I like how your substack pieces are even more distilled and clearer than your book.
In your book, you mention (if my memory is correct) that you couldn't stand analysis and couldn't make yourself study it. Instead you were drawn to geometry. Do you think that is because you had some (genetic?) affinity for shapes or were you unable to find the right mental "moves" to understand analysis intuitively or you had stumbled upon the right mental "moves" for geometry and didn't want to leave that comfort zone (or some other reason)?
Your question is a fair one, let me try to answer it as honestly as I can. While I can't "prove" anything about it, I see no reason to suspect that my relative affinity for geometry (vs analysis) has a substantial *direct* genetic base. Here's my (subjective) line of thoughts:
1/ Early childhood
As discussed in my book, I have poor uncorrected vision (my correction is about -6D for both eyes) and had to wear glasses since preschool. This created an irrational fear of becoming blind, and I devised "survivalist strategies" for navigating the world with my eyes closed. I suspect this is what gave me a substantial head start in geometry. Short-sightedness has a strong genetic component, but the whole chain of events is pretty random. Had I no spend that much time as a solitary latchkey kid, I may not have played the same cognitive games.
(By the way, this moment wasn't necessarily my key "Thurston moment". A much more profound health challenge took place during my first months of life, which I'll discuss in a future book, triggering the need for intense & high stakes pre-verbal cognitive activity which — it seems — altered my thinking patterns, and created much deeper emotional insecurity that what "normal" kids experience. The best evidence I have is my hypermnesia (my most profoundly divergent cognitive ability) which covers detailed visual memories (cross-checked decades later) of places and objects associated with events that took place *much before* the conventionally admitted horizon of childhood amnesia.)
2/ As a student
During my first two years of studies, I had a mild preference for algebra and geometry but was also reasonably at ease with analysis. It's only during my first year at the École normale supérieure, where we had a fast-track program packing academic years 3 and 4 into single calendar year, that I bailed out of everything analysis/measure theory/statistics/PDE — and even number theory. The last analysis course I took was the mandatory "Complex Analysis 1", in my first Fall term at the ENS.
The reason was that it was simply impossible to study everything, and I was more interested by the proposed courses in algebra (Galois Theory, Coxeter Groups, Enumerative Combinatorics) and computer science (Lambda Calculus, Complexity, Cryptography)...
This "fast-track" was actually a poorly designed program that specialized too hard and too early, and it has now been reverted by the ENS.
Not maintaining my analysis skills (I've barely manipulated an Ɛ since then) was a bad idea, and it almost created in an impairment where there previously was none.
In retrospect, my biggest regret is number theory, which I ignored because I hated the sloppy analytic notations with tiny indices, and hadn't fully realized that these notations too, just like clean categorical notations, were just shorthand for much more intuitive things.
So, yes, I made the mistake of staying within my comfort zone and I'm not proud of that (the flip side being that I'm quite proud of my algebraico-geometrico-homotopico-category-theoretic intuition.)
That's fascinating. Thank you for the detailed response.
It would be an interesting project for someone with some mathematical knowledge to interview high achieving mathematicians and ask them about their formative experiences with math. When did they start to find the mental "moves" to really do math? What were those mental moves?
Somewhat related is the following on stackoverflow. I don't understand the answers, but these are cool ideas.
I'm convinced much more of us could drastically improve our math ability through enough dedication, study, practice, the right attitude, etc.
But I'm still left with the question of how do we explain the mathematical geniuses.
Am I to take away from this article that there isn't really a "ceiling" and almost anyone could theoretically be able to do math at a very high level of only they just did the right things? (Whatever those things may be)
My take is indeed that this potential explanation of math geniuses is more credible than the competing ones. It comes with important fine prints though:
1. "Almost anyone": my claim is a bit weaker than that.
- First, unfortunately, far fewer people than "almost anyone" are free from substantial genetic and congenital hindrances (think: nutrition, fetal alcohol exposure, viruses, brain hypoxia, etc.).
- Second, when you want to explain a "one in a billion" genius like Terry Tao, it's natural to assume that there is a good alignment on many levels. So I wouldn't be that surprised if Terry happened to be in the top 10% or even the top 1% from a genetic potential standpoint. But 1/100 hardly explains a 1/1000000000 phenomenon.
- My take is that the genetic potential gap between "genetic 1 in 10 (or 100)" and "genetic 1 in a million (or billion)" can't credibly explain a 1/1000000000 phenomenon like Terry (because genomic potential is likely to be Gaussian, while Terry differs from other excellent mathematicians by *orders of magnitude*.) Even if Terry was 1/100 genetically, the main contributor to the incredible "1 in a billion" rarity would be *something else*.
2."could theoretically be able to do math at a very high level of only they just did the right things": the caveat here is with the meaning of "could theoretically if"
- Given 1/ the proven extreme precocity of math genius (with Terry, it was very visible at 2 yo), 2/ the extreme cognitive reconfiguration that is required, it is quite likely that the events that take you on a "math genius" path must occur very early in life, at peak neuroplasticity, probably before language is acquired.
- This means that math genius isn't teachable nor coachable, at least not in the traditional sense: you can't coach a pre-language infant on how they should manipulate pre-language concepts in their head.
While we might observe some pattern of particular life events that have a higher chance of triggering an idiosyncratic cognitive development that is conducive to math genius (think: Ted Kaczinsky health incident in infancy that, according to his mother, left a permanent emotional scar), it's likely that the same events would lead to completely different outcomes in a great number of cases. We'd be far from a reproducible explanation.
In a way, my take isn't really about *explaining math genius* (as I'm missing crucial details on what is actually taking place in the brain), but rather about *explaining where to look for and NOT look for when trying to explain math genius*.
„ 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.”
Huh? Twin studies estimate heritability by comparing fraternal twins with identical twins. „all of these interactions get assigned to and inflate the genetics/heritability bucket“ NO. They get assigned the shared environment bucket. This quote completely misunderstands how twin studies work. :( like, it couldn’t be more wrong. Otherwise this is a great article, thanks for writing it!
I agree that Sasha's sentence is a bit confusing due to the missing "identical" that is implicit here (identical twins are known to spend more time together, be more frequently in the same section at school, etc.). At least that's how I interpret it, as an explanation for why EEA violation shouldn't be dismissed as anecdotal.
(Since Sasha is obviously aware of how twin studies work. By the way, I also recommend his piece on how twin studies can yield entirely different results if you relax some assumptions:
"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."
My understanding was that it's twins raised apart, and how close in similarity they are to twins raised together, that explains the heritability view?
As I'll explain in a future post, "twins reared apart" studies are flawed in a much deeper way than "twins reared together" studies, which explains why most hereditarians have now switched to the latter to justify their claims.
I really enjoyed reading this piece. I’ve had a similar but also totally different experience of transitioning out of hereditarian views (prompted by Turkheimer as well) through thinking a lot about polygenic autism. In a piece I wrote, I try to decipher why genetics are stressed in some cases (autism) and not in others (oppositional defiance disorder). If I could be so bold as to suggest my own writing.
Loved this post - found it in your twitter thread and I’m feeling sad I hadn’t come across your work sooner.
Thurston’s ‘third thing’! Yes!
I’m convinced that there’s a huge blindspot over imagination, simply thinking about something in a certain way, that ‘wreckless process’ done free of judgement (in our own heads), and what can grow out of this. The repetitive mental habits that compound over time, shaping skills in ways that aren’t obvious til later.
Seeing a person juggle for the first time might look like sorcery. But they simply tried to do something seemingly impossible until something clicked and suddenly they’ve chained a series of simple tasks into a spectacle.
What’s happening in a child’s imagination, while they stare vacantly at a leaf might be causing all kinds of miraculous leaps, of those ‘clicks’ happening and being chained together. Two years later they appear so inexplicably gifted in that thing that it *must* be an aberration.
The example of visual impairment leading to 1 finding an alternative pathway to visualising depth 2 finding this pathway makes it easier to vidualise 4th and 5th etc..
Is perfect. Limitation in one domain leading to excelling elsewhere.
In Oliver Sacks book Island of the colourblind, the tribes of Pinglap with achromatopsia there thought that those without it were worse at foraging because they were distracted by colour, and had a very undeveloped sense of material (plant) texture. If we look at b&w photo we can tell from how light diffuses if something’s velvet, metal, etc.
The key word here being ‘developed’. These people were hugely atuned to texture due to a huge limitation, a visual ‘disability’. They had no physical gift, just an absence of one that lead to the nurturing of another talent. They could tell one red berry from another at a distance from its texture alone, which would astound outsiders.
I’m also convinced that even seemingly unrelated factors like being afraid of one’s parents, finding it hard to sleep, or simply being very bored a lot will lead to a child finding imaginative or habitual exercises that keep them distracted from psychic pain. Whether due to positive or negative external forces, habits of the imagination are going to lead to *something* being developed and it might not reveal itself.
It seems fair to say that that exceling in one area often coincides with being unusually deficient elsewhere. Which of course, because there is an opportunity cost to fixating on one thing, especially as a one year old.
Do you think it’s partly to do with the difficulty/impossibility of studying thought, especially in the early years? Ie until there’s evidence of ‘grasping’ something, it’s as though there’s nothing much going on up there except the usual standard development, at differing speeds. It’s like they’re missing the fractal aspect to thought, no?
The question of whether it's possibly to articulate these processes, explain them and teach them (even though they'll be less effective when taught after infancy's peak neuroplasticity) is a very interesting one.
In a way, math is an approach to transcribe inner mental activity — it's both a success (math rocks) and a failure (people don't get it.)
I do think that we have a huge margin of progression, as individuals and as a species, in how we can share more effectively these "unseen actions" (that's precisely the topic of my book.)
Great post, David! I have a practical question: what steps could the French government take to encourage greater interest in mathematics and science in general - often perceived as "scary" subjects - among young people, especially young women?
My impression is that France's math teaching problem is critical in primary school, where teachers are almost exclusively hired among humanities graduates with zero affinity with math and zero understanding. A first step would be to make these positions substantially more attractive and competitive (not sure this is doable under current constraints), or to launch a pragmatic plan to reconcile the existing teachers with math.
The third factor portion of this piece reminded me of the relative age effect: how, just by being born earlier in relevant selection periods, the small head start provides some benefits that compound and naturally develops a virtuous cycle—much like 'idiosyncratic cognitive development'. Though a less individual example, it's interesting to see how random events can have a lasting impact on achievement. It leaves one wondering how much of the nonshared environment goes unnoticed or is simply taken for granted.
"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."
This is the general sense I've gotten from reading about and around influential mathematicians and mathematics. As you've alluded to, the layman (myself included) cannot truly understand the nature of mathematics without unlearning many misconceptions and engaging with it properly. From what I've gathered, these processes can inform and/or reinforce each other.
I'm looking forward to reading your book.
*Edited for clarity, brevity, and adjusted opinion*
Reading some of the answers you really see "This bizarre, almost childish attitude..." in mathematical thinking.
It also seems like Bill Thurston explored a local instance of this broader cognitive phenomenon in the world of mathematics:
"It's important to ask yourself 'Why?' when others don't comprehend something of this nature. Usually it's because they have competing mental models where your words are nonsense. It requires some excavation of the old before you can embrace the new."
I'm going to remember the William Thurston 2-D to 3-D to 4-D & 5-D example for the rest of my life. A super important proof of concept of the fundamental wrongness of the way we normally think about the interplay of genetics and environment. Love it love it love it!!!!
Will I ever understand *any* of the things he proved using his ability to conceive higher dimensions? lolno I'm way too dumb for that. Just reading his Wikipedia page makes my eyes cross.
You didn't mention motivation, which is inherited by environment, not genes.
Few people have the motivation to dedicate their energy to mathematics, and those who do earn very little social rewards from it (money, success, attractiveness.), and when they do earn anything, the reward isn't immediate
Your article assumes that being good at math is a valuable talent, while human experience shows the opposite in most cases
you just kick the can down the road, since now the question is whether a person is in fact capable of doing the kinds of effortful 'mental moves' that result in genius-approximating or at least relatively hugely surprising/significant progress. if so, what explains that? the same old, insanely, ridiculously complex entanglement of genetics + environment (any & everything that isn't genetics) — neither of which is controlled by anyone. regardless of our extensive limitations researching all the variables umbrella-termed as 'environment', they are extremely numerous, dishomogeneous, varied & complex. it's really any & everything that aren't genetic processes. (+ consider the fact that reality is such that it's never possible to inquire what any traits or behavior 'would've been like' sans environment or sans genes.) the 'shared family environment' + 'non-shared individual environment' partitioning's operationalization is extremely coarse-grained and more so for the latter part, so we don't access a plethora of relevant environmental mechanisms.
it's cool if there are reliable methods of improving particular skills, but none of this is 'simple', in that none of the mental moves, the creation & sustained exploitation of novel affordances are like some irreducable, primitive, simple 'things' that you 'just do'. any instance of executing some method decomposes into immensely complex, dynamic clusters of traits & behaviors, which still just are entirely functions of genes+environment. there is no 'beyond' nature and nurture, since these are just levels of analysis of the one world. just like 'my geometry with square circles' is not anything, beyond the world, ie. what exists, isn't anything.
i'm also not quite sure why or how you want to cash out stochasticity. in my current understanding, it's a modelling approach having to do with *describing* (modeling, representing...) phenomena with random probability distributions, but you claimed that the brain & life (!!!) (??) are inherently stochastic, which seems (trivially) technically wrong and also extremely contentious and empirically unwarranted.
true randomness may exist. it may even occur in brains, sure, though we can never empirically establish that, since it'd just be events that are untethered from any kind of already existing states of affairs; a brute, unexplainable occurrence. no 'apparent' patterns intelligible by our evolved cognition and/or no predictability in some informational structure don't warrant inferring true randomness.
Thanks a lot for your comment. Let me try to address its core aspects:
1/ Kicking the can down the road?
It all depends on the end goal. Is it nurturing more Gauss-level geniuses? In that case, I admit I have nothing to offer: I have no idea how to do that. Or is it helping the hundreds of millions of kids globally who can't make sense of high school math or calculus? Well, in that case, I do think that they'd be in much better shape if the noxious idea that it's mostly a matter of genes wasn't so widespread (among them, among their peers, among their parents and teachers.) This would create more awareness and more willingness to make progress on the next topic:
2/ "it's cool if there are reliable methods of improving particular skill"
My book is my best attempt at documenting "secret math", the inner cognitive and emotional approach that is conducive to the development of a stronger mathematical intuition. I'm not saying that my book is perfect, but I do hope that it's a substantial step forward. At least a good number of credible people think so, and I receive a lot of reader feedback from students and teachers who claim it's been extremely helpful to them.
I'm confident that many more people will advance these ideas (most of which predate my book), adapt them and implement them at scale.
I honestly believe that our current incapacity to bring most of a generation to, say, understand calculus in the same way they can read or write (which is far from perfect) is a transient historical situation, and that 50 years from it'll quite different.
Which brings us back to the crucial question of addressing the noxious hereditarian myths. Check out the Hacker News discussion on my Quanta interview (https://news.ycombinator.com/item?id=42200209) and you'll see how prevalent these myths are, to the point where it's almost impossible to hold a reasonable conversation on these topics.
3/ stochasticity & co
Honestly this is a casual exposition piece, and all notions of randomness should be understood in the weakest epistemological sense. I would have phrased it differently if I had wanted to open yet another epistemological front on causality/agency/indeterminacy/etc. — honestly I have more than enough on my plate with the topics at stake in my book, so I won't elaborate.
Not at this point. I expect that it will take some time (= several years) for something structured & quantified to take place. Right now I'm receiving a lot of individual feedback from teachers (and students) who tell me the book has influenced them (I've also given some invited lectures at teacher training conferences), but I have no hard figures to show about applicability at scale and controlled studies.
I'm not an education science professional and don't claim to be one. My book is about the essence of math and the subjective process of making sense of it — which is already enough on my plate. My book aims at culture change and lacks practical implementation details for teachers (I chose to focus on the students' perspective, as it concerns more people, and it is closer to that of research mathematicians).
I'm not competent to craft intervention scripts, which will have to vary a lot depending on grade and environment. But I do hope that my book will inspire enough teachers to turn the core ideas into concrete actions (a difficult but extremely important job).
It's a long game — we've been wrong about math for millennia, and it'll take decades for us to reposition it. Whatever the interventions, they'll succeed only if students accept that math isn't what it's long been considered to be.
I volunteer as the hands on mathematics designer/builder/facilitator for several STEM after school programs and summer camps. We offer our programs as helping to reduce "math anxiety" which appears to be a measurable thing in social science. This also keeps the math teachers happy that we aren't advocating to change their curriculum.
There is some innate variance in the efficiency of the neuron-machinery, but the talent gap primarily stems for the monstrous variance in attitude, exacerbated by the absence of a proper cultural framework to communicate this essential aspect of math.
I like how your substack pieces are even more distilled and clearer than your book.
In your book, you mention (if my memory is correct) that you couldn't stand analysis and couldn't make yourself study it. Instead you were drawn to geometry. Do you think that is because you had some (genetic?) affinity for shapes or were you unable to find the right mental "moves" to understand analysis intuitively or you had stumbled upon the right mental "moves" for geometry and didn't want to leave that comfort zone (or some other reason)?
Thanks for the feedback!
Your question is a fair one, let me try to answer it as honestly as I can. While I can't "prove" anything about it, I see no reason to suspect that my relative affinity for geometry (vs analysis) has a substantial *direct* genetic base. Here's my (subjective) line of thoughts:
1/ Early childhood
As discussed in my book, I have poor uncorrected vision (my correction is about -6D for both eyes) and had to wear glasses since preschool. This created an irrational fear of becoming blind, and I devised "survivalist strategies" for navigating the world with my eyes closed. I suspect this is what gave me a substantial head start in geometry. Short-sightedness has a strong genetic component, but the whole chain of events is pretty random. Had I no spend that much time as a solitary latchkey kid, I may not have played the same cognitive games.
(By the way, this moment wasn't necessarily my key "Thurston moment". A much more profound health challenge took place during my first months of life, which I'll discuss in a future book, triggering the need for intense & high stakes pre-verbal cognitive activity which — it seems — altered my thinking patterns, and created much deeper emotional insecurity that what "normal" kids experience. The best evidence I have is my hypermnesia (my most profoundly divergent cognitive ability) which covers detailed visual memories (cross-checked decades later) of places and objects associated with events that took place *much before* the conventionally admitted horizon of childhood amnesia.)
2/ As a student
During my first two years of studies, I had a mild preference for algebra and geometry but was also reasonably at ease with analysis. It's only during my first year at the École normale supérieure, where we had a fast-track program packing academic years 3 and 4 into single calendar year, that I bailed out of everything analysis/measure theory/statistics/PDE — and even number theory. The last analysis course I took was the mandatory "Complex Analysis 1", in my first Fall term at the ENS.
The reason was that it was simply impossible to study everything, and I was more interested by the proposed courses in algebra (Galois Theory, Coxeter Groups, Enumerative Combinatorics) and computer science (Lambda Calculus, Complexity, Cryptography)...
This "fast-track" was actually a poorly designed program that specialized too hard and too early, and it has now been reverted by the ENS.
Not maintaining my analysis skills (I've barely manipulated an Ɛ since then) was a bad idea, and it almost created in an impairment where there previously was none.
In retrospect, my biggest regret is number theory, which I ignored because I hated the sloppy analytic notations with tiny indices, and hadn't fully realized that these notations too, just like clean categorical notations, were just shorthand for much more intuitive things.
So, yes, I made the mistake of staying within my comfort zone and I'm not proud of that (the flip side being that I'm quite proud of my algebraico-geometrico-homotopico-category-theoretic intuition.)
That's fascinating. Thank you for the detailed response.
It would be an interesting project for someone with some mathematical knowledge to interview high achieving mathematicians and ask them about their formative experiences with math. When did they start to find the mental "moves" to really do math? What were those mental moves?
Somewhat related is the following on stackoverflow. I don't understand the answers, but these are cool ideas.
https://mathoverflow.net/questions/25983/intuitive-crutches-for-higher-dimensional-thinking
I'm convinced much more of us could drastically improve our math ability through enough dedication, study, practice, the right attitude, etc.
But I'm still left with the question of how do we explain the mathematical geniuses.
Am I to take away from this article that there isn't really a "ceiling" and almost anyone could theoretically be able to do math at a very high level of only they just did the right things? (Whatever those things may be)
Thanks for your feedback!
My take is indeed that this potential explanation of math geniuses is more credible than the competing ones. It comes with important fine prints though:
1. "Almost anyone": my claim is a bit weaker than that.
- First, unfortunately, far fewer people than "almost anyone" are free from substantial genetic and congenital hindrances (think: nutrition, fetal alcohol exposure, viruses, brain hypoxia, etc.).
- Second, when you want to explain a "one in a billion" genius like Terry Tao, it's natural to assume that there is a good alignment on many levels. So I wouldn't be that surprised if Terry happened to be in the top 10% or even the top 1% from a genetic potential standpoint. But 1/100 hardly explains a 1/1000000000 phenomenon.
- My take is that the genetic potential gap between "genetic 1 in 10 (or 100)" and "genetic 1 in a million (or billion)" can't credibly explain a 1/1000000000 phenomenon like Terry (because genomic potential is likely to be Gaussian, while Terry differs from other excellent mathematicians by *orders of magnitude*.) Even if Terry was 1/100 genetically, the main contributor to the incredible "1 in a billion" rarity would be *something else*.
2."could theoretically be able to do math at a very high level of only they just did the right things": the caveat here is with the meaning of "could theoretically if"
- Given 1/ the proven extreme precocity of math genius (with Terry, it was very visible at 2 yo), 2/ the extreme cognitive reconfiguration that is required, it is quite likely that the events that take you on a "math genius" path must occur very early in life, at peak neuroplasticity, probably before language is acquired.
- This means that math genius isn't teachable nor coachable, at least not in the traditional sense: you can't coach a pre-language infant on how they should manipulate pre-language concepts in their head.
While we might observe some pattern of particular life events that have a higher chance of triggering an idiosyncratic cognitive development that is conducive to math genius (think: Ted Kaczinsky health incident in infancy that, according to his mother, left a permanent emotional scar), it's likely that the same events would lead to completely different outcomes in a great number of cases. We'd be far from a reproducible explanation.
In a way, my take isn't really about *explaining math genius* (as I'm missing crucial details on what is actually taking place in the brain), but rather about *explaining where to look for and NOT look for when trying to explain math genius*.
„ 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.”
Huh? Twin studies estimate heritability by comparing fraternal twins with identical twins. „all of these interactions get assigned to and inflate the genetics/heritability bucket“ NO. They get assigned the shared environment bucket. This quote completely misunderstands how twin studies work. :( like, it couldn’t be more wrong. Otherwise this is a great article, thanks for writing it!
Thanks for your feedback!
I agree that Sasha's sentence is a bit confusing due to the missing "identical" that is implicit here (identical twins are known to spend more time together, be more frequently in the same section at school, etc.). At least that's how I interpret it, as an explanation for why EEA violation shouldn't be dismissed as anecdotal.
(Since Sasha is obviously aware of how twin studies work. By the way, I also recommend his piece on how twin studies can yield entirely different results if you relax some assumptions:
https://theinfinitesimal.substack.com/p/twin-heritability-models-can-tell )
"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."
My understanding was that it's twins raised apart, and how close in similarity they are to twins raised together, that explains the heritability view?
Very thought-provoking article, thank you.
Thanks!
As I'll explain in a future post, "twins reared apart" studies are flawed in a much deeper way than "twins reared together" studies, which explains why most hereditarians have now switched to the latter to justify their claims.
Looking forward to it.
I really enjoyed reading this piece. I’ve had a similar but also totally different experience of transitioning out of hereditarian views (prompted by Turkheimer as well) through thinking a lot about polygenic autism. In a piece I wrote, I try to decipher why genetics are stressed in some cases (autism) and not in others (oppositional defiance disorder). If I could be so bold as to suggest my own writing.
Great to hear that! Looking forward to reading your posts.
Loved this post - found it in your twitter thread and I’m feeling sad I hadn’t come across your work sooner.
Thurston’s ‘third thing’! Yes!
I’m convinced that there’s a huge blindspot over imagination, simply thinking about something in a certain way, that ‘wreckless process’ done free of judgement (in our own heads), and what can grow out of this. The repetitive mental habits that compound over time, shaping skills in ways that aren’t obvious til later.
Seeing a person juggle for the first time might look like sorcery. But they simply tried to do something seemingly impossible until something clicked and suddenly they’ve chained a series of simple tasks into a spectacle.
What’s happening in a child’s imagination, while they stare vacantly at a leaf might be causing all kinds of miraculous leaps, of those ‘clicks’ happening and being chained together. Two years later they appear so inexplicably gifted in that thing that it *must* be an aberration.
The example of visual impairment leading to 1 finding an alternative pathway to visualising depth 2 finding this pathway makes it easier to vidualise 4th and 5th etc..
Is perfect. Limitation in one domain leading to excelling elsewhere.
In Oliver Sacks book Island of the colourblind, the tribes of Pinglap with achromatopsia there thought that those without it were worse at foraging because they were distracted by colour, and had a very undeveloped sense of material (plant) texture. If we look at b&w photo we can tell from how light diffuses if something’s velvet, metal, etc.
The key word here being ‘developed’. These people were hugely atuned to texture due to a huge limitation, a visual ‘disability’. They had no physical gift, just an absence of one that lead to the nurturing of another talent. They could tell one red berry from another at a distance from its texture alone, which would astound outsiders.
I’m also convinced that even seemingly unrelated factors like being afraid of one’s parents, finding it hard to sleep, or simply being very bored a lot will lead to a child finding imaginative or habitual exercises that keep them distracted from psychic pain. Whether due to positive or negative external forces, habits of the imagination are going to lead to *something* being developed and it might not reveal itself.
It seems fair to say that that exceling in one area often coincides with being unusually deficient elsewhere. Which of course, because there is an opportunity cost to fixating on one thing, especially as a one year old.
Do you think it’s partly to do with the difficulty/impossibility of studying thought, especially in the early years? Ie until there’s evidence of ‘grasping’ something, it’s as though there’s nothing much going on up there except the usual standard development, at differing speeds. It’s like they’re missing the fractal aspect to thought, no?
Thanks for the feedback!
The question of whether it's possibly to articulate these processes, explain them and teach them (even though they'll be less effective when taught after infancy's peak neuroplasticity) is a very interesting one.
In a way, math is an approach to transcribe inner mental activity — it's both a success (math rocks) and a failure (people don't get it.)
I do think that we have a huge margin of progression, as individuals and as a species, in how we can share more effectively these "unseen actions" (that's precisely the topic of my book.)
For example, you used the term "rockstar" for Laurent Lafforgue and Ngo Bo Cho, why not "mathstar"?
That tells you everything you need to know about the social value of mathematical talent
Fair point!
Great post, David! I have a practical question: what steps could the French government take to encourage greater interest in mathematics and science in general - often perceived as "scary" subjects - among young people, especially young women?
Wow, not sure I'm competent to answer that one.
My impression is that France's math teaching problem is critical in primary school, where teachers are almost exclusively hired among humanities graduates with zero affinity with math and zero understanding. A first step would be to make these positions substantially more attractive and competitive (not sure this is doable under current constraints), or to launch a pragmatic plan to reconcile the existing teachers with math.
Thank you writing it, very inspiring.
This was a good read (and reread).
The third factor portion of this piece reminded me of the relative age effect: how, just by being born earlier in relevant selection periods, the small head start provides some benefits that compound and naturally develops a virtuous cycle—much like 'idiosyncratic cognitive development'. Though a less individual example, it's interesting to see how random events can have a lasting impact on achievement. It leaves one wondering how much of the nonshared environment goes unnoticed or is simply taken for granted.
"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."
This is the general sense I've gotten from reading about and around influential mathematicians and mathematics. As you've alluded to, the layman (myself included) cannot truly understand the nature of mathematics without unlearning many misconceptions and engaging with it properly. From what I've gathered, these processes can inform and/or reinforce each other.
I'm looking forward to reading your book.
*Edited for clarity, brevity, and adjusted opinion*
Thank you Jonathan!
A related thread: https://mathoverflow.net/questions/38639/thinking-and-explaining
Reading some of the answers you really see "This bizarre, almost childish attitude..." in mathematical thinking.
It also seems like Bill Thurston explored a local instance of this broader cognitive phenomenon in the world of mathematics:
"It's important to ask yourself 'Why?' when others don't comprehend something of this nature. Usually it's because they have competing mental models where your words are nonsense. It requires some excavation of the old before you can embrace the new."
We will continue to try our best
Intuitive>>>>>>
I'm going to remember the William Thurston 2-D to 3-D to 4-D & 5-D example for the rest of my life. A super important proof of concept of the fundamental wrongness of the way we normally think about the interplay of genetics and environment. Love it love it love it!!!!
Will I ever understand *any* of the things he proved using his ability to conceive higher dimensions? lolno I'm way too dumb for that. Just reading his Wikipedia page makes my eyes cross.
Nature sets the upper limit/potential, nurture/circumstance determines the actual outcome.
You didn't mention motivation, which is inherited by environment, not genes.
Few people have the motivation to dedicate their energy to mathematics, and those who do earn very little social rewards from it (money, success, attractiveness.), and when they do earn anything, the reward isn't immediate
Your article assumes that being good at math is a valuable talent, while human experience shows the opposite in most cases
you just kick the can down the road, since now the question is whether a person is in fact capable of doing the kinds of effortful 'mental moves' that result in genius-approximating or at least relatively hugely surprising/significant progress. if so, what explains that? the same old, insanely, ridiculously complex entanglement of genetics + environment (any & everything that isn't genetics) — neither of which is controlled by anyone. regardless of our extensive limitations researching all the variables umbrella-termed as 'environment', they are extremely numerous, dishomogeneous, varied & complex. it's really any & everything that aren't genetic processes. (+ consider the fact that reality is such that it's never possible to inquire what any traits or behavior 'would've been like' sans environment or sans genes.) the 'shared family environment' + 'non-shared individual environment' partitioning's operationalization is extremely coarse-grained and more so for the latter part, so we don't access a plethora of relevant environmental mechanisms.
it's cool if there are reliable methods of improving particular skills, but none of this is 'simple', in that none of the mental moves, the creation & sustained exploitation of novel affordances are like some irreducable, primitive, simple 'things' that you 'just do'. any instance of executing some method decomposes into immensely complex, dynamic clusters of traits & behaviors, which still just are entirely functions of genes+environment. there is no 'beyond' nature and nurture, since these are just levels of analysis of the one world. just like 'my geometry with square circles' is not anything, beyond the world, ie. what exists, isn't anything.
i'm also not quite sure why or how you want to cash out stochasticity. in my current understanding, it's a modelling approach having to do with *describing* (modeling, representing...) phenomena with random probability distributions, but you claimed that the brain & life (!!!) (??) are inherently stochastic, which seems (trivially) technically wrong and also extremely contentious and empirically unwarranted.
true randomness may exist. it may even occur in brains, sure, though we can never empirically establish that, since it'd just be events that are untethered from any kind of already existing states of affairs; a brute, unexplainable occurrence. no 'apparent' patterns intelligible by our evolved cognition and/or no predictability in some informational structure don't warrant inferring true randomness.
Thanks a lot for your comment. Let me try to address its core aspects:
1/ Kicking the can down the road?
It all depends on the end goal. Is it nurturing more Gauss-level geniuses? In that case, I admit I have nothing to offer: I have no idea how to do that. Or is it helping the hundreds of millions of kids globally who can't make sense of high school math or calculus? Well, in that case, I do think that they'd be in much better shape if the noxious idea that it's mostly a matter of genes wasn't so widespread (among them, among their peers, among their parents and teachers.) This would create more awareness and more willingness to make progress on the next topic:
2/ "it's cool if there are reliable methods of improving particular skill"
My book is my best attempt at documenting "secret math", the inner cognitive and emotional approach that is conducive to the development of a stronger mathematical intuition. I'm not saying that my book is perfect, but I do hope that it's a substantial step forward. At least a good number of credible people think so, and I receive a lot of reader feedback from students and teachers who claim it's been extremely helpful to them.
I'm confident that many more people will advance these ideas (most of which predate my book), adapt them and implement them at scale.
I honestly believe that our current incapacity to bring most of a generation to, say, understand calculus in the same way they can read or write (which is far from perfect) is a transient historical situation, and that 50 years from it'll quite different.
Which brings us back to the crucial question of addressing the noxious hereditarian myths. Check out the Hacker News discussion on my Quanta interview (https://news.ycombinator.com/item?id=42200209) and you'll see how prevalent these myths are, to the point where it's almost impossible to hold a reasonable conversation on these topics.
3/ stochasticity & co
Honestly this is a casual exposition piece, and all notions of randomness should be understood in the weakest epistemological sense. I would have phrased it differently if I had wanted to open yet another epistemological front on causality/agency/indeterminacy/etc. — honestly I have more than enough on my plate with the topics at stake in my book, so I won't elaborate.
are you aware of any research or pedagogical interventions based on your book?
Not at this point. I expect that it will take some time (= several years) for something structured & quantified to take place. Right now I'm receiving a lot of individual feedback from teachers (and students) who tell me the book has influenced them (I've also given some invited lectures at teacher training conferences), but I have no hard figures to show about applicability at scale and controlled studies.
I'm not an education science professional and don't claim to be one. My book is about the essence of math and the subjective process of making sense of it — which is already enough on my plate. My book aims at culture change and lacks practical implementation details for teachers (I chose to focus on the students' perspective, as it concerns more people, and it is closer to that of research mathematicians).
I'm not competent to craft intervention scripts, which will have to vary a lot depending on grade and environment. But I do hope that my book will inspire enough teachers to turn the core ideas into concrete actions (a difficult but extremely important job).
It's a long game — we've been wrong about math for millennia, and it'll take decades for us to reposition it. Whatever the interventions, they'll succeed only if students accept that math isn't what it's long been considered to be.
I volunteer as the hands on mathematics designer/builder/facilitator for several STEM after school programs and summer camps. We offer our programs as helping to reduce "math anxiety" which appears to be a measurable thing in social science. This also keeps the math teachers happy that we aren't advocating to change their curriculum.
Thanks for the great article and the book.
Thank you Ed, that's fantastic news!
Here is the summary of an RCT on math anxiety by John Hopkins
"Indexing Crazy 8s: Measuring the Effects of an Extra-Curricular Math Experience
on Children’s Math Attitudes"
https://s39684.pcdn.co/wp-content/uploads/2023/04/Crazy8sFinalReport_revised.pdf
Spot on!
There is some innate variance in the efficiency of the neuron-machinery, but the talent gap primarily stems for the monstrous variance in attitude, exacerbated by the absence of a proper cultural framework to communicate this essential aspect of math.