt to Kontorovich, are careful to note that there is something in mathematics that goes beyond theorem proving. Yet this is usually framed as a passing remark, not a major shift in the narrative.

Well, this is a major shift in the narrative, and the public will never take it seriously if it isn’t properly explained. Right now, it looks like an excuse.

You can’t reposition a plurimillennial brand without being explicit about what you are doing, why you are doing it, and which additional value it will bring to everyone. It is necessarily a big deal, necessarily difficult, and the time-window to get it right is extremely narrow.

This might be the largest product recall in memory—we sold a bogus idea of math to billions of people, and the bill is coming due.

The only way is to lead with the core message and hammer it over and over again. Let me do my share by citing another Thurston quote (also picked by Avigad as the epigraph to his recent essay on Mathematical Understanding):

We are not trying to meet some abstract production quota of definitions, theorems and proofs. The measure of our success is whether what we do enables people to understand and think more clearly and effectively about mathematics.

Long term, that could force a complete rethinking of the ethos and evaluation criteria of the profession. Mathematicians will have to revoke the honor code and the second law of the Intuition Club—“if you really want to talk about intuition, make it sound casual and accessory, because we ain’t the psychology department.” If Thurston is correct, then we are the psychology department.

All that is easier said than done. The mathematical community isn’t a structured organization that can convene an emergency offsite and put together a crisis response.

It doesn’t even share a unified view of mathematics. Some genuinely think the whole point is to “solve previously unsolved problems”, while others object to any serious discussion of intuition. I disagree with both of these views, but my opinion is nothing but that—an opinion.

The absence of a clear definition and mission statement keeps the community on the back foot.

Christian Szegedy, a researcher who has been a co-founder of xAI and Math Inc, made the prediction that a “super-human mathematician AI” would emerge by June 2026. This is the kind of declaration that is either empty (electromechanical calculators have been superhuman for a century) or maximalist in the worst possible manner. Not surprisingly, Szegedy also wrote that he found Terry Tao “out of touch with the reality that was coming up quickly.”

The problem isn’t with the prediction itself (Szegedy has real technical legitimacy and the right to his opinions, however grandiose) but with the general climate that makes it very difficult for a response to be heard.

One simple idea I keep returning to is that we need something like a Mathematical Intelligence Scale, or Mathematical Automation Scale, like there is for self-driving cars.

That would address the pressing danger of objective benchmarks that, because of their very nature, fail to account for intelligibility and concept-building. When AI crushes the next iteration of First Proof, the press release should read: “AI solves First Proof and achieves level 3 (out of 10) on the Mathematical Intelligence Scale”.

Mathematicians shouldn’t be afraid of subjectivity. It is not about cheating. It is about being fair and not committing collective seppuku. In fact, any act of problem-setting is already subjective—there is nothing objective about the First Proof problems, Erdős problems, or Millennium Prize problems, apart from the fact that (in theory) we can objectively assess whether any of them has been solved.

This quality of being objectively verifiable is missing from the more profound aspects of mathematical creativity, such as problem-setting itself, canonization and concept-building, reframing, simplifying, conjecturing, up to major theory-building and the creation of entirely new branches of mathematics—yet that doesn’t mean these activities do not exist and aren’t central to mathematics.

There is even an objective dimension to mathematical subjectivity. When Descartes connected algebra to geometry, when Newton and Leibniz invented calculus, it was pretty clear to their contemporaries that something major had just taken place. In a sense, our entire technostructure, including AI startups, lives on the dividends of those inventions.

An ideal scale would be open-ended—or have seemingly inaccessible upper rungs—since math isn’t something to be “solved.”

Mathematics is known to be inexhaustible and infinitely hard. This implies that there is room for infinite progress. The one urgency is to make it clear to everyone that it isn’t a trophy that is up for grabs.

The future of mathematics

Sadly, a lot of the damage is already baked in. Even with perfect execution, it is impossible to conduct an epistemic revolution under the pressure of an external timeline. This one will have to take place in a broader context that is extremely unfavorable to core science, where every nuance is instantly dismissed as gatekeeping.

These will be difficult years for mathematicians. Students will desert. Money will dry up. Postdocs will ask pressing questions and senior faculty will have to come up with answers. At parties, people will start giving them weird looks, as if they had become living fossils.

Yet, at some point, the pressure will alleviate.

The industry will have to normalize and the AI-for-math startups will have to find their business models. Which, in all likelihood, won’t be about solving Erdős problems.

To me, this might be the most striking aspect of the current moment. For all the flex and fury, the AI labs are barely engaging with mathematics itself, its core problems and significance. Which is entirely normal, since this is none of their business.

When the dust settles, if it ever does, we will find out that human mathematics has survived, transformed.

I won’t make any prediction on the timeline, because I have no idea. I don’t know which capabilities will be achieved when, which breakthroughs they will enable, and which limitations will appear in hindsight.

What I can say is that the change is already taking place. This really is a scientific revolution and the research community has started actively engaging with it. As I was writing this article, I saw remarkable developments every single day in my timeline. My initial plan was to cite the most striking ones, but there are just too many and I am overwhelmed.

A pessimist would argue that this is a fear-induced response and a symptom of the disruption. We are seeing the same pattern across all industries. When capital gets reallocated so fast, on a scale so massive, there may be no rational way to prepare for the future.

It is not clear to me how I would approach this if I hadn’t quit active research. The low-hanging fruit would be to vibe-code a formalization of my yet-unwritten cellular decomposition theorem. Then I could test a few wild ideas and conjectures. But all this is old-world thinking, and I suspect that even the collaboration modes and deliverables of mathematical research will have to change.

My view, which is hardly original, is that it will become unthinkable to do math without AI assistance, just like it has become unthinkable to do math without set theory and LaTeX.

But this says nothing about the second-order effects. I tried to imagine them as a forecasting exercise, and came up with a dozen or so ideas, not all profound and not all credible. Here are three that I find interesting:

1. A clearer cut between pure and applied mathematics. While both deserve to be treated as first-rate citizens, I was never happy with the traditional way of drawing the line. Algebra is presumed to be pure because it is abstract, yet it can be immensely practical. On the other hand, “applied” domains tend to have their fair share of theoretical offshoots.

My take is that pure and applied should be regarded as different sweet spots in the intelligibility-vs-applicability trade-off. It is entirely rational to invest 10 billion dollars of compute in the equations of plasma containment, if it fast-tracks the design of grid-scale fusion reactors, even if intelligibility has to wait for another century or two. I wouldn’t say the same about the Riemann hypothesis.

2. The rise of intuition-maxxers. To me, that would be the clearest sign of a new Golden Age. We are not seeing them right now because the few people with the rare combination of math and agentic skills are poached by the AI labs. But this is a market anomaly that is likely to correct in the near future (the combination is becoming less rare, and the AI labs can only absorb so many of them).

I suspect AI will enable the same transformation that samplers and sequencers enabled in music. The technical barrier to grasp frontier concepts will drop, and brave young mathematicians will deploy unorthodox tactics to survey entire new continents at a pace that will mystify their elders.

They will see further than anyone before, standing on the shoulders of giant machines.

3. A renewed interest in the philosophy and neuroscience of mathematics. In his 1979 essay, Reuben Hersh noted that Platonism and formalism were both untenable, and added this comment:

The problems of truth and meaning are not technical issues in some recondite branch of logic or set theory. They confront anyone who uses or teaches mathematics. If we wish, we can ignore them... It would be surprising if this had no practical consequences.

AI is confronting us with a practical consequence that no one10 saw coming, and that is already impacting the cognitive well-being of the general population. If we fail to articulate the neuroplastic value of intelligibility, there will be no justification whatsoever for teaching mathematics, given that the solutions to all the mathematical problems encountered by 99.99999% of the population over the course of their lifetime will be available at their fingertips.

People are vaguely aware that this would result in idiocracy, but they can’t really explain why, and we have yet to find teaching and evaluation modalities that are adapted to the new reality.

What concerns me is the false belief that there is a “normal” level of mathematical competency, as if certain concepts were any more “real” or “concrete” than the others, and advanced mathematics was gratuitously abstract. This creates the illusion that the current (and wildly unsatisfactory) abilities of the general public somehow reflect a “natural” state—when they are in fact the result of massive (yet frustratingly inefficient) pedagogical investment—and represent some solid ground. The truth is that there really is no bottom, and nothing prevents humans from functioning in a culture where there is no vocabulary for numbers above 5 (as it is the case for some hunter-gatherer tribes).

Some prophesy that mathematics will eventually resemble Chess, a sport that a few eccentrics practice with passion and the general public can safely ignore.

This is a shortsighted view that fails to account for the meaning-making and truth-seeking dimensions of mathematics, as well as the massive cognitive benefits of mathematical training. The world makes a lot more sense once you have an intuitive understanding of numbers, sets, statistics, linear algebra, calculus, differential geometry, machine learning, combinatorics, dynamical systems, model theory, group theory, singularity theory, homotopy theory, lambda calculus, complexity, category theory,… (there is no reason to stop at any given place, even though it is probably unwise to order the full menu)

The situation isn’t unique to mathematics. AI adoption creates the same exact problems in many other domains. But mathematics has a unique status and a unique aura, owing to its ancient companionship with the very notion of intelligence.

This is why the “AI for math” situation is so significant, and why its proper resolution matters so much for the future of our species.

Many people hate math, and they tend to put AI in the same bag. They view anything with mathematics or computers in it as a threat to their humanity and subjectivity, their raw experience of being alive, immersed in this world, trying to make intuitive sense of it.

They might be surprised—and thankful—to find out that mathematicians are actually on their side, fighting the good fight. As Carl Jacobi once put it, “the object of mathematics is the honor of the human spirit.”

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The result is stated in the last sentence of the last slide of my presentation:

Complex Reflection Groups Are Somehow Real

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One morning about a week ago, I opened my direct messages tab on X and found this inbound request:

I am someone finishing up a PhD in [a subject that isn’t pure math]. How should I cope with the claims that agi is here in 1-3 years and that all intellectual labor will be done. It seems many are doing great research right now but is it all for naught?

Scrolling my timeline, I found a viral post (falsely) claiming that “the top physicists already surrendered” to AI superiority. All this took place while I was on a train to Paris-Saclay University, to speak on “math in the age of AI” at a workshop for machine-learning graduate students.

I returned to my direct messages tab, thought about a quick response, then realized that it deserved a much longer one. I continued ruminating throughout the day and, the next morning, I reached out to ask permission to write a long-form public response.

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Dear Student,

I should start with a disclaimer—I don’t know much about your domain, and my reply will draw heavily from my own experience (an N=1 sample). But I suspect this is part of what you’re looking for, since you reached out to an actual human.

An easy answer would be that true AGI is unlikely to be here in 1-3 years. But that would miss the fundamental reality that some incomplete form of superhuman AI is already here, already disrupting the value chain of intellectual creation.

A friend of mine, an excellent mathematician who won several high-profile prizes, recently confessed to me that even she was a bit afraid—not for herself, as her career is firmly established, but for her students.

I do not think “all intellectual labor will be done” any time soon. But I do think that we are entering a mass extinction event where legacy career strategies for intellectuals will become demonetized. This is how I am reinterpreting your question—not as one about coping, which never works, but as one about surviving and thriving in a radically new environment.

The problem is, nobody knows what the world will look like 20 years from now. I don’t have a clear idea either, and cannot give you a prepackaged answer. What I can do, however, is share my perspective on the true nature and core value creation of intellectual work.

What’s the point of all that? Is it ever a good idea to embark on an academic career?

These questions predate the rise of LLMs. By some accident of history, the recent generations of academics could afford to mostly dodge them. This is no longer the case and AI is only the last straw.

I do not think these questions have generic answers. They are contextual, from one person to another, from one moment of your life to another.

Selfish answers are entirely OK when you’re a PhD student or postdoc: “The point is right now I’d rather do that than finding a real job. I’m learning tons of cool stuff and it’s fun to cosplay as a career intellectual. I do feel like a fraud, but I’m curious to see if I can outgrow this feeling.”

I actually think this is the best answer for most PhD students. This should be the time of peak impostor syndrome, and if that’s not the case, then it’s usually a bad sign.

Impostor syndrome, in itself, is a good reason to not get distracted by the AGI discourse, not that there isn’t something to it, but that you’re at the point in your trajectory where you’re likely to vastly underestimate your true capabilities. This skews your perceptions. Eg, you’ve just framed AI as a threat with which you have to cope, rather than an opportunity, whereas one could rationally argue that you’re much better positioned to benefit from it than your PhD advisor and all the academics you look up to.

My go-to career advice for PhD students and postdocs is to never ever let fear discourage you, and to never set any limits to what you can accomplish. It’s not that you won’t encounter practical limits in the future—you will—but you currently have no idea when and where and no way of knowing except for trying.

That certainly means that you should complete your PhD. But that doesn’t necessarily mean that you should seek a postdoc and/or a tenure-track position. It might be a good idea or a bad idea, depending on your personal situation, the credible opportunities on the visible horizon, your constraints and priorities, and how you perceive your optionality.

At this point, it’s still OK to be pragmatically selfish. Do you think a few more years of academia will make you stronger and more confident? Do you think that you still have important things to learn and discover? Or are you mostly concerned about the risks to your value on the job market? These are tough, private decisions, and they are hard to get right when impostor syndrome clouds your judgment—but fantasies about long-term prospects shouldn’t serve as excuses.

To me, your question about AGI concerns the next phase, when your motivation should gradually shift from “will it make me stronger?” to “which value am I really bringing to the world?”

This is the question that made me decide, at 35, to quit my permanent research-only position. And it’s the same question that, 15 years later, made me decide to resume full-time intellectual work, as an independent author.

I was never comfortable with the attitude of many of my peers. I remember a conference dinner where a brilliant young mathematician, a few years older than me, declared that that he didn’t think his research had any social value, but was nevertheless extremely happy to have the privilege of focusing full time on it.

At least he was “doing great research” (to quote your words).

It is a sad reality that not all tenured academics are doing great research. Before I try to clarify what is at stake, let me go back to the functioning of academia and the unique circumstances that led to the current situation.

University professors like to think that their primary job is to advance science. But except in specific fields like engineering or biomedical sciences, there is very little demand for the science they produce, and academics aren’t really hired for that part of the job.

The fabulous expansion of academia in the second half of the 20th century was for a large part a response to the fabulous expansion of undergraduate education, itself driven by the demographic boom and the democratization of college. And the current crisis of academia, which predates LLMs, is primarily explained by the nascent demographic decline and the stagnating demand for higher education.

What about research, in this context?

Sure, academic research seeks to push the frontiers of human understanding. But to a large extent it is a by-product of the academic technostructure, where it serves both as a perk and a career evaluation framework.

The opening lines of David Lodge’s Changing Places: A Tale of Two Campuses, published in 1975, capture the “perk” dimension at its historical peak (back then, international air travel was a rare luxury):

High, high above the North Pole, on the first day of 1969, two professors of English Literature approached each other at a combined velocity of 1200 miles per hour. They were protected from the thin, cold air by the pressurized cabins of two Boeing 707s, and from the risk of collision by the prudent arrangement of the international air corridors. Although they had never met, the two men were known to each other by name. They were, in fact, in process of exchanging posts for the next six months, and in an age of more leisurely transportation the intersection of their respective routes might have been marked by some interesting human gesture: had they waved, for example, from the decks of two ocean liners crossing in mid-Atlantic, each man simultaneously focusing a telescope, by chance, on the other, with his free hand…

Meanwhile, the “evaluation framework” aspect has gained pre-eminence, creating massively distorted incentives. Non-tenured academics publish to survive, tenured academics publish to maintain their symbolic status—and this goes a long way to explaining the monstrous scientific overproduction and its associated pathologies, from futility to self-plagiarism, from predatory journals to outright fraud.

In pure mathematics, where deciphering a single paper outside of your core expertise can require weeks of effort, the fragmentation of knowledge and comprehension is so profound that becoming familiar with two distinct micro-domains with cross-applicability can grant you a career-long monopoly.

It’s never entirely clear whether overspecialization and technical inaccessibility are bugs or features. On the one hand, these seem to be unavoidable consequences of the unstoppable expansion of knowledge. On the other hand, they are so miraculously aligned with the interests of the stakeholders that you can’t help wondering.

This aligns surprisingly well with Peter Thiel’s mantra: “competition is for losers: if you want to create and capture lasting value, look to build a monopoly.”

Will AI replace intellectual labor? Well, if intellectual labor consists of scanning gazillions of preprints and data sources, spotting hidden connections, applying known techniques, and writing boilerplate copy, it surely will—and, remarkably, this will suffice to produce original research.

But original research isn’t the same as great research. Contrary to a delusion that many academics have bought into, you can spend your life expanding an obscure corner of the cathedral of knowledge, and still create no meaningful value.

Here’s how Larry McEnerney articulates it in his unforgettable lecture on The Craft of Writing Effectively, a must-watch for anyone with intellectual or creative ambitions:

Faculty come into my office, and forgive me for the drama, but in my office, there’s two chairs in the writing corner. I have a chair at my desk and then there’s two chairs over here. There’s my chair and then there’s the writer’s chair and next to the writer’s chair is a box of Kleenex. And I’m not kidding.

Because I have people coming to me saying, “they’re not accepting my proposal, they’re not accepting my draft.” I get faculty who come in and say “they’re not publishing my work.” And of course there’s Kleenex there because like, you know, careers are depending on it. And sometimes it’s because it’s not clear. And sometimes it’s because it’s not organized. And sometimes it’s because it’s not persuasive. But overwhelmingly it’s because it’s not valuable.

And the other stuff doesn’t matter. If it’s clear and useless, it’s useless. It’s organized and useless, it’s useless. It’s persuasive and useless, it’s useless.

That’s the way it is.

This pain was already there and it will only increase. LLMs crush humans on clarity, organization, persuasiveness. They still struggle with rigor but are making steady progress on that front.

Yet the core question of value creation remains intact, and in the foreseeable future it will remain a question for humans. Why are we studying this or that? Are we happy or not with the results? What are we really trying to figure out? How will it help us make sense of the world?

Your job as an intellectual is to own these questions, not to hide behind the smoke screen of technical mastery.

This may be disconcerting, because these aren’t well-posed questions. They are part of the annoying fluff your advisor forces you to add to get your project financed, and you know that the real work takes place downstream, once they have been transcribed into better-posed technical questions. Or does it?

It’s precisely because these questions aren’t well-posed that we need humans to think about them. If the real job of radiologists was to spot tumors on X-rays, then this 2016 recommendation by Geoff Hinton would have been wiser:

We should stop training radiologists now. It’s just completely obvious that within five years, deep learning is going to do better than radiologists.

(Ten years later, the US is facing a major shortage of radiologists.)

Or, if you prefer a more thoroughly time-tested example—if the real job of painters was visual accuracy, then academic painter Paul Delaroche would have been right to declare in 1839, upon seeing a daguerreotype:

From today, painting is dead.

Interestingly, the quote has no traceable source and it is likely to be apocryphal. In any case, Delaroche peacefully pursued his own creative career until his death in 1856.

I like the photography parallel because it illustrates the nature of our blindspots. From “photography will kill painting” to “AI will kill intellectual work”, we’re being fed barbarians-vs-luddites tales of technology disruption. But at a deeper level, innovation triggers cascading cognitive disruptions, whose long-term effects are much harder to predict. To someone like Paul Delaroche, the art of Monet, van Gogh, Cézanne, Pollock, or Warhol, wasn’t even the stuff of nightmares—it was plainly unthinkable.

To be honest, this framing would have terrified me as a PhD student, when I had no serious scientific ambition, no expectation that my work would ever matter, and profound insecurities about my own capabilities and future. Technical mastery was hard enough, I would have hated to see that rug pulled from under my feet.

But you could see it the other way around: by becoming less dependent on technical mastery, research could become way more exciting.

I don’t think you should make too much of it in the short term. In any case, you still have to operate within the existing framework, and technical mastery still is on the critical path. Your investment won’t be lost, not because you will continue to practice the same skills for the next thirty years, but because they will rewire your brain and elevate your worldview, which you will need to competently interact with whatever technology exists in future. (It’s the same reason why art students continue to study the old masters, and continue to learn portrait techniques and perspective.)

But to guide you over the long term, I do think you should keep two open tabs at the back of your mind and revisit them every year or so, in a sort of mental retreat:

1. Technology. You can’t pretend it’s not there. No field will be immune to it, although the penetration and time dynamics will vary a lot. What’s tricky is that this will never be a stable situation. The hardest thing will be to refrain from passing definitive judgments, and forcing yourself to regularly experiment with new tools, new versions and new use cases (this will confront you to difficult explore/exploit dilemmas).

2. Sincerity/alignment. Are you really working on stuff that matters? If successful, will your research create meaningful value beyond your peer network? Again, it’s OK to be pragmatic and even a bit cynical in the short term, especially when the institution is imposing impossible demands on you. But you have to be very careful and not let this sincerity gap widen over time—ideally it should shrink at every major step of your career.

It’s much easier to embrace technological change once you have a clear idea of what you’re trying to achieve. There’s always been good and bad reasons to engage in intellectual work, and AI will make the bad reasons infinitely worse.

If the sincerity/alignment aspect feels too abstract or grandiose, here’s a practical proxy: are you genuinely interested in what you are working on? If you didn’t have to earn a living, and if you didn’t have any social insecurities, would you still be doing that?

In a way, this isn’t much different from Rilke’s advice on writing:

No one can advise and help you, no one. There is only one way. Withdraw into yourself. Explore the reason that bids you write, find out if it has spread out its roots in the very depths of your heart.

I’ve been using this framework for many years and continue to find it extremely helpful. I suspect it touches something fundamental about intellectual creation, which will become even more critical in the age of AI—if someone is ever to hold a durable cognitive edge, it can only be proprioceptive and attentional.

Indeed, your intellectual sincerity is arguably your most unique and defensible cognitive resource. You have to trust your subjectivity, because it is the only place where everything comes together, the official knowledge that AI can already process infinitely faster than you, and your embodied intelligence, your life experience, the things you’ve learnt from the world and cannot yet articulate.

Human creativity takes place at this interface, which is also where it is received and valued by others.

I see no hard reason why machines couldn’t some day emulate our subjective, carnal experience of being alive, and out-human us in every possible way. But this purely speculative question has no relevance to your career prospects, as the current technology cycle is absolutely not about that. It is plausible that the academic job market could become tighter in the coming years, but our urge to better understand the world isn’t going to vanish—and, at its core, this is a human endeavor.

Whether in AI or in any other discipline, the truth is we’ve barely scratched the surface. Technology is progressing at an intimidating pace, but we don’t get to choose and the fundamental question remains what we’re going to do with it.

So if you find your research genuinely exciting, and if you have the opportunity to pursue it in favorable conditions, I see no reason why you should give up.

In particular, I have no doubts that your generation and the generations to come will continue to produce successful intellectuals, in all domains of knowledge, and that their work will continue to be recognized and valued. Some of this work will even be considered revolutionary, even though I have no idea what shape it will take.

In any case, doing what you truly care about is the best way to minimize regret, an essential objective in times of uncertainty.

All the best,
David.

Some people are smarter than others—and some are fabulously smart. We may disagree on many things, but can we at least agree on that?

The origin of cognitive inequality is a hot, divisive topic. The existence of cognitive inequality is hardly controversial. We do struggle to come up with a satisfactory definition of intelligence, especially in its most creative and socially valuable forms, but most agree that people like Leonardo da Vinci, William Shakespeare, Nikola Tesla, or John von Neumann were extraordinarily intelligent.

This aligns well with my experience in pure math academia, where careers and even personal identities are shaped by the shared perception of a mesmerizingly wide cognitive gap between mathematicians. This is a central theme in their discussions and a source of both fear and admiration.

Yet for all the culture wars on the origins of cognitive inequality, there is little debate on its very nature.

When someone is unusually smart, what is going on in their brain? This is a different question from “where does cognitive inequality come from?”, and also from “can we really measure it?” (another controversial topic).

It is a basic, foundational question that stems from two widely accepted facts—that some people are smarter than others, and that intelligence is a manifestation of brain activity—and asks for the natural follow-up: what is really going on? What are the physical differences between the brain of a super smart person and the brain of a normal person?

The garden of bad metaphors

Despite its adjacency to some of the most contentious issues in science, this topic has remained relatively quiet. It might be because of this very proximity.

Or maybe the question is too simple and natural. Five-year olds love to ask brutal questions like this one, but grown-ups are usually more hesitant, as they don’t want to come across as unsophisticated.

Or perhaps this is because there were historical precedents where people went looking for answers and came back empty-handed. The brain of Carl Friedrich Gauss sits in a jar on a storage shelf somewhere at the University of Göttingen, and is so unremarkable that it remained mislabeled for over a century without anyone noticing. After Albert Einstein died in 1955, the pathologist removed his brain without the permission of the family, preserved it in formaldehyde, and dissected it into hundreds of slices—some researchers claimed to have observed peculiarities in the samples, but these findings are generally regarded as unconvincing.

If we stick to solid science, there is remarkably little we know on the subject.

But here’s the catch. While the simplest and most natural question remains unaddressed, people continue to rely on the same flawed brain metaphors and the same dubious biological assumptions. This only adds to the confusion around IQ and the heritability of intelligence.

For example, people marvel at the “speed” and “working memory” of von Neumann’s brain, as if it were a CPU with a unique clock speed and L1 cache. But from a biochemical perspective, what could this even mean?

Genes code for proteins, which assemble into cells, which assemble into tissues. Within-species genetic variability isn’t about new genes coding for new proteins with new functional roles that open the possibility for radically new architectures—it is about minute variations in the quantity and conformation of the same proteins, which retain the same overall structures and functional roles.

It is true that these minute variations compound over embryogenesis and life, resulting in sizable physical and physiological differences between individuals. Yet it is also true that all human beings are built with the same architecture and the same functional blueprint. It’s not like some people have normal hearts while others have 500 horsepower turbopumps. And it’s not like some have Snapdragon 8 smartphone processors in their brains while others have NVIDIA H200 Tensor Core GPUs.

To stick with this terrible CPU metaphor, it’s more like some people have a Snapdragon 7 while others have a Snapdragon 8—the performance gap is noticeable, but everybody runs the same apps.

There is one caveat to that. Genetic anomalies can derail brain development and hamper its normal functioning. A Snapdragon 8 with a defunct thermal management probe will not operate properly. Phenylketonuria, also known as PKU, is an example of a recessive genetic condition that, if untreated, can lead to microcephaly and severe learning disabilities.

By contrast, there is no accidental change to a Snapdragon 8 that magically turns it into an NVIDIA H200. This probably explains why we have never found any specific mutation that causes a drastic increase in cognitive ability.

Yet, still, there MUST be a physical difference between the brain of a super smart person and that of a normal person. If not, where would the cognitive difference come from?

There are no miracle people

Throughout my journey as a pure mathematician, it often struck me that progress wasn’t only about mathematics itself, but also metacognition and emotion control.

Like many math students, I started out with a strong hereditarian prior, because I couldn’t think of an alternate model for the shocking inequality pattern around me—which, clearly, couldn’t be explained by social determinism.

Then, over time, I realized that genetic determinism was equally problematic, not least because it couldn’t account for my own progress trajectory.

Later on, I found out that many prominent mathematicians had tried to articulate the idea that their talent was first and foremost a cognitive attitude. A famous example is that of Descartes who, in the opening lines of his Discourse on Method, insisted that he wasn’t particularly gifted:

For myself, I have never presumed my mind to be any way more accomplished than that of the common man.

Instead, he attributed his successes to his chance discovery of a miraculous “method” that, for all practical matters, is an assortment of metacognitive techniques aimed at improving the clarity and reliability of one’s intuition. If you were to generate a tag cloud of his favorite words, intuition, clear and distinct would stand out with massive fonts. The same themes can be found over and over again in the writings of Poincaré, Hadamard, Thurston, Grothendieck, and many others.

This is part of a broader pattern where some of the biggest names in science have rejected the notion that they were born with unique abilities, insisting instead on their curiosity and stubbornness:

• Isaac Newton: “But if I have done the public any service this way it is due to nothing but industry & a patient thought.”

• Albert Einstein: “I have no special talent. I am only passionately curious.”

• Richard Feynman: “I was an ordinary person who studied hard. There are no miracle people. It just happens they got interested in this thing, and they learnt all this stuff.”

• Alexander Grothendieck: “This power is in no way some extraordinary gift—like an uncommon cerebral strength, (shall we say). . . . Such gifts are certainly precious, worthy of the envy of people (like me) who haven’t been blessed with them at birth, ‘beyond measure.’”

Needless to say, these bold declarations have barely made a dent in the hereditarian carapace of the general public.

To be clear, I completely get why most people find these statements absurd and hypocritical. I remember hearing Einstein’s “no special talent” quote when I was fifteen, and instantly hating it. Seriously, who wants to hear supermodels lecturing us on the importance of inner beauty? Then I became a professional mathematician and Einstein’s quote gradually felt less and less irritating. And, after I crossed a certain threshold in my career, it even appeared profound.

If all these brilliant people failed to communicate their point, it might have been because they weren’t explicit enough about what they meant by industry, patient thought, or curiosity.

This is a central theme of my book Mathematica, a Secret World of Intuition and Curiosity, which opens with Einstein’s “no special talent” quote and tries to fill in the missing details—what it means to do math, the unseen actions that we perform in our heads, the slow and gradual build-up of our intuition.

We have no good words to discuss these things. No one ever explained to us what it means to think, meditate, imagine, reason, or dream.

What is the neurological basis of these activities? What do they have in common? What differentiates them? What are their long-term effects on our brains? Could it be that some people practice them the wrong way, while others have stumbled upon remarkably effective techniques that radically transformed their capabilities?

I am a mathematician, not a neuroscientist, and I didn’t get to these questions through theoretical considerations. Rather, I took the practical path of trying to become better at math, experimenting with myself, tinkering with my mental imagery, and doubling down on what worked best.

On the surface, this is just a story about mathematics—and this part is fairly consensual.

Most mathematicians agree on the phenomenology of “doing math”, and they often feel that traditional discourses about mathematics miss or misrepresent something fundamental about their inner experience.

The consensus breaks down when discussing the consequences. Surprisingly, some agree that math talent develops through a specific mental practice that school never really explains, yet stick to the idea that it is primarily innate.

This is where the relatively niche story of extreme mathematical talent collides with the much broader cultural and scientific issue of cognitive inequality.

Mathematics is widely considered to be the purest and most “g-loaded” form of intelligence—the dispute about the actual thought processes of one-in-a-million outlier mathematicians reveals, at its core, a dispute about the fundamentals of human cognition.

This puts us mathematicians in an awkward spot, as our autobiographical knowledge often conflicts with popular hereditarian myths.

I have written on this before (on Turkheimer’s Third Law, on the catastrophic science of Twins Reared Apart) and will continue the series. But debunking bad science is one thing, and explaining math talent is another.

Since I have no expertise on the core neuroscience, I have avoided this topic as much as possible.

However, after seeing the same bad brain metaphors popping up again and again in my social media timelines, I am coming to the conclusion that hereditarian myths are so tightly intertwined with flawed models of human cognition that it is impossible to debunk the former without fixing the latter.

Everyone needs a theory, especially when established science hasn’t yet provided one. If I could move away from my past hereditarian beliefs, it was because I cobbled together my own alternate model of cognitive inequality, informed by my life experience and my (imperfect) understanding of modern neuroscience.

Before I share this private, conjectural theory, and before everyone jumps to the comment section to call out my recklessness, let me clarify its epistemic status:

First, it is nothing but a theory—that is, a coherent set of unproven hypotheses. I have no proof, no empirical evidence, and no legitimacy on the subject.

Second, the theory is clearly simplistic and incomplete. It has gaps, blindspots, ambiguities, some of which are intentional (this is a Substack post, not a peer-reviewed publication) and others result from my ignorance and naivety.

This is just my private attempt to reconcile my own experience of becoming better at math with this fundamental lesson from neuroscience: the brain is a learning device, not a computing device. I am sharing it chiefly because of its value as a mental model.

This theory isn’t perfect, but it is arguably less biologically absurd than “von Neumann was a mutant with a crazy fast CPU.”

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A conjectural theory of cognitive inequality

Conjecture 1. Some differences in the structure, volume, speed, and efficiency of neural tissue result from genetic variability, but they cannot account for the magnitude of the observed cognitive inequality.

One known issue with IQ is that it artificially compresses the upper end of the distribution to fit a bell curve. Subjectively, the cognitive gap in math (and many other domains) feels much wider than that. Here is how I put it in my book:

Math is so unequal that it’s as if some people could run the one-hundred-meter dash in under a second, while the majority wouldn’t make it in a week. While it’s conceivable that some people may genetically be endowed with a neuronal metabolism that is more efficient and powerful, making them, let’s say, twice as capable in math, or why not ten times as capable in math, it’s hard to believe that genes alone could explain such an absurd level of inequality.

As detailed in an earlier post, the heuristic is that highly heritable polygenic traits are typically Gaussian—the canonical example being height, where people cluster in a narrow ±25% band around the mean—while the distribution of math talent has a much fatter tail, like a Pareto distribution.

The phrasing of the conjecture is intentionally informal, but it can be turned into a quantified, falsifiable prediction:

Consider the population P of one-in-a-million overachievers in the most cognitively demanding fields. You could pick active chess grandmasters, International Math Olympiads gold medallists, Fields medallists & invited speakers at the International Congress of Mathematics, Nobel laureates in physics, or whichever variation you like. There is some subjectivity in these choices, but I suspect most people will find them relevant. My prediction is that we will never find a within-family polygenic score S such that the median value of S on the population P lies in the top centile for S with respect to the general population.

I wouldn’t actually be surprised if P turned out to be indistinguishable from the top decile, but the safest top centile prediction is already substantial: to go from top centile polygenic scores to one-in-a-million outcomes, there would be four orders of magnitude unexplained by genetics.

This falsifiable version has two unfortunate issues: 1/ it relies on achievement as a proxy for talent, 2/ it is elitist. This is because the conjecture is fundamentally about the non-Gaussian aspects of cognitive inequality, which are more readily visible in the tail part of the distribution (a region where there exists no well-calibrated measurement of cognitive talent).

A less explicitly testable but more relatable version would be: if you’re genetically average, you still have a non-trivial chance of landing in the top 1% for any meaningful dimension of cognitive achievement.

The next conjecture is harder to turn into a quantified, falsifiable statement. But it is a natural counterpart to Conjecture 1, since the interconnected structure of our brain is its one aspect that could credibly exhibit extreme, Pareto-like variability.

Conjecture 2. At any given moment, the measured cognitive differences between individuals are primarily explained by differences in their synaptic connectomes.

This aligns with the conventional view that human intelligence emerges from the interconnected network of neurons. The Parietal-Frontal Integration Theory (P-FIT)—a classical biological model—explains cognitive inequality by the relative quality of the integration between different regions of the brain, emphasizing the importance of the connectome.

Of course, the human brain has some fixed, heritable features, and its morphogenesis is controlled by regulatory genes that vary from one person to another. But, again, this feels more like Snapdragon 7 vs Snapdragon 8 nuances.

What matters here is that the fine interconnected structure is continuously modified in response to stimuli. In fact, this plasticity is the prevalent explanation for learning and memory formation, as captured by the Synaptic Plasticity and Memory Hypothesis:

Activity-dependent synaptic plasticity is induced at appropriate synapses during memory formation and is both necessary and sufficient for the information storage underlying the type of memory mediated by the brain area in which that plasticity is observed.

The only bit of Conjecture 2 that might be original is its emphasis on measurement. It claims, in particular, that IQ tests primarily measure differences in the connectome.

Let me motivate this by sharing my personal reaction to Raven’s Progressive Matrices, one of the most common IQ tests, and supposedly one of the most g-loaded ones. Here’s a typical question:

I have no idea how “normal” people approach a question like this one, but it is very hard for a mathematician like me to take it as a satisfactory test of “working memory,” “processing speed,” or “logical thinking”—whatever these expressions mean.

I hardly have to think to get the right answer. When looking at the picture, I get the eidetic perception of three superimposed permutation matrices (one for the background geometric shape, one for the color of the foreground rectangle, one for the angle of the foreground rectangle.) And, thanks to this eidetic perception, I perceive this question as requiring the same cognitive load as computing 132 + 37.

Subjectively, it feels that by projecting mathematical structure onto the picture, I am able to reduce the need for “working memory”.

Now you could object that I was born with a special brain equipped with eidetic perception of permutation matrices. But I know for a fact that this wasn’t the case. I only learnt about this notion when I was eighteen, and I continued to develop my intuition for it over the following decades, practicing undergraduate linear algebra, then graduate-level group theory, then producing original research on structures that generalize permutations. Over the years, these patterns have become as natural and concrete to me as whole numbers are to laypeople. I feel them like visual rhythms.

While my degree of algebraic synesthesia might be extreme, I suspect that many software engineers—and other professionals who are regularly exposed to tabular structures—develop a similar feel. The increasing penetration of tabular structures in our cognitive environments might actually explain why Raven’s Progressive Matrices “has demonstrated a rate of IQ gain of 7 points per decade, more than double the rate of the Flynn effect as manifested on WAIS, SB, and other multifactorial intellectual tests.”

The Flynn effect applies to all mathematical structures, including the ones that feel obvious to “normal” people. Two thousand years ago, in Ancient Rome, my comparison “trivial” problem (132 + 37 = 169) would have constituted a more challenging IQ test question (CXXXII + XXXVII = CLXIX.)

Our number sense has substantially evolved over the past millennia. We live in a world where (almost) everybody understands Hindu-Arabic numerals, (almost) everybody understands zero, and (almost) everybody understands negative numbers—three notions that had long been out of reach of ordinary people. This large-scale anthropological transformation, which cannot be explained by changes in our genomes, is directly attributable to changes in our cultural and technological environment.

Note that Conjecture 2, in itself, is entirely compatible with the hereditarian worldview. Indeed, it could be that some people are born with substantially better learning mechanisms, producing over time a vastly superior connectome. It could be, for example, that my unusual ability to develop eidetic perceptions of mathematical structures is entirely explained by genetic factors.

However, this will become less credible in light of the next three conjectures.

Conjecture 3. People have vastly different mental habits and metacognitive approaches.

This one is very concrete, despite the unfortunate softness of its language. This makes it the most easily testable.

The reason for the soft language is one we already discussed—we have no biologically meaningful words to express the specific unseen actions that we can perform in our heads when we think, meditate, imagine, reason, daydream…

Whenever neuroscientists observe brain activity with EEG or MRI, they notice wildly different activation patterns. This is in fact the very reason why they study brain activity: to see how it varies in reaction to certain stimuli, or from one person to another. Yet, somehow, we lack the cultural scaffolding to take notice of our distinct “brain modes” and train ourselves to use them strategically.

We know about wakefulness and sleep, and within sleep we know about paradoxical sleep—but there are many flavors of wakefulness. Mental habits and metacognitive approaches are my attempts to make up for this missing vocabulary.

The conjecture predicts that, if we make a reasonable effort to categorize these “brain modes” and the distinct strategies one could employ to switch from one to another when accomplishing certain tasks, and if we produce statistics of how much time different people spend on distinct “brain modes” over weeks and months, we will find shocking differences.

Metaphorically, my prediction is that some are walking 100,000 steps a day while others are constantly doomscrolling on their sofas.

Some existing empirical evidence is directionally aligned with the conjecture. In a fascinating 2016 fMRI study, Marie Amalric and Stanislas Dehaene observed that professional mathematicians process complex mathematical statements by recruiting brain areas that aren’t typically mobilized in language processing, specifically in the inferior temporal and parietal regions. This happens regardless of whether the statements belong to their specific domain of research. By contrast, the control population of non-mathematicians with equal academic standing activate the expected language-processing regions when exposed to the same mathematical statements.

This aligns well with my subjective experience. When confronted with a mathematical statement that isn’t immediately obvious to me, I consciously switch to a special “math mode” where I try to “see” things in a certain way, to “feel” their presence and “interact” with them as if with imaginary body parts.

This peculiar way of channeling my attention developed over time. At first it was quite demanding, then it became second nature. When it involves mathematical objects that are familiar to me, it can be effortless.

The Amalric-Dehaene study only concerns reactive brain activity, when prompted with a mathematical statement. But mathematicians preemptively practice all day long, through peculiar rumination techniques that slowly enhance their intuitive comprehension.

When Einstein insists that he is only “passionately curious,” when Grothendieck declares that “the quality of the inventiveness and the imagination of a researcher comes from the quality of his attention, listening to the voice of things,” I believe that they are talking about activities of this very nature.

Strikingly, Descartes’s entire approach can be reinterpreted in those terms. This passage from his Rules for the Direction of the Mind makes it clear that the method was really about attention control:

The whole method consists entirely in the ordering and arranging of the objects on which we must concentrate our mind’s eye if we are to discover some truth.

Conjecture 3 is again compatible with the hereditarian worldview. On paper, this ability to access these special brain modes and maintain them with sufficient intensity and duration could very well be innate.

Interestingly, many people spontaneously default to this genetic interpretation when they are presented with measurable biological differences between brains.

This reveals a double standard. When hearing that the brains of professional mathematicians have specific activation patterns, people feel vindicated in their hereditarian beliefs. Yet they accept non-genetic causes for the biceps of a bodybuilder or the liver of an alcoholic. Because cognitive activity takes place under a veil of invisibility, people underestimate the magnitude of the practice gap.

Conjecture 4. Most of these differences in mental habits and metacognitive approaches are acquired.

An essential caveat: acquired isn’t equivalent to nurtured. Many things are acquired that are extremely hard, if not impossible, to intentionally nurture or self-develop. By acquired, I simply mean that most of these differences cannot be attributed to genetic variance and are best understood as developmental outcomes—the divergence emerges and gradually widens throughout the long and messy process that starts with the egg’s fertilization and continues into adulthood. (Which doesn’t mean that genetic variability has no influence—it is very likely to have a substantial influence, yet not predominant.)

The conjecture is intentionally vague and hard to test, and should rather be viewed as a generic pattern for building more modest, falsifiable predictions.

A concrete approach would be to create longitudinal versions of the Amalric-Dehaene study, especially ones covering the peak neuroplasticity period of early infancy. My hunch is that there are multiple neurodevelopment pathways, and each child follows a singular trajectory shaped by a unique combination of genetic, cultural, and emotional factors, and also freak events—with long-term consequences on their cognitive style.

But these early childhood studies would come with many ethical issues, as well as practical ones.

A simpler version would be to follow older students through predictably decisive stages of their progression. The French preparatory classes system is a good example. It takes “gifted” 17-18 year-olds and submits them for two years to an incredibly ambitious boot-camp-style math curriculum, a make-or-break experience that produces remarkable outcomes as well as disastrous ones (with less than 1% of the world population, France has received nearly 20% of all Fields medals ever awarded; but many students are left traumatized.)

These were transformative years for me, in the most profound sense. Something radical changed in my attention mechanisms, imagination techniques, and ability to suppress intellectual fear. The only expression that I can think of is cognitive transmogrification.

My prediction is that the progress trajectories of individual students throughout these two years would be significantly correlated with the intensification and/or more frequent use of the specific Amalric-Dehaene “math brain” activation patterns.

Conjecture 5. Differences in mental habits and metacognitive approaches compound, over time, into measurable differences in cognitive ability. 

This one basically says that Einstein was right, and that this really is about developing the right kind of curiosity.

A fair objection is that Conjecture 5 is too vague. Another fair objection is that it is circular, unless one finds a reliable way of untangling the causality. Do people become smarter because they’ve discovered the secret curiosity mode, or do they discover it because they’re already smarter? (There is no reason to expect a binary answer, as cognitive development is an ongoing process with obvious feedback loops.)

While it is vague and speculative, I personally find it clarifying.

From a machine-learning perspective, the conjecture can be viewed as a natural consequence of the prior ones. Indeed, our synaptic connectome reconfigures in response to not just primary stimuli—the raw sensory signal we receive from the world—but also secondary stimuli, the stream of mental imagery that we continuously elaborate.

When you read a book, the primary stimulus is the ink on the page. If some books make you much smarter, the cause lies not just in the ink on the page, but also in the associated secondary stimuli, the drifting thoughts and mental images triggered by the book and lingering for minutes, hours, days, years.

This is the fundamental reason why teaching is so difficult, and why successful educational interventions rarely scale well. While it is possible to deterministically control the primary stimuli, their impact on the secondary stimuli is always indirect and contingent on many uncontrolled factors, most notably the existing connectome and emotional state of the student.

This suggests bolder, reinforced versions of Conjecture 5:

Conjecture 5 bis. In humans, advanced cognitive development is primarily mediated by secondary stimuli. Genetics do define a ceiling that varies from one person to another, but very few people ever approach their ceiling and, in practice, much of the observed cognitive variance is attributable to differences in the intensity and quality of secondary stimuli.

Conjecture 5 ter. Extreme cognitive talent is the exceptional outcome of unusually effective metacognitive approaches sustained over long periods of time, producing unusually intense and beneficial secondary stimuli.

This is the fundamental intuition behind the entire framework. Why do we dream? Why do we constantly imagine things? Why do we experience this bizarre thing that people call a stream of consciousness? These capacities are metabolically expensive and, on the face of it, entirely reckless—we are constantly retraining our brains on our own hallucinatory slop.

Why did our ancestors evolve these singular behaviors, if not as cognitive reinforcement mechanisms? At the core, this is a machine-learning question. If we’re feeding this much synthetic data into our cortex, then there must be a good reason for that.

Conjecture 5 ter might be overambitious, but it provides a non-magical explanatory framework for extreme talent, consistent on many levels:

1. The brain activity that produces these secondary stimuli is a physiological reality with concrete, measurable manifestations.

2. From a theoretical machine-learning perspective, secondary stimuli are expected to have a massive impact on cognitive fitness.

3. According to Conjecture 3, there should exist massive measurable differences in how these secondary stimuli streams are produced from one individual to another—so cognitive inequality would at last have a physiological explanation based on something where people would exhibit massive physical variability.

4. As secondary stimuli aren’t part of the shared experience, and their effects compound over decades, it would also explain why cognitive inequality has this magical feel of “coming from nowhere” (hence the common hereditarian misinterpretation.)

5. It would align with the accounts of historical “geniuses” who insisted their abilities stemmed from practice and attention (Einstein, Descartes, Grothendieck)—and it would explain why they struggled so much to find the right words.

The 20% full glass

Returning to solid ground, the natural question is this: assuming that the framework is directionally correct, what are the practical consequences?

If you dislike hereditarianism because it doesn’t leave enough room for human agency, then my answer might not leave you entirely happy.

The human brain is a fabulously complex, non-linear device with multiple layers of feedback loops, which makes its behavior inherently hard to control.

There is ample evidence that key aspects of cognitive inequality have already crystallized by the time children acquire the language capability to make sense of metacognitive guidance. Whether it’s the genetic lottery, the socioeconomic lottery, or the idiosyncratic neurodevelopment lottery, the conclusion is basically the same: you cannot train your kid to become the next Einstein.

There is, however, a crucial nuance. Mental habits and metacognitive approaches cannot be directly transferred from one brain to another, but it is possible to discuss them—or at least try to—and everyone has a meaningful margin for improvement.

This is the 20% full glass and, however depressing it may look from afar, it is worth your time and effort.

I am familiar with this bizarre, seemingly contradictory feeling. Like many math students, I was shocked by the immensity and apparent immutability of the talent gap. My standing in the math cognition hierarchy seemed determined once for all in a relatively boring place, far above “normal people” and far below “the geniuses”. And, in a sense, this is where I have always remained—far above “normal people” and far below “the geniuses”.

But the talent gap is so wide that this isn’t a well-defined spot. With a three-decades hindsight, it is also clear that the math cognition hierarchy of my peer group wasn’t truly static. In fact, there have been some impressive upward and downward trajectories.

For me, the hardest part wasn’t to identify the right mental habits—I had a clear enough idea from my early 20s—but to cobble together the courage and resilience to practice them with more discipline.

No theory of cognitive inequality would be complete without a discussion of fear. Why did we evolve cognitive inhibition, this bizarre response to intellectual difficulty, this visceral panic, this brain fog that prevents us—among other things—from even engaging with mathematical abstractions?

Interestingly, Conjecture 5 bis contains a tentative explanation. Learning from our own imaginary stimuli is a risky business that isn’t supposed to work unless the right conditions are met. It can improve our cognitive fitness, otherwise we would have gone extinct. But we know of many circumstances where it has the opposite effect (superstitions, cults, paranoid delusions…)

This final conjecture might be overambitious and, in any case, it is hard to validate or falsify:

Conjecture 6. Cognitive inhibition is an adaptive protection against learning from unreliable mental imagery, which must be overcome to unlock creative thinking and mastery. It is partially regulated by social feedback, resulting in a self-reinforcing cognitive stratification that crystallizes with age.

It mirrors a subjective experience shared by many mathematicians, who insist on the prominent role of self-confidence. Here is a striking passage from Bill Thurston’s celebrated essay On Proof and Progress in Mathematics:

It regularly happens that someone who was in the middle of a pack proves a theorem that receives wide recognition as being significant. Their status in the community—their pecking order—rises immediately and dramatically. When this happens, they usually become much more productive as a center of ideas and a source of theorems. Why? First, there is a large increase in self-esteem, and an accompanying increase in productivity. Second, when their status increases, people are more in the center of the network of ideas—others take them more seriously. Finally and perhaps most importantly, a mathematical breakthrough usually represents a new way of thinking, and effective ways of thinking can usually be applied in more than one situation.

This is a perfect illustration of the 20% full glass. We do perceive fairly rigid cognitive hierarchies and, to a certain extent, this perception is correct. Yet these hierarchies are far less rigid when we look at individual trajectories over long periods of time, which is the right perspective for thinking about our future. Progress involves reinforcing self-confidence, finding the right sparring partners and ecosystems, and developing new ways of thinking.

If intelligence is primarily embodied in our connectomes (Conjecture 2), then we should think about it in the same way we think about wealth, as the continuing outcome of a non-deterministic capitalization process.

The 80% empty glass is that life is unfair and we cannot replay the past. There are no miracle people, but there are miracle trajectories.

The 20% full glass is that the one thing that is truly ours—our attention, the focus of our curiosity, how we navigate our stream of consciousness—may matter more than we ever dreamed.

A ball and a bat cost a total of $1.10. The bat costs $1 more than the ball. How much does the ball cost?

I borrowed this problem from Daniel Kahneman’s Thinking, Fast and Slow. I encourage you to try it with your friends. Most people will answer that the ball costs 10¢, but that’s not the right answer. If the ball cost 10¢, then the bat costs $1.10—since it costs $1 more than the ball—and the ball and the bat together cost $1.20.

If you tell them why their answer is wrong, your friends will get it easily enough. But they won’t necessarily know the right answer. They’ll even find tons of excuses to not look for it: it’s hard to do the calculations, they’d have to write down the equations but they don’t have a pen, they can’t be bothered, whatever…

The correct answer is 5¢.

The ball and bat problem is a perfect illustration of the theory of cognitive biases, which Kahneman developed with Amos Tversky. According to this theory, we have two distinct cognitive systems, System 1 and System 2.

System 1 allows you to give immediate and instinctive responses, without even trying. When someone asks you how much is 2 + 2, what year you were born, or which weighs more between an elephant or a mouse, you don’t even have to think. But it’s also System 1 that makes you answer, incorrectly, that the ball costs 10¢.

System 2 is what you have to use when you’re asked to calculate 47 x 83, or how many days have passed since your birth. You know how to get the answer, but you’d have to think. You probably need pencil and paper. One thing is certain: you don’t really want to do it. Even if System 2 is more reliable and rigorous, you only use it when you have no other choice, because thinking hard, doing calculations, and logical reasoning are all tiresome.

The theory can be summed up as follows:

• Each time our System 1 gives us an answer, we’re tempted to use it without calling on System 2, not even to verify that the answer is correct. Because System 2 uses a lot of mental energy and resources, we primarily rely on our instinct. Biologically, we’ve developed a preference for intellectual laziness.

• In certain situations, our System 1 is systematically wrong. We all make the same mistakes, all the time, as if the wiring schematic in our brain was defective. These are the “cognitive biases” that the theory aims to study. We all want to say that the ball costs 10¢.

Kahneman’s book became a best-seller in part because it went beyond the simple theoretical observation and proposed a concrete method to avoid falling into the trap.

He has a simple recommendation: learn the list of cognitive biases presented in his book by heart, and each time you recognize one of the typical situations, fight your inclination and use your System 2 while trying to ignore your System 1.

I think there is a better way of doing it, which I’ll explain.

“That’s cheating!”

The first time I heard the story of the ball and the bat, it was from a friend who was studying cognitive science at Princeton. She had just read Kahneman’s book and wanted to do the test with me.

Like most people, I gave an instinctive response. I listened to my System 1 without knowing it was called System 1. Without thinking, without doing any calculations, I gave the first answer that popped into my head: “5¢.”

I felt that my answer annoyed my friend but I didn’t immediately know why. She took the time to explain what was up. I was supposed to answer “10¢,” or at least take a few seconds before answering “5¢.” At any rate, there was no way I was supposed to answer “5¢” immediately, without taking the time to think about it. That was simply not allowed. A guy had even won the Nobel Prize for showing it was impossible.

Quickly enough, right before changing the topic of conversation, my friend did however come up with an explanation—a simple, pragmatic, and not entirely false one:

That’s cheating! You’re a mathematician!

When I had my friends and colleagues take the test, I was sincerely surprised to find that so many of them answered 10¢, and even more surprised at their difficulty in finding the right answer after admitting that their initial response was wrong. The most incredible thing was that everyone spoke to me about “doing the calculations,” as if there were any need for that, as if it weren’t visually evident that the right answer was 5¢.

A or B

This ball and bat story began to really intrigue me. I tried to understand what was stopping my friends from seeing the right answer when it was so obvious. I did my own little investigation and I believe I have found an explanation.

After asking my friends about the price of the ball, I followed up with this question:

Imagine that you have to make an important decision in your life. You have the choice between option A and option B. Your intuition tells you to choose A, but your reason tells you to choose B. What do you do?

I presented this question to more than a dozen of my non-mathematician friends, and almost all of them answered, without hesitation, that they’d follow their intuition and choose A. Only one person chose B. Another hesitated for a long time, without ever giving a clear response.

Nothing guarantees that you’d get such a high percentage of people choosing A if you tried this experiment on your own. My protocol suffers from what is called selection bias, in that my friends aren’t necessarily representative of the general population, and it may very well be that people who listen to their intuition have greater chances of becoming my friends.

The exact proportion of A and B didn’t really interest me. What I wanted to know was whether someone would come up with the same response I would have given myself. No one did.

My hypothesis is that my unusual response to the question is the key that allowed me to become good at math and, along the way, to reeducate many of my cognitive biases.

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This isn’t a reasonable assumption

Kahneman says that thousands of American students took the ball and bat test, and that “the results are shocking.” At the lower-tier universities, the error rate was over 80%. Even students at Harvard, MIT, and Princeton gave the wrong answer more than 50% of the time.

Kahneman’s book is fascinating, but I’m confused whenever I see him opposing “the right answer” and “the intuitive answer,” as if there were only one intuitive response possible, and it was necessarily false. For example, he writes:

It is safe to assume that the intuitive answer also came to the mind of those who ended up with the correct number—they somehow managed to resist the intuition.

In other words, Kahneman finds it safe to assume that I shouldn’t exist. My opinion, understandably, is that this isn’t a reasonable assumption.

But beyond the relatively minor question of my own existence, this anecdote reveals a major disconnect between Kahneman’s theory and what all mathematicians know deep in their bones. It’s up to you to decide who’s in the best position to give you advice on mental calculations.

Kahneman finds it shocking that 50% of the students at Harvard, MIT, and Princeton blindly relied on a manifestly false intuition, and I am as shocked as he is.

But I am equally shocked by something that Kahneman apparently finds completely normal: how is it that 50% of the students at Harvard, MIT, and Princeton managed to get accepted despite having such faulty intuitions?

Having studied and taught at highly competitive universities, I know that students who can directly “see” the correct answer in their head have an enormous competitive advantage. I don’t understand how the others can even compete. I imagine that they compensate by intensive cramming, something I’m completely incapable of and the very thought of which gives me a headache.

Kahneman’s advice consists of identifying the situations where we should “resist” our intuition and submit ourselves to System 2. It’s strange advice coming from someone who’s spent his life documenting our aversion to effort, our preference for instinctive and immediate responses, our immoderate love for System 1—and our hatred of System 2.

This idea that we should resist our bad instincts and entirely submit to a robotic mode of thought was once the prevalent paradigm in education. Kahneman is well enough placed to know why it can’t work.

Another point bothers me. It’s true that we should be wary of our System 1. But what are we to make of our System 2? Personally, I stopped trusting mine after ninth grade, when I found out I wasn’t able to string together three lines of calculations without making a mistake.

But the most troubling aspect is that Kahneman reasons as if our intuition were hardwired, with no possibility for us to reconfigure or reprogram it. Had he lived in the ancient Roman era, he would almost certainly have said that it was impossible to represent mentally the result of the operation 1,000,000,000 - 1, because the number greatly exceeded the capacities of human intuition.

System 3

When I need to make an important decision in my life, if my intuition tells me to choose option A and my reason tells me to choose option B, I tell myself there’s something going on and I’m not ready to make the decision.

That’s the moment to resort to what I call System 3.

System 3 is an assortment of introspection and meditation techniques aimed at establishing a dialogue between intuition and rationality. You use it each time you try to recall your dreams, to put words to the fleeting impression that left a strange taste in your mouth, to sort out your most confused and contradictory ideas.

When I was eighteen and I discovered that the stupid images in my head had a tendency to correct themselves once I made the effort to describe and name them, when I got into the habit of lending an ear to the dissonance between my intuition and logic, I put System 3 at the center of my strategy for learning math. The results exceeded my wildest expectations.

We all know System 3 and we all use it, at least from time to time. My mathematical journey taught me that a voluntary and radical use of System 3 is not only possible, it augments our intuitive capacities well beyond the supposed limits of human cognition.

Through the years, the systematic search for a better alignment between my intuition and logic has become my way of understanding the world, others, and even myself.

In practical terms, here’s what that means. When my intuition tells me A and rationality tells me B, I put myself in the position of a referee. I force myself to translate my intuition into words, to tell it like a simple and intelligible story. Vice versa, I try to picture what logical reasoning is actually expressing, to experience it in my body, to hear what it’s trying to say. I ask myself if I really believe it. I fumble about. It takes time but it’s not a real effort. It’s more like a meditation on running water, something going on in the background that might stop and start, then all of a sudden become clear days, months, or even years later.

The goal is to understand where things are going wrong. Are my intuition and logic even speaking the same language? Are they even talking about the same things?

My intuition is never perfect. It’s often relevant, but sometimes it’s just rubbish. The good news is that it’s generally fixable. As for logic, that’s never wrong. At least officially. Except that it doesn’t necessarily say what I think it’s saying.

In the end, it’s almost always my intuition that wins. When I force it to listen to what logic is saying, it takes that into account and adjusts its position. Logic is something inert, like a pebble. My intuition is organic, it is living and growing.

It’s obviously stupid to call this approach System 3. It should simply be called thinking or reflecting. But the meaning of these words has been hijacked by a tradition that wants to make us believe that we should think contrary to our intuition. We’re told that our intuition is the mortal enemy of reason, that any dialogue between the two is impossible, and thinking means you have to submit blindly to System 2.

I’m personally incapable of thinking against my intuition and I have serious doubts as to the sincerity of people who claim they can.

Your intuition really is your most powerful intellectual resource. However, at the risk of spoiling your dreams, I must be honest with you—your intuition isn’t a magical elixir, or your lucky star, or the hand of God on your shoulder. It’s much more trivial than that. It’s the tangible manifestation of something that is invisible yet perfectly concrete and material: the entanglement of synaptic connections between your neurons that your brain continuously constructs and reorganizes, as it has done since you were in the womb. Your brain contains roughly as many neurons as there are stars in the Milky Way. Each of these neurons is, on average, tied to thousands of other neurons. This fabric of hundreds of trillions of interconnections is the network of your mental associations. Its structure is your way of giving meaning to the raw information continually flooding into your brain. This is, literally, your vision of the world. All that you have seen, heard, felt, imagined, or desired, all of your experience, all that you know, all that you remember, is encoded in this web. When your intuition speaks, that’s where it’s speaking from.

Your intuition will always be more powerful and better informed than the most sophisticated of language-based reasonings. For all that, it’s not infallible. If your intuition tells you the ball costs 10¢, it’s plainly wrong.

My intuition isn’t any less fallible than yours. It’s always getting things wrong. I have, however, learned never to be ashamed of it. I don’t disdain my mistakes, I don’t push them aside, because I don’t think that they betray my intellectual inferiority or some cognitive biases hardwired in my brain. On the contrary. Nothing’s more exciting than a big glaring error: it’s always a sign that I’m not looking at things in the right way, and that it’s possible to see them more clearly. When I’m able to put my finger on an error in my intuition, I know it’s good news, because it means that my mental representations are already in the process of reconfiguring themselves.

My intuition has the mental age of a two-year-old—it has no inhibitions and always wants to learn. If you stop mistreating your own, you’ll see that it’s exactly like mine, only asking to be allowed to grow.

The price of a ball

Because I have terrible handwriting, and because I’m easily distracted, I have a tendency to make mistakes in calculations.

I discovered in ninth grade that the only way to get around that was to verify after every three lines that what I was writing still made sense and that I really believed it. In other words, I learned how to use my System 1 to supervise the work of my System 2. From this time on, I was incapable of manipulating mathematical objects that I had no intuition for.

At what moment did I stop primarily visualizing numbers through their written form? I don’t recall. But it undoubtedly goes back to the same period. Decimal writing of numbers is useful for written calculations but it is certainly less practical when you want to form an intuitive idea about the validity of those calculations. This is where System 1 has an edge: it isn’t bound by the constraints of language and writing.

Depending on the context, I have many different ways of visualizing numbers. I have, for example, a tendency to visualize price in terms of length. When my friend told me that the ball and bat together cost $1.10, I immediately translated her words into a mental image that looked something like this:

When she told me that the bat cost $1 more than the ball, here’s how I saw it:

Then the two images came together in my head and morphed into something like this:

If this is how you visualize the problem, it doesn’t take a genius to figure that a ball costs 5¢.

A mental image is neither good nor bad in and of itself. Its value lies in what it allows you to understand. There are countless ways to visualize the problem and I don’t pretend that mine is better. My numeric intuition isn’t all that remarkable. I wouldn’t be able to do the calculation if the ball and bat together cost $2,734.18 and the bat cost $967.37 more than the ball.

I have these pictures in my head because, in my life, I have made a lot of calculation errors. Instead of concluding that I was terrible at math, I simply looked for simpler ways to see things, to grasp what I was writing.

In time, with this approach, I constructed a great variety of mental images that help me today to better understand the world—including the particular one that enabled me to find the price of the ball.

If you want to learn to find it obvious that the ball costs 5¢, I recommend proceeding like a mathematician would when faced with a new and incomprehensible idea. Rather than learning my pictures by heart, train yourself to construct pictures that work for you. The most important messages, the ones you should always bear in mind, are these:

1. You can reprogram your intuition.

2. Any misalignment between your intuition and reason is an opportunity to create within yourself a new way of seeing things.

3. Don’t expect it all to come at once, in real time. Developing mental images means reorganizing the connections between your neurons. This process is organic and has its own pace.

4. Don’t force it. Simply start from what you already understand, what you can already see, what you find easy, and just play with it. Try to intuitively interpret the calculations you would have written down. If it helps, scribble on a piece of paper.

5. With time and practice, this activity will strengthen your intuitive capacities. It may not seem like you’re making progress, until the day the right answer suddenly seems obvious.

You’ll need a number of training sessions. Exactly how many, I don’t know. It’s not worth tiring yourself out—better to split it up into short five-minute sessions, and think about it in the shower or while on a walk. Some people told me they could visualize the ball-and-bat problem in under a week, but your mileage may vary.

Above all, take your time. It’s good to think about it only once a week or once a month. Most important, keep at it and don’t let it drop. It will come eventually.

Solving a problem is only ever a pretext. The important thing is that you have the power to reeducate your intuition, to gain confidence in your body and thoughts.

Nothing about this should surprise you. Solving the problem of the ball and the bat is like standing up on a surfboard. Kahneman says that the first time you stand up on a surfboard, you’ll fall in the water, and concludes that humans are born with a defective sense of balance and that getting up on a surfboard can never become intuitive. His advice is to get out of the water and learn the laws of physics by heart. My advice is to get back up on the board.

Thinking fast, slow, and super slow

Our culture conveys false beliefs about how our brain functions, and that these false beliefs keep people away from the simple actions that would allow them to become better at math.

When you say to people that certain truths are by nature counterintuitive, you tell them that they can never really understand. It’s a way of discouraging them. Nothing is counterintuitive by nature—something is only ever counterintuitive temporarily, until you’ve found means to make it intuitive.

Understanding something is making it intuitive for yourself. Explaining something to others is proposing simple ways of making it intuitive.

None of this takes away from the value of Kahneman’s work. The cognitive biases that he documents are striking human realities of great social importance. We all have biases, even if they aren’t hardcoded and vary from one person to another, and certain biases happen to be more widespread and problematic than others.

His distinction between System 1 and System 2 has the merit of being simple. In a sense, it picks up the classic opposition between left brain and right brain, but in a modern version, without the anatomical nonsense. It’s just a basic model, but it’s appealing, and helps us become aware of our different ways of mobilizing our mental resources.

Here is a quick summary of the characteristic principles of System 3, the big oversight in Kahneman’s theory. It is a good model for the real nature of mathematical work.

System 1 is our intuitive capacity. We all like to describe it using electrical metaphors: with our intuition, we say that we think with the speed of lightning. It’s not entirely false. Our brain isn’t properly speaking an electrical circuit, but the signal that’s transmitted along the neurons is electrical in nature.

System 2 is our capacity for rigorous reasoning. We imagine it in mechanical terms, with gears or something of the sort. That doesn’t correspond to any biological reality. What we are biologically capable of is to pretend we’re robots and mechanically apply a preset series of instructions. With the right set of instructions, we can make logical conclusions and valid calculations. But it’s so disagreeable and against our nature that we usually give up after a few seconds, or at most a few minutes. In the end we’re rather sorry robots: we make too many mistakes and we can’t go the distance.

System 3 is so entirely ignored by our culture that I can’t find the right word to characterize it. As I said above, I would like to say that System 3 simply corresponds to our capacity for thinking. But the verb to think doesn’t mean much since it’s been used as an injunction to submit to System 2.

The activity of System 3 is a special kind of meditation, but this word is also much too vague. Not all meditation is an activity of System 3. System 3 specifically aims at establishing a dialogue between Systems 1 and 2, in order to understand their misalignments and resolve them. Rather than a free meditation, it’s one constrained by the principle of noncontradiction. Its ultimate goal is to revise and update System 1 while taking into account the results of System 2.

It is also necessary to distinguish System 3 from the capacity of our System 1 to revise itself without any deliberate action on our part. Our mental plasticity results from the constant reconfiguration of our synaptic network: our mental circuits evolve in response to our experiences. You can imagine neurons as minuscule plants that grow and sink their roots deeper and deeper.

Every time we practice a given activity we habituate our System 1 to the specifics of that activity. When we try to stand up on a surfboard, we habituate our System 1 to the hard realities of Newtonian physics and construct our surfing instincts. With System 3, we habituate our System 1 to the hard realities of logical consistency and construct our instincts for truth.

The great misunderstanding of math teaching stems from the fact that all the tangible manifestations of mathematics—its confusing language, its incomprehensible notation, its bizarre and rigid reasoning—seem to tie it to System 2.

Most people take that at face value. They become discouraged in a few minutes, or throw themselves into a masochistic effort that has zero chance of succeeding.

But a few people choose to rely on their System 3. They’re not aware that they’re doing anything special. Math just feels easy to them. It doesn’t even feel like work. They’re just seeing pictures in their heads, and spending a couple of minutes a day looking at these pictures and asking themselves naïve questions.

To them, it all looks completely normal. It certainly doesn’t feel like they have a gift.

This post is adapted from Chapter 11 of Mathematica: A Secret World of Intuition and Curiosity.

This story began two years ago, during the pre-launch of my book on the cognitive techniques of mathematicians. As authors do in those circumstances, I sent a handful of copies to selected high-profile people, in the hope that it would spark the public conversation.

I had a fantastic success rate with mathematicians. I sent a copy to Pierre Deligne, who wrote back saying he loved the book, “especially the insistence that learning or doing math is about changing one’s brain.” I sent a copy to Terry Tao, who blurbed the book. It was harder to get my foot in Steve Strogatz’s door, as he initially declined to even receive the book. But I insisted and, after a few days, he sent back a warm thank you note. He then blurbed the book and posted a big shout-out on the launch day.

Outside of academia, the pattern was quite different. Most people never responded. One exception was Paul Graham, who kindly replied to my two-liner cold email.

I knew it was a long shot, but I couldn’t not try. First, because I admire him and enjoy his writing on software and startups—and even more so his writing on writing itself and its transformative powers, which is remarkably aligned with central ideas from my book (and this essay.) This made me feel a genuine intellectual kinship.

Second, because I noticed that he was regularly posting on mathematics. It all clicked into place when I found out that his father was a mathematician—the alignment was no coincidence.

This made me excited and confident that he would love the book. But, unfortunately, I never heard from him again.

Let it be clear that I view this as completely normal. He is a busy person with a million competing interests and, no doubt, a massive reading backlog. Put simply, he owes me nothing.

But this was still frustrating as I had a third, more important reason for sending him the book: I wanted to engage with his thinking on a topic where I fundamentally disagree with him—genetic determinism.

Hereditarian statements are quite common in Paul Graham’s essays, though most of them are casual remarks that are peripheral to the core message. A good example is this passage from his otherwise great post on determination:

A good deal of willfulness must be inborn, because it’s common to see families where one sibling has much more of it than another. Circumstances can alter it, but at the high end of the scale, nature seems to be more important than nurture.

There isn’t much to object here, as he is using the appropriate amount of hedging (“must be”, “seems”). But the passage does reveal a strong hereditarian prior.

I brushed this one off as an “accident”, and filed it under the category of “unexamined hereditarianism”. The logical gap in the argument was so blatant that it was clear to me that he hadn’t put much thought into it. Indeed, it is not because a trait is striking, precocious, hard to nurture, and unexplained by the social context that it is necessarily innate.

This kind of hereditarianism was highly prevalent in the pre-genetics era and I’ve learnt not to take it at face value. A famous example is Henri Poincaré’s quip that “Mathematicians are born, not made”, which always struck me as being equally rigorous to Brillat-Savarin’s earlier quip that “One becomes a cook, but one is born a roaster.”

My fundamental disagreement isn’t about that—it’s about his sharing on X of pseudoscientific content produced by hereditarian blogger Cremieux, offering him a massive platform.

This is where this story starts for good—two years ago, I stumbled upon this powerful tweet by Paul Graham and almost fell off my chair:

According to this data assembled by @cremieuxrecueil, the difference in IQ between identical twins reared separately is only half a point greater than the difference between two tests of the same person.

This wasn’t a casual remark. This was hardcore genetic determinism, coming from a respected public figure, and backed by a striking visual that cited five independent scientific studies.

This tweet was viewed 3.5 million times. I can’t speak for the other 3,499,999, but I found it disturbingly persuasive.

The takeaway, it seemed, was pretty simple: when you separate two identical twins at birth, raise them in two random families, and test their IQs in adulthood, the two results are barely more divergent than two different tests of the same person.

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Fifty shades of correlations

I was genuinely shocked. My “centrist” views on heredity and intelligence—that genes matter but are only one factor—were being challenged by clear-cut empirical evidence. So were my views on mathematics, the core ideas in my book, and my own understanding of my life journey.

To explain why this visual was so destructive to my worldview, I need to get into the technicalities and, more specifically, into the notion of heritability, a statistical measure of how much genetics influence a given trait.

While heritability is an imperfect notion—with nasty caveats that are beyond the scope of this post—it has become a de facto standard and, for better or worse, the complex debate on cognitive inequality is often reframed as a one-dimensional debate on the heritability of IQ.

A pure blank-slatist would put it at 0%. A pure hereditarian would put it at 100%. Any reasonable person would put it somewhere in the middle, leaving two questions unresolved: where exactly?, and, what does the figure even mean?

The three simulations below illustrate three potential values for the heritability of IQ, 30%, 50%, and 80%. In each case, the dots represent 1000 random people, each placed according to their genetic potential for IQ (horizontal axis) and actual IQs (vertical axis). Heritability measures how close the dots are to fitting on a line. Mathematically, it is defined as the R^2 of the linear regression.

At 30%, one does observe a faint correlation between genetic potential and IQ. The correlation becomes clearer at 50%, while remaining quite noisy. This is an essential aspect to keep in mind: 50% may sound like a solid heritability figure, but the associated correlation is rather modest. It’s only at 80% that the picture starts to “feel like” a line.

Heritability is, by construction, a population-level aggregate. Before it can inform policy-making (or even personal decision-making), it must be interpreted at the level of individuals. This is where things get interesting and counterintuitive.

Let’s say, for example, that you are a genetically average person. How much does that affect your prospects?

• Surprisingly, at 30%, it’s as if your genes didn’t matter at all. With an average potential, you still have a decent chance of landing at the top or bottom of the IQ distribution. Actually, in this specific random sample, one of three smartest people around (the top 0.3%) happens to have an almost exactly average genetic make-up, and the fourth dumbest person has a slightly above-average potential.

• At 50%, being genetically average starts to limit your optionality, but the spread remains massive. Had you been marginally luckier—say, in the top third for genetic potential—you’d still have a shot at becoming one of the smartest people around.

• At 80%, though, your optionality has mostly vanished. It’s still possible to move a notch upward or downward, but the game is mostly over. In this world, geniuses are born, not made.

This discussion is generally omitted by hereditarians, which is unfortunate, because it is the only way to clarify the stakes. There is a fundamental asymmetry in the debate. Heritability matters a lot when it is extremely high, because it then supports genetic determinism, but for the rest of the range the exact figure has little practical significance.1

This is why “heritability centrists” rarely pay attention to the active research in behavioral genetics. Whether the heritability of IQ ends up being 15%, 30%, or 50%, this is “interesting” but not “Earth-shattering”. Of course, if the true figure is 80%, then it’s an entirely different story.

Speaking as a mathematician and author of a book on mathematical cognition, I should add two important remarks.

First, while social media thrives on memes about mathematical genius and the “IQ red pill”, mathematicians themselves aren’t that interested in IQ. This might be because they understand how the concept is defined. By design, IQ is a statistical construct that captures some common denominator, the “g factor”, among a variety of cognitive tasks. It’s as if someone had pooled a large number of biomarkers to extract a general health score, the “h factor”.

That doesn’t mean the construct is useless. The concept of good health is a meaningful one and simplistic proxies like “biological age” do make sense in particular contexts. IQ was designed as a cheap, scalable way of identifying students with learning disabilities and, for this specific use case, it is reasonably pragmatic.

Do note that the “g factor” is an abstraction that is impossible to directly measure. All you can get is a proxy that has some degree of correlation (say, 80%) with the actual thing you care about. Given that meaningful real-life cognitive abilities such as “mathematical talent” (whatever the expression means) are themselves only correlated with the “g factor”, you end up with a complex chain of more-or-less tight correlations that, as a whole, has no credible reason to be tight.

That is, unless the “g factor” captures a fundamental biological reality, in the same sense a CPU has hardware-defined characteristics like clock speed or L1 cache.

This is where the asymmetry becomes abysmal:

• If the heritability of IQ is 50%, then I’d expect the correlation chain to be loose, and I wouldn’t be surprised if the heritability of math talent ended up around 20% or 30%.

• By contrast, if the heritability of IQ is 80%, then I would eat my hat and concede that mathematical talent is primarily innate and mostly immutable. This is because 80% heritability would be indicative that the “g factor” is a core metric of the human body, hardcoded at the proteomic level. And, given that Raven’s Progressive Matrices, one of most g-loaded IQ tests, has such a strong smell of undergraduate algebra (it’s all about 3-cycles and permutation matrices), I would have to admit that this hardcoded ability is a close cousin of mathematical talent.

The second essential remark concerns the non-genetic part of cognitive inequality. A blank-slatist would claim that this is all social determinism. I absolutely don’t believe that and this is precisely why I insist on telling this story.

For all their performative battles, hereditarians and blank-slatists operate within the same deterministic frame (genetic determinism vs social determinism), denying any meaningful role for human agency and the messy, noisy process we call life.

This is where I strongly align with Paul Graham’s hacker ethos, clearly recognizable in this passage from his piece on intelligence:

I predict the gap between intelligence and new ideas will turn out to be an interesting place. If we think of this gap merely as a measure of unrealized potential, it becomes a sort of wasteland that we try to hurry through with our eyes averted. But if we flip the question, and start inquiring into the other ingredients in new ideas that it implies must exist, we can mine this gap for discoveries about discovery.

My book is literally about this “interesting place”. I have spent two decades mining the metacognitive gap between my (seemingly limited) innate ability and actual mathematical talent. Along the way, I became capable of producing research in algebra and geometry at a level I had never dreamt of.

From a behavior genetics perspective, the wiggle room between genetic and social determinism is captured by Turkheimer’s Third Law: “A substantial portion of the variation in complex human behavioral traits is not accounted for by the effects of genes or families.”

Not all this “substantial portion” is amenable to mining, and it could be that extreme mathematical talent results from idiosyncratic cognitive development that is impossible to replicate. But, on the other hand, there is a long tradition of mathematical “geniuses” such as Descartes, Einstein, or Grothendieck, who insisted that they had no special talent and owed their success to particular habits they happen to have developed (Descartes goes as far as calling this a “method”.)

I completely get why most people would assume that mathematical talent is innate: the subject is confusing and hard to teach, and some random kids are fabulously brilliant for reasons no-one fully understands. Yet there is an unheard, latent consensus within the mathematical research community that one can get better by adopting the right mindset and attitude, becoming fearless in the face of one’s limitations, and practicing the right “unseen actions” in one’s head with an inordinate amount of patience, curiosity, and persistence.

This mindset and attitude are the central topics of my book, and I believe that there are important takeaways for everyone—most notably, that anyone can make stellar progress. As Deligne puts it, “learning or doing math is about changing one’s brain”. This implies that cognitive transformation is possible, but it doesn’t imply that it is easy, nor that everyone has an equal chance.

But I don’t get to choose the heritability of IQ so that it fits my existing beliefs. A key lesson of mathematics is that if your beliefs are wrong, then there’s no point waiting to revise them.

If the true figure was proven to be 80%, then I would readily admit that I was delusional—and that Deligne was delusional with me, alongside Descartes, Einstein, and Grothendieck.

This is why Cremieux’s visual was so shocking to me. Taken at face value, it presented unobjectionable evidence that the heritability of IQ was extremely high, north of 80%.

But was this interpretation even correct?

The unravelling

Past the initial shock, I noticed the scale on the left and realized where the curious pixelated appearance was coming from—it was simply showing the actual granularity.

All five studies are very small, with a cumulative sample size well under 200 pairs. My analysis below focuses on the three largest ones (Burt, Minnesota, and Shields), all based on datasets of about 40-50 pairs. The remaining two studies are too tiny to justify a separate discussion, and they suffer from the same issues anyway.

The second thing that caught my attention was the brutal cliff, which felt unnatural. I would have expected a smoother, bell-curve-like drop.

Upon closer inspection, the cliff was primarily attributable to the bizarre distribution of the yellow dataset (Burt).. This was the first study I searched for, and I was in for a big surprise.

As a non-specialist and a long-term heredity centrist with little interest in the detailed picture, I hadn’t paid much attention to the troubled history of the IQ wars. And, in particular, I was unaware that Cyril Burt had been at the center of possibly the biggest scientific fraud scandal of the 20th century, precisely because of this very study. Most of his “twins reared apart” never existed, in the literal sense, as his dataset was a fabrication (hence the abnormal cliff).

I was left with two non-fraudulent, non-microscopic studies of “twins reared apart” whose primary publications were easy to retrieve:

• Shields, J. (1958), Twins brought up apart, Eugenics Review, 50 (2), 115-123.

Shields Twin Brought Up Apart

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• Bouchard, T. J., Lykken, D. T., McGue, M., Segal, N. L., & Tellegen, A. (1990), Sources of human psychological differences: The Minnesota study of twins reared apart, Science, 250 (4978), 223-228.

Bouchard The Minnesota Study 1990

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I am including the actual papers because I found them both readable and eye-opening. As my conclusion below will be pretty damning, I want everyone to be able to check for themself.

Here’s the bottom line: I have no idea what motivated Cremieux to include Burt’s fraudulent data, but even without it his visual is highly misleading, if not manipulative.

It is very unfortunate that 3.5 million people were exposed to it. I suspect that only a tiny fraction went past the initial impression, and the vast majority went away with the incorrect impression that these studies prove something massive about the heritability of IQ.

As I’ll argue, the raw data might be correct, but it is entirely inconclusive, for reasons that the authors of these studies were perfectly aware of and that clearly transpire from an honest reading of their publications. There is no reason to doubt that the twin pairs studied by Shields and Bouchard were actual “twins reared apart”, in a lax sense. But the mythical “twins reared apart” that enable a clean separation of nature and nurture simply do not exist.

Shields himself was extremely cautious in the concluding section of his 1958 article:

As I pointed out at the beginning, the study is not quite complete and the conclusions so far must be regarded as tentative.

… it must be agreed that in some respects the social differences between the families in which most of our twins were brought up were not very large. It would, therefore, be unwise when we come to evaluate the final results, to claim that genetical factors would show up as clearly if the environments of the twins were radically different.

Here he was alluding to a well-known shortcoming of his approach, the range-restriction of adoptive families. But this is only the last in a list of four major biases that affect his study and that of Bouchard.

Bias #1: Shared congenital factors

There is a remarkable 1981 video interview of Thomas Bouchard (conducted by Niels Juel-Nielsen, the author of one the tiny studies), where Bouchard explains how he fell in love with the approach:

I just thought that was such a beautiful experimental design. I love simple designs that are cutting and definitive.

I agree that there is something irresistible in the idea of studying identical twins reared apart. It feels like a “natural experiment” that would magically tell nature and nurture apart. In fact, this appearance of simplicity, of being “cutting and definite”, is precisely what made Cremieux’s visual so compelling and viral.

But even under unrealistically perfect conditions, with twins truly separated at birth and placed in truly random families—as we’ll see, that was far from being the case—the experiment would still be confounded by a substantial bias: twins share a womb for nine months.

Bouchard was perfectly aware of this issue, as he noted in his publication:

MZA twins share prenatal and perinatal environments, but except for effects of actual trauma, such as fetal alcohol syndrome, there is little evidence that early shared environment significantly contributes to the variance of psychological traits.

But, disappointingly, his study satisfies itself with the “little evidence” argument and doesn’t even attempt to estimate this bias.

To a modern reader, it is surprising that the referees found that acceptable. One likely explanation is that the importance of prenatal and perinatal environments wasn’t fully understood at the time.

Since 1990, we have collected massive evidence that prenatal and perinatal environments are major contributors to the population-level IQ variance, and anyone who’s ever been around a pregnant woman is aware of that.

Indeed, pregnancy and birth are critical periods for cognitive development, with a multitude of well-documented risk factors:

• fetal alcohol syndrome, which remains today one of the top contributors to population-wide IQ variance. The situation was likely much worse in the time of Bouchard, as his cohorts were born before the US Surgeon General advised women not to drink during pregnancy, and decades before FAS was recognized as the tip of a much broader spectrum. It was only recently agreed that even occasional drinking is damaging (these labels on wine bottles are there for a reason).

• maternal smoking (which, again, was much more prevalent in Bouchard’s cohorts.)

• lead exposure (not anecdotal, again, given the time period)

• preterm births (fairly common for twin pregnancies)

• birth asphyxia (twins have a higher risk; this might concern one twin, or both)

• congenital infections (rubella, cytomegalovirus and many others)

• nutrient deficits (folic acid, etc.)—just take a look at the recommended supplements for pregnant women.

These non-genetic factors are far from negligible and, in fact, they dominate the list of known direct causes of cognitive disabilities, alongside a few genetic and chromosomic disorders. Once you aggregate their contribution, plus a provision for unquantified prenatal factors that are still under the radar, this actually makes it hard to believe that heritability could be above 80%, as this would leave very little variance left for everything else—that is, for the whole post-uterine experience (what we call life.)

Regardless of their prevalence in the general population, these risks are compounded for twins reared apart, as 1/ twin pregnancies have a much higher chance of complications, and 2/ pregnancies resulting in adoptions have a much higher chance of being associated with inadequate prenatal care.

Note that the existence of confounders isn’t necessarily a death blow to the approach. By estimating their impact, one might manage to correct for them.

In fact, it does seem that Bouchard and his team attempted to do just that, and this is where the story becomes truly bizarre. But, wait, we have three more biases to cover before we get there.

Bias #2: Shared early life

On average, the twins in the Bouchard study were separated at 5.1 months. In the Shields study, 25 pairs were separated during their first year, but 6 were separated during the second year, 4 between ages two and four, and individual pairs were separated as late as seven, eight, and nine years of age.

In other words, the phrase “reared apart” should be taken with a substantial grain of salt. It subliminally suggests “separated at birth”, while it effectively means “separated in early childhood or later.”

Whatever can be said about bias #1 applies verbatim here. In The Blank Slate, Steven Pinker does argue that our societies overestimate the importance of the first few months. He might be correct, but it is difficult to believe that these months do not matter at all. (I’m writing this as the father of two kids, who was genuinely surprised by the intensity of the emotional bound and communication abilities that develop from the very first days.)

Some kids are properly fed, loved, cared for, talked to, and some are strapped in bouncers in front of Cocomelon. And, in all likelihood, this isn’t a cosmetic nuance.

To make things worse, the kids in the Bouchard and Shields studies are unlikely to have experienced uneventful early childhoods—a telltale sign is that they were eventually put up for adoption. Which brings us to the next bias:

Bias #3: Shared cause for separation

Shields gives a fascinating account of what triggered these adoptions:

The reasons for these twins being parted were various. Illegitimacy accounts for six pairs, the death of the mother for eight. The most frequent reason was that the mother was ill, pregnant again or considered unable to look after both twins [...] In two cases the children were removed from home by the Poor Law authorities. In one case the father sold one twin to settle his debts.

For all their diversity, these twin pairs all had one thing in common: they experienced something out of the ordinary and—except in the last example—they experienced it together.

Whether their mother was putting booze in the baby formula, whether she had paranoid schizophrenia, whether she died from a rare monogenic disease, or whether both parents died in the sinking of the family yacht, these extraordinary events constitute a massive selection bias for a multitude of confounders that will mess up the analysis if you fail to control for them.

Bias #4: Range-restriction of adoptive families

The first three biases are easy to grasp, because it’s intuitively obvious that congenital lead exposure or dysfunctional postnatal care would harm both twins in ways that would then be misattributed to genetics.

This fourth bias is more subtle because it is fundamentally about population-wide statistics.

For the experimental design to be “cutting and definitive”, as Bouchard dreamt it to be, the twins must be selected at random (bias #3), incubated in 100%-consistent artificial wombs (bias #1), and placed right after birth (bias #2) in truly random families—which never happens in real-life.

This is bias #4. In our societies, adoptive families are the opposite of random. They tend to have specific family structures, live in specific neighborhoods, and share a specific set of values and cultural habits. The simply act of wanting to adopt a kid—and being entrusted with one—filters out the most dysfunctional environments.

If you take two seeds from the same apple and plant them in separate pots, you might be able to derive meaningful insights on the relative importance of the environment. That is, unless you place them side-by-side on the same balcony.

In The Blank Slate, in the chapter where he argues that adoption studies (not just of twins) suggest that family environments matter much less than we used to think, Pinker adds this essential caveat:

An important proviso: Differences among homes don’t matter within the samples of homes netted by these studies, which tend to be more middle-class than the population as a whole. But differences between those samples and other kinds of homes could matter. The studies exclude cases of criminal neglect, physical and sexual abuse, and abandonment in a bleak orphanage, so they do not show that extreme cases fail to leave scars.

What if these extreme cases were the childhood equivalent of fetal alcohol syndrome, maternal smoking, cytomegalovirus, and extreme preterm births—rare occurrences that contribute an inordinate share of the population-wide IQ variance?

Keep in mind, here, that the variance is the average of the square of the distance to the mean: a single individual who is three standard deviations away from the mean contributes as much as 36 people who are half a standard deviation away from the mean.

These aren’t new insights brought by our modern sophistication. Shields was fully aware of bias #4 as early as in 1958 and, in fact, this is precisely why he was so prudent in his conclusions. He had good reason to be prudent, for his dataset was catastrophically confounded by range-restriction. Just read his own account:

In twelve cases the twins were brought up in families that were unrelated to one other. In the remainder they were brought up by different branches of the same family, in twenty cases one twin staying with his own mother.

We should praise Shields for his insight and honesty. He tried something, it didn’t work, and that’s fine—this is what good science often looks like.

But, frankly, it doesn’t make any sense to repackage his results into shiny visuals, as if they held any significance. At this point, from an insights perspective, this is pretty much game over for the Shields dataset.

We are left with only the Bouchard study and this is where things are going to get… how to put it?… “embarrassing”.

As he often told (for example in the above interview with Juel-Nielsen), Bouchard started the massive Minnesota Study of Twins Reared Apart after he got word of the “Jim Twins.”

James Arthur Springer and James Edward Lewis were born in 1940. They had been separated soon after birth and placed in separate adoptive families in Ohio. Their reunion, in early 1979, became an instant media sensation, owing to the crazy “coincidences” in their story. All this, alongside Bouchard’s early interest in their case, is documented in this December 1979 article in the New York Times:

• Each married and then divorced a woman named Linda. Their second wives were both named Betty.

• Springer named his first son James Allan. Lewis named his first son James Alan.

• Each man grew up with an adopted brother named Larry.

• During childhood, each owned a dog named Toy.

• Both twins had law‐enforcement training and had worked part time as deputy sheriffs in their Ohio towns 70 miles apart.

• They shared many common interests, such as mechanical drawing, block lettering and carpentry.

• ….

When the Jim Twins were reunited, Bouchard was a 41 years old academic. His instinct told him that this was his moment—his unique opportunity to leave a durable mark in science—and he went on to bet the rest of his career on the project.

It’s a pity that Bouchard had such a poor statistical intuition, as it would have been the right time to pause for a serious reassessment of the Jim Twins miracle.

In statistics, when coincidences are too good to be true, there’s a good chance that something nasty is going on in the background. True coincidences do happen, acts of God or random “flukes” from which there is absolutely nothing to learn. But I would never use this argument to brush off the Jim Twins story, as I have never seen a fluke of this magnitude.

Alternatively, it could be that the data was fabricated. But I found no evidence that the Jim Twins story wasn’t authentic.

Are we all good? Can we move on to accepting the Jim Twins story as a compelling case of genetic determinism? Not quite, because we still haven’t ruled out the nastiest of all scenarios—that the Jim Twins story might have been authentic, but cursed, and Bouchard should have run away from it because it meant the exact opposite of what he thought it meant.

After I quit my permanent academic job in pure mathematics, I went on to launch a tech startup that specialized in mining the latent sociological information present in seemingly innocuous consumer data such as first names, zipcodes, transaction histories and minute behavioral proxies.

I never dated any Linda, nor any Betty, but I doubt this says much about my genes. I rather suspect that it says something about my year of birth and social background (and the fact that I grew up in Paris.)

Likewise, I’m inclined to think that, if both Jim Twins had an adopted brother named Larry, this is telling us something about the psychosocial proximity of their adoptive families, rather than the telepathic influence of genes.

Interestingly, just the latent feature space of pet names already captures incredibly subtle information about the cultural backgrounds and values of their owners.

The Jim Twins story might be meaningless from a population genetics perspective, but still carry a mesmerizing lesson on the social conformism of the white family-oriented middle-class in suburban Ohio in the 1940s and 1950s.

The Sullivan County Community College gene

One common weakness of the Shields and Bouchard studies is their reliance on “naturally occurring” pairs of “twins reared apart”. What if we could deliberately create such pairs?

This sounds like a nazi experiment, yet a rogue adoption agency did try to pull it off. This incredible story is related in Three Identical Strangers, a 2018 documentary about three identical triplets who were deliberately placed, respectively, in a “working-class” family, a “middle-class” family, and an “upper-class” family.

The documentary is both hilarious and tragic, and holds several fascinating lessons. First, of course, that facial features and overall appearance are strongly determined by genes—but we already knew that.

Second, that genetics alone can’t predict life outcomes. The documentary does mention that all three triplets suffered from mental health problems in their youth—which is hardly exceptional in the general population and, in fact, fairly common among adoptees. Yet David and Robert, two of the triplets, seem to have moved forward reasonably well into middle-age, while Edward sadly took his life at age 34.

Third, that true randomization is hard. Indeed, even this deliberate, unethical manipulation wasn’t immune to range-restriction. The documentary opens with the account of the triggering event to the reunion, a farcical story of mistaken identity in the dorms of Sullivan County Community College, where two of the triplets have been students.

What a striking coincidence, isn’t it? Either the triplets carried a yet-undiscovered Sullivan County Community College gene, or something went horribly wrong in the randomization process.

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The dog ate my control group

It’s time for the most troubling part of the analysis.

As he led the Minnesota study throughout the 1980s, it isn’t plausible that Bouchard was unaware of bias #1 (mentioned in his paper), bias #2 (the paper provides the mean and standard deviation of the separation age), and bias #4 (already discussed by Shields in 1958.)

It isn’t plausible either that he was unaware that there was a simple method for estimating these biases and filtering them out, since he did exactly what one is supposed to do to apply the method.

Indeed, at the end of above excerpt from his 1981 discussion with Juel-Nielsen, Bouchard reveals that he is simultaneously building a dataset of fraternal twins reared apart. This is the first step of the process, after which one can proceed to a differential analysis of the IQ correlations between MZA (short for monozygotic twins reared apart) and DZA (short for dizygotic twins reared apart) pairs.

The DZA sample serves as a control group. It is subject to the same exact biases as the MZA dataset, but DZA twins share only 50% of their genome while MZA share 100%. From there, one can recover a debiased estimate for the heritability. This makes use of Falconer’s formula, a statistical trick based on simplifying assumptions that are never satisfied in real-life, but provides a reasonable first-order approximation.

The technique is standard in behavior genetics. It is the bread-and-butter technique of “twins reared together” studies, and everyone agrees that biased correlations cannot be taken at face value.

It is impossible that Bouchard, publishing in 1990, could have ignored any of this. The entire discussion—from the imperious need to correct for biases to the standard methodology for doing so—was fully laid out in Douglas Scott Falconer’s seminal book, Introduction to quantitative genetics, whose first edition was published in 1960:

Now imagine that you have spent the past ten years assembling the best available sample of MZA and DZA pairs. You are about to publish a landmark study aiming to provide a “cutting and definitive” estimate of the heritability of IQ, resolving a major scientific debate. Which of these options would you choose?

1. Base your estimate on the standard methodology, a differential analysis of MZA and DZA pairs.

2. Present the MZA data, then argue through ad hoc, complex reasoning that biases are under control—without ever showing any DZA data to back that claim.

Strangely, Bouchard and his team went for option 2. Moreover, they made the fabulously awkward decision to acknowledge the existence of a DZA control group while simultaneously withholding the data, citing a dog-ate-my-homework excuse:

Due to space limitations and the smaller size of the DZA sample (30 sets), in this article we focus on the MZA data (56 sets).

No, you’re not dreaming—Thomas Bouchard just pulled a Fermat right before your eyes!

“I have a marvelous proof but the margin is too narrow” is a notoriously weak argument in mathematics. In experimental science, it is beyond egregious, since readers can’t even try to reconstruct the missing data by themselves.

The “space limitations” argument is highly implausible. First, because it would have taken essentially zero space to provide the one missing metric that could have bolstered, or tarnished, the credibility of their heritability estimate. Second, because this shouldn’t have been about adding something to the publication—it should have been about writing it using an entirely different approach.

Indeed, if the IQ correlation within MZA pairs was statistically significantly higher than the correlation within DZA pairs, then this silver bullet argument should have been put at the center of the article. By contrast, if no meaningful differential signal was found, then this essential information (that would have put the entire approach into question) shouldn’t have been withheld.

As for the “smaller size” argument, it isn’t convincing either. Statistical noise varies according to the inverse of the square root of the sample size, so the difference between the two series isn’t massive: √56/√30 ≈ 1.37, and a 37% precision gap hardly justifies to trust one sample while throwing away the other one.

It could be that the article was just poorly written, that Bouchard and his team had a silver bullet argument and “forgot” to use it. But who can seriously believe that?

Piecing everything together, a more credible scenario would be that Bouchard set out to conduct a proper study, then found out that there was no meaningful signal. That, in itself, wouldn’t have been surprising, as the study was statistically underpowered and exposed to multiple confounders. With 56 MZA pairs and 30 DZA pairs, both R^2 estimates were highly vulnerable to noise, and a few outliers here or there could have messed up the entire analysis.

Jay Joseph (whose blanket rejection of any role for genes does not align with my views) claims to have located the missing DZA data in subsequent publications by the Minnesota team, and that the two R^2s did not differ at a statistically significant level—meaning that the study had failed to identify any genetic contribution. That is a very serious objection2, and Bouchard’s evasive response is even more concerning (he reiterates the sample size excuse without ever addressing the specific claim about R^2.)

As an outsider with no quarrel or vested interest, I find it incomprehensible that such a fundamental question could remain unanswered 35 years after the study was published: was there any meaningful differential signal between the MZA and DZA pairs, or not?

To be clear: this issue isn’t to second guess who did what, when and why, but to clarify the actual content of the raw data. Pre-replication crisis psychology was operating under an entirely different set of norms, and practices that we now identify as fraud used to be considered “fair game”. For example, it wouldn’t be that surprising if a little bit of “cleaning” had taken place between the primary data collection and the tabulated results, as this used to be common practice. Extracting compelling results from microscopic datasets is a delicate art and there are many documented instances where a well-intended human hand was present to “help nature.” Or, to put it bluntly: there is no way a non-preregistered observational study of 56 MZA pairs would make it to Science in 2025.

Bouchard notes that “The study of IQ is paradigmatic of human behavior genetic research.” I can only imagine how it would have felt to come out empty-handed after a ten-year effort to deliver a clear-cut answer. Had this happened to me, I have no idea how I would have reacted. I assume it would have been extremely tempting to reframe the story in a way that wasn’t technically false—but wasn’t entirely true either.

What infuriates me is the repackaging of five zombie datasets into a shiny visual that fools 3.5 million people, including smart people like Paul Graham. Out of curiosity and integrity, I accepted the challenge to my worldview. This sent me down a three-month rabbit hole to learn everything I could on the subject, only to discover that the whole thing was a mirage.

Epilogue

As I eventually found out, all this was well known to those with a strong interest in the topic. The study of “twins reared apart” is widely regarded as a scientific dead-end and behavioral geneticists have entirely moved on to new strategies.

A telltale sign is that no-one since Bouchard has tried to replicate his approach. Charles Murray, the co-author of The Bell Curve—hardly a blank-slatist—effectively pronounced a funeral eulogy for Bouchard’s vision in his 2020 book:

For many people, “twin studies” brings to mind the famous Minnesota Study of Twins Reared Apart. It got so much publicity because it produced so many dramatic examples of similarities in adults who had never met each other. When the separated twin brothers who inspired the study were reunited in their 30s, it was discovered that as children both had had a dog named Toy. As adults, both had been married twice, first to wives named Linda and then to wives named Betty…

…but the method’s potential was limited. Separation of identical twins at birth happens so seldom that large sample sizes are impossible. The range of environments in which separated twins are raised is narrow—adoption agencies don’t knowingly place infants with impoverished or dysfunctional parents. In contrast, it is not difficult to assemble large samples of twins who have been raised together.

In a nutshell, the current debate revolves around this:

• The classic “twins reared together” protocol leads to heritability estimates in the 50-70% range. But everyone knows that the assumptions in Falconer’s formula do not hold in real-life and there is no general consensus on how to fix this.

• Direct genomic studies (GWAS) lead to much lower estimates (around 20% or below). But they are subject to their own data and modelling limitations.

• Most people agree that twin studies overestimate heritability while genomic studies underestimate it, and the truth must lie somewhere in-between. People are working to resolve the gap with new techniques and meta-arguments. As far as I understand, the frontline seems to be stabilizing around the 30-50% range. Sasha Gusev argues for the lower end of that band, but not everyone agrees.

This is an intricate scientific debate that I enjoy watching from the sidelines, although my personal stake is now pretty low. As discussed, the nuance between 30% and 50% heritability is scientifically interesting but, at the individual level, I don’t find it very informative (especially since no current technology is capable of measuring one’s true genetic potential.)

I am not a behavioral geneticist and, paradoxically, this is precisely why I felt the need to write this detailed account of my thought process. Despite my solid background in mathematics and data analysis, it cost me serious effort to build a watertight debunk, and I thought it was worth sharing it with the other 3,499,999.

As I emerged from the rabbit hole, this is what struck me—this wasn’t really about Cremieux’s slide, this was about eradicating the “twins separated at birth” trope that had infected me as a teen and distorted my expectations of what was scientifically plausible.

It is fundamentally hard to believe that such a simple and beautiful design could be so profoundly flawed. And yet, people have tried it over and over again, and failed over and over again.

From a public conversation perspective, there are three important takeaways:

1/ There is no “cutting and definitive” method for estimating the heritability of IQ. This is the saddest and most annoying part of the story. Bouchard’s dream is dead and we have nothing to replace it with. All current methods and estimates, from “twins reared apart” to GWAS and RDR, make use of statistical models whose underlying assumptions are eminently debatable. I have my own reading of the current debate, based on my own mathematical intuition of how these models behave under mild and moderate violations of their core assumptions, but it is very hard to communicate and, of course, there is no one-stop referable source for the layperson.

For all practical purposes, this is Brandolini’s law on steroids. Every time my book is debated on social media, I’m getting insulted by people who are firmly convinced that I am a lunatic, an ideologue, or a grifter, for asserting that one can get better at math. My interview with Quanta Magazine resulted in a flame war on Hacker News, where my supporters were called “flatearthers” and “secular creationists.” My favorite insult to date is one that I received on X: “you actually are half commercial person half prophet selling his gene pool experience as a standard worldview."

My sad conclusion, at this point, is that there is no short and simple way to respond to these people.

2/ Statistical nuances and caveats are very ineffective arguments. This I already knew, but the reminder was quite brutal. In the early stages of my journey, I was effectively battling a steelman version of Cremieux’s visual, since I had been really impressed by the data and was genuinely thinking to myself: “wow, this looks big!” The first sources I found did mention some “biases” in Bouchard’s study—but, come on!, who cares about biases when the data speaks for itself?

As we learned in the schoolyard, nitpicking is for losers, and no one is going to listen to technical jargon in the face of glaring evidence.

This explains why this piece is so long. I don’t think one can effectively argue against the Jim Twins narrative. To fully understand what was going on, I had to build my own counterstory, with secondary characters and subplots, that transformed the abstract concept of “biases” into concrete details that were memorable and real.

3/ People overestimate the importance of 30%-50% heritability. This is the one optimistic conclusion, because it is easy to address. As the debate is narrowing down to the 30-50% band, some hereditarians are tempted to reframe this as a victory for their side, as if they had always been fighting against blank-slatists. But even if the end figure happens to be at the higher end of this band, that would still constitute a decisive victory for heredity centrists.

The question never was about whether or not genetic differences contribute to the spread of intellectual talent—they obviously do.

The question always was about the “interesting place” Paul Graham talked about, the meaningful space between genetic potential and actual achievement, and whether or not it really existed. And, at 30% or 50%, this place surely exists.

It is large, wild, and messy. Some regions are accessible to nurture and some are not. Some are entirely governed by chance. But it is open for mining, and there’s plenty of room for human agency.

Andrew Wiles became an instant superstar on 23 June 1993, when he announced a proof of Fermat’s Last Theorem in front of a packed conference room at the Isaac Newton Institute for Mathematical Sciences in Cambridge. But during that summer, his Princeton colleague Nick Katz spotted a gap in the proof and, by September 1993, it had become clear that there was no easy fix.

Just imagine. You’ve spent the past seven years working secretly on a three-hundred-year-old mathematical problem, one of the hardest in human history. You’re not a ‘fake it till you make it’ grifter. You’re a serious, genuine, respected, high-integrity professional.

How would you react?

It’s hard to imagine all the emotions he must have gone through. But this later interview offers a glimpse. With a teary voice, Wiles recalls the “most important moment of [his] working life”, when he had this “incredible revelation” on 19 September 1994 and finally found a fix.

Wiles did what any experienced mathematician would do in such circumstances: he slept on it.

It is a well-known fact, abundantly documented in mathematical folklore, that new proofs have a pretty high mortality rate in the first few days. But Wiles’ “incredible revelation” did survive the night:

I checked through it again the next morning and, by 11 o’clock, I was satisfied. I went down and told my wife: “I’ve got it, I think I’ve got it! I’ve found it!” It was so unexpected, I think she thought I was talking about a children’s toy or something. She said “Got what?” and I said “I’ve fixed my proof, I’ve got it!”

But why, in the first place, is it possible to “fix” a proof? Isn’t a proof supposed to be either valid or invalid? What does it mean for a proof to be broken?

To see the puzzle more clearly, remember that mathematics is supposed to be a logical endeavor and, from a formal logic standpoint, there isn’t much difference between a proof, a computation, or a theorem. A proof is nothing but a peculiar type of theorem—one that asserts that these specific steps constitute a valid logical derivation of statement X.

When rephrased in terms of theorems, the questions become mesmerizing:

Why, in the first place, is it possible to “fix” a theorem? Isn’t a theorem supposed to be either true or false? What does it mean for a theorem to be broken?

Wiles’s example may not feel like a big deal, because his proof was “fixed” long before it was published. But there are instances where broken proofs and broken theorems have slipped into the mathematical corpus and remained there for decades without anyone noticing.

My personal favorite is a 1961 “theorem” in homological algebra that ended up being “disproved” in 2002 article by Amnon Neeman, an Australian mathematician.

Aussie mathematical humor is a niche genre that rarely breaks into the mainstream, but Neeman’s deadpan title has a shot at eternal glory: “A counterexample to a 1961 ‘theorem’ in homological algebra.” The opening paragraph is in the same vein:

Abstract. In 1961, Jan-Erik Roos published a “theorem”, which says that in an [AB4∗] abelian category, lim^1 vanishes on Mittag–Leffler sequences. See Propositions 1 and 5 in [4]. This is a “theorem” that many people since have known and used. In this article, we outline a counterexample.

Now *this* is a pretty big deal, because of a fundamental theorem in formal logic that asserts, basically, that any mathematical theory containing a single contradiction instantly disintegrates into a giant atomic cloud of nonsense, where every statement is both true and false.

According to Google Scholar, Jan-Erik Roos’s 1961 paper was cited 231 times (as of this morning), by articles that have themselves been cited thousands of times, by articles that have themselves been cited a gazillion times, and so forth...

So why is it that Amnon Neeman’s paper didn’t annihilate mathematics?

My take is that this paradox encapsulates a profound aspect of mathematics that has been overlooked for centuries by philosophers and logicians alike and—once properly accounted for—provides a compelling evidence in favor of the dissenting “conceptualist” view on the foundations of mathematics.

Before I articulate the case, let me be clear that this example from homological algebra, albeit extreme, represents a general pattern: published mathematical research is riddled with errors.

My friend Raphaël Rouquier—an outstanding mathematician who was awarded several serious prizes and is a professor at UCLA—maintains an Errata summary that currently references sixteen papers, about 25% of his published research. I am not singling him out because he is sloppy, but because he is ego-less and I know he won’t be offended.

As for the great Terry Tao, he authored four “Erratum” papers correcting four of his earlier works. This is the mathematical equivalent of a security patch—when an error is so significant that a dedicated article must be urgently published to set the record straight.

The mathematical community has been aware of these issues for ages and, in fact, the entire field of mathematics exists as an attempt to get rid of them. This is why proofs were invented. This is why formal logic was invented. This is why research papers are long and tedious. This is why academic journals take ages to get them refereed, accepted, and published.

The current incarnation of this millennial quest is the titanic effort to formalize mathematics, that is, to re-implement existing theorems and proofs within formal “proof assistants” such as Lean. Indeed, while extant mathematics may seem formal to outsiders, this isn’t really the case—it is merely written in a “mock” formal language that mixes abstract symbols with handwaving, and it is a running joke that most proofs are full of gaps. That wouldn’t eliminate the need for human-readable proofs (that cannot span gigabytes), but it would ensure that they are backed by machine-verified formal derivations.

Kevin Buzzard and his team at Imperial College have embarked on a multi-year project to formalize the proof of Fermat’s Last Theorem and, interestingly, they have already been confronted with an “Amnon Neeman moment” of their own.

Kevin reflects on this in a fascinating blog post that goes right to the heart of our topic. Rather than following Wiles’s original reasoning, they adopted a broader approach relying on an abstract machinery called crystalline cohomology, introduced by Alexander Grothendieck and developed in the 1970s by his student Pierre Berthelot.

In the final months of 2024, as Kevin and his team had just set off on this project, they received a message from a French mathematician, Antoine Chambert-Loir, who had noticed some strange behaviors with Lean—it was constantly complaining about the standard human argument behind one key lemma of the theory.

As Antoine later found out, this proof was actually broken. Here’s how Kevin frames it:

According to the way I currently view mathematics (as a formalist), the entire theory of crystalline cohomology vanished from the literature at the moment Antoine discovered the issue, with massive collateral damage (for example huge chunks of Scholze’s work just disappeared, entire books and papers vaporised etc).

But, again, the asteroid ended up not hitting the Earth and not wiping out the dinosaurs: they found a fix!

What is going on here? Why is it never a disaster? What makes mathematicians so incredibly lucky?

Kevin explains it later in his post, revealing along the way that he isn’t a true formalist. True formalism—the technical term is ontological formalism—posits that mathematics is just a meaningless game of meaningless symbols, detached from any semantics humans might project onto it.

For a true formalism, there is no such thing as “fixing” a proof, as the whole edifice shatters into unsalvageable pieces at the exact moment a contradiction is introduced. I have yet to come across a career mathematician who really thinks that.

Kevin is more likely to be a methodological formalist, someone who thinks that computer-aided formalization is the correct framework for implementing mathematical proofs—a position I’m aligned with.

This doesn’t prevent him from attributing meaning to mathematics. In the midst of the crisis, as crystalline cohomology had been cleansed out of formalized mathematics, it continued to exist in Kevin’s brain:

The thing I want to stress is that it was absolutely clear to both me and Antoine that the proofs of the main results were of course going to be fixable, even if an intermediate lemma was false, because crystalline cohomology has been used so much since the 1970s that if there were a problem with it, it would have come to light a long time ago.

This illustrates the plight of the working mathematician, an expression coined by Reuben Hersh and discussed in my earlier piece on conceptualism: the philosophical debate on the foundations of mathematics has been hijacked by an artificial opposition between Platonism and formalism while, in fact, practical mathematics has to borrow from both sides.

Some career mathematicians self-identify as Platonists. Some don’t. But all of them are methodological Platonists, in the sense that no-one can do math without attaching intuitive meaning to the abstract symbols in front of them. You have to actively imagine that the symbols mean something, that they refer to actual “objects”, that these objects do “exist”, somewhere, somehow, in a certain way.

In essence, conceptualism is the synthesis of methodological Platonism with methodological formalism. It posits that mathematics is a human activity that relies on a formal game of “meaningless” logical deduction, as a means of strengthening and solidifying our ability to build new concepts and make intuitive sense of them.

It’s very hard to argue against ontological Platonism, because the only nuance is whether or not these imaginary cognitive objects “really” exist, in the ethereal realm of ideas—which by definition we can’t access.

A pop math trope asserts that ontological formalism was defeated by Kurt Gödel, which I’ve always found unconvincing. Sure, incompleteness was terrible news for Hilbert’s program—but terrible news in itself isn’t supposed to have ontological consequences.

Personally, I only started to view ontological formalism as untenable after I observed the failure modes of mathematics.

Meaningless formal systems should break down like Prince Rupert’s drops, these toughened glass beads that are unbelievably resistant, until you snap them at the right place and they explode into powder.

By contrast, real-world mathematics breaks down like pottery. As a whole, the mathematical corpus is a giant, collaborative work of kintsugi, the Japanese art of pottery-fixing. The whole thing started off as unstructured clay and was made hard by our very hand.

“Fixing” a theorem or a proof isn’t about building entirely new ones. It is about filling small gaps in arguments, or replacing technical lemmas that happen to be false, or replacing a theorem with a similar one that is mildly more restrictive. All these notions—small gaps, technical lemmas, similar theorems with mildly more restrictive conditions—are impossible to make sense of outside of the meaning layer that humans project onto formal systems. Fracture lines don’t propagate too far because they tend to be meaningless. Mathematicians even have a word for this—a statement might be false, yet morally true.

While the formal layer is how mathematics is expressed, the meaning layer is where the real action is taking place—and what holds it together at large scale.

My dear Serre,
Thanks for the various papers you’ve so generously sent me, as well as for your letter. Nothing new here. I’ve finished my ridiculous piece on homological algebra (but it’s the only way I have of understanding, through sheer persistence, how things work)…

As Kevin Frey and I sat together to resolve outstanding issues in the draft of the English language edition of Mathematica, we spent twenty minutes debating this tiny passage from a 1956 letter from Alexander Grothendieck to Jean-Pierre Serre.1

We ended up agreeing on “ridiculous piece”, although we were perfectly aware that it failed to capture the full expressivity of “emmerdante rédaction”, the original phrase used by Grothendieck. Here’s how Claude puts it:

The phrase “emmerdante rédaction” is particularly colloquial - using a somewhat vulgar term to describe the writing as annoying or tedious, which gives insight into the informal, friendly relationship between the correspondents.

The dismissive language is stunning. Here was Grothendieck, arguably the most influential mathematician of the past century, disparaging his own Tôhoku paper, a groundbreaking work that was about to revolutionize homological algebra.

I used this quote as the opening to Chapter 7, which discusses the specificities of mathematical writing through the lens of Grothendieck’s philosophy of discovery. To Grothendieck, the essence and value of mathematics cannot be captured in writing. It doesn’t lie in formal logic or cryptic formulas (“it’s not in this form that you find the soul of understanding mathematical things”), but in the inner experience of making sense of them:

What drives and dominates my work, its soul and reason for being, are the mental images formed during the course of the work to apprehend the reality of mathematical things.

Yet, in an apparent paradox, Grothendieck was an excruciatingly careful writer, obsessed with details and exhaustivity. He spent the last decades of his life in complete isolation, producing 70,000 pages of notes on everything — his mathematical ideas, his dreams, the entire universe... He even wrote a 10-page treatise on kimchi:

Kimchi Grothendieck

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The paradox is resolved once you realize that Grothendieck viewed writing, and specifically writing within the constraints of mathematical formalism and logical consistency, as an apparatus for “gathering intangible mists from out an apparent void” and solidifying them into stable mental images.

Writing isn’t optional — it is central to the process. In Grothendieck’s own words:

The role of writing is not to record the results of research, but is the process itself of research.

or, circling back to his 1956 letter:

it’s the only way I have of understanding, through sheer persistence, how things work.

Short form vs long form vs excruciating form

How much of this applies outside of mathematics? Most of it, if not all — but with a fundamental caveat that has become more and more apparent to me over the past few months.

Writing math is a remarkably well-defined activity and, if you have practiced it at any level, you will certainly have a feel for what Grothendieck is alluding to. Mathematical formalism is monstrously difficult to manipulate, to the point where it feels like torture. Yet its rigidity is unavoidable, as mathematical intuition can only develop through is confrontation with cold and hard logic.

In fact, forcing yourself to write things down rigorously, especially when you’re still confused, is an insanely effective life-hack routinely practiced by top mathematicians.

But outside of mathematics, writing means many different things.

My oldest son is entering primary school and learning to write. At his level, I can see that the simple act of forming a few letters requires intense focus.

For adults, we usually reserve the word writing for activities that go beyond the mere act of forming letters — or typing on a keyboard — and involve crafting original sentences and paragraphs. But even this definition remains loose, as I was recently reminded at my own expense.

After the English language edition of Mathematica was published, I became much more active on X. For all its flaws, I loved the reach and diversity, and it forced me to get better at the art of the aphorism.

Once you practice X for a while, especially if you experiment with viral content, the limitations of short form become pretty obvious. I started this Substack out of this frustration, as there were many important topics adjacent to my book that couldn’t be reduced to X threads.

If you had asked me six months ago, I would have told you that writing for Substack felt more similar to writing books than to writing for X. I’ve changed my mind on this.

After taking a three-month hiatus from Substack to complete the first draft version of my next book, I am amazed at how difficult it was. If X posts are short form and Substack is long form, then we need a different category for books — excruciating form.

To me, the experience of writing books feels very similar to writing math. This is probably because I operate with a similar mindset: starting with something that is fuzzy in my head, but feels important, and trying to “nail” it in a way that will withstand the test of time.

Math comes with a pre-existing framework that guarantees long-term validity (provided that ZFC is consistent — so far so good 😅). This is why writing math is always excruciating and always transformative, regardless of the format: it forces you to create new mental categories and stabilize their meaning in your head.

Outside of math, there is no pre-existing standard for quality writing. But you can create a makeshift standard by raising the stakes around your own production:

• Will it make sense to people who know nothing about you?

• Can it survive translation? Can it be read in Japan and Turkey, in Texas and India, in Kenya and Bulgaria?

• Can gen Z offer it to their boomer grandparents, and vice-versa?

I don’t claim to have a clean methodology to answer these questions, and I am far from certain that my next book will pass the cut (at this point, I’ve identified several issues that need to be fixed).

But in the process of asking myself these questions, I did notice that they acted a bit like mathematical formalism: they were external constraints forcing me to be clearer and more articulate and, in the end, this reshaped my worldview.

There is something unique in the book publication process, in its archaism and solemnity, and especially in the fact that there is no way to “hot-fix” it once it is in press, no way to “quick-deploy” a follow-up piece that will respond to the objections prompted by the first one.

I’m a deep admirer of writers like Scott Alexander and Gwern who manage to thrive in long form while entirely skipping excruciating form.2 But, personally, I haven’t yet found a way of elevating my thought without resorting to it from time to time. This might be due to differences in personality, in cognitive style, or in the nature of what we’re trying to create.

To me, the main practical difference between long form and excruciating form comes down to this:

• When writing a Substack post, I know beforehand what the post is about and what I want to say.

• When writing a book, I have an intense but confused perception of what I want to say, but I’m initially incapable of articulating it, and the months-long process of trying and failing gradually makes me capable of writing it.

Here’s a more cynical way of looking at it: I can see how Claude 6 or ChatGPT-7 could be leveraged to fast-track the writing of this piece, but if I hadn’t written Mathematica in the first place, I would have no clue how to prompt for it.

Over the past few months, as I unplugged from X and Substack to focus on my next book, I was struck by how difficult it was for me — cognitively, emotionally, physically. It was much harder than anything else, much harder than I expected. I felt stupid and miserable the entire time, and I hated it. And yet — this ordeal changed me.

Whether you’re a reader or a writer, don’t give up yet on books. This is the No Free Lunch Theorem of cognitive development: at this point, we have no painless technology for collectively elevating our worldview.

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Alongside wealth inequality, math talent inequality is one of the most brutal forms of inequality on this planet. Like wealth inequality, it is brutal at every step of the ladder.

It’s like fractals. The 1% is insanely stronger than the 99%. The 0.1% is yet again insanely stronger. And yet again the 0.01%. The 0.0001% is a whole different species. I have friends in the 0.000001% and they scare the hell out of me. And then there’s Terry Tao.

When Usain Bolt set his 9.58 world record in the 100m dash, he ran 1.5% faster than the silver medalist, and only 3% faster than the last-place finisher. Math talent inequality feels radically different from that.

When I discussed the issue with Hugo Duminil-Copin, the 2022 Fields medalist, he shared his perception that Terry Tao is at least 10x faster than he is when it comes to processing new mathematical concepts and making sense of them.

Hugo’s view is far from marginal. In fact, most career mathematicians I know have similar perceptions of the talent gap.

Like all creative endeavors, mathematical research builds on passion, drive, luck, and the willingness to venture deep into uncharted territory. Yet it does place cruel and unusual demands on the brain, and some people do seem to possess a “natural” advantage.

How do you explain John von Neumann who, at age six, could divide eight-digit numbers in his head?1

How do you explain Gauss, born to poor and illiterate parents, who proved at age 19 that regular heptagons aren’t constructible with compass and straightedge, a problem that had resisted the smartest people for two millennia?

How do you explain Terry Tao? How do you explain Ramanujan? How do you explain Galois? How do you explain Grothendieck?

How do you explain, on the other hand, the millions of kids who struggle with elementary math?

To anyone who has engaged with advanced mathematics, or to anyone familiar with its history, it is obvious that social determinism can only explain a fraction of what is actually taking place. This is why so many people instinctively default to the opposite explanation: innate ability.

Yet, as we’ll see, the jarring magnitude of math talent inequality is a compelling evidence against genetic determinism.

The uncalibrated world of genius

Before going any further, I should clarify the scope and semantics of my assertion.

I am obviously NOT claiming that genes play no role in this story, as there is massive scientific evidence of the contrary. Human cognition emerges from the human brain, a physical organ whose capabilities are clearly affected by genetic variability. There are known genetic conditions that cause severe learning disabilities and this is just the extreme end of a spectrum.

My point is that genetics alone cannot “explain” genius, at least not in the casual sense most people give to these words.

Wiktionary defines talent as “marked natural ability skill.” When this marked ability becomes extraordinary, we enter the territory of genius:

I am perfectly aware that these are fuzzy notions and I won’t attempt to fix this.

Neither talent nor genius are well-defined scientific concepts, yet they are pragmatic constructs that capture an essential feature of our daily experience: it does feel that “intelligence” and “creativity”—whatever these words mean—are distributed in a markedly uneven manner, and that a few people are quite extraordinarily endowed.

I ran a quick unscientific experiment on how my audience perceives the intelligence gap:

Please give your gut-feel answer, without thinking too much: compared to a “normal person”, how much smarter was Einstein?

Out of 585 votes, 14.5% went to “20x smarter”, and a full 25% of respondents selected the top option, “on a different planet”.2

This framing is ridiculously over-simplistic, as there is no calibrated scale of intelligence that adequately captures the uniqueness of people like Einstein. IQ, which was designed as a low-cost instrument for spotting learning disabilities, is notoriously uncalibrated at the top.

Yet, as we saw with the example of Hugo and Terry, there are indirect ways of probing the magnitude of the talent gap, and they do point toward massive differences. In math and related domains, genius does feel like extreme wealth, with a distribution that resembles a Pareto distribution.

This is the phenomenon we call genius. In our experience, the cognitive talent gap feels monstrous, abysmal, orders of magnitude wider than typical within-species variability.

Mathematical genius is especially striking as it often manifests itself from an early age, outside of any obvious social explanatory pattern. I completely get why most people would presume that it is innate—I used to think that way too.

This is the heuristic behind the genius myth: faced with extraordinary talent, people look for an extraordinary explanation, and the only one they can think of is extraordinary genetics.

This heuristic is at play in Hans Bethe’s famous characterization of John von Neumann (Bethe was the 1967 physics Nobel prize laureate, and he had worked alongside von Neumann on the Manhattan project):

I always thought von Neumann’s brain indicated that he was from another species, an evolution beyond man.

Yet this heuristic is mathematically wrong: when you are faced with Pareto-like, species-redefining inequalities, the correct heuristic is to assume that they are NOT primarily caused by genetic variability.

While genetic variability is certainly a contributor to cognitive inequality, it cannot be the primary driver of the genius phenomenon, at least not in the near-deterministic manner envisioned by Bethe.

The Gaussian world of polygenic heritability

The correct framework for analyzing these heuristics is the notion of heritability, a statistical measure of the fraction of the variance of a trait that is associated with genetic variability.

Real-world genetics never quite follows the textbook pattern of Mendelian genetics, where one single gene deterministically controls a specific trait. Even eye color, often presented as a prototypal example, is influenced by a multiplicity of genes.

If there was a “math gene”, a single gene with a paramount influence on math talent, it would have been located decades ago.

In fact, the norm is rather the opposite: all high-level human traits exhibit a certain degree of heritability, but they are almost always highly-polygenic, that is, under the influence of a great number of genes whose individual contribution is always tiny. For example, despite hundreds of studies involving millions of participants, no single genes has even been found to cause 1% or more of the population variance of any complex human behavorial trait.

Wealth is a good example of a highly polygenic trait with low heritability. While there are massive non-genetic factors influencing one’s net worth, it is also true that it helps to be smart, good-looking, and generally healthy. These factors are influenced by our entire genome, through fabulously complex pathways.

Height is the prototypical example of highly-heritable highly-polygenic trait, where genome-wide studies have identified hundreds of contributing loci and heritability is estimated to be close to 80%.

Now there’s something striking about the distribution of height: it looks like a bell curve, aka a Gaussian or a normal distribution.

In fact, the self-reported body height of US men would fit a near-perfect Gaussian if it wasn’t for the cheaters claiming to be 6’, when they’re only 5’11.

The explanation is simple. Gaussians typically occur when you model the cumulative impact of large numbers of independent coin flips, which is precisely what you’d expect to see with a highly-heritable highly-polygenic trait:

• under a linear model of genome expression, the genetic contribution is obtained by adding the contributions of each individual genes, which behave like independent coin flips;

• this results in a polygenic score that obeys a Gaussian law;

• if the heritability is high, one would expect this Gaussian input signal to remain prevalent in the distribution of phenotypic outcomes.

Here’s a simulation of a Gaussian polygenic trait with 80% heritability:

Note that there are no massive outliers at the top of the phenotype distribution.

In other words, the scatterplot doesn’t exhibit any genius phenomenon. It mathematically can’t. In the Gaussian world of highly-heritable polygenic traits, there is no room for Pareto outcomes.

Like most hereditarian fallacies, the genius myth stems from our misguided bias toward deterministic “explanations”, in the face of phenomena whose nature is inherently stochastic.

Bethe’s quote is typical of this deterministic mindset: when he suggests that von Neumann might be “from another species”, he isn’t alluding to 80% heritability, but rather to 99% heritability or more, something akin to genetic determinism: von Neumann had a unique talent because he had a unique genetic make-up.

This contrasts with the above scatterplot, where the “most talented” individual isn’t even the one with the highest polygenic score. Contrary to a common misconception, 80% heritability still is very far from genetic determinism.

But the higher you go with polygenic heritability, the closer you get to a true Gaussian distribution of outcomes, and the harder it gets to account for the extraordinary spread of cognitive talent, the very phenomenon that made you look for a genetic explanation.

In the end, this is a shape mismatch: you can’t fit Terry Tao under a bell curve.

Smells like Yule process

By contrast with Gaussian distributions, Pareto distributions and power laws typically emerge from sequential drawing processes where each step builds on prior results.

The standard mechanism3 is called preferential attachment process, aka Yule process, aka “rich-get-richer” scheme:

• start with a given population and hand out $1 to everyone

• at each step, randomly allocate an additional dollar to one person in the population, where everyone’s chance to win is proportional to how much they already have,

• when repeated indefinitely, this process converges to a Pareto distribution.

Vilfredo Pareto first identified power laws in the context of wealth inequality, but these statistical distributions are ubiquitous in nature and society, whether you model the number of likes on social media posts, the frequency of words and first names, the population of cities, the magnitude of solar flares, or the mass of asteroids. And they always reflect an underlying preferential attachment phenomenon.

Conversely, I don’t know of any quantified trait that follows a Pareto-like distribution and has a well-documented high heritability (say, in the 70%-100% range)—and I’m still waiting for anyone to propose a biologically credible mechanism that yields Pareto-like phenotypes while maintaining high heritability (see the Q&A below for more on this.)

Where do we go from here?

If your prior is that cognitive talent is mostly a matter of genes, it’s unlikely that my argument will 100% convince you. But I do hope that it will help open your eyes.

This is what it did for me.

In my early 20s, I was a struggling pure math PhD student, trapped in a negative feedback loop. I was firmly convinced that I was gifted, but not gifted enough for research-level algebra and geometry, and this was making me quite depressed.

Meanwhile, some of my peers were experiencing unbelievable success dynamics.

The intuition that the two aspects might be connected—the obscene talent gap and the striking feedback mechanisms—prompted me to update my hereditarian priors. It led me to work specifically on rebuilding my self-confidence and place long-term strategic bets on my cognitive development.

When you experience the full journey from struggling PhD student (the cursed way of belonging to the top 0.1%) to proving a conjecture that is older than you (a 0.0001% or 0.00001% event), you acquire a unique perspective on cognitive inequality. To me (and to me only), the simple fact that my position on the math talent ladder changed so much over the years is a compelling proof that this position wasn’t inscribed in my genes.

But I had no way of knowing that at 25.

In retrospect, this intuition that genius isn’t an essence but a state, the outcome of a trajectory, proved foundational to me. It helped me look at the right places and served as a gateway to life-altering insights on how mathematics actually functions, why top mathematicians think in such bizarre ways, and what I could learn from them.

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Questions and answers

This post builds on an earlier X thread that went viral and sparked thousands of reactions. I thank everyone who commented on it, enabling me to map out the most frequent questions and objections:

Is this an actual mathematical proof that genius can’t be genetic?

Of course not. It can’t be, since genius isn’t a well-defined notion.

However, it does qualify as a strong heuristic argument, based on solid math. It belongs to the broader category of morphogenetic arguments which, as a mathematician, I tend to find pretty compelling. It reflects how us mathematicians look at the world:

• When we see that soap bubbles are round and honeycombs are hexagonal, we think: “OK, these are minimal surfaces.”

• When we see elliptic orbits around the Sun, with Kepler laws, we think: “OK, there must be a differential equation that yields exactly that.”

• When we see a bell curve, we think: “OK, these are independent coin flips."

• And when we see a Pareto distribution, we think: “OK, there’s no way this is caused by genetics.”

Does this contradict the scientific consensus on the heritability of IQ?

Not quite. I’m discussing extreme cognitive talent which, as mentioned above, lies on an open-ended scale that isn’t captured by IQ.

It is important to keep in mind that, while advanced math may feel like a perversely difficult IQ test, this is a massive oversimplification, and there is no reason to assume that the heritability of “math talent”, whatever it means, is strongly tied with that of IQ. As far as I know, there is zero direct evidence supporting the widespread belief that extreme math talent has a genetic origin.

Note also that, at the moment, there is no scientific consensus on a definite figure for the heritability of IQ. Twin models did suggest high figures, based on disputable modeling assumptions (see the next Q&A item below), but direct genomic studies have failed to confirm this and the missing heritability problem is far from resolved.

Alex Young is one of the leading experts on the topic. When I asked him for his gut feel, here’s what he replied:

My sense is that heritability of IQ is in the range of 30-70% with very high confidence.

My argument is entirely compatible with that.

You’re assuming that genome expression is linear, correct?

Correct. The argument that highly heritable highly polygenic traits can only produce Gaussian phenotypes is indeed based on this assumption.

Note however that the vast majority of the heritability literature makes the same exact assumption. It is known to be incorrect, yet it is often considered to be a reasonable first-order simplification when studying large-scale polygenic phenomena.

In particular, linearity is a core modeling assumption of twin studies, essential to their high estimates for the heritability of IQ. If you relax the assumption and update the twin study framework to account for a simple model of pathway-dependent genetic interaction, you end up with much lower figures.

I won’t go into the mathematical details, but my argument is quite robust to mild violations of the linearity assumption, as long as these violations remain “local”.4

While real-world genomic expression is never strictly linear, you’d have to come up with a pretty violent nonlinear expression mechanism if you want to produce Pareto-like outcomes. Upon doing that, you will have introduced a massive amount of non-genetic expression noise that will instantly eat away the heritability5 — that is, even before you realize that high-dimensional nonlinear systems are viciously unstable and stochastic chaos at the molecular level is probably incompatible with life.6

What if genius was 99% heritable but in a quasi-Mendelian way, through ultra-high-impact ultra-rare variants?

Why not, on paper. But the counter-argument only works if these variants are powerful enough to break down the Gaussian math of polygenic heritability, and 1/ there is zero evidence for such variants, 2/ you’d observe speciation-like events where “mutant progenitors” would yield “genius dynasties.”

You’re assuming that cognitive talent follows a Pareto distribution, which is questionable.

Correct. However, the argument doesn’t really need talent to follow a Pareto distribution. I framed it this way for the sake of the exposition, but what actually matters is the existence of orders of magnitude differences between individuals, like in my Hugo vs Terry example.

If you agree that there are abysmal gaps in cognitive ability between individuals, then there’s no point nitpicking about the exact shape of the distribution.

If you disagree, then you’re raising a profound and legitimate objection against the very concept of genius. However, this is beyond the scope of the current post, which is specifically concerned with the genius fallacy, the invalid heuristic that abysmal gaps in cognitive ability require abysmal gaps in genetic potential.

Height isn’t a true Gaussian and, in fact, it’s a bit tail-heavy.

Correct. Robert Wadlow, the tallest person in recorded history, was literally too tall to fit under a bell curve: at 2.72m (or, if you prefer, 8’11), he was 14 standard deviations above the mean.

Yet he was only about 60% above the average human height, which brings us back to the prior objection: this isn’t about true Gaussians vs true Pareto distributions, it’s about whether or not near-deterministic genetics can produce orders of magnitude inequalities between individuals of the same species.

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1

EDIT (November 2025): Removing this apocryphal anecdote about six year old Johnny, which I had cited from Wikipedia without checking the primary source. See Erik Hoel’s debunking of the legends around von Neumann:

The Intrinsic Perspective

John von Neumann Shot Lightning From His Arse

The movie Braveheart has a great scene where, by whispers and tales, the legend grows of the Scottish rebellion leader William Wallace (played by Mel Gibson in one of his best roles…

Read more

9 months ago · 117 likes · 56 comments · Erik Hoel

The paperback edition of Mathematica: A Secret World of Intuition and Curiosity will go on sale on May 20th in the US (and July 22nd in the UK).

I received my author copies today in the mail and wanted to celebrate this by giving away 5 signed copies. Then I realized that I didn’t know of any clean protocol to select the winners.

My first instinct was to make it a speed contest, selecting the first five people commenting on this post. But do I really want to reward social media addiction?

My second idea was a needs-based allocation, favoring students and people from countries where imported books are unaffordable. I thought about asking people to apply by commenting on this Substack post with a quick summary of who they are, what they do, and where they’re coming from.1

But am I really competent to judge who “needs the most”, or “deserves the most”, a free copy of my book? A possible fix would have been to make it a beauty contest, by selecting the most liked comments, but this felt even more unhealthy.2

I wrote my book for everyone, parents and teenagers, undergrads and PhD candidates, dilettantes and Fields medalists, preschool teachers and hedge fund managers, and I see no reason to exclude anyone.

This led me to my third attempt and final choice: a random draw, but a smart one!

Here’s the application form—if you’re simply interested in entering the contest, you can stop reading here. But if you’re curious about my ridiculously over-engineered protocol, you’ll find all the details below.

[JUNE 1st UPDATE: the contest is now over, see the results at the end of this post.]

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The rules

1. You can apply from any country where it’s possible to send books to. If you’re one of the five winners, I’ll ship your copy to the address of your choice. (I’ll take care of the shipping fees, but depending on your country you may have to pay some import duties.)

2. Please apply with only ONE email address. Multiple entries by the same person with multiple email address strings will result in elimination. (It’s OK to apply multiple times with the same exact email address, but this won’t affect your chances of winning.)

3. I’ll keep the form online until June 1st, after which I’ll select the winners using the protocol below.

The drawing protocol

I never liked public giveaways where “winners will be chosen at random”, especially when the stakes are low and no-one is monitoring the process, as they provide the perfect environment for cheating. Can you really trust the organizers? And even if they don’t intentionally cheat, can you trust them to understand what it means to pick winners “at random”, and not just “through a random method”?

I pictured myself drawing the winners “at random” and realized how hard it would be not to cheat, if only by re-drawing with a different seed if I didn’t like the first batch of “winners” (= if the list exhibited a weird pattern that conflicted with my confused sense of fairness.)

The clean solution is to commit beforehand to the drawing protocol, including any random seed that will be used in it, and make all this information public in advance. But for this to solve the trust issue, participants need to have reasonably good reasons to believe that the actual drawing did follow the published protocol.

This is the kind of problems that people typically aim to solve with the blockchain.

But I wanted to keep things simple, and thought it was a really bad excuse to launch my own coin.

The first solution I came up with was to pick the five participants with the lowest values for a computed field based on a cryptographic hash function like SHA256:

SHA256(email_address)

On surface, this looks like a fair protocol, but it has two problems.

First, it would have killed all the fun. Indeed, anyone can check the SHA256 of their email address. If your address is “alexander@grothendieck.biz”, then there’s no point entering a contest that you have virtually zero chance of winning:

>> SHA256("alexander@grothendieck.biz")
f964bbacea738d84b00d457110b88eabc13af4005484d72ce2ce4ff69533fcbf

Worse, the contest is extremely easy to hack: you simply have to create alternate addresses until you find one that drastically increases your chances:

>> SHA256("alexander1@grothendieck.biz")
0082860981eb275a1f715e17aa563992210614a7b9c62f1811f18b706d5da728

A basic protection against this is to rely on salted hashes:

SHA256(email_address + salt)

where the salt is a predefined string.

This is the protocol that I will use and, of course, I’ve already picked my salt.

I can’t disclose my salt in advance, as this would recreate all the flaws of the unsalted protocol, but I’ll reveal it after the drawing is done and, to allow you to verity this information, here is the SHA256 of my salt:

>> SHA256(salt)
6d2317c887b9a6c805bf8395c19042faee02bf495503ccde67d7b730272d4c44

Is this bullet-proof?

No.

Using rainbow tables or brute-force/dictionary attacks, there is a possibility that someone will invert the hash and crack my salt.

You can also try mind-reading.

My position on that is fairly simple: if you really want it that hard, then you really deserve a free copy of my book.

[JUNE 1st UPDATE — the contest is now over]

The salt was “196883”, the dimension of the smallest faithful complex representation of the monster.

>> SHA256("196883")
6d2317c887b9a6c805bf8395c19042faee02bf495503ccde67d7b730272d4c44

It would have been easy to choose a long random string. But I liked the idea of hiding an optional skill contest behind a chance contest, so I picked a “meaningful” short string which appears three times in the book (hence the “You can also try mind-reading”—the salt was amenable to both psychological insight and full-blown dictionary attack.)

There were 111 participants and only 2 of them, William and Lachezar, have implausibly “lucky” email addresses. Erika too may not have cracked the salt, but it’s less certain: her “luck” was a high but not implausible, her email address looks a bit weird but might be genuine, and she posted only a few hours before the end of the contest. Ahmet Cagri’s and Pietro’s email addresses look 100% genuine.

FIRSTNAME      SHA256(email_address + salt)
William        000000006e2c7c6d28fc583506f3ce...
Lachezar       00000000d6b4342927411d8ed46efc...
Erika          005ad209262eb41f8ae71b25851126...
Ahmet Cagri    01053152f43cb2a39902b51ed83920...
Pietro         01ebe3ed20cc1c711da7b1e3c40d98...

I didn’t know what to expect, but I like this outcome which exhibits a good balance of skill and luck. Had it turned into a pure hashrate battle, my plan was to allocate more books to the giveaway and split the contest into two separate rankings, one for “natural luck” and another for “hacker’s luck” — that wasn’t even necessary.

Thanks everyone for participating, and congrats to all the winners, the lucky ones and the hackers.

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You don’t really need to think. You can see the answer in your head: 999,999,999. The answer is easier to picture than it is to pronounce.

It seems obvious, and yet it wasn’t always like that. To someone living in ancient Rome, for example, it wouldn’t have been obvious at all.

In classical Latin, the word billion didn’t exist (neither did million). To communicate the idea, the easiest thing would have been to call it the product of “a thousand times a thousand times a thousand.” A Roman during the time of Julius Caesar should have been able to understand that, even if it might have given them a bit of a headache. But if you had told them that you were capable of taking this number, subtracting one from it, and picture the answer immediately in your head, they wouldn’t have been able to follow.

They would have taken you for some kind of math whiz.

You’d be hard pressed to write 999,999,999 in Roman numerals. If Roman numerals are the only numbering system you know, 999,999,999 is much more than a big number you don’t run into every day. It’s a number that you can’t even “look” at. It’s so terrifying that it makes your head spin. The idea that someone could instantly “see” it clearly and without any effort is absurd.

But there’s nothing extreme about the ancient Romans. Their understanding of numbers was really quite advanced. The traditional way of counting among certain aboriginal Australian people is based on parts of the body. You count from 1 to 5 on the fingers, then move up the arm: 6 is the wrist, 7 the forearm, 8 the elbow, 9 the biceps. When you get to 10 (the shoulder), you keep going up the body—12 is the earlobe. Yet if each number needs a corresponding body part, how do you get to a billion?

In the Amazon, Yanomami languages have an even more restricted numeral system: there’s a word for “one” and another for “two,” but there’s no word for “three,” just a catchall word that basically means “a lot.”

For someone who sees the world in this way, discovering that there’s a clear distinction between 25 and 26 that can be perceived in a split second must come as something of a revelation, comparable to what math students experience when they learn that there are many different sizes of infinity that can be precisely described.

A Complete Sham?

An inhabitant of ancient Rome would be able to grasp immediately the difference between XXV and XXVI. But your agility with big numbers would lead them to believe that you’re a math whiz. That idea makes you smile, because you know for certain that you’re no math whiz.

But are you sure about that?

If you think a math whiz is some kind of mutant with supernatural powers, if you think that they have some kind of computer in their head that lets them do calculations super quickly using the same methods that you know, then you’re wrong.

In the end, math whizzes are kind of like Santa Claus: they don’t really exist. When you think you’ve seen Santa, it’s never really Santa, just someone dressed up like him. When you think you’ve seen a magician, it’s never really a magician, it’s always an illusionist, someone who knows tricks that can create the illusion that they have magical powers.

And when you think you see a math whiz, it’s never really a math whiz, it’s always just someone who has a way of seeing numbers that turns calculations that you find complex and scary into something easy and even obvious.

The truth is that we’re all basically bad at mental calculation, except when we have an intuitive way of radically simplifying the calculation and “seeing” the result.

The decimal system based on Hindu-Arabic numerals is a “trick” that lets us see certain results as obvious. The main difference between a math whiz and you is that their bag of tricks is bigger than yours and they’re more used to playing with them.

Real Understanding

The decimal system of writing numbers seems so obvious to you that you can’t even remember learning it. It’s just like using a spoon. You use it without really thinking about it, like it’s an extension of your own body. When you see 999,999,999, you think you’re seeing the number directly, without realizing that you’re seeing it with the help of a tool.

Decimal writing is a purely human invention. More than simply a system of writing, it’s a door into a state of consciousness where whole numbers, however big they may be, become concrete and precise objects. At the same time, the infinitude of whole numbers becomes commonplace.

Something previously unimaginable suddenly becomes commonplace: this is exactly the type of effect mathematics produces in your brain. It’s a marvelous sensation, a great delight.

When you were a child, you were proud to be able to count to 10, then 20, then 100. It gave you bragging rights at recess. In order to brag some more, you would have wanted to know the biggest number.

To tell the truth, your awareness of numbers wasn’t that far off from those people who can count to 2 or 5 and are firmly convinced that the next number, the number many, is the biggest number.

One day, you realized that no number was the biggest. Even if you might have arrived at this conclusion some other way, decimal writing gave you a shortcut. You know that every number is followed by another. You know how to see the succession of numbers like a counter that turns, and you know that this counter can turn indefinitely. There’s no limit, there’s no special number after which the counter stops working.

Yet for 99 percent of human history, no one had been able to picture a number counter turning in their head.

The number counter turning in your head is the collective work of great mathematicians who, from prehistory until the Middle Ages, fashioned the image of numbers that we share today.

This image isn’t natural. It wasn’t inscribed in your body the day you were born. It’s partially arbitrary: we might have chosen another system for writing numbers, and you would see them differently.

More than four thousand years ago the Babylonians invented a sexagesimal system: they wrote their numbers in base 60 rather than base 10. Babylonian mathematicians were the most advanced of their time. Your mental image of hours, minutes, and seconds remains profoundly influenced by their vision of numbers.

What is natural, however, is your capacity to assimilate abstract mathematics and to really understand them, to modify your brain so that this math really becomes part of you.

You believe you can see the number 999,999,999. What you’re really doing is deciphering a complex and abstract mathematical notation. You decipher it instantly, fluently, without even realizing it. Whole numbers may not be your mother tongue, but you’ve become bilingual.

Successful math becomes so intuitive that it no longer feels like math. If the example seems stupid to you, it’s precisely because you understand it at the deepest level.

Real Magic Doesn’t Exist

At the start of their careers, young mathematicians often feel like imposters.

It’s a feeling I know well, and in my case it seemed entirely justified. The results contained in my PhD thesis were so obvious that it was almost like a trick. My theorems were always simple, and their proofs never contained any real difficulties.

Everyone around me seemed to be better at math. They were working on profound stuff that was way out of my league. They were writing papers that were extremely difficult to read, with proofs that seemed incredibly complex and technical. If I managed to understand a few of them, it’s only because they happened to be easier than usual.

I wanted to know how to do real math, difficult math. But all that I was able to learn was the easy math, the math for dummies.

It seems silly to say this, but it really took me years to realize it was only an optical illusion. The horizon was shifting with me. It was always staying at my level.

Real magic doesn’t exist. When you learn a magic trick, it ceases being magical. That may be sad, but you’d better get used to it.

If you find that the math you do understand is too easy, it’s not because it’s easy, it’s because you understand it.

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Dear David,

Hello again!

We have now received three reader reports for the English language edition of Mathematica/Secret Math. I am attaching all three for your consideration.

The first report is what I would characterize as ambivalent/not very helpful. I think that reader wants a different kind of book and does not consider himself a target reader for it. The second and third reports are supportive, and the two readers “get” what your goals are for the book.

I opened this email from my editor, Jean, and my heart sank. Something was wrong but I couldn’t tell what.

A report that was “ambivalent/not very helpful”? What could it possibly mean?

I jumped to the attachments and found the first reader report. It took me about 10 seconds to scan through it. The report wasn’t “ambivalent/not very helpful”. It was nuclear.

The worst part was this horrible passage in the concluding paragraph:

I am struggling to think who would want to read this book […] Should Yale University Press publish this? That’s a tough call for me.

In reader report parlance, the passage is far less ambivalent than it purports to be. It functions as a “kiss of death” that annihilates all prior positive words from the reviewer.

It’s like a reference call at the end of a hiring process, where a former employer would say something like this: “David has many qualities and there are many reasons to like him. But I’m struggling to think who would want to work with him. Should you hire him? That’s a tough call for me.”

Long story short, we overcame this stressful moment. Jean was persistent enough and luck was on our side. Had one of the other two reports been slightly less enthusiastic, it would have been a completely different scenario.

The book ended up being published and Yale University Press found out that there are quite a number of people who “want to read it”.

However, I’ve continued to think a lot about this report, which correctly anticipated the central themes of the online debate around my book.

This negative report is the perfect entry point to this Substack and to the themes that I will be discussing in the coming months.

I have no idea who this reviewer might be—Jean’s email only leaked that he’s a “he”—and I am not interested in finding out. I hold no grudge. Reader X, as I'll call this anonymous mathematician, was obviously sincere in his assessment. What’s most interesting is what he liked and disliked:

The central theme of this book is one with which I half-heartedly agree.

The agree part is this: The formal language in which mathematics is written is essential, but it obscures how mathematicians create their art.

Where I disagree with the author is his rejection of the role of innate talent. When I was younger I could play the piano, but I was no musician. I certainly practiced.

It’s a pity that the author takes an (my opinion) extreme view against any notion of innate ability.

It’s a pity because it dilutes the important message of this book: There are ways to think about mathematics based on intuition and those ways can (should!) be nurtured as part of mathematics education.

Reader X thinks that my criticism of hereditarian myths is a useless distraction from the valuable part of my project. I think the exact opposite: that his unscientific views on the origin of mathematical talent prevent him from seeing the bigger picture and understanding my core message about mathematics itself.

Yet Reader X is factually correct on one thing: the heritability debate is indeed a distraction from the important debate on the nature of mathematics.

I just think that this is unavoidable. The math genius myth is an essential component of our millennial misconceptions on mathematics and human cognition, and a major obstacle to their resolution.

This is why I chose to confront the problem head on and open my book with this famous Einstein quote:

I have no special talent. I am only passionately curious.

The quote serves as a leitmotif throughout the book, alongside strikingly similar quotes by René Descartes and Alexander Grothendieck. We collectively refuse to take these great geniuses seriously when they deny that their talent is innate, and this attitude has a major side-effect: we also refuse to learn from them.

A joke in bad taste

When Einstein talks about curiosity, he’s not picking a random word. He's not trying to be polite, or modest, or politically correct. He is simply attempting to articulate his unusual mental posture, a cognitive attitude that is extremely difficult to convey with words, but is nevertheless learnable and, to a certain extent, coachable—if only we care to listen.

Reader X thinks he knows better than Albert Einstein. He offers this counterargument:

My “counterexample” to this talent-doesn’t-matter assertion is Terry Tao; I offer this photograph of a ten-year-old Tao discussing mathematics with Paul Erdős. In my opinion, Tao is an innate mathematical genius. What other ten-year-old could pal around doing mathematics with Uncle Pál?

There is a double irony here. First, Terry Tao has publicly stated that he doesn’t believe that mathematical talent is a matter of genes:

Are there people who are naturally good at math and others who are hopelessly bad at math? I don’t think so. I think everyone has an innate mathematical talent.

Second, Terry ended up blurbing my book, which makes him a less-than-ideal counterexample to my ideas.

But Reader X was remarkably prescient in the sense that similar objections popped up in dozens of online conversations around the book. My interview with Quanta Magazine sparked an absurd shouting match on Hacker News with hundreds of comments focused on one single aspect of the interview: my claim that mathematics does not require an innate gift.

On Twitter/X, the most creative insults have been thrown at me. I’ve been called a “midtard,” an “insane flat-earther,” a “half commercial person half prophet selling his gene pool experience as a standard worldview.”

I was expecting such reactions, but I wasn’t expecting them to be so common and emotionally loaded. Some people apparently feel personally insulted by the very notion that mathematical genius isn’t innate.

Ironically again, this was anticipated by Alexander Grothendieck in his mathematical autobiography Harvests and Sowings, in a passage that was actually quoted in my book:

When you dare say such things, you see on everyone’s face, from the dullest who are sure they’re dull, to the smartest who are certain they’re smart and well above common mortals, the same smiles, part embarrassed, part knowing, as if someone had made a joke in bad taste.

A common pattern in the backlash is the notion that career mathematicians who challenge hereditarian myths must be either delusional or indoctrinated.

For some reason, people imagine that we’ve spent our entire lives in the ivory tower. We weren’t born in “normal families”, we never went to “normal schools”, we never met “normal people”, and we never had to teach kids who were struggling with math.

They also assume that we are ignorant of the entire IQ heritability debate, and that we bluster out utter nonsense for the mere pleasure of being insulted on the internet.

I can’t speak for Terry Tao, but I would never venture into such contentious territory without serious evidence up my sleeve.

Returning to Reader X’s “counterexample”, here are a few comments:

First, I never denied the existence of a massive talent gap in mathematics. This is in fact the starting point of my book, as illustrated by this excerpt from Chapter 1:

A striking aspect of mathematics is the regular occurrence of incredibly talented individuals who appear out of nowhere and, from a young age, demonstrate abilities vastly superior to that of their peers. At the other end of the spectrum, many people struggle with high school math or even primary school math. In the absence of a better explanation, it is natural enough to attribute this extreme level of inequality to “innate” talent.

Second, I am not denying that this gap is partly attributable to genetics. Again from Chapter 1 of my book:

There’s no doubt that natural aptitude in math, like natural aptitude in any other physical activity, isn’t equally distributed among individuals.

What I am disputing is the supposedly genetic origin of mathematical talent, this mysterious ability that some kids do possess in spectacular amounts while many otherwise intelligent kids seem to be entirely lacking.

It’s not because extreme mathematical talent manifests itself from an early age that it is necessarily innate. I understand why one might be tempted to make that jump. But this indeed is a jump and, in my opinion, an invalid one.

The nuance between “precocious + unexplained” and “innate” is subtle yet essential. The culprit is our traditional Nature vs Nurture framework, which makes us presume that whatever cannot be explained by social determinism must necessarily be explained by genetic determinism.

This preconception is invalid, as captured by Turkheimer’s Third Law: “A substantial portion of the variation in complex human behavioral traits is not accounted for by the effects of genes or families.”

In this earlier post, I discussed the example of Bill Thurston, a true mathematical “genius” with precocious and seemingly “superhuman” capabilities for which there is well-documented explanation that isn’t genetic.

In fact, the existence of spectacular geniuses like Terry Tao and Bill Thurston is a strong argument AGAINST hereditarianism. Indeed, as argued in the first chapter in my book, the math talent gap is too extreme for genetics to be the primary driver:

Human beings do exhibit innate biological differences, but overall we are a fairly homogeneous species. People differ in height, muscular strength, cardiac output, and lung capacity, and part of this variability can be traced to genetic factors. Yet those differences never encompass multiple orders of magnitude.

To use a metaphor that we’ll develop in the next chapter, math is so unequal that it’s as if some people could run the one-hundred-meter dash in under a second, while the majority wouldn’t make it in a week. While it’s conceivable that some people may genetically be endowed with a neuronal metabolism that is more efficient and powerful, making them, let’s say, twice as capable in math, or why not ten times as capable in math, it’s hard to believe that genes alone could explain such an absurd level of inequality.

Indeed, highly-heritable polygenic traits like height follow a Gaussian distribution of outcomes, where people cluster around the mean, while mathematical talent resembles more a Pareto distribution. In layman’s terms, here is how to think about this argument:

• Despite persistent rumours of the contrary, almost all human beings measure the same exact height, 170cm, plus or minus 30%. Size does matter, but not to the point where different people have to use different doors, different stairs, different cars and different chairs.

• If mathematical talent resembled anything like this, no-one would be talking about it, and you wouldn’t have continued reading this post.

This isn’t a trivial point, but a statistical smoking gun. The stunningly uneven distribution of mathematical talent bears the unmistakable signature of a non-deterministic cumulative process, where one capitalizes on past successes and the initial genetic signal gets diluted in an ocean of noise.

This argument triggered thousands of objections on social media: What about IQ? What about twin studies? What about adoption studies? Are you making a heuristic argument or a rigorous proof? Are you a radical leftist, a wokester, a blank-slatist?

If you’re asking yourself the same questions, please make sure that you’re subscribed to this Substack, as I’ll address them in future posts.

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But, as Reader X rightfully pointed out, the heritability debate can become a distraction from the more important issue on the table: our profound misconceptions on the nature of mathematics.

An attempt to change the world

Despite his reservations, Reader X concedes that my book contains valuable insights on how mathematics should be presented to students and the general public:

There are ways to think about mathematics based on intuition and those ways can (should!) be nurtured as part of mathematics education.

Many of the ideas [presented in this book] should be read by K-12 and college mathematics instructors.

The repeated use of “should” is a striking indicator of cognitive dissonance:

• Reader X agrees that math is presented in a dysfunctional way,

• he agrees that “K-12 and college mathematics instructors” would benefit from reading my book,

• yet he assumes that no-one will want to read it.

This cognitive dissonance is a pretty common symptom among career mathematicians. It leads to a fatalist worldview where mathematicians feel perpetually misunderstood. They complain that mathematics is unfairly ridiculed as a sterile, anal retentive discipline—and they see no way out.

Alas, if only the public knew that math is ultimately about intuition and creativity! If only they could see its beauty! If only they could experience the rapturous joy of feeling math!

We mathematicians love to lament about the state of mathematical education, yet we’re also good at dodging the tough question: what if we were ultimately responsible for the disaster?

This is where hereditarian myths come back into the picture: they provide us with the perfect excuse. If “normal people” are genetically incapable of understanding math, what can we do about it?

I spent my entire mathematical career stuck in this dissonance and found it deeply depressing. I decided in my late 20s, when I was an assistant professor at Yale, that I wanted to do something about it.

I became obsessed with the idea of writing a popular book about math. I tried every year or so, and systematically failed. I was always giving up after a page or two. There was a profound mismatch between what I wanted to write, and what I was capable of writing.

Most popular math authors are driven by the same frustration. Quite often, they’re successful mathematicians who are fed up with the atrocious public image of math, and look for ways to repair the injustice. Beyond the actual mathematical content of their books, they’re also trying to share their inner experience of math, in an attempt to change the world.

Just look at these titles: “poetry”, “joy”, “love”—these are attempts to reposition math itself.

But here’s the problem. A century ago, Henri Poincaré wrote fabulous books on mathematical intuition. G.H. Hardy wrote his famous Mathematicians’s Apology. There’s even a 1930 book called The Enjoyment of Math, by Rademacher and Töplitz.

And none of this changed anything.

If all these great mathematicians failed to resolve the dissonance, then one needs to try an entirely different approach.

My opening came when I taught an introductory course on mathematics to humanities students at the École normale supérieure in Paris. It was a free form course, with no preset curriculum and an audience mostly consisting of philosophy students eager to learn about foundational aspects.

I knew a bit about these topics from my student years, but having to teach them in front of a class placed them in an entirely different perspective.

This was a deeply unsettling experience. The foundational myths of mathematics, which I had accepted when they were presented to me in my 20s, stopped making any sense at all. The philosophical debate around Platonism, formalism, and intuitionism appeared entirely artificial and meaningless.

I had never heard of Reuben Hersh and wasn’t aware that, back in the 1970s, he had faced the same exact issue in the same exact context, finding himself in the awkward position of having to teach the “foundations” of mathematics and coming to the conclusion that the standard material made no sense at all:

Here I’m teaching this course to these poor students and I get halfway or more through it and I tell them: “That’s it. We mathematicians failed. All of us.”

And now what? That’s terrible! I was really embarrassed… You don’t want to appear before innocent hopeful students and tell them that’s all you can say. No, they want more, they’re entitled to more…

“Russell said this, Hilbert said that, Brouwer said this other thing, and they were all mistaken.”

What kind of answer is that? It’s really pathetic. I’ve been worrying about it ever since.

This moment came at a turning point in my career. I was in my early 30s and had just proved a substantial conjecture in my domain, which boosted my mathematical self-confidence. Had I taught this course a few years earlier, I would probably have concluded that I was the one not getting it right.

Mathematica has its roots in my personal notes for this experimental course. Back then, I knew what was broken, but I wasn’t yet capable of fixing it.

My ideas continued to mature for another 15 years, yet the fundamental intuition traces back to this primal ahah moment: what is to me the most important aspect of mathematics, what makes it so unique and enjoyable and what enabled me to become a successful mathematician, is something that no-one ever told me about, and that I stumbled upon entirely by chance.

The project crystallized in the summer of 2020, when I made an accidental discovery. As I was re-reading Descartes in parallel with Grothendieck’s Harvests and Sowings and Thurston’s famous essay On Proof and Progress in Mathematics, it struck me that the three of them, coming from entirely different backgrounds, were basically telling the same exact story—a story that was resonating with mine. Each one of them, with his own language and idiosyncrasies, was trying to articulate the peculiar cognitive attitude that is unique to mathematics, a special way to playing with one’s imagination.

While high-brow philosophy has never managed to produce a compelling account of the true nature of mathematics, there has been a remarkably consistent consensus among mathematicians as to what it means to do math, what it feels like and how one gets better at it.

A good way of framing this is by distinguishing “official math” from “secret math”:

• “Official math” is the aspect of mathematics that is traditionally taught in school: the rigid definitions, the logical reasonings, the abstract symbols on the page. It has all the attribute math is famous for: it is cold, objective, rigorous, obscure.

• “Secret math” is about the unseen actions that one performs in one’s head, the mental images through which one attempts to make sense of the abstract symbols. “Secret math” was never part of the curriculum, because it wasn’t taken seriously. It lacks all the attributes that one expects from math: it is naive, fuzzy, subjective, hard to articulate and hard to replicate.

• Yet career mathematicians know that “secret math” is where the real action takes place. In private, they freely discuss their intuitions, explain how they “see” things, admit what puzzles them and their naive attempts at making sense of the many things that they don’t yet understand.

“Secret math” is a martial art that relies on “official math” as equipment. Logic and formalism aren’t tools for thinking, but tools for finding out where we’re thinking wrong and gradually honing our intuitions.

This is where the two threads merge together: the hereditarian myths around mathematical talent take their roots in our profound and ancient misconceptions on the nature of mathematics.

Once you realize that mathematicians rely on mental techniques that are never even mentioned at school, it’s no longer surprising that math is uniquely hard to teach. But if you are firmly convinced, like Reader X, that mathematics requires a special innate talent, then any discussion of the inner experience of making sense of math will seem like a complete waste of time.

In his Rules for the direction of the Mind, Descartes has this crazy conspiracy theory that the ancient Greek philosophers/mathematicians intentionally concealed the “true” mathematics in an attempt to protect their prestige. If the secret of mathematics was disclosed to the general public, it would become readily apparent that mathematicians aren’t really that special:

I have come to think that these writers themselves, with a kind of pernicious cunning, later suppressed this mathematics as, notoriously, many inventors are known to have done where their own discoveries were concerned. They may have feared that their method, just because it was so easy and simple, would be depreciated if it were divulged; so to gain our admiration, they may have shown us, as the fruits of their method, some barren truths proved by clever arguments, instead of teaching us the method itself, which might have dispelled our admiration.

Mathematica1 is my attempt to document these secrets and “leak” them to the public. It is based on the accounts of Descartes, Grothendieck and Thurston, and also on my own experience as a mathematician.

From pre-school to PhD programs, all students of mathematics confront the limits of their cognitive abilities. Career mathematicians confront theirs every day. No-one is born with an innate understanding of high-level abstractions, and no-one is immune to the fear of not being smart enough. But there are techniques to engage with mathematics in a more enjoyable and more effective manner.

Knowing that my book was venturing into several ideological minefields, I chose to remain practical and down-to-Earth. My goal wasn’t to win debates with hereditarians or specialists of the philosophy of the mind, but to deliver concrete value to casual readers. I know that the book leaves many questions unanswered and many stones unturned, and will use this Substack for objection-handling and material that couldn’t find its place in the print edition.

In two prior posts, this one and this other one, I discussed the conceptualist approach to the philosophy of mathematics which, in my view, is the proper way to account for the “official math” vs “secret math” tension. This other post opened a thread on the heritability debate that I’ll pursue in the future.

Of course, I will also pursue the exploration of “secret math”.

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With the possible exception of a few lost tribes somewhere in the Amazon or Andaman Islands, career mathematicians are the last true animists in this world.

It’s no secret that something weird is going on with mathematical research. But when you experience it from the inside, it’s orders of magnitude weirder than anything you could have imagined.

Here’s how Srinivasa Ramanujan described his thought process:

While asleep, I had an unusual experience. There was a red screen formed by flowing blood, as it were. I was observing it. Suddenly a hand began to write on the screen. I became all attention. That hand wrote a number of elliptic integrals. They stuck to my mind. As soon as I woke up, I committed them to writing.1

Maybe Ramanujan doesn’t count. He was an outlier and his story may have been exaggerated. So let’s take another example. This influential research article, published in 1990 and cited over a thousand times, was coauthored by Bob Thomason and a ghost:

The first author must state that his coauthor and close friend, Tom Trobaugh, quite intelligent, singularly original, and inordinately generous, killed himself consequent to endogenous depression. Ninety-four days later, in my dream, Tom’s simulacrum remarked, “The direct limit characterization of perfect complexes shows that they extend, just as one extends a coherent sheaf.” Awaking with a start, I knew this idea had to be wrong… I had worked on this problem for 3 years, and saw this approach to be hopeless. But Tom's simulacrum had been so insistent, I knew he wouldn't let me sleep undisturbed until I had worked out the argument.2

OK, but what if Thomason doesn’t count either? What about the truly famous mathematicians, those who went beyond proving great theorems and redefined mathematics itself?

Alexander Grothendieck, often regarded as the greatest mathematician of the 20th century, wrote extensively about his obsession with dreams. He left an unpublished manuscript, The Key to Dreams, where he expressed his belief that God himself—whom he called “the Dreamer”—was dreaming inside of his head.

In Harvests and Sowings, his 1000 pages autobiographical essay, he insisted on the importance of dreaming and daydreaming in mathematical discovery. He attributed his unique creativity not to extraordinary intellectual abilities, but to his uncommon dreamlike approach, “gathering intangible mists from out of an apparent void”, which he cultivated in transgression of an ancient taboo:

It would seem that among all the natural sciences, it is only in mathematics that what I call ‘the dream’ or ‘the daydream’ is struck with an apparently absolute interdiction, more than two millennia old.3

But in all honesty, Grothendieck was also some kind of weirdo. Let’s try a more grounded mathematician, Bill Thurston, the 1982 Fields medallist who wrote several influential essays on the nature of mathematics. He too was an avid dreamer:

I have decided that daydreaming is not a bug but a feature.4

What is going on here? Are we observing the deliquescence of mathematics, a postmodern collapse fuelled by New Age and psychedelics?

What about classical mathematicians? What about the 17th century? What about René Descartes, the greatest mathematician of his time and the father of modern rationality?

His early biographer Adrien Baillet had access to a manuscript, now lost, where Descartes explained how his philosophy—what we call rationalism—was revealed to him directly by the Spirit of Truth. The event took place on the fateful night of November 10, 1619, when 23 year old Descartes fell asleep near an overheated stove and had a series of three dreams. After a first dream involving a melon, Descartes woke up and fell asleep again:

A new dream immediately came to him, in which he thought he heard a sharp and resounding noise, which he took for a clap of thunder. The fright he felt woke him up immediately; and having opened his eyes, he perceived many sparks of fire scattered throughout the room… The thunder, the clap of which he heard, was the signal of the Spirit of Truth descending upon him to possess him.5

Ramanujan, Thomason, Grothendieck, Thurston, Descartes: it looks like we have a pattern.

But what could drive serious mathematicians to view their dreams as a credible source of insight? Aren’t dreams supposed to be the opposite of rationality, the opposite of what mathematics relies upon?

As we will see, there is a way to resolve the apparent contradiction between the rationality of mathematics and the irrationality of mathematical creativity, but it will require us to reset our fundamental beliefs about reality, human cognition, and language.

The spiritualist worldview

In an earlier post, I argued that we’ve been wrong about math for 2300 years, operating under a flawed metaphysics that is incompatible with the practical aspects of doing math.

Our fantastic assumptions on the nature of math reflect our continued adherence to the spiritualist worldview, the system of beliefs that we inherited from our hunter-gatherer past.

The spiritualist worldview posits that there are two “layers” in reality, the physical world, inhabited by material objects, and the spiritual world, inhabited by spiritual entities, whichever way you want to categorize them: gods, spirits, sprites, souls, ideas, abstractions, “truth” and “meaning”, qualia…

Remarkably, many people who view themselves as “rationalists” stick to this archaic worldview. Most of them are oblivious to their own spiritualist beliefs, which they have conveniently swept under a very large rug which they call “mathematics”.

This particular tradition goes back to Galileo’s famous pronouncement that the universe is written in the language of mathematics.

We’re so used to this framework that it takes a while to defocus and notice its bizarre implications. Language is a behavioral trait of Homo Sapiens and other animals—is Galileo trying to tell us the universe is sentient and speaks mathematics? Or that it was created by a mathematical demiurge? Except for the substitution of mathematics in place of Word, his pronouncement isn’t much different from the opening line of the Gospel of John: “In the beginning was the Word, and the Word was with God, and the Word was God.”

Galileo’s vision enables scientists and laypeople to go about their daily lives undisturbed by spirits and sprites, except for the few people whose job is to operate under the rug: mathematicians. This is where the framework breaks down, and it breaks down spectacularly.

The issue is that Galileo never explained how humans can develop this curious ability to figure out and manipulate the language of the universe. If you take his words at face value, mathematicians are modern shamans who interpret nature by entering a dialogue with cosmic entities. No wonder, mathematical talent is presumed to be innate and manifests itself through trance-like illuminations.

This also leads to Descartes’s dualism, his belief that the human brain is both a physical organ and a spiritual organ created by God in his image. Descartes’s method, in his own words, amounts to paying attention to the “seeds of Truth” that God has planted in each of us.

For the working mathematician, this creates a really awkward situation. Mathematics is the pinnacle of rationality, and yet the mathematical experience itself feels supernatural.

Some mathematicians go full-blown mystics and attribute their creativity to divine intervention, such as Ramanujan who claimed that his results were revealed to him by his personal goddess and, as we’ve seen, Grothendieck who thought that God himself was dreaming inside his head.

Platonism, the notion that mathematical objects “exist” in the ethereal realm of ideas, is the soft-core and socially acceptable version of this belief. It is extremely prevalent in the mathematical community, although most mathematicians are vaguely aware that it is somehow problematic. Of course, no-one has a clue as to how mathematicians’ brains can magically “access” the ethereal world of ideal entities.

Contrary to a widespread misconception, formalism doesn’t offer a viable alternative. Formalism is a methodological framework for unifying mathematics and analyzing proofs,6 but it isn’t a credible ontology for mathematics. No-one studies ZFC set theory without projecting “meaning” onto it: why would you care about specific axioms, if they are meaningless garbles? In the real world, self-proclaimed formalists are just Platonists in the closet.

The conceptualist worldview

Once we give up on magical entities, the situation becomes much clearer:

If we stick to what we are certain of, we can observe that:

1. Mathematics is written using formal systems, whether legacy ones (numerals, symbolic expressions, geometric constructions) or modern ones (axiomatic set theory, category theory, proof assistants).

2. To make sense of these formal systems, we project “meaning” onto them: we imagine that symbols represent “real objects”, just like we think of a person when we read their name.

3. To develop our intuitive ability to “perceive” the mathematical “reality”, we have to spend considerable time interacting with it with an active exploration mindset, playing with examples, following reasonings and calculations, and thinking intensely about it in a meditative, dreamlike fashion.

This leads to my proposed conceptualist characterization of mathematics as a mental activity based on manipulating formal systems and imagining that they hold absolute and immutable truths about real objects.

Many mathematicians, even Platonists, agree that—for all practical matters—doing mathematics is just doing that. But most object to the metaphysical deflationary step of declaring that mathematics is just that.

They refuse to let go of the “transcendence” and “objectivity” of mathematics, which they view as its most precious aspects. As Grothendieck pointed out, there is an ancient taboo against imagination and subjectivity in mathematics, and no serious mathematician is supposed to primarily rely upon them.

But by insisting that the objects they imagine to be real are actually real, they lock themselves in a much broader delusion.

Yet Platonists do have a point: it is humanly impossible to do mathematics without imagining that mathematical objects are real. Grothendieck is absolutely clear about that in Harvests and Sowings:

All my life I’ve been unable to read a mathematical text, however trivial or simple it may be, unless I’m able to give this text a ‘meaning’ in terms of my experience of mathematical things, that is unless the text arouses in me mental images, intuitions that will give it life.

Like formalism, Platonism is an ontological dead-end but a methodological necessity.

Platonists are also right to insist that mathematics isn’t arbitrary. The superhuman attributes of mathematics, its “transcendence” and “objectivity”, aren’t lost in the conceptualist approach: they are merely contained to where they belong, within meaningless formal systems. The meaning of mathematics, our intuitive interpretation of symbolic expressions as “statements” about “mathematical objects” that hold a “truth value”, is a human cognitive phenomenon that can’t possibly have these qualities.

Truth and meaning aren’t objective attributes of the physical reality. They only exist in the intelligibility layer that we project onto it. Through its reliance on consistent formal systems, mathematics enables us to fantastically expand our intelligibility layer and create broad cognitive alignment at species level.

This explains the unique and indisputable Platonic feel of mathematics—but it’s not an excuse for magical thinking.

The reality of mathematics

When we couldn’t make sense of lightning and thunder, we attributed it to the wrath of gods. When we couldn’t make sense of mathematical cognition, we were stuck with the spiritualist worldview.

This is no longer the case and, in my view, time has arrived for career mathematicians to seriously reexamine their position. Mathematical Platonism isn’t just metaphysically wrong, it is also harmful in very practical ways.

First, it obscures the actual cognitive processes involved in mathematical thinking and creates an unnecessary barrier to entry. Imagination shouldn’t be a dirty word. If mathematicians never explain the special mental tactics that drive neuroplasticity and help them solidify their mathematical intuition, if no-one tells kids that they should actively engage their imagination, it shouldn’t come as a surprise that most people don’t get math and just hate it.

The conceptualist perspective makes it possible to have an honest conversation on how one should process math in the secret of their head. This is the approach that I have followed in my book, and the feedback received proves that this conversation can be illuminating at all levels, even for people who always thought that they hated math.

The second reason concerns mathematicians themselves.

I started this post with striking examples of mathematicians flirting with the limits and entertaining a bizarre proximity with dissociative modes of thoughts. These reckless acts of imagination are characteristic of high-level research and often produce stunning breakthroughs.

But the story doesn’t always end well.

There is an inherent danger in suggesting that mathematicians have a direct, shamanic access to higher-level truths. When performed outside of mathematics, without the ultimate guidance of formal systems, reckless acts of imagination rarely produce valuable insights.

When John Nash was first committed to a psychiatric institution for his paranoid delusions, he received the visit of his colleague George Mackey, who asked him this blunt question:

How could you, a mathematician, a man devoted to reason and logical proof… how could you believe that extraterrestrials are sending you messages? How could you believe that you are being recruited by aliens from outer space to save the world?7

Nash had this stunning reply:

Because the ideas I had about supernatural beings came to me the same way that my mathematical ideas did. So I took them seriously.

As Nash’s case dramatically illustrates, the spiritualist worldview isn’t just confusing to outsiders: it is also confusing to mathematicians themselves.

There should be no secrets and no taboos regarding the underpinnings of rationality. The power of mathematics lies not in supernatural access to eternal truths, nor in unbridled imagination, but in the constant back-and-forth between an apparently sterile formalism and our continuing effort to make intuitive sense of it.

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In a 2015 post, The Parable of Talents, Scott Alexander recalled his struggles with piano and mathematics and their role in shaping his hereditarian beliefs:

Without some notion of innate ability, I don’t know what to do with this experience.

The solution to me being bad at math and piano isn’t just “sweat blood and push through your brain’s aversion to these subjects until you make it stick”. When I read biographies of Ramanujan and other famous mathematicians, there’s no sense that they ever had to do that with math.

These words resonate with me, although my experience of mathematics was from the other side: that of “gifted” kids.

My “gift” wasn’t quite genius-level, but it was sufficient for me to coast up near the top of the most elite section of the most elite high school in my country, without conscious effort. This made me well aware of the unfathomable abyss beneath me. Before long, I also found out about the towering heights above me. Studying at the École normale supérieure alongside future rockstars like Ngô Bảo Châu and Vincent Lafforgue was an unforgettable lesson in humility.

Like Scott, I couldn’t make sense of my direct experience without assuming a major role for innate ability. At a time when blank-slatism was still fashionable, I convinced myself that mathematical talent had a strong genetic component. Later on, when I heard about twin studies’ estimates of IQ heritability, I accepted them without much critical thinking: I felt vindicated.

I stuck to these hereditarian beliefs for a very long time, until they were run over by my own progress trajectory.

Experiencing the full journey from “talented but not stellar student” to “hey, I proved a conjecture that was older than me”, I discovered that my mathematical intelligence and intuition were malleable to a degree that I had never anticipated. This wasn’t a theoretical insight, but a by-product of years of systematic practice and deliberate experimentation, leading to a drastic shift of my approach to mathematics.

As my book documents, it is quite common for elite mathematicians to express strong anti-hereditarian views. One of the most striking occurrences on record is nearly four hundred years old:

I have never presumed my mind to be any way more accomplished than that of the common man. Indeed, I have often wished that my mind was as fast, my imagination as clear and precise, and my memory as well stocked and sharp as those of certain other people… But I venture to claim that since my early youth I have had the great good fortune of finding myself taking certain paths that have led me to reflections and maxims from which I have fashioned a method by which, it seems to me, I have a way of adding progressively to my knowledge and raising it by degrees to the highest point that the limitations of my mind and the short span of life allotted to me will permit it to reach.

In short: there is a “method” for becoming a mathematical genius, and the author of these lines claims to have discovered it by chance. The excerpt is, of course, from the opening page of René Descartes’s Discourse on Method.

It is easy to dismiss this quote as delusional, or disingenuous. Yet it captures the essence of one of history’s most influential books, and aligns remarkably well with comparable quotes by Newton, Einstein, Feynman, and Grothendieck.

The Nature vs Nurture fallacy

In retrospect, my earlier hereditarian beliefs owed a lot to my naïve reliance on the traditional “Nature vs Nurture” framework.

To unpack its logic, let’s look at Carl Friedrich Gauss, the poster child of “innate talent” and one of the greatest mathematicians in history.

Gauss was born in poverty in 1777. His father worked odd jobs and his mother was an illiterate chambermaid who didn’t even register his date of birth. In primary school, his teachers were so impressed by his prodigious capabilities that they appealed to the Duke of Brunswick to finance his education. As a teenager, Gauss solved a foundational problem in geometry on which no progress had been made since the ancient Greeks.

What could explain Gauss’s striking genius? Since it can’t be his social background and upbringing, it has to be his genes... right?

Well, no.

While there might be a genetic dimension to Gauss’s story, it’s not at all certain and the reasoning itself is invalid. It was the exact reasoning that made me jump—like Scott did—from “math talent is weird and precocious” to “math talent is genetic”. The fallacy is rooted in our bad habit of using nature as a shorthand for “genetic determinism”, nurture as a shorthand for “social determinism”, and working under the implicit assumption that these two forces suffice to explain all differences in talent.

A much better framework is provided by the “three laws of behavior genetics” proposed in 2000 by geneticist Eric Turkheimer:

• The first law says that genes always matter: Nature > 0.

• The second law says that, as far as behavior is concerned, the family in which you are raised matters less than your genes: Nature > Nurture.

• The third law points out the gap in our reasoning: “A substantial portion of the variation in complex human behavioral traits is not accounted for by the effects of genes or families”—or, in other words, Nature + Nurture ≪ 100%.

What if Gauss's genius was better explained not by nature, not by nurture, but by the mysterious remainder in Turkheimer's third law—that substantial portion unaccounted for by genes or family environment?

But what is this third factor, and why would it matter so much?

The Third Factor

Geneticists often refer to it as the nonshared environment—the unique context in which a person develops. As Turkheimer puts it:

Perhaps the appropriate conclusion is not so much that the family environment does not matter for development, but rather that the part of the family environment that is shared by siblings does not matter.

This helps understand why “twins reared together” studies overestimate heritability: identical twins share a uniquely broad chunk of common environment, considerably broader than that of regular siblings or even fraternal twins.1

This notion of nonshared environment is meaningful, but fails to convey the full power of the third factor, especially when it comes to explaining extreme ability.

When I look at my own trajectory and the unique events that shaped my mathematical destiny, the most salient features are better captured by another phrase: idiosyncratic cognitive development.

If these words sound abstract, they’ll become much clearer with our next example.

Bill Thurston (1946-2012), one of the most celebrated mathematicians of the recent period, received the Fields medal for his pioneering work in geometry. He fascinated his peers with his ability to visualize geometric structures in 4 and 5 dimensions—a cognitive miracle that screams “innate talent” like nothing else.

Yet Thurston wasn’t born with a perceptive superpower, but with the exact opposite: an actual impairment. Due to a severe congenital squint, he couldn’t superpose the visual fields of his two eyes and was deprived of the natural ability to see the world in 3 dimensions. He had to reconstruct this ability through intense practice, stretching his imaginative capabilities to “stitch” together the 2-dimensional images from his eyes and develop an intuitive perception of depth:

[His mother] worked with him for hours when he was two, looking at special books with colors. His love for patterns dates at least to this time.2

But why stop there?

As a first-grader he made the decision “to practice visualization every day.” Asked how he saw in four or five dimensions he said it is the same as in three dimensions: reconstruct things from two-dimensional projections.

It’s an open secret among career mathematicians that mathematical intuition indefinitely progresses through brave, persistent, reckless exercises of the imagination. Thurston’s idiosyncratic cognitive journey traces back to his chance discovery of this phenomenon.

His story holds several essential lessons:

1. By age 2, you can be launched toward mathematical brilliance through chance events that are invisible to the traditional Nature vs Nurture framework.

2. By the time you reach primary school, your cognitive attitude and advantage will set you apart. You’ll be so distinctly overtrained that your talent will appear effortless.

3. From the outside—and possibly to yourself—the exact chain of events will be hidden, and it’ll look like you were born that way.

When Descartes invoked his “great good fortune”, he opted out of deterministic explanations, a remarkably prescient move at a time when probabilities weren’t yet properly conceptualized. The human brain is an immensely complex system, with multiple operating modes and interwoven layers of high-dimensional nonlinear dynamics. As such, it is inherently stochastic. This is why we never found any genius genes, nor any recipe for “nurturing” people like Gauss: deterministic fairytales can’t account for black swan events.

While we can pinpoint a specific trigger event in Thurston’s early life, there is no reason to expect all mathematical biographies to reveal such clear moments. Life is stochastic, get over it.

The miracle of Ramanujan

So what’s in it for people like Scott Alexander? If you struggle with math, does it make any difference whether you lost at life’s lottery or the genetic lottery?

Yes, for several reasons.

The first one is obvious: math itself is hard enough. Everyone is intimidated, including the most “talented” students. Adding a false notion that it requires a rare biological make-up only makes it more intimidating and creates the perfect excuse for not trying.

Overcoming my own inhibitions and my fear of being biologically inferior to the “geniuses” was the dominant challenge of my early years as a mathematician.

Here’s the irony: when you spot an impossibly huge talent gap above you, it signals a massive opportunity to grow—without ever closing the gap. You can’t become Gauss by deciding to, but you can become immensely better at math than you currently are.

A second reason is less obvious but more profound.

Consider Srinivasa Ramanujan, whom Scott refers to in his post. Ramanujan was “gifted” in such an incredible manner that you might suspect that his entire story was made up. In a nutshell, he was a self-taught mathematician who lived in poverty and hallucinated sophisticated theorems in his dreams, without ever being able to provide logical proofs.

It’s so tempting to cast Ramanujan as superhuman. But this misses something essential, captured here by Misha Gromov (himself a “genius-level” mathematician):

A misuser of statistics may reject [the] Ramanujan phenomenon as “a fluke of chance”, but, in fact, this miracle of Ramanujan forcefully points toward the same universal principles that make possible mastering native languages by billions of children.3

It’s only once you bring back Ramanujan to biological normalcy that you can ask the one question that truly matters: how did he do it?

As a first year student at the École normale supérieure, I followed a semester-long combinatorics course given by Xavier Viennot, aiming to construct a “nonverbal proof” of a messy formula by Ramanujan. The approach used intuitive objects such as combinatorial trees, colored beads, dominos, and hexagonal tiles, very much in the spirit of this video but with much more detail. This course permanently altered my worldview: I realized how fertile it was to assume that, in the end, Ramanujan could only come up with his formulas because there was a path to finding them obvious.

I’ve written above that my hereditarian beliefs were run over by my progress trajectory in mathematics. I should add that this was inevitable, as those beliefs were actively blocking my progress.

If you believe mathematics requires superhuman computing abilities, you’ll miss its true essence—finding simple, intuitive patterns in what initially seems unintelligible.

Throughout my scientific journey, I was regularly struck by the realization that math was more of an attitude, which eventually consolidated into an aptitude, rather than the other way around. It almost felt like a special brain mode, something that was initially hard to access but gradually became instinctive—a special way of playing with my imagination, even when it didn’t make sense, and asking stupid questions without fear of judgment.

This view is fairly consensual among career mathematicians. Here is how Thurston framed it in his MathOverflow profile:

Mathematics is a process of staring hard enough with enough perseverance at the fog of muddle and confusion to eventually break through to improved clarity. I'm happy when I can admit, at least to myself, that my thinking is muddled, and I try to overcome the embarrassment that I might reveal ignorance or confusion. Over the years, this has helped me develop clarity in some things, but I remain muddled in many others.

This bizarre, almost childish attitude is extremely hard to communicate to outsiders. Which makes it, as of today, extremely hard to teach. But that doesn’t make it innate.

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For everyone’s convenience, here’s the full set of illustrations and tables from the print edition. A few comments:

• All the drawings and mathematical diagrams were commissioned for the original French edition and created by Éléonore Lamoglia — I thank her for her patience, precision, and dedication. The elephant looks easy, but it took a while to get it right.

• The photograph selection varies a bit from one print edition to the other, due to copyright technicalities and also each publisher's specific art guidelines. E.g., the Russian edition use a different Ben Underwood picture, the French edition a different Fosbury picture, and the US edition a different Tom Hales picture.

• In the print editions, the pictures are free-floating without any legend, as their content is always obvious from context.

• The print editions are all black & white. Whenever possible, I’m including here the original color photographs.

• Some historical photographs are only available in low resolution, and some lack credits. It took some effort (and AI upscaling) to convince my publishers that these pictures were still worth printing.

Chapter 2: The Right Side of the Spoon

A cognitive transformation program with no historical precedent:

Chapter 5: Unseen Actions

A shape-sorting toy:

A tail-walking dolphin:

A tail-walking human:

Chapter 6: Refusing to Read

Bill Thurston:

Chapter 7: The Child’s Pose

Alexander Grothendieck right before he left mathematics:

Alexander Grothendieck as a child:

Chapter 9: Something’s Going on Here

An icosahedron:

Cartesian coordinates:

A 2-dimensional shadow of an hyper-icosahedron (aka 600-cell, aka hexacosichoron.)

Chapter 10: The Art of Seeing

Bill Thurston as a child:

Ben Underwood:

Chapter 11: The Ball and the Bat

How the correct answer to the ball and bat problem can be intuitive, despite Kahneman’s claim of the contrary:

Thinking fast, slow, and super slow:

Chapter 12: There Are No Tricks

How to visualize and compute the 1 + 2 + … + 99 + 100 sum (note: there’s more than one way to do, and mine is in no way superior nor canonical.)

“Staircase” visualization of the sum (shown here for 1 + 2 + … + 17 + 18, but you get the idea.)

How to get to 5000 (Thurston’s initial approximate answer): it’s the surface area of the white triangle. To get to 5050, one must add the area of the small shaded triangles.

An alternate way to get to the correct answer: it’s half the surface of this 100x101 rectangle. This is where the double-sum “trick” comes from.

Different domains of mathematics, studying different types of intuitive objects:

Chapter 14: A Martial Art

René Descartes after Frans Hals. Note the “tough-guy” gaze of this former mercenary, ready to jump for his sword. This is fairly typical of 17th century Dutch art and compares well to modern gangsta rap iconography.

Chapter 15: Awe and Magic

An infinite grid:

An infinite line:

A trefoil knot:

A different way to draw the trefoil knot:

The unknot:

Another way to draw the unknot:

Yet another way to draw the unknot:

A very messy way of drawing the unknot (can you mentally untangle it?):

What makes you so sure that the trefoil can’t be deformed into the unknot? Would you bet your life on it?:

A maximum density stacking of oranges:

Tom Hales:

The densest way to arrange coins:

Maryna Viazovska:

Chapter 16: Controlling the Universe

Ted Kaczynski at age 26 (as the youngest assistant professor in the history of UC Berkeley) and 53 (at the end of the longest search in the history of the FBI):

Chapter 18: The Elephant in the Room

Two languages, two sets of rules:

Chapter 19: Abstract and Vague

My favorite page in the entire book is one that is plainly impossible to adapt for audio. It’s a good reason to reproduce it in full:

A neuron:

Epilogue

G.H. Hardy (left) and Srinivasa Ramanujan (right).

The infamous page 379 from Russell & Whitehead’s Principia Mathematica:

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If you’re not familiar with the issue, please take a look at this quote from Wikipedia:

There is no general consensus among mathematicians about a common definition for their academic discipline.

Just think about it: billions of kids are forced to study math, yet we can’t even agree on what it’s supposed to be. And guess what? It doesn’t go well.

The stakes are clear: this is a weird animal, a metaphysical issue with real societal repercussions. If mathematicians hadn’t skipped Marketing 101, they would have taken the problem more to heart. You can’t teach a subject you can’t define, for the same reason you can’t sell a product you can’t explain.

Historically, there have been two competing approaches to defining mathematics:

1. Through what it studies: numbers, shapes, structures,…

2. Through how it functions: axioms, theorems, proofs,…

These two approaches align with the two prevailing “philosophical schools”:

1. Platonism: mathematical objects “exist” in the ethereal realm of ideas.

2. Formalism: mathematics is a mechanical game of syntactic deduction with zero transcendent semantics.

I put quotation marks around “philosophical schools”, because this framing is a misnomer. Indeed, both positions are known to be equally untenable. They’re more like “polarities” between which mathematicians have to navigate, as perfectly captured by this brilliant quip from Reuben Hersh (from his great 1979 article1, a recommended read):

the typical “working mathematician” is a Platonist on weekdays and a formalist on Sundays.

There’s something really weird going on here: you can’t do math without imagining that the cryptic symbols on the page “mean” something, that they “represent” actual “objects” that “exist” and “have properties”; yet you can’t ground this activity on anything else than meaningless formalism.

Like many of my fellow mathematicians, I’ve experienced this “philosophical plight” first hand and found it deeply unsettling. My personal inability to make sense of math was a key driver of my interest in it and, in a way, I became a research mathematician because I couldn’t figure out what math was really about.

At some point, I even studied the arcane field called “philosophy of mathematics”, which left me even more frustrated. What Hersh writes about it resonates with my own experience:

Mathematicians themselves seldom discuss the philosophical issues surrounding mathematics; they assume that someone else has taken care of this job. We leave it to the professionals.

But the professional philosopher, with hardly any exception, has little to say to the professional mathematician. Indeed, he has only a remote and inadequate notion of what the professional mathematician is doing.

In the end, Hersh continues, mathematicians tend to give up and just “do math”:

Thus, if we teach our students anything at all about the philosophical problems of mathematics, it is that there is only one problem of interest… and that problem seems totally intractable.

Nevertheless, of course, we do not give up mathematics. We simply stop thinking about it. Just do it.

Active “philosophers of mathematics” may very well object, and rightfully point to the meaningful developments that took place in the 45 years since Hersh’s article. Yet the prevailing attitude of career mathematicians hasn’t changed much, and many of them continue to live by these practical guidelines:

• Math is hard to define and it’s a fact of life.

• Philosophy of math is for cranks and losers.

• We don’t really need to define math, because we know what we do when we do math.

Many of them don’t even realize that the last item contains a strong, latent philosophical stance as well as an implicit definition of math: “mathematics is what mathematicians do.”

Hersh’s conclusion is that this third approach is the way out of the fake Platonism vs formalism debate:

The alternative of Platonism and formalism comes from the attempt to root mathematics in some nonhuman reality. If we give up the obligation to establish mathematics as a source of indubitable truths, we can accept its nature as a certain kind of human mental activity.

Thurston takes a similar path in his celebrated essay On proof and progress in mathematics2 (another great read).

While I agree with both Hersh and Thurston, I do think that we need to go one step further and actually characterize this “certain kind of mental activity” (which, in my view, none of them has convincingly done.)

An essential starting point is the latent consensus within the mathematical community as to what it means to do math and what it feels like.

I have attempted to document this consensus in a general audience book3, based on the writings of Descartes, Grothendieck, and Thurston, and also my own experience as a mathematician. While the book is certainly not perfect (and not everyone agrees with all of it), the overwhelmingly positive feedback (from Abel Prize winners and Fields medallists to high school teachers and students) is a strong indicator of the presence of this consensus.

Before I repackage the book’s core message into a tentative definition of math, let me make a digression through one of the hottest philosophical debates in medieval Europe: the Quarrel of Universals.

In a nutshell, “universals” are concepts, abstractions that are detached from any particular individual: beauty, roundness, youth… On which level of reality do these things “exist”? The debate is as old as philosophy itself and, in a way, it mirrors the Platonism vs formalism debate:

• Realism is akin to Platonism: it asserts that universals really exist, in the ethereal realm of idealities.

• Nominalism is akin to formalism: it asserts that universals are mere conventions of language.

These positions have been documented since at least Plato and Aristotle, but the debate was revived in the 12th century with the emergence of a third position:

• Conceptualism asserts that universals are products of human cognition.

Conceptualism was introduced by polymath & firebrand Peter Abelard, “the Descartes of the 12th century”, and later championed by William of Ockham.

While conceptualism is often portrayed as a flavor of nominalism (because it rejects the realist/Platonist belief in transcendent entities), it does bring a major departure from vanilla nominalism: words are more than mere utterances, more than ink on a page, they actually map to things that exist in our brains.

I was personally unable to make sense of math until my personal encounter with deep learning, latent features and sparsity turned me into a radical conceptualist.

Let me explain what this viewpoint brings to the debate on the foundations of math.

If you want to characterize math as a “certain kind of mental activity”, you have to account for all the specifics of this activity, which everyone tends to agree upon:

• the reliance on formal deduction systems,

• the notion that statements are either true or false, and proofs must be 100% bullet-proof, in a curiously rigid manner that doesn’t match any “real-world” property of language,

• a constructive approach to crafting new concepts from existing one, as if playing Lego with definitions.

For millennia, the smartest people around have struggled to make sense of the entire combo. In addition, there was this weird “Platonic feel” that mathematical concepts (numbers, shapes, structures, etc.) are actual “objects” that “exist”.

This messy situation has fostered a variety of interpretations, all coming with their own specific blindspots — from Descartes’ dualism to Galileo’s take that “math is language of the universe”, from Frege’s logicism to Brouwer’s intuitionism, from Hilbert’s formalism to Ramanujan’s mysticism.

My central claim is that we’re at a turning point in this story, not because we’re smarter, but because recent progress in neuroscience and machine learning help us remove centuries-old stumbling blocks. Once you view cognition as a dynamic learning process through which concepts “emerge” in the brain, everything becomes clearer.

Without further ado, here’s my proposed “conceptualist consensus” on what math is really about:

1. Math is a human mental activity based on playing the “Game of Truth”: when we do math, we make as if notions had precise definitions that were perfectly stable over time, as if statements had binary “truth” values, as if one could play Lego and deduce new true statements from existing ones.

2. It is an imaginary activity supported by symbolic writing systems encoding the rules of the Game of Truth.

3. As an imaginary activity, math is a driver of neuroplasticity. This explains the indisputable “Platonic feel” of math: when we spend enough time imagining mathematical abstractions, we end up “perceiving” them as if they really existed.

Interestingly, this take is both 100% consensual (no-one challenges the specifics of what it means to do math) and 100% provocative (most people have a knee-jerk reaction to the notion that math is an “imaginary” activity.)

Many will object that math can’t be just that, and that it must have some sort of “transcendent” dimension. My intention isn’t to troll anyone, but rather to bring clarity on a number of aspects which, in my view, can only be resolved through a radical conceptualist stance. Let me add these important precisions:

1. It’s not derogatory to say that math is based on imagination. People like Einstein, Descartes, Grothendieck, and Thurston have all insisted on the importance of imagination in math, and they meant it in a positive way.

2. Likewise, it’s not derogatory to say that “meaning” and “truth” are cognitive phenomenons. It’s not woke either. It’s just a pragmatic, science-driven conceptualist view on human cognition. Language has two sides, a symbolic side (utterances or ink on a page) and a semantic side. At the core, a conceptualist is someone who interprets semantics as a cognitive process rather than a shamanic access to the world of ideas (thus breaking from the traditional interpretation of semantics by Platonists, realists and spiritualists).

3. Imaginary doesn’t mean arbitrary: the rules of the Game of Truth aren’t random and they might actually be the greatest invention in history. When we follow these rules, our mental images consolidate and crystallize into a coherent worldview that is insanely powerful.4

While the philosophical (and controversial) part of the conversation is fascinating, it shouldn’t become a distraction. The pragmatic (and consensual) part is where the real substance lies, as anticipated by Hersh in this visionary remark:

The problems of truth and meaning are not technical issues in some recondite branch of logic or set theory. They confront anyone who uses or teaches mathematics. If we wish, we can ignore them… It would be surprising if this had no practical consequences.

If you agree with me on the specifics of what “doing math” actually entails, you may start to notice the elephant in the room: while we were stuck in the pointless Platonism vs formalism debate, we failed to communicate that active imagination was an essential step of mathematical comprehension.

What if the people who “suck at math”, or “have no intuition”, were simply those who never had a chance to practice the right imagination techniques, the ones that stimulate neuroplasticity in the right way? Or who gave up too early, not knowing that it’s a slow process, that requires a certain degree of insistance? It is an open secret among career mathematicians that feeling lost at sea is the normal first step of learning a new mathematical concept, and you have to persist for a while before it dissipates.

Along the way, we also failed to articulate the most compelling value proposition of mathematics: math is an imagination technique that makes you smarter.

This is what really struck me as I was writing my book: once you read Descartes, Grothendieck, and Thurston in parallel, you start to notice that they speak almost only of imagination. Each describes the use of imagination in new modalities, discovered by accident and breaking with what they’d been taught. They all view this as the secret to their success. If we don’t communicate this dimension of math, we’re missing an essential piece.

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