David Bessis
@davidbessis
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Rogue mathematician. "The product of mathematics is clarity and understanding. Not theorems, by themselves." — Bill Thurston
Joined January 2012
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"Mathematica, a Secret World of Intuition and Curiosity" is out tomorrow — phew!
It took me 20 years to fail to write it and 1 to actually do it.
Thanks
@stevenstrogatz
, Terry Tao, Hugo Duminil,
@DrEugeniaCheng
,
@benorlin
, Ian Stewart for the great blurbs!
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A simple math proof that math talent isn’t primarily driven by genetics. ⤵️
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My most ambitious post ever: we've been *wrong* about math for 2300 years and I’m going to *fix this* in a single Twitter thread.
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This Grothendieck quote is the best-kept secret about mathematics:
Always start with your naive intuition, even if it’s plainly dumb, and then refine it by asking a barrage of “stupid” questions. Waiting in silence until you “get it right” will only lead to paralysis.
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The
#1
reason why we fail to teach math: we present it as knowledge without telling kids it's a motor skill developed by practicing unseen actions in your head. Passive listening is useless, yet we never say it. We’re basically asking kids to take notes during yoga lessons.
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Why you don't understand math — and what you can do about it. ⤵️
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I’m very sad to learn that Pierre Cartier passed away yesterday.
A long time ago, when I was a depressed 22 yo math student who was going through very hard times and had just abandoned his PhD, Cartier helped me get back on track. He tutored me during a full transition year,
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Math geniuses rarely believe that math talent is innate. Just look at these quotes. You have 3 choices:
1. They're clueless.
2. They're joking.
3. They're revealing a profound truth, and we're ignoring it.
What's your vote? Poll right below. ⤵️
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It’s like fractals. The 1% is insanely better at math than the 99%. The 0.1% is yet again insanely better. And yet again the 0.01%. The 0.0001% is a whole different species. I have friends in the 0.00001% and they scare the hell out of me. And then there’s Terry Tao.
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Incompetence in one figure: UN 2024 projection for India TFR:
- fertility has declined at constant slope for 60 years
- all recent data points toward a continuing decline
- no other country succeeded at stopping TFR decline
=> yet UN predicts TFR to magically stabilize overnight
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“You can't fit Terry Tao on a bell curve. You need a power law. This alone proves that math talent can’t be primarily driven by genetics. Indeed, to get a power law, you need a Yule process, which genetics will never give you — all you’ll ever get with genes is a bell curve.”
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Math is about reconfiguring our brains and reprogramming our intuitions. By ignoring this, we effectively refuse to teach math. We teach the cryptic symbols and convoluted formulas, but we never teach the secret art of making intuitive sense of them.
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Both philosophies are dead-ends, as expressed by Reuben Hersh brilliant quip that “the working mathematician is a Platonist on weekdays and a formalist on Sundays.”
(Quote is from his beautiful 1979 paper:
)
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But unlike wealth, math talent manifests itself from an early age (Gauss proved crazy theorems in his teens) and is only loosely correlated with social background (Ramanujan was the self-taught son of a sari vendor.)
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People typically assume innate talent for *others*, rarely for themselves. What did Einstein, Descartes, Feynman and Grothendieck think of innate talent? Check out these fascinating quotes!
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Besides wealth, math talent is the most unbelievable form of inequality I’ve ever been faced with.
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Math is based on a meditation technique called “logic.” It’s an imaginary game where we pretend that words have precise meanings and “truth” can be absolute. In this game, you can combine simple “truths” into more complex ones, just like with Legos.
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In medieval Europe, people thought most humans couldn't swim, so they didn't even try teaching it. In the modern world, delusions around IQ make math almost unteachable.
Yet almost everyone succeeds at groking what used to be advanced research level math (Hindu-Arabic numerals,
@QiaochuYuan
I have taught kids before, and you have to understand that half of the population is below 100 IQ. Getting them to memorize some mechanical techniques and then apply them to pass a state test is the best you are ever going to do with them.
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“The product of mathematics is clarity and understanding. Not theorems, by themselves.”
— Bill Thurston
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Math works because when we play with our mental images using the fantasy rules of logic, our intuition becomes crisper, sharper, and more powerful. The real foundation of math isn’t axioms but neuroplasticity.
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First, there was my personal journey: how I progressed from undergrad to grad student to career mathematician, the epiphanies along the way, how I broke through my perceived glass ceiling. In the end, it no longer felt innate, it felt more like there was an untaught “method”.
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It's time to come clean: when we *do* math, we’re not truly accessing Platonic entities that free-float in the fantasy world of perfection; we’re just *imagining* them. Math is a mental activity, a cognitive hack that transforms our intuition and makes us smarter.
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IQ only follows a bell curve because it was arbitrarily defined to do so. It was designed to spot intellectual disability (which it does decently well) & the top end of the scale is uncalibrated and irrelevant (Mensa is for losers in Birkenstocks!)
My article on IQ.
Turning out to be a small booklet.
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Here's a rational explanation: the math "geniuses" are those who stumble upon the right mental habits, the special imagination techniques that drive neuroplasticity. When this chance discovery takes place in infancy, it sets you on a "superhuman" cognitive trajectory.
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A long feature article on Grothendieck in the
@guardian
, with some actual reporting (which is quite rare on this subject) and photos I had never seen before.
h/t
@OlivierReims
and
@coulmont
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These definitions reflect the two prevailing philosophies of mathematics:
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.
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From the time of Euclid, there has been two competing approaches to defining math:
1/ through what it studies: numbers, shapes, structures…
2/ through how it functions: axioms, theorems, logical deduction…
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Like wealth inequality, math inequality never stops stunning you as you move up the ladder.
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Height is the prototypal example of a highly heritable polygenic trait. And guess what? Real world distribution of height fits an actual bell curve! (Although some cheaters claim to be 6’0 when they’re actually 5’11.)
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Math isn't just hard; it’s also confusing. Some people struggle with it. Others are stunningly good and can’t figure out why.
Meanwhile, the “unreasonable effectiveness of math” fascinates everyone.
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Where do power laws show up in nature? Whenever there’s a “rich-get-richer” cumulative process over time — the technical term is “preferential attachment”, aka Yule process: wealth, fame, word frequencies, viral threads, city populations…
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"Math is the act of refining your intuition by constantly exposing its weakness."
Great aphorism, I wish I had written it this way!
starting to learn about grothendieck's model of practice via
@davidbessis
' phenomenal "mathematica" --
math is the act of refining your intuition by constantly exposing its weakness
no surprise then that the main challenge can be the pain
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This is why so many people assume that math talent is primarily innate. When I was a 20 yo math student, this is also what I used to think (which is a good thing: I now respect everyone’s opinion.) Here’s what made me change my mind.
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This frustration is a key driver of our desire to engage the general public and attempt to redefine the public perception of math. This is why we insist on using words (such as “poetry”, “joy”, or “love”) that aim to reset the expectations (ping
@stevenstrogatz
).
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Why would such a process be mistaken for innate talent? It takes place entirely in your head.
No-one can copy what you do in your head, your secret mental tricks. You just get better and better. No-one understands why and they just think you’re gifted.
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What a splendid irony: the extreme level of math inequality, which is the very reason why we’re tempted to explain it by genetics, is also the very reason why it can’t be explained by genetics.
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My main gripe with Platonism is that it perpetuates a false narrative. If math is about accessing mystical entities in a parallel world, then mathematicians are essentially shamans. Platonism turns math into an unexplainable & unteachable gift reserved for the privileged few.
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Before I dive into the details, let me be clear: I’m not making the stupid claim that genes play no role at all. They obviously must play *some* role. I’m just saying that genes can’t be the primary driver and, in particular, that they can’t explain why math is so deeply unequal.
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Math inequality more resembles wealth inequality, which is the prototypal example of a power law, aka Pareto law.
Highly heritable polygenic traits cannot obey a power law. It’s mathematically impossible, period. QED
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In my experience, progressing in math involved hard work and a series of epiphanies: learning to listen to my intuition, discovering new ways of playing with my imagination & new tactics to overcome my inhibitions.
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A shocking number of people believe that mathematical intuition is an innate talent. If kids believed that floating requires a special gift, swimming would be unteachable as lessons would turn into chaos. Kids would freak out and drown, or mutiny.
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3. Be curious. Seek clarity. Leave no stone unturned. Whenever there’s something that troubles you, something weird or seemingly incoherent, something unintuitive, try to articulate exactly what troubles you. Think freely, seek mischief & then only use logic as a referee.
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Most high-level traits are polygenic: they’re controlled not by one gene, but by a multitude of genes, each contributing a little. The total contribution obeys a normal law, aka a bell curve, because it’s what you get when you roll many independent dice.
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Twin science is junk science.
Worse, it's *viral* junk science — it takes 5 seconds to believe the IQ heritability figures from twin studies, but you need to read the actual papers and understand the math to figure out why it's horseshit.
It's Brandolini's law on steroids.
All these names blocking
@jayjoseph22
are losers, academic leftovers/psychologists, or just bloggers/propagandists using statistical tools tools they have no idea about, only able to fool one another.
See more rigorous research by
@SashaGusevPosts
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The top 3 reasons why these updates don’t get 10M views:
1. People aren’t interested in facts
The global collapse in fertility rates is a perfect topic for ideological rants — you can lament the decline of religious values, critique capitalism or smartphones, celebrate the fall
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Math talent absolutely does NOT fit a bell curve. With bell curves, everyone is pretty much *average*. Most people’s height is, give or take, within 25% of the mean. Next time you run into two people whose heights differ by a 10x factor, give me a call.
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The math genius myth persists because people are fascinated by the extreme inequality of math talent. However, as explained in this separate thread, this extreme inequality forcefully points toward an *acquired* talent, not an innate one.
A simple math proof that math talent isn’t primarily driven by genetics. ⤵️
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2. Embrace your cluelessness. Toddlers love math because they're used to being wrong all the time. They take no offense. They can’t figure out why the square block won’t fit in the round hole, yet they swallow their pride and go back to it, month after month after month.
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Thurston was born with a severe squint. He worked hard to learn to perceive the world in 3D. When he entered primary school, he made the decision to practice his imagination every day. He taught himself to "view" in 4D and 5D, then became the best geometer of the 20th century.
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I find it amazing that the UN's entire "Population Division" is capable of MISSING the global collapse of fertility rates — a world-redefining phenomenon that anyone with a Wikipedia access can figure out in 10 minutes.
It paints a scary picture of how useless UN really is.
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@stevenstrogatz
Mathematicians can’t yet agree on a definition of math, but they already share a latent consensus about what it means to do math and what it feels like.
“Mathematica, a Secret World of Intuition and Curiosity” is my attempt to document this consensus.
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If you’re confused by math, whether you don’t get it or you get it but don’t get why, I have good news for you: you're a victim of a 2300 year old error in metaphysics, and the fix is around the corner.
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Misplaced pride is the leading obstacle to mathematical learning.
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It started in early childhood, with naive games such as navigating my home with eyes closed, by visual memory. This set me on a course of developing a strong geometric intuition which, decades later, helped me prove nice theorems.
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@stevenstrogatz
When we do math, we play the “Truth Game”: we make *as if* words had a stable meaning, statements had an absolute truth value and we could infer new valid statements from existing ones by pure logical deduction, without any empirical evidence.
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Math is hard to teach because it relies on unseen mental actions that are beyond words. But by denying the true nature of math, we’re making it 100x harder. Just consider these 3 core lessons that most mathematicians view as critical, yet are widely ignored by the general public:
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By the way, you’ve probably read about twin studies “proving” that IQ is highly heritable. I've read the actual papers and found them quite lacking from a core reasoning and statistical literacy viewpoint. By contrast, the most recent GWAS studies found very little heritability.
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“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.”
— Bill Thurston
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Si vous entendez dire que Maryna Viazovska a reçu la médaille Fields parce qu'elle est une femme et ukrainienne (comme dans le commentaires affligeants sous ce post de
@lemondefr
), voici quelques arguments pour répondre: 🧵
La mathématicienne ukrainienne Maryna Viazovska a reçu la médaille
#Fields
, qui récompense ses travaux ayant trait à la meilleure façon d’empiler des sphères. Elle devient la deuxième femme lauréate du prestigieux prix | par
@dlarousserie
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@stevenstrogatz
In Chapter 4 of my book, I use the example of “a billion minus one”: everybody can see the result in their head. But 2000 years ago nobody could see it, because it’s incredibly hard to write with Roman numerals.
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For 2300 years, we've been working with flawed definitions of math. We defined it as the science of numbers and shapes, or that of logical deduction, but none of these definitions make any sense and they obscure the true nature of math.
My most ambitious post ever: we've been *wrong* about math for 2300 years and I’m going to *fix this* in a single Twitter thread.
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No-one succeeds at riding a bike on their first attempt: riding feels entirely impossible, until it becomes easy as pie. Every new math concept confronts you with the same exact learning process: it feels abstract and insanely hard, until it becomes concrete and obvious.
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@stevenstrogatz
What does it mean in practice? You can’t define math if you don’t recognize that cognition is a dynamic learning process.
Math is a particular way of orchestrating the interplay between language and intuition.
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Why should you care? Aren’t metaphysical debates supposed to have zero impact on your daily life? Well, buckle up, as this may be the one exception. I'm not making this up: millions of kids cry every day because we screwed up the ontology of math.
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@stevenstrogatz
Successful math is math that no longer feels like math. Nobody feels that whole numbers are abstractions, yet they are.
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When you’re in pure math, intelligence feels much more like a power law.
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1. Be persistent. How much is a billion minus one? You instantly see the answer in your head, as if you could touch it. But this relies on your intuition of numbers, which took years to build. This timescale is typical of what ultimate mathematical comprehension requires.
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Like many mathematicians, I had this weird feeling that something was deeply broken in the traditional way of presenting math.
I thought there was nothing I could do about it, because the issue seemed ancient and unfixable, and it almost felt like a structural feature of math.
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But what sealed the deal was when I noticed this short math argument:
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@stevenstrogatz
But like Hersh, I’m of the opinion that the issue isn’t just a matter of public perception: we need to **fix** the definition of math, and the only way to do so is to get rid of deeply ingrained yet untenable philosophical traditions.
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The more you look at it, the clearer it gets: math talent doesn't follow the rules of highly heritable traits. These traits are highly correlated among siblings—you rarely see one brother who’s 7 feet tall & another who’s 5 feet. But with math, such differences are pretty common.
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The IQ supremacist's dilemma:
- This thread annoys me but I can disparage it for assuming linearity of genome expression. 😎
- No, wait, *all* twin studies rest on linearity. 😱 If I reject linearity, there's zero science left to back my BS claim that IQ is 80% heritable. 😱😱
A simple math proof that math talent isn’t primarily driven by genetics. ⤵️
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It’s easy to be impressed by people who solve tough math problems in an instant. The successful outcome of a neuroplastic learning process often feels like magic, but the learning process itself is all but magical — it’s painfully slow.
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That's all for today!
If you’ve enjoyed this thread, you’re going to enjoy my book that explores these questions in much greater detail.
"Mathematica, a Secret World of Intuition and Curiosity" is out tomorrow — phew!
It took me 20 years to fail to write it and 1 to actually do it.
Thanks
@stevenstrogatz
, Terry Tao, Hugo Duminil,
@DrEugeniaCheng
,
@benorlin
, Ian Stewart for the great blurbs!
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Think about it: you’ve been taught math by people who held a nonsensical view of math. These people had themselves been taught math by people who held a nonsensical view of math… and so on for close to 100 generations — no wonder it was pretty confusing 🙃!
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Mendelian inheritance is when a single gene is involved. It gives rise to binary outcomes: you have it or not. Famous examples are the HERC2 gene (primary predictor of eye color) and the HBB gene (responsible for sickle-cell anemia.)
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@stevenstrogatz
About 1, let me simply say this:
Just like you can’t market a product if you can’t explain it, you can’t teach a subject if you can’t define it.
If we can’t define math, it’s no surprise we’re having so much trouble teaching it.
In the end it’s **our** responsibility.
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That’s all for today — thanks for your attention! Let me know what you think in the comments and the quick poll below.
If you’re intrigued by these questions, you'll find them explored in greater depth in my book:
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If math was Mendelian, be assured that the involved gene would have been located decades ago. You can forget about it.
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Argentina... Please explain the physical law that makes a falling knife stop falling in mid-air.
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"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."
– Bill Thurston
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@stevenstrogatz
My hunch is that the “unreasonable effectiveness of math” follows from a machine-learning theorem: the Truth Game is an effective way of accelerating the convergence of a deep learning network.
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@stevenstrogatz
I entirely agree with Reuben Hersh’s deep conclusion that math can only be defined as “certain kind of human mental activity”. This also very well aligns with Bill Thurston’s take that “mathematics is that which mathematicians study”.
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@stevenstrogatz
Why am I saying it might be possible to fix it now? Because we’re reaching a time where the legacy philosophical nonsense about thought, cognition and language is finally dissolving.
In the age of ChatGPT and Zoloft, unexamined dualism can only remain fashionable for so long.
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@stevenstrogatz
I feel that we are at an interesting moment in the history of math where we're actually going to find a valid definition:
1/ because it needs to be done,
2/ because it may finally become possible.
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This seems like generic self-help advice that you're free to overlook. But once you realize that math isn't about numbers or shapes, nor logical deduction, but about rewiring your brain, you'll understand why these 3 basic lessons are on the critical path to success.
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Now the details. First, genetic inheritance can be either Mendelian or polygenic.
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It seriously makes no sense to me. How can the UN release such absurdly stupid models without their science leadership losing their job overnight?
Didn't they have an undergrad intern curious enough to plot the curves and spot the issue?
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The gist of the argument is simple: Usain Bolt ran in 9.58s, and it’s reasonable to assume that genetics played a key role in his performance — had he run in 1 millisecond, this would no longer be a reasonable assumption.
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@stevenstrogatz
In the end, mathematicians are Platonists on weekdays because math works. It makes their intuitions so crystal clear that they get the feeling that they can **touch** the abstractions they’re manipulating.
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@stevenstrogatz
My book was heavily influenced by my personal encounter with Deep Learning and my feeling that it offers an empirical way of resolving the Quarrel of Universals: Abelard’s Conceptualism (a flavor of Nominalism) has won.
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Of course, math talent isn’t something that you can accurately measure, so it’s a bit of stretch to say that it obeys a power law. But it doesn’t have to. My argument works as long as math inequality is too extreme to fit a bell curve — which I think we agree on.
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I'm a hardcore anti-conspiracy "don't assume evil when it can be explained by incompetence" kind of guy... but this is stretching my limits. I knew bureaucrats could be incompetent, but I never thought they could be THAT incompetent.
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@stevenstrogatz
This weird and seemingly crazy way of playing Legos with definitions is unique to math and provides its foundational Formalist flavor.
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@stevenstrogatz
Personally, I wasn’t able to make sense of math while I was stuck in the Platonism vs Formalism debate and its “Quarrel of Universals” counterpart: the Realism vs Nominalism debate.
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You might object that math talent is correlated with IQ, which does follow a bell curve. But this isn't a valid objection.
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Grothendieck was so far ahead of his time that psychiatry will take 100 years to catch up.
(The picture is from his 10 page essay on kimchi.)
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If math talent was anything close to a bell curve, it’d be boring as hell & you wouldn’t be reading this thread.
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