Beyond nature and nurture

How mathematics changed my view on talent

Young Bill Thurston, at the onset of his mathematical journey.

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.

How Bill Thurston saw the world: left and right, illustrations from his graduate course on The geometry and topology of 3-manifolds, center, illustration from his 1974 research article on The theory of foliations of codimension greater than one.

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.

Thurston’s childish imagination at play: this drawing from his graduate course on 3-manifolds is a mathematically accurate introduction to his concept of “train-tracks”, an abstract notion that enabled him to tackle previously intractable problems in geometry.

1. Heritability estimates from twin studies vary tremendously according to which assumptions are put into the model, as explained here by Sasha Gusev. See also this recent interview of Sasha: “My guess is that hundreds or even thousands of interacting environments accumulate over a person’s lifetime, many of which are not even measurable. […] When we look at twins, who share their rearing environment extremely closely (are literally born at the same time), all of these interactions get assigned to and inflate the genetics/heritability bucket.”

2. This quote and the next one are from the obituary notice published in the Notices of the American Mathematical Society. Thurston’s story and approach to mathematics are discussed in much greater detail in my book.

3. Misha Gromov, Math Currents in the Brain, 2014 manuscript.