"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."
But in sprinting that is huge. His world record is extraordinary. It is 0.11 seconds faster than the 2nd fastest 100m of all time (by Tyson Gay). And to explain how huge that is, to get a 0.11 second gap from Tyson Gay, you have to go down the next 13 fastest men. From Tyson Gay down, it resembles how fastest times normally appear. If you look at other athletics events, you don't see the same pattern with the top 10 athletes. The gaps from top to 2nd are no more than the next 2 or 3.
We could talk about type 2 fibres or Jamaican sprinting but Yohan Blake and Asafa Powell have those. What Blake and Powell don't have is a height of 6'5". And until Bolt, no-one who was 6'5" was a sprinter. 6'2" was as far as it went. 6'5" has disadvantages. But Bolt has been studied. He has one leg shorter than the other (probably from scoliosis), and runs asymetrically and he twists as he runs and it's thought this fixes the problem. He is mostly about this strange combination that is genetic.
That's exactly my point. If cognitive talent was primarily constrained by genetics and molecular pathways, you'd observe something similar to what you observe in the 100m dash.
But that's not what I'm getting from what you said. You are implying that 1.5% extra speed is small in a 100m final, when it isn't. Olympic finals are normally won by something between 0.01 and 0.05 seconds, not 0.11 seconds. 1.5% in a lot of fields would be normal but it's huge here. The normal variance of the best sprinters, of runners like Blake, Powell, Gay is tiny. Because the genetics are very similar, the dedication is very similar, the support is very similar. Bolt didn't train as hard as these other men and still won by an enormous time because of his genetic traits.
But that's also the variance of 100m runners. Who not only have the genetics for it but then apply themselves to improving themselves at it. As someone who was hopeless at running I not only have bad genetic times, but I also don't practice running. It's not fun to do. I swim, I do some weights. I'm genetically good at math, but I also then applied myself to doing math puzzles as a child. Which I think that children with poor maths genes don't do. If they're storytellers they write stories, if they're artists, they paint.
And we notice the math differences because it concerns us. Math is very useful in the modern world. In 1000 BC Namibia, running would have been of greater concern. You have to hunt, you fight battles on foot.
No no, 1.5% is HUGE for the 100m dash. This is my whole point. A percentage that is huge for the 100m dash is tiny for math talent. This means that math talent builds up through an entirely different process.
Note that I'm not saying that the 100m speed is just about genetics. What I'm saying is that, from the distribution alone, you cannot say whether it's 10% or 50% or 99% genetic. The genetic signal is Gaussian, but many real-life non-genetic signals are Gaussian too, and a Gaussian + a Gaussian is again a Gaussian. That's the case for height, where the non-genetic part is Gaussian too.
It's pretty clear that top sprinters need to have had a healthy childhood without invalidating conditions, and need to practice immensely.
My point is causal in the other way: when the outcome isn't Gaussian, then you can rule out the 99% heritability *without even knowing more about the details of how it works in real-life*.
Despite your response to this, the answer here is that the genetic effect is non-linear. The fact that standard models of heritability assume linearity is irrelevant. The reason they do that is because nonlinear genetic interactions are too hard to observe: in the sort of data we have access to they are indistinguishable from non-genetic environmental effects. They are also insensitive to selection. But the fact that it’s unrewarding to model them doesn’t mean they don’t exist.
When a talent requires multiple distinct capacities, each of which is heritable via linear and additive genetic factors, but where the facility with the talent is a nonlinear function of the capacities, you can very easily get a pareto distribution. And that seems like a pretty plausible model for math aptitude.
BTW I think that this is compatible with there being a very substantial non-genetic component in mathematical achievement.
Don't get me wrong, I really think the "cognitive phenotype" is highly non-linear.
What I'm skeptical of is whether one can build a highly non-linear polygenic score with a high-covariance (above 70%) with the "cognitive phenotype".
In other words: I'm 100% with you on "talent is a nonlinear function of the capacities", but I suspect that the bulk of the non-linearity comes from life itself, and not from epistasis per se.
4. It doesn't matter whether all of genetic lit makes the linear genome expression assumption, it makes no sense. You cannot talk about power laws and assume that the violations will remain “local”.
About 1 and 2: it might as be log-normal, in fact it is too uncalibrated for us to tell these options apart. The wealth metaphor does reflect my perception of math as a capitalization journey, but I won't argue for it beyond the metaphor as, again, I don't want to overstate the (inexisting) calibration of math talent.
3: Yes, Tao (like Descartes, Newton, Einstein, Grothendieck) has taken powerful stances against the innate talent myth. Thanks for the link.
4: I think the broader issue is whether "math talent" (whatever it means) is a proper "phenotype" in the biophysical sense. The "roughly linear" model makes sense at the molecular and tissue level. It make no sense at the level of the neural connectome.
Let's explore what it means to say that genes can or can't 'explain' genius.
I interpret 'explain' here to imply that the game changing competence we call genius stems directly from some genetic contribution.
But that in turn leads to recognizing that explaining 'genius' is really explaining the outcomes of a competence we call 'genius.'
We can't see genius. We can just recognise its outomes.
So the question becomes - is saying genes can't explain genius the same as saying genes can't explain the outcomes we recognize as markers of what we call genius?
In a trivial sense this is obviously true. There is no direct link between genes and a proof.
So we can only mean that genes have endowed someone with some competence or competences that enabled those outcomes.
Or perhaps a propensity to acquire such competence.
And a makeup that embraces curiosity and hard work.
And...
Do we even really know what those competences are?
What was it about Morphy that resulted in chess skills that were light years ahead of his peers?
What is it about the latest chess engines that makes them crushingly stronger than any human players.
"We can't see genius. We can just recognise its outcomes. So the question becomes - is saying genes can't explain genius the same as saying genes can't explain the outcomes we recognize as markers of what we call genius?"
Indeed. Once you define genius as a pragmatic category, there is no such thing as "unrealized genius" — there are only "manifestations of genius".
So what is the explanation for people like Von Neumman or Einstein? Are you saying that it's a result of a whole bunch of different factors, so that there's no monocausal explanation?
What I like about Thurston's story is that it illustrate why it might be both monocausal (in the sense that one anomaly triggered a whole chain of events) and multifactorial and messy (Thurston's mother had a unique personality and mindset, and Thurston reacted in a unique way).
The propensities of the seed depend on the soil it lands on. Most children ask questions and are ignored. It is a wonder they acquire speech . Some parents teach their children writing and arithmetic alongside speaking. Von Neumann and Tao had such parents. Ramanujan and Gauss didn't benefited from that. Maybe the propensities of their consciousness received what they needed when they needed it. I mean in terms of knowledge and emotional environment. No matter how persistent a new consciousness is if it lands in a unwelcoming land it will perish or will grow malignant. Mathematicians are renown for being evil cold sadistic even. The skill they acquired was hard work and represented an advantage.
"Mathematics" originates from the Greek word "máthēma" (μάθημα), which means "what one learns," "what one gets to know". So it is a skill '' a bag of tricks'' in relation with the regularities of the reality. Because reality behaves with regularity. So knowing mathematics is like having money. Having a skill is the difference between death and life hence the character of the most mathematicians - tenacity and Asperger-ism. ''Moscow does not believe in tears'' and neither most mathematicians because mathematics is the skill that is learned in direct relation with survival.
This though does not mean that mathematicians understand what they are actually doing.
It is a matter of acceptance of the way of the reality to be an almost competent mathematician.
Do you have any thoughts on how to work the Yule dynamics to your own favor? How do you attract more skill to yourself so to speak? Is this broadly the core topic of your book, that we should develop much better representational capacities so that we can grok concepts and ideas much faster, and so exploit the rich-get-richer dynamics?
Yes, this is in a broad way what my book is about.
I think *realizing* that there is a Yule process is already a major step, as you can use this as a motivator: the talent gap looks unsurmountable, and it probably is, yet you have an *insane* margin of progression and it'll get easier over time, like all capitalization processes.
Then one should look at how smart people think and behave, and how they explain their attitude, as their advantage is often there: a peculiar way to process information and react to it. Eg, this is the way I read Descartes's Discourse on Method, and Grothendieck's Harvests and Sowings: as self-improvement / martial art / yoga books.
It is also essential to take our emotional state and personal environment very seriously, as self-confidence is one of the key dimensions that gets capitalized along the way. Here again, there are actual lessons to learn from top mathematicians, on how they face their doubts and incomprehension, and how they learn to welcome their own mistakes.
That's what I mean when I say that the Yule process intuition (which I only formalized much later) helped me open my eyes: it made me take seriously the advice that smart people were giving, instead of brushing it off on the grounds that they were "different".
This is a really interesting topic that I don't fully feel comfortable commenting on, But I have at least a couple of things to say: First, there are at least two types of distributions that exhibit the multiple orders of magnitude differences you mention, true power-law distributions and log-normal distributions. I've dabbled in this a bit, and the one area I've taken a serious look at (distribution of earnings on Patreon) is almost certainly a log-normal distribution (rpubs.com/frankhecker/1025912). The typical explanation for log-normal distributions is that they result from the multiplicative effect of many relatively independent factors (each of which may be normally distributed to one extent or another).
A couple of other commenters have mentioned factors that might go into forming mathematical genius other than raw IQ, including the ability to visualize and "creativity" broadly defined. I can think of analogies in other fields; for example, for women, becoming a top supermodel depends on not just height (lots of women are relatively tall), weight (lots of women are relatively thin), and general attractiveness (lots of women are relatively attractive) but on having a particular combination of bodily dimensions both large-grained and fine-grained, along with a particular type of personality, way they can move, conformation to what is considered "beautiful" in a particular society and a particular time, and so on.
So, I'm provisionally prepared to treat mathematical "genius" as the result of multiple factors operating relatively independently, some genetically determined with IQ-level heritability and some not, with the resulting orders of magnitude differences sometime being apparent at a relatively early age and sometimes developing over time. I can also see the worth of your argument that mathematicians can through particular practices raise themselves to higher levels than one might expect from their raw IQ, by concentrating on other things (like visualization) that factor into overall success. (Another real-world analogy: Taylor Swift is a mid-level vocal talent, but she exhibited strong songwriting abilities at an early age, and she was a quick and proficient student when it came to learning the ins and outs of the music business.)
It is correct that power laws create massive gaps like the ones observed in the genius phenomenon.
But when you multiplicatively combine multiple factors, the outcome is unstable and the deterministic-signal-to-noise ratio collapses, especially since the factors combine through a "journey" life process.
Your example of Taylor Swift points exactly at the logic of Yule processes: the right combination of skills and luck, of access and charm, of being at the right place at the right time with the right abilities, the right support and the right ethos, can lead to superhuman outcomes.
this does not imply determinism is false. it may be true that in order to become a field medalist(or at least have a reasonable shot at being one) you would have to whatever top percentile in math talent but it isn't sufficient to guarantee anything. but without that necessary threshold you basically will fail even if you have all the other right ingredients. maybe from a pure prediction pov, that's not determinism, but for the sense in which people talk about ability it is! without it you fail!
"to become a field medalist you would have to whatever top percentile in math talent"
But this may have little to do with genetics. It is unquestionable that math talent can manifest itself from an early age, but a lot has already happened back then in terms of cognitive development.
Plus, no-one really knows in which percentage of math talent one lies until this talent has had a chance to manifest itself.
One thing I noticed from your book is that you put a lot of emphasis on visualizing. For example, you propose that the "simplest" way to prove 1 + 2 + ... + n = n(n+1)/2 is to visualize blocks and mentally manipulate them until you get a rectangle, and you have several examples like this.
What about people with aphantasia (https://en.wikipedia.org/wiki/Aphantasia)? This seems like it could be a very real impediment to understanding mathematics, especially if you think the key is to make mental images, and it seems like it is genetic.
I do emphasize visualization, because it's easy to communicate and memorable — so memorable that the non-visual examples of the book are less easily remembered (eg, the simpler, probabilistic proof of the same result). But mathematical intuition doesn't have to be visual.
As for aphantasia, from what I understand, its origin still is unclear, and it seems to be retrainable to a certain extent.
[Full disclosure: throughout my math career, I feel that I substantially expanded by phantasic abilities by systematic training—which again explains my skepticism against the notion that it should be hard-coded.]
I feel like most of us are unable to tap into our capabilities due to our inherent nature of our minds beautifully depicted in a verse https://asitis.com/6/34.html from my favorite book Bhagwat #Gita :
For the mind is restless, turbulent, obstinate and very strong, O Krsna, and to subdue it is, it seems to me, more difficult than controlling the wind.
but there is a solution #yoga provided there as well which does help but then again mind rebels 🤪
I'm not sure I understand the argument. Suppose mathematical progress tends to be exponential: it's proportional to what you already understand. Now suppose that the maximum possible rate at which people can make mathematical progress is distributed according to a Gaussian. Then we'd expect to see exactly what we do see: anyone in the tail will accumulate knowledge at speeds which are orders of magnitude higher than others.
When you talk about your own personal experience, I can understand where you're coming from. It's true that no one should be a determinist when thinking about their own life (or at all, in my humble opinion). You'll never know what your full potential is until you just take a chance and try to reach it. But that doesn't mean that there aren't very real inequalities between different individuals' cognitive potential. It sure looks like it's determined by *something,* whether or not it's purely genetic.
Thanks for the feedback. Let me try to explain this as it is a core aspect.
In this perspective, what we perceive (the measured phenotype) is the exponential mathematical understanding, not the genetic Gaussian learning rate. But the expression mechanism (the lifelong capitalization journey of expanding your understanding) is by nature a messy process influenced by many external factors and chance events. This expression noise will substantially reduce the heritability.
This is illustrated by Joseph Bronski's simulation (https://x.com/BronskiJoseph/status/1888315043353153642) which starts from an unrealistically high 100% heritability (before the lifelong learning mechanism) and ends up with 64% heritability.
Moreover, this perspective doesn't even capture a major feature, illustrated by the Hugo vs Terry example: for mathematicians, it feels like the *learning speed* itself is influence by what you have learned (both math itself and acquired confidence / metamathematical insights / methodology) — because a Gaussian learning speed would never yield a 10x factor.
Which means the real life is much messier than your perspective and Joseph Bronski's simulation.
So, yes, it's possible for a genetically-influenced trait to show Pareto-like variability, but the expression mechanism from polygenic score to phenotype introduces a LOT OF NOISE that drastically reduces heritability. In the end, the top performers will be the ones with the luckiest expression journey, rather than the luckiest polygenic score.
It's a messy process, yes, fair enough. And yet, certain people seem to have a mysteriously consistent level of "luck" throughout their careers. I'm not convinced it's wrong to attribute that to something "in" them--talent, drive, calling, whatever. As you say, this is all very difficult to talk about precisely. In any case, if your goal is to get people to be less fatalist about their chances of success in math, I am totally on board with that.
It's not that they have a mysteriously consistent level of "luck" throughout their careers — it's indeed that they have become "mathematically smarter" in a persistent manner. So, yes, it is really "in them".
My point is that this thing "in them" can't be a biochemical parameter (like hormone level or neurotransmitter sensitivity) or a biomechanical parameter (like the twich fiber ratio of their muscles or the size of this or that region of their brain), because these things have the wrong scaling laws.
This thing must rather be associated with their "mental habits" and the internal organization of their brain. It only happened in them only because the brain is immensely malleable, especially in the early years of life, and they went through an unusual developmental pathway (see https://davidbessis.substack.com/p/beyond-nature-and-nurture).
In the example of Terry Tao, it is well document that he was already mind-blowing at age 2. So the one in a million chain of events that made him a *candidate* for becoming a one in a billion mathematician had already occurred.
This fluke of chance didn't systematically "re-happen" throughout his life, it had happened once was permanently capitalized in him.
There are things we can learn and imitate from top mathematicians, that will make us better at math, but that can't transform us into top mathematicians unless they occur in prime infancy.
"In the example of Terry Tao, it is well document that he was already mind-blowing at age 2. So the one in a million chain of events that made him a *candidate* for becoming a one in a billion mathematician had already occurred."
But this doesn't help us better understand what the genetic contribution was does it? The fact we have no idea what that contribution was doesn't mean there wasn't one. Why should Tao not be genetically gifted in a way that works for math just as Bolt was genetically gifted in a way that worked for running.
I'm not sure I'm following what you are really claiming.
My claim is based on the shape of the "genius distribution".
We don't know the genetic factors that contribute to genius, and we don't understand the chains of post-fecundation events that result in genius. But if the dominant factors were genetic, the *statistical distribution* of extreme capability would follow a Gaussian distribution, not a Pareto distribution.
While this doesn't help pinpointing the exact causal chain, it gives a better understanding of what's at stake (and what's not). Specifically, the distribution of outcomes "looks like" it is caused by a capitalization process. This is very important aspect of math cognition, which most career mathematicians agree is essential to understand (and most students fail to fully recognize.)
I think the problem is that you are reading the statement
'he's a genius'
as a statement of wonder that implies it stems 'solely' from some genetic gift
but is that fair?
that isn't what is being said
it is a description of a present state not an explanation
I'm sure that many, perhaps most, people think that inheritance plays a major part, perhaps to some the gift is so alien that it is wholly a gift from god/inheritance] so basically the whole part
but this isn't a necessary interpretation]
if you explored the math of springers what distribution would you expect to find?
I don't understand why claims about genius are incompatible with high heritability of IQ. After all genius is not just IQ. You know Einstein and Grothendieck were huge "geniuses", even moreso than von Neumann who had the highest IQ.
I wrote "probably incompatible with the absurdly high figures (above 80%)" that some assert to be the scientific consensus. I don't think it's a relevant topic, but thought I had to mention this aspect as it would otherwise have come as an objection. Here's the reasoning behind my comment:
If IQ really was 94% heritable (which the linked post assert is a possibility), then IQ would be established as a mind-blowingly robust measure of cognitive capacity, as it would have a direct biological interpretation (at this point, IQ is only a mildly relevant measure of cognitive capacity). Now if the population variance of a mind-blowingly robust measure of cognitive capacity is 94% explained by genetics, it's honestly hard to imagine that other measures of cognitive performance wouldn't be massively determined by genetics.
This is not hard for me to imagine at all! We know von Neumann's daughter is an accomplished scientist, but Grothendieck's five kids weren't. I strongly suspect that the non-g things like creativity are far less heritable than g/IQ. We know personality is significantly less heritable than IQ. To give you an analogy, height is extremely heritable. The other parts of being a good basketball probably have some heritability too but not nearly as much. Work ethic, say? The time of the year you are born? Perhaps that affects a lot of things. There are things like that for soccer. Birth order effects? The neighborhood you grow up and how much basketball is played there? It would make a lot of sense that g is 80% heritable in adulthood but the non-g aspects of mathematical ability are only 20% heritable. In fact I would guess that something like that is the case.
I personally would have a hard time combining these two facts (high cognitive functioning is indeed a prerequisite for Grothendieck-level creativity, albeit a far from sufficient one.)
But I won't die on this hill, especially since this aspect isn't central to the discussion.
I think everything you say is in fact compatible with the most genetic determinist positions in behavioral genetics. Just as discussion of non-height aspects of basketball ability are correct even though heritability of height is very good.
You don’t want to tether your correct claims about mathematical ability to possibly true and possibly false claims about behavioral genetics.
"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."
But in sprinting that is huge. His world record is extraordinary. It is 0.11 seconds faster than the 2nd fastest 100m of all time (by Tyson Gay). And to explain how huge that is, to get a 0.11 second gap from Tyson Gay, you have to go down the next 13 fastest men. From Tyson Gay down, it resembles how fastest times normally appear. If you look at other athletics events, you don't see the same pattern with the top 10 athletes. The gaps from top to 2nd are no more than the next 2 or 3.
We could talk about type 2 fibres or Jamaican sprinting but Yohan Blake and Asafa Powell have those. What Blake and Powell don't have is a height of 6'5". And until Bolt, no-one who was 6'5" was a sprinter. 6'2" was as far as it went. 6'5" has disadvantages. But Bolt has been studied. He has one leg shorter than the other (probably from scoliosis), and runs asymetrically and he twists as he runs and it's thought this fixes the problem. He is mostly about this strange combination that is genetic.
That's exactly my point. If cognitive talent was primarily constrained by genetics and molecular pathways, you'd observe something similar to what you observe in the 100m dash.
But that's not what I'm getting from what you said. You are implying that 1.5% extra speed is small in a 100m final, when it isn't. Olympic finals are normally won by something between 0.01 and 0.05 seconds, not 0.11 seconds. 1.5% in a lot of fields would be normal but it's huge here. The normal variance of the best sprinters, of runners like Blake, Powell, Gay is tiny. Because the genetics are very similar, the dedication is very similar, the support is very similar. Bolt didn't train as hard as these other men and still won by an enormous time because of his genetic traits.
But that's also the variance of 100m runners. Who not only have the genetics for it but then apply themselves to improving themselves at it. As someone who was hopeless at running I not only have bad genetic times, but I also don't practice running. It's not fun to do. I swim, I do some weights. I'm genetically good at math, but I also then applied myself to doing math puzzles as a child. Which I think that children with poor maths genes don't do. If they're storytellers they write stories, if they're artists, they paint.
And we notice the math differences because it concerns us. Math is very useful in the modern world. In 1000 BC Namibia, running would have been of greater concern. You have to hunt, you fight battles on foot.
No no, 1.5% is HUGE for the 100m dash. This is my whole point. A percentage that is huge for the 100m dash is tiny for math talent. This means that math talent builds up through an entirely different process.
Note that I'm not saying that the 100m speed is just about genetics. What I'm saying is that, from the distribution alone, you cannot say whether it's 10% or 50% or 99% genetic. The genetic signal is Gaussian, but many real-life non-genetic signals are Gaussian too, and a Gaussian + a Gaussian is again a Gaussian. That's the case for height, where the non-genetic part is Gaussian too.
It's pretty clear that top sprinters need to have had a healthy childhood without invalidating conditions, and need to practice immensely.
My point is causal in the other way: when the outcome isn't Gaussian, then you can rule out the 99% heritability *without even knowing more about the details of how it works in real-life*.
Despite your response to this, the answer here is that the genetic effect is non-linear. The fact that standard models of heritability assume linearity is irrelevant. The reason they do that is because nonlinear genetic interactions are too hard to observe: in the sort of data we have access to they are indistinguishable from non-genetic environmental effects. They are also insensitive to selection. But the fact that it’s unrewarding to model them doesn’t mean they don’t exist.
When a talent requires multiple distinct capacities, each of which is heritable via linear and additive genetic factors, but where the facility with the talent is a nonlinear function of the capacities, you can very easily get a pareto distribution. And that seems like a pretty plausible model for math aptitude.
BTW I think that this is compatible with there being a very substantial non-genetic component in mathematical achievement.
Don't get me wrong, I really think the "cognitive phenotype" is highly non-linear.
What I'm skeptical of is whether one can build a highly non-linear polygenic score with a high-covariance (above 70%) with the "cognitive phenotype".
In other words: I'm 100% with you on "talent is a nonlinear function of the capacities", but I suspect that the bulk of the non-linearity comes from life itself, and not from epistasis per se.
Hi David, have replied to your original tweet:
1. If what you're writing is true, then it 'seems' that "At the extreme right tail, math genius might behave like wealth".
2. Have you considered log-normal?
3. Here's Tao's dismissing the notion of genius and 'innate ability' in Math.
https://terrytao.wordpress.com/career-advice/does-one-have-to-be-a-genius-to-do-maths/
4. It doesn't matter whether all of genetic lit makes the linear genome expression assumption, it makes no sense. You cannot talk about power laws and assume that the violations will remain “local”.
Thank you Pavan.
About 1 and 2: it might as be log-normal, in fact it is too uncalibrated for us to tell these options apart. The wealth metaphor does reflect my perception of math as a capitalization journey, but I won't argue for it beyond the metaphor as, again, I don't want to overstate the (inexisting) calibration of math talent.
3: Yes, Tao (like Descartes, Newton, Einstein, Grothendieck) has taken powerful stances against the innate talent myth. Thanks for the link.
4: I think the broader issue is whether "math talent" (whatever it means) is a proper "phenotype" in the biophysical sense. The "roughly linear" model makes sense at the molecular and tissue level. It make no sense at the level of the neural connectome.
Let's explore what it means to say that genes can or can't 'explain' genius.
I interpret 'explain' here to imply that the game changing competence we call genius stems directly from some genetic contribution.
But that in turn leads to recognizing that explaining 'genius' is really explaining the outcomes of a competence we call 'genius.'
We can't see genius. We can just recognise its outomes.
So the question becomes - is saying genes can't explain genius the same as saying genes can't explain the outcomes we recognize as markers of what we call genius?
In a trivial sense this is obviously true. There is no direct link between genes and a proof.
So we can only mean that genes have endowed someone with some competence or competences that enabled those outcomes.
Or perhaps a propensity to acquire such competence.
And a makeup that embraces curiosity and hard work.
And...
Do we even really know what those competences are?
What was it about Morphy that resulted in chess skills that were light years ahead of his peers?
What is it about the latest chess engines that makes them crushingly stronger than any human players.
You can't pin it down to one thing.
I'm tying myself up in knots here.
It's hard.
We need a genius to figure it out for us.
"We can't see genius. We can just recognise its outcomes. So the question becomes - is saying genes can't explain genius the same as saying genes can't explain the outcomes we recognize as markers of what we call genius?"
Indeed. Once you define genius as a pragmatic category, there is no such thing as "unrealized genius" — there are only "manifestations of genius".
But there is certinly such a thing as unrealized talent :)
So what is the explanation for people like Von Neumman or Einstein? Are you saying that it's a result of a whole bunch of different factors, so that there's no monocausal explanation?
I don't know enough about them, but here's my favorite model: https://davidbessis.substack.com/p/beyond-nature-and-nurture
What I like about Thurston's story is that it illustrate why it might be both monocausal (in the sense that one anomaly triggered a whole chain of events) and multifactorial and messy (Thurston's mother had a unique personality and mindset, and Thurston reacted in a unique way).
The propensities of the seed depend on the soil it lands on. Most children ask questions and are ignored. It is a wonder they acquire speech . Some parents teach their children writing and arithmetic alongside speaking. Von Neumann and Tao had such parents. Ramanujan and Gauss didn't benefited from that. Maybe the propensities of their consciousness received what they needed when they needed it. I mean in terms of knowledge and emotional environment. No matter how persistent a new consciousness is if it lands in a unwelcoming land it will perish or will grow malignant. Mathematicians are renown for being evil cold sadistic even. The skill they acquired was hard work and represented an advantage.
"Mathematics" originates from the Greek word "máthēma" (μάθημα), which means "what one learns," "what one gets to know". So it is a skill '' a bag of tricks'' in relation with the regularities of the reality. Because reality behaves with regularity. So knowing mathematics is like having money. Having a skill is the difference between death and life hence the character of the most mathematicians - tenacity and Asperger-ism. ''Moscow does not believe in tears'' and neither most mathematicians because mathematics is the skill that is learned in direct relation with survival.
This though does not mean that mathematicians understand what they are actually doing.
It is a matter of acceptance of the way of the reality to be an almost competent mathematician.
Do you have any thoughts on how to work the Yule dynamics to your own favor? How do you attract more skill to yourself so to speak? Is this broadly the core topic of your book, that we should develop much better representational capacities so that we can grok concepts and ideas much faster, and so exploit the rich-get-richer dynamics?
Yes, this is in a broad way what my book is about.
I think *realizing* that there is a Yule process is already a major step, as you can use this as a motivator: the talent gap looks unsurmountable, and it probably is, yet you have an *insane* margin of progression and it'll get easier over time, like all capitalization processes.
Then one should look at how smart people think and behave, and how they explain their attitude, as their advantage is often there: a peculiar way to process information and react to it. Eg, this is the way I read Descartes's Discourse on Method, and Grothendieck's Harvests and Sowings: as self-improvement / martial art / yoga books.
It is also essential to take our emotional state and personal environment very seriously, as self-confidence is one of the key dimensions that gets capitalized along the way. Here again, there are actual lessons to learn from top mathematicians, on how they face their doubts and incomprehension, and how they learn to welcome their own mistakes.
That's what I mean when I say that the Yule process intuition (which I only formalized much later) helped me open my eyes: it made me take seriously the advice that smart people were giving, instead of brushing it off on the grounds that they were "different".
This is widely recognized in chess.
Kasparov often referred to it:
“Hard work is a talent. The ability to keep trying when others quit is a talent.”
“I am a great believer in luck and I find the harder I work, the more I have of it.”
“Steady effort pays off, even if not always in an immediate, tangible way.”
This is a really interesting topic that I don't fully feel comfortable commenting on, But I have at least a couple of things to say: First, there are at least two types of distributions that exhibit the multiple orders of magnitude differences you mention, true power-law distributions and log-normal distributions. I've dabbled in this a bit, and the one area I've taken a serious look at (distribution of earnings on Patreon) is almost certainly a log-normal distribution (rpubs.com/frankhecker/1025912). The typical explanation for log-normal distributions is that they result from the multiplicative effect of many relatively independent factors (each of which may be normally distributed to one extent or another).
A couple of other commenters have mentioned factors that might go into forming mathematical genius other than raw IQ, including the ability to visualize and "creativity" broadly defined. I can think of analogies in other fields; for example, for women, becoming a top supermodel depends on not just height (lots of women are relatively tall), weight (lots of women are relatively thin), and general attractiveness (lots of women are relatively attractive) but on having a particular combination of bodily dimensions both large-grained and fine-grained, along with a particular type of personality, way they can move, conformation to what is considered "beautiful" in a particular society and a particular time, and so on.
So, I'm provisionally prepared to treat mathematical "genius" as the result of multiple factors operating relatively independently, some genetically determined with IQ-level heritability and some not, with the resulting orders of magnitude differences sometime being apparent at a relatively early age and sometimes developing over time. I can also see the worth of your argument that mathematicians can through particular practices raise themselves to higher levels than one might expect from their raw IQ, by concentrating on other things (like visualization) that factor into overall success. (Another real-world analogy: Taylor Swift is a mid-level vocal talent, but she exhibited strong songwriting abilities at an early age, and she was a quick and proficient student when it came to learning the ins and outs of the music business.)
It is correct that power laws create massive gaps like the ones observed in the genius phenomenon.
But when you multiplicatively combine multiple factors, the outcome is unstable and the deterministic-signal-to-noise ratio collapses, especially since the factors combine through a "journey" life process.
Your example of Taylor Swift points exactly at the logic of Yule processes: the right combination of skills and luck, of access and charm, of being at the right place at the right time with the right abilities, the right support and the right ethos, can lead to superhuman outcomes.
this does not imply determinism is false. it may be true that in order to become a field medalist(or at least have a reasonable shot at being one) you would have to whatever top percentile in math talent but it isn't sufficient to guarantee anything. but without that necessary threshold you basically will fail even if you have all the other right ingredients. maybe from a pure prediction pov, that's not determinism, but for the sense in which people talk about ability it is! without it you fail!
"to become a field medalist you would have to whatever top percentile in math talent"
But this may have little to do with genetics. It is unquestionable that math talent can manifest itself from an early age, but a lot has already happened back then in terms of cognitive development.
Plus, no-one really knows in which percentage of math talent one lies until this talent has had a chance to manifest itself.
Pretty sure that's true in most sport.
For elite results, if at first you don't succeed, give up
One thing I noticed from your book is that you put a lot of emphasis on visualizing. For example, you propose that the "simplest" way to prove 1 + 2 + ... + n = n(n+1)/2 is to visualize blocks and mentally manipulate them until you get a rectangle, and you have several examples like this.
What about people with aphantasia (https://en.wikipedia.org/wiki/Aphantasia)? This seems like it could be a very real impediment to understanding mathematics, especially if you think the key is to make mental images, and it seems like it is genetic.
I do emphasize visualization, because it's easy to communicate and memorable — so memorable that the non-visual examples of the book are less easily remembered (eg, the simpler, probabilistic proof of the same result). But mathematical intuition doesn't have to be visual.
As for aphantasia, from what I understand, its origin still is unclear, and it seems to be retrainable to a certain extent.
[Full disclosure: throughout my math career, I feel that I substantially expanded by phantasic abilities by systematic training—which again explains my skepticism against the notion that it should be hard-coded.]
I feel like most of us are unable to tap into our capabilities due to our inherent nature of our minds beautifully depicted in a verse https://asitis.com/6/34.html from my favorite book Bhagwat #Gita :
For the mind is restless, turbulent, obstinate and very strong, O Krsna, and to subdue it is, it seems to me, more difficult than controlling the wind.
but there is a solution #yoga provided there as well which does help but then again mind rebels 🤪
I'm not sure I understand the argument. Suppose mathematical progress tends to be exponential: it's proportional to what you already understand. Now suppose that the maximum possible rate at which people can make mathematical progress is distributed according to a Gaussian. Then we'd expect to see exactly what we do see: anyone in the tail will accumulate knowledge at speeds which are orders of magnitude higher than others.
When you talk about your own personal experience, I can understand where you're coming from. It's true that no one should be a determinist when thinking about their own life (or at all, in my humble opinion). You'll never know what your full potential is until you just take a chance and try to reach it. But that doesn't mean that there aren't very real inequalities between different individuals' cognitive potential. It sure looks like it's determined by *something,* whether or not it's purely genetic.
Thanks for the feedback. Let me try to explain this as it is a core aspect.
In this perspective, what we perceive (the measured phenotype) is the exponential mathematical understanding, not the genetic Gaussian learning rate. But the expression mechanism (the lifelong capitalization journey of expanding your understanding) is by nature a messy process influenced by many external factors and chance events. This expression noise will substantially reduce the heritability.
This is illustrated by Joseph Bronski's simulation (https://x.com/BronskiJoseph/status/1888315043353153642) which starts from an unrealistically high 100% heritability (before the lifelong learning mechanism) and ends up with 64% heritability.
Moreover, this perspective doesn't even capture a major feature, illustrated by the Hugo vs Terry example: for mathematicians, it feels like the *learning speed* itself is influence by what you have learned (both math itself and acquired confidence / metamathematical insights / methodology) — because a Gaussian learning speed would never yield a 10x factor.
Which means the real life is much messier than your perspective and Joseph Bronski's simulation.
So, yes, it's possible for a genetically-influenced trait to show Pareto-like variability, but the expression mechanism from polygenic score to phenotype introduces a LOT OF NOISE that drastically reduces heritability. In the end, the top performers will be the ones with the luckiest expression journey, rather than the luckiest polygenic score.
It's a messy process, yes, fair enough. And yet, certain people seem to have a mysteriously consistent level of "luck" throughout their careers. I'm not convinced it's wrong to attribute that to something "in" them--talent, drive, calling, whatever. As you say, this is all very difficult to talk about precisely. In any case, if your goal is to get people to be less fatalist about their chances of success in math, I am totally on board with that.
It's not that they have a mysteriously consistent level of "luck" throughout their careers — it's indeed that they have become "mathematically smarter" in a persistent manner. So, yes, it is really "in them".
My point is that this thing "in them" can't be a biochemical parameter (like hormone level or neurotransmitter sensitivity) or a biomechanical parameter (like the twich fiber ratio of their muscles or the size of this or that region of their brain), because these things have the wrong scaling laws.
This thing must rather be associated with their "mental habits" and the internal organization of their brain. It only happened in them only because the brain is immensely malleable, especially in the early years of life, and they went through an unusual developmental pathway (see https://davidbessis.substack.com/p/beyond-nature-and-nurture).
In the example of Terry Tao, it is well document that he was already mind-blowing at age 2. So the one in a million chain of events that made him a *candidate* for becoming a one in a billion mathematician had already occurred.
This fluke of chance didn't systematically "re-happen" throughout his life, it had happened once was permanently capitalized in him.
There are things we can learn and imitate from top mathematicians, that will make us better at math, but that can't transform us into top mathematicians unless they occur in prime infancy.
"In the example of Terry Tao, it is well document that he was already mind-blowing at age 2. So the one in a million chain of events that made him a *candidate* for becoming a one in a billion mathematician had already occurred."
But this doesn't help us better understand what the genetic contribution was does it? The fact we have no idea what that contribution was doesn't mean there wasn't one. Why should Tao not be genetically gifted in a way that works for math just as Bolt was genetically gifted in a way that worked for running.
I'm not sure I'm following what you are really claiming.
My claim is based on the shape of the "genius distribution".
We don't know the genetic factors that contribute to genius, and we don't understand the chains of post-fecundation events that result in genius. But if the dominant factors were genetic, the *statistical distribution* of extreme capability would follow a Gaussian distribution, not a Pareto distribution.
While this doesn't help pinpointing the exact causal chain, it gives a better understanding of what's at stake (and what's not). Specifically, the distribution of outcomes "looks like" it is caused by a capitalization process. This is very important aspect of math cognition, which most career mathematicians agree is essential to understand (and most students fail to fully recognize.)
I think the problem is that you are reading the statement
'he's a genius'
as a statement of wonder that implies it stems 'solely' from some genetic gift
but is that fair?
that isn't what is being said
it is a description of a present state not an explanation
I'm sure that many, perhaps most, people think that inheritance plays a major part, perhaps to some the gift is so alien that it is wholly a gift from god/inheritance] so basically the whole part
but this isn't a necessary interpretation]
if you explored the math of springers what distribution would you expect to find?
I don't understand why claims about genius are incompatible with high heritability of IQ. After all genius is not just IQ. You know Einstein and Grothendieck were huge "geniuses", even moreso than von Neumann who had the highest IQ.
I wrote "probably incompatible with the absurdly high figures (above 80%)" that some assert to be the scientific consensus. I don't think it's a relevant topic, but thought I had to mention this aspect as it would otherwise have come as an objection. Here's the reasoning behind my comment:
If IQ really was 94% heritable (which the linked post assert is a possibility), then IQ would be established as a mind-blowingly robust measure of cognitive capacity, as it would have a direct biological interpretation (at this point, IQ is only a mildly relevant measure of cognitive capacity). Now if the population variance of a mind-blowingly robust measure of cognitive capacity is 94% explained by genetics, it's honestly hard to imagine that other measures of cognitive performance wouldn't be massively determined by genetics.
This is not hard for me to imagine at all! We know von Neumann's daughter is an accomplished scientist, but Grothendieck's five kids weren't. I strongly suspect that the non-g things like creativity are far less heritable than g/IQ. We know personality is significantly less heritable than IQ. To give you an analogy, height is extremely heritable. The other parts of being a good basketball probably have some heritability too but not nearly as much. Work ethic, say? The time of the year you are born? Perhaps that affects a lot of things. There are things like that for soccer. Birth order effects? The neighborhood you grow up and how much basketball is played there? It would make a lot of sense that g is 80% heritable in adulthood but the non-g aspects of mathematical ability are only 20% heritable. In fact I would guess that something like that is the case.
Fair enough.
I personally would have a hard time combining these two facts (high cognitive functioning is indeed a prerequisite for Grothendieck-level creativity, albeit a far from sufficient one.)
But I won't die on this hill, especially since this aspect isn't central to the discussion.
=> I removed the paragraph.
Thanks for the feedback.
I think everything you say is in fact compatible with the most genetic determinist positions in behavioral genetics. Just as discussion of non-height aspects of basketball ability are correct even though heritability of height is very good.
You don’t want to tether your correct claims about mathematical ability to possibly true and possibly false claims about behavioral genetics.
I remembered that I watched a few videos saying something close to what you're saying about a year ago
(https://youtu.be/xSL7wSJNURI?feature=shared)
In this video from 2016 he discusses much more deeper about it, but it's exclusive for channel members (https://youtu.be/u9dXkSmfldo?feature=shared)
I think you can activate auto-translate captions to see it.