I strongly believe that the average human being can be exceptional in any niche topic given enough time, dedication and focus.

The author of the book has picked out mathematics because that was what he was interested in. The reality is that this rule applies to everything.

The belief that some people have an innate skill that they are born with is deeply unhelpful. Whilst some people (mostly spectrum) do seem have an innate talent, I would argue that it is more an inbuilt ability to hyper focus on a topic, whether that topic be mathematics, Star Trek, dinosaurs or legacy console games from the 1980’s.

I think we do our children a disservice by convincing them that some of their peers are just “born with it”, because it discourages them from continuing to try.

What we should be teaching children is HOW to learn. At the moment it’s a by-product of learning about some topic. If we look at the old adage “feed a man a fish”, the same is true of learning.

“Teach someone mathematics and they will learn mathematics. Teach someone to learn and they will learn anything”.

This gives you vicious and virtuous cycles: Students' learning speed increases with time and past success. So "talented" students learn quickly and have extra time to further explore and improve, leading to further success. Students who struggle with the time constraint are forced to take shortcuts like memorizing "magic formulas" without having time to really understand. Trying to close that gap is very hard work.

Reminds me of my personal peeve that "studying" should not be "being taught", studying is pursuit of understanding, "being taught" is what happens in primary school (and I'm aware I'm simplifying here).

This sets them up for fantastic success very quickly. [1] shows a great example of this.

I'm fond of saying "You can do anything you want, but wanting is the hard part", because to truly be a grandmaster, genius-level mathematician, olympic athlete, etc, requires a dedication and amount of preparation that almost nobody can manage. Starting late, with emotional baggage, kids, and having to spend 5 years relearning how to learn? Forget it.

1. https://danielkarim.com/how-to-become-a-genius-the-polgar-ex...

I was having a problem agreeing with this subthread, and I have you to thank for putting it into words that I can finally formulate my disagreement against.

Have you never met one of those people for whom they did not need to "want"? They could literally phone it in and still do better than anyone else, no matter how dedicated they were. Even should practice/study be necessary for them, they benefited from it to some absurd proportion that I couldn't even guess to quantify. I've known more than one of these people.

I think most believe they don't exist for two reasons. The first is the ridiculous number of television shows and movies that depict motivation as being the key to success. We're just inundated with the (unsupported by evidence) that this is the means to extraordinary genius. Second, I would say that this is the most comforting theory. "Why yes, I could have been a gifted whatever or a talented something-or-other if I had put the time in, but I chose this other thing instead."

Maybe some would say we all need to believe this, that a society that doesn't believe in it is harsher or more unkind.

They're just quick. But the ones I've met, at least, are quick to make associations. When I really dig and ask them to explain themselves or a concept, they usually make analogies to things they know, but I don't. Then I have to go learn that thing. Then they try the analogy again, but I haven't fully learned it from years of making analogies about it.

Years of grad school experience was painful like this, until I got to a point 10 years after grad school, after a PhD, and well into research, that I "just got" things (in my subfield) as well. It's these experiences that made me feel that it's 80% preparation and perspiration (both of which are dominated by time), and 20% "other" mythology. Don't get me wrong, that 20% is what makes a 2 year old read earlier than others, and getting started reading at 2 (and continuing it!) for 4 years before starting school will make you light years ahead of your peers. The same goes for chess, math, etc etc. There is something legendary about Oppenheimer learning enough dutch in 6 weeks to deliver a lecture. Or perhaps learning to translate his lecture and memorizing it. Who knows.

Do we really believe there's a magical "genius" such that they can do anything? No, so what are the limits to their genius? The limits are defined by what they are a genius at. This is a tautological definition.

I'm not saying "Anyone at any time can become a genius at anything". I'm saying "If you take a kid, start early, and cultivate them just right so that you have time to realize compounding effects, - you can let them grow into basically anything" (probablistically speaking - there are learning disabilities and physical issues etc).

I add third (okay, 2b) - because the pain of coming up with the fact other people are better than you at a deep, fundamental level is too overwhelming.

"We can't on one hand believe in the plasticity and retrainability of the mind, while simultaneously believing that speed is something only a few are born with."

Sure we can, the initial orientation of neurons differs between people, so some people need less "plasticity and retrainability" to be good at a task. Plasticity is physical characteristic like height and varies between people.

Initial speed usually isn't that important, but speed of learning is important and makes the difference between possible and impossible within a human lifetime.

Yes - there will be 10x-ers. And that group will have a 10x-er iside it, and so on given exponential dropoff of frequency of talent. Bobby Fisher is a few std dev above even the best, perhaps.

Generally speaking, "You can do anything you want, but wanting (enough, and naturally) is the hardest part" might need a three standard deviation limit.

Have you heard the phrase: Being average among those who practice makes you 9X% among the population? I think that's what I'm saying - you can be a top performer if you dedicate yourself, especially early enough, but almost nobody will.

Likely so, but is suggest that personality, drive and motivation are also very important factors. I know from experience that stuff I had little interest in as a youngster and that I've still little in I still know little about.

Yes, my interests have grown and broadened over the years but simply I regard some stuff so irrelevant to my life that it's not worth a second thought and I am much better off applying my limited number of neurons to matters of greater importance and enjoyment.

Of course, no one has the luxury of just learning about what one finds interesting and or enjoyable, life's knocks and experiences along with utilitarian-like imperatives force one to learn stuff they'd rather not know about.

I respectfully, but strongly, disagree. There's a reason most NBA players are over 2 meters tall, and one does not become taller with time, dedication nor focus.

It might be different for intellectual skills but I am not that sure.

Almost anyone can become decent at almost anything though. Which is good already!

Being tall isn't a skill. I suspect you could be skillful enough at basketball to overcome the hight disadvantage. However, I think most people who might become that skillful see the high disadvantage (plus the general difficulty of becoming a pro basketball player) and take a different path through life. It's also possible that the amount of time that would be needed to grow your skill past the height disadvantage is too long, so it's not feasible to do it to gain a position in the NBA.

Careful with that "all", even the most highly intelligent humans still have peaks and deficits in different domains.

The interesting finding is that this universal correlation is strong, real, and durable. Of course people in general have cognitive domains where they function better or worse than their g factor indicates, and that's before we get into the fact that intellectual task performance is strongly predicated on knowledge and practice, which is difficult to control for outside of tests designed (successfully, I must add) to do so.

Track and swimming are where innate physical attributes have the most obvious benefits. Michael Phelphs had the perfect body for swimming. There is no amount of trainingg that 99.999% of the population could do to get close to what Usain Bolt ran. Most humans could not train to run under 4 minutes in a mile or under 2:30 in a marathon. They just don't have the right muscular and cardiovascular physiology.

Team sports are of course more complicated as other qualities come into play that aren't as directly physiological.

Of course, but I don't think anyone was seriously suggesting that. The vast majority of humans can become pretty good at swimming though. And that was my interpretation of the original claim about cognitive tasks, mathematics, etc.

https://www.lines.com/guides/average-height-nba-players/1519

A paid promotion of International Median is not Average Association.

https://en.m.wikipedia.org/wiki/Frank_Lentini

The fact that he has a wiki page, and that many folks with born without or who have lost legs (~500,000/year Americans experience limb loss or are born with a limb difference https://amputee-coalition.org/resources/limb-loss-statistics...) do not, suggest that the number of people with < 2 is far greater than the number of people with > 2. So the average is still less than 2.

For better or worse, number of legs (or number of arms) is canonical example people use to demonstrate the statistical principal a significant majority of a population can be above average of some metric.

> Lentini was so respected among his peers that he was often called "The King".

Because it turns out that, in the NBA, height does not correspond with ability! This of course makes sense, because all the players are filtered by being NBA professional basketballers. A shorter player simply has more exceptional ability in another dimension, be that dodging reflex, ability to visualize and then hit a ball trajectory from the three point line, and so on. Conversely, a very tall player is inherently useful for blocking, and doesn't have to be as objectively good at basketball in order to be a valuable teammate.

Despite this lack of correlation, when you look at an NBA team you see a bunch of very tall fellows indeed. Simpson's Paradox.

We see the same thing in intellectual pursuits. "I'm not nearly as smart as the smartest programmer I know, but I get promoted at work so I must be doing something right. Therefore anyone could do this, they just have to work hard like I did". Nope. You've already been selected into "professional programmer", this logic doesn't work.

[0]: https://en.wikipedia.org/wiki/Simpson's_paradox

Being interested in a subject is massively helpful to learning it. But interest arises circumstantially, it's an emotion. The grim reality that it would be really useful to you to learn a certain subject does not necessarily make you interested in the subject, unfortunately. (Perhaps "financially interested", but that's something else.)

Why does curiosity being natural necessarily mean some people are born without it? It could also mean everyone (or every average human) is born with it, and overtime it gets pushed out of people.

So the minimum might not be zero, but it isn’t some fixed quantity.

People do studies on weeks old babies, it seems unlikely curiosity has been restrained at that point.

I think the case you mentioned is explained by an idea covered in attachment theory. Children explore when they feel safe and secure. Safety and security come from the caregivers, the parents. When that is absent, because the parents' emotional state makes the children feel insecure, then the children are restrained by their own emotions.

And this also gives the proponent (you in this case) an excuse to blame a person for not focusing hard enough or not being dedicated enough if they don't grasp the basics, let alone excel.

Nonsense!

The brain you are born with materially dictates the ceiling of your talent. A person with average ability can with dedication and focus over many years become reasonably good, but a genius can do the same in 1 year and at a young age.

We have an education system which gives an A Grade if you pass the course, but 1 person may put on 5 hours a week and the other works day and night.

Edit: I don't mean to argue that there aren't genetics involved in determining aptitude on certain tasks, of course, but the assumption that genius is born and never made feels like a very shallow understanding of the capacity of man.

Cool, come and have a coffee with me :) I have older and younger siblings and was the one randomly blessed.

Whereas most recognised talents are associated with hard work and so there is then this visible link, I am a good example as I did the bare minimum throughout education (and beyond...).

The way my brain processes and selectively discards/stores the information it receives is very different to majority of the population. I have no control over it.

I take zero credit for any of my achievments - I regularly meet intelligent people near to retirement who have been to a tier 1 university, may have PHDs, worked 60 hours a week since they were born, been on course and what not and cannot reach the levels I can.

My nurturing was no different to siblings/peers (and was terrible!)

Note: I have my weaknesses too, but as a whole, I am exceptional. Not through effort!! Completely random - neither of my parents are intelligent and nothing up the ancestary tree as far as I know.

I met the guitar guy a few years later outside his house. He always had just one guitar but now owned something like 20, something like a hundred books about music. Quite the composer. It looked and sounded highly sophisticated. The dumb guy didn't exist anymore.

That said, some people received a smaller starting container, and might need some help cracking it. That’s the work of those who think they’ve found a bigger one.

If we spent 50% of time thinking productively - inborn thinking speed would matter. But in my estimate even 5% is generous.

So it matters far more what kind of feedback you have to filter out the wrong results, and how much time you spend thinking - than how quickly you can do it.

Also practice helps with speed.

Absolutely correct. And that begins with getting their interest, thus their attention; and that's a whole subject in and of itself.

In practice your views result in stinting access of non-existent (in your opinion) talented children to a faster education track. They don't exist therefore they don't need different treatment (finer points get lost when the idea disseminates). Quite a hot theme in American education two (or so) years ago.

I think the only thing in autism that I'd call an innate talent is detail-oriented thinking by default. It'd be the same type of "innate talent" as, say, synesthesia, or schizophrenia: a side effect of experiencing the world differently.

A side effect for which there is a substantial, lifelong, and most importantly wide cost, even if it occasionally confers usually small, usually fleeting, and most importantly narrow advantage.

I'm not sure what you are trying to say.

I'm saying, if it doesn't ruin lives to the point of preventing reproduction, then it stays in the gene pool.

Basically, I'm saying this:

> The trait is still here and still prevalent meaning people are still getting laid and starting families and presumably leading fulfilling lives.

My first thought when the article got to the dialog between logic and intuition bit was that the same is true for school level physics.

I see you're familiar with Ryan North.

I would argue something different. The "skill" angle is just thinly veiled ladder-pulling.

Sure, math is hard work, and there's a degree of prerequisites that need to be met to have things click, but to the mindset embodied by the cliche "X is left as an exercise for the reader" is just people rejoicing on the idea they can needlessly make life hard for the reader for no reason at all.

Everyone is familiar with the "Ivory tower" cliche, but what is not immediately obvious is how the tower aspect originates as a self-promotion and self-defense mechanism to sell the idea their particular role is critical and everyone who wishes to know something is obligated to go through them to reach their goals. This mindset trickles down from the top towards lower levels. And that's what ultimately makes math hard.

Case in point: linear algebra. The bulk of the material on the topic has been around for many decades, and the bulk of the course material,l used to teach that stuff, from beginner to advanced levels, is extraordinarily cryptic and mostly indecipherable. But then machine learning field started to take off and suddenly we started to see content addressing even advanced topics like dimensionality reduction using all kinds of subspace decomposition methods as someting clear and trivial. What changed? Only the type of people covering the topic.

I don't agree. First of all, ladder-pulling in math is observed at all levels, not only cutting-edge stuff. Secondly, it's in applied mathematics where pure math takes a queue onto where to focus effort. See how physics drives research into pure math.

It’s a well-established effect in pedagogics that learning vs. difficulty has a non-monotonic relationship, where you don’t learn efficiently if the material is either too hard or too easy compared to your current level. There is an optimum learning point somewhere in-between where the material is “challenging” – but neither “trivial” nor “insurmountable” – to put it that way.

I've taught math to psychology students, and they just don't get it. I remember the frustration of the section's head when a student asked "what's a square root?" We all know how many of our fellow pupils struggled with maths. I'm not saying they all lacked the capability to learn it, but it can't be the case they all were capable but "it was the teacher's fault". Even then, how do you explain the difference between those who struggled and those who breezed through the material?

Or let's try other topics, e.g. music. Conservatory students study quite hard, but some are better than others, and a select few really shine. "Everyone is capable of playing Rachmaninov"? I don't think so.

So no, unless you've placed the bar for "mathetical skill" pretty low, or can show me proper evidence, I'm not going to believe it. "Everyone is capable of..." reeks of bullshit.

I do broadly agree with your position that some people are going to excel where others fail. We know there trivially exist people with significant disabilities that will never excel in certain activities. What the variance is on "other people" (a crude distinction) I hesitate to say. And whatever the solution is, if there is even a solution, I'd at least like for the null hypothesis to be "this is possible, we just may need to change our approach or put more time in".

On a slightly more philosophical note, I firmly believe that it is important to believe some things that are not necessarily true -- let's call this "feel-good thinking". If someone is truly putting significant dedicated effort in and not getting results, that is a tragedy. I would, however, greatly prefer that scenario to the one in which people are regularly told, "well, you could just be stupid." That is a self-fulfilling prophecy.

Not really. There's nothing inherently special about people who dedicated enough time to learn a subject.

> "Everybody can do everything." Well, that's simply not true. I'm fairly sure you (yes, you in particular) can't run the 100m in less than 10s, no matter how hard you trained.

What a bad comparison. So far in human history there were less than 200 people who ran 100m in less than 10s.

I think you're just reflecting an inflated sense of self worth.

"You didn't work hard enough." People really blame you for that, not for lacking talent.

> So far in human history there were less than 200 people who ran 100m in less than 10s.

And many millions have tried. There may be 200 people who can run it under 10s, but there are thousands that can run it under 11s, and hundreds of thousands that can run it under 12s. Unless you've got clear evidence that those people can actually run 100m in less than 10s if they simply try harder, I think the experience of practically every athlete is that they hit a performance wall. And it isn't different for maths, languages, music, sculpting (did you ever try that?), etc. Where there are geniuses, there also absolute blockheads.

That's not to say that people won't perform better when they work harder, but the limits are just not the same for everyone. So 'capable of mathematical reasoning' either is some common denominator barely worth mentioning, or the statement simply isn't true. Clickbait, if you will.

Interestingly, the 100m metaphor is extensively discussed in my book, where I explain why it should rather lead to the exact opposite of your conclusion.

The situation with math isn't that there's a bunch of people who run under 10s. It's more like the best people run in 1 nanosecond, while the majority of the population never gets to the finish line.

Highly-heritable polygenic traits like height follow a Gaussian distribution because this is what you get through linear expression of many random variations. There is no genetic pathway to Pareto-like distribution like what we see in math — they're always obtained through iterated stochastic draws where one capitalizes on past successes (Yule process).

When I claim everyone is capable of doing math, I'm not making a naive egalitarian claim.

As a pure mathematician who's been exposed to insane levels of math "genius" , I'm acutely aware of the breadth of the math talent gap. As explained in the interview, I don't think "normal people" can catch up with people like Grothendieck or Thurston, who started in early childhood. But I do think that the extreme talent of these "geniuses" is a testimonial to the gigantic margin of progression that lies in each of us.

In other words: you'll never run in a nanosecond, but you can become 1000x better at math than you thought was your limit.

There are actual techniques that career mathematicians know about. These techniques are hard to teach because they’re hard to communicate: it's all about adopting the right mental attitude, performing the right "unseen actions" in your head.

I know this sounds like clickbait, but it's not. My book is a serious attempt to document the secret "oral tradition" of top mathematicians, what they all know and discuss behind closed doors.

Feel free to dismiss my ideas with a shrug, but just be aware that they are fairly consensual among elite mathematicians.

A good number of Abel prize winners & Fields medallists have read my book and found it important and accurate. It's been blurbed by Steve Strogatz and Terry Tao.

In other words: the people who run the mathematical 100m in under a second don't think it's because of their genes. They may have a hard time putting words to it, but they all have a very clear memory of how they got there.

As for my own half-baked opinion, I want to say that the Church-Turing Thesis and Chomsky's innate theory of cognition have something to add to the picture. Homo sapiens as a species essentially has the capacity to think analytically and mathematically; I want to argue this is a universal capacity loosely analogous to the theory of universal Turing machines. So what matters is people's early experiences where they learn how to both practice and, critically, to play, when they learn difficult ideas and skills.

Also, as an amateur pianist, most people don't know that modern piano teaching emphasizes not fixed limits of the student but that many students learn the wrong techniques even from well-meaning piano coaches. Just the other day I was watching a recent YouTube Julliard-level masterclass where the teacher was correcting a student on her finger playing technique, presumably this student had been taught the wrong technique since childhood. With music or sports a coach can visually see many such technique problems; with math teaching it of course harder.

Any comments on the "10x programmer"?

Current education is _extremely_ biased to concrete arithmetic and a bit of algebra. If you have a predisposition to either you will do extremely well. If you don't you won't.

Those have little to do with how math is done by mathematicians.

In short: education needs to catch up to what's happened since the 1920s in maths. Parents are conservative and don't want their kids to learn something they themselves don't understand, so we're stuck with what we have until enough generations pass and 20th century math is absorbed by osmosis into the curriculum.

> A good number of Abel prize winners & Fields medallists have read my book and found it important and accurate. It's been blurbed by Steve Strogatz and Terry Tao.

Sounds like people mostly living in different bubbles? What do they know about the world?

They aren't hanging out with the kids who fail in school because maths and reading and writing is to hard, and then start selling drugs instead and get guns and start killing each other.

> [they] don't think it's because of their genes

Do you think someone would tell you, if he/she thought it was?

I mean, that can come off as arrogant? Wouldn't they rather tend to say "it was hard work, anyone can do it" and prioritize being liked by others

> Pareto-like distribution like what we see in math

Unclear to me what you have in mind. If there's a graph it'd be interesting to have a look? I wonder whats on the different axis, and how you arrived at the numbers and data points

Well, they do know something about math — in particular that it requires a certain "attitude", something that no-one told them about in school and they felt they only discovered by chance.

Starting from Descartes and his famous "method", continuing with Newton, Einstein, Grothendieck all these guys insisted that they were special because of this "attitude" and not because of what people call "intelligence". They viewed intelligence as a by-product of their method, not the other way around. They even wrote books as an attempt to share this method (which is quite hard to achieve, for reasons I explain in my book.)

Why do you bring "kids who fail in school" and "start selling drugs" into this conversation? What does it have to do with whether math genius is driven by genetics or idiosyncratic cognitive development?

And why would a mathematician be disqualified from discussing the specifics of math just because they're not hanging out with lost kids? Are you better qualified? Did you sequence the DNA of those kids and identified the genes responsible for their learning difficulties?

>> [they] don't think it's because of their genes

> Do you think someone would tell you, if he/she thought it was?

Well, an example I know quite well is mine. I was certainly "gifted" in math — something like in the top 1% of my generation, but not much above and definitely nowhere near the IMO gold medallists whom I met early in my studies.

A number of random events happened to me, including the chance discovery of certain ways to mentally engage with mathematical objects. This propelled me onto an entirely different trajectory, and I ended up solving tough conjectures & publishing in Inventiones & Annals of Math (an entirely different planet from the top 1% I started from)

My relative position wrt my peer group went through a series of well-delineated spikes from 17yo (when I started as an undergrad) to 35yo (when I quit academia), associated with specific methodological & psychological breakthroughs. I'm pretty confident that my genes stayed the same during this entire period.

And as to why I was initially "gifted", I do have some very plausible non-genetic factors that might be the explanation.

I don't claim this proves anything. But I see no reason why my account should be disqualified on the grounds that I'm good at math.

Usually, competency in one domain is presumed to make you a bit more qualified than the random person on the internet when it comes to explaining how this domain operates. Why should math be the exception?

Of course. That does not mean that intelligence doesn't play a (big) role.

> Starting from Descartes and his famous "method", continuing with Newton, Einstein, Grothendieck all these guys insisted that they were special because of this "attitude" and not because of what people call "intelligence"

That doesn't make sense. Back when they were active, intelligence, IQ tests and the heritability of intelligence hadn't been well studied. They didn't have enough information, like we do today: https://en.wikipedia.org/wiki/Heritability_of_IQ#Estimates "Various studies have estimated the heritability of IQ to be between 0.7 and 0.8 in adults and 0.45 in childhood in the United States."

And, evolution and genetics weren't these peolpe's domains. Does it make sense to assume they were authorities in genetics and inheritance, because were good at maths and physics?

Sometimes they were wrong about their own domains. Einstein did say "Genius is 1% talent and 99% hard work" (I can understand how it makes sense from his own perspective, although he didn't know enough about this animal species, to say that).

But he also said "God does not play dice" and was wrong about his own domain.

> Why do you bring "kids who fail in school" and "start selling drugs" into this conversation?

It was an example showing that the researchers live in bubbles.

That they're forming their believes about humans, based on small skewed samples of people. There's billions of people out there vastly different from themselves, whom they would have left out, if thinking about about others' abilities to learn.

In fact, now it seems to me that you too live in a bubble, I hope you don't mind.

> Usually, competency in one domain is presumed to make you a bit more qualified than the random person on the internet when it comes to explaining how this domain operates.

1) Maths and 2) evolution, DNA, genetics, intelligence, learning and inheritability are not the same domains.

Anyway, best wishes with the book and I hope it'll be helpful to people who want to study mathematics.

I'd guess the person you're responding to has thought more carefully about this issue than the median HN commenter has.

He was wrong in his guesses about me and what I've read, and wrong about the quote too (see sibling comment).

> is from 2015 ... what we've learned since 2018

You're saying the graph is somehow invalid, because of newer GWAS related research?

The blog he links to looks biased to me. Are there two camps that don't get along: looking at DNA (GWAS), vs looking at twin studies etc ... yes seems so. I'll reply to both of you in another comment

I know this reply may not suffice to convince you, but unfortunately I won't be able to argue forever.

Did you ever consider the possibility that you might be the one living in a bubble?

FYI, the concept of innate talent predated IQ tests and twin studies by many millenia. Two of the authors I'm citing in my book (Descartes and Grothendieck) believed that innate talent existed and they both declared they would have loved to be naturally gifted like these or these people they knew.

You're declaring that these incredibly smart people were wrong about their own domains, which is a pretty bold claim to make. What do you have in support of this claim? A fake Einstein quote?

It's a sad fact of life that most quotes attributed to Einstein are fabricated. Next time, please check "The Ultimate Quotable Einstein", compiled by Alice Calaprice.

This may come as a shock to you, but Google page 1 isn't always a reliable resource. Nor is Wikipedia, even though it's quite often correct. As it happens, there's a pretty large "Heritability of IQ" bubble on the internet. It's active and vocal, but it's also quite weak scientifically — the page you're citing is a typical symptom, and it absolutely doesn't reflect the current scientific knowledge.

The IQ heritability claims that you're citing are based on twin studies and they have taken in serious beating in the past decade, especially in light of GWAS.

It's true that a number of people have been fooled by twin studies, most notably Steven Pinker, in Chapter 19 of the Blank Slate (did you read it?)

You see, Pinker is a linguist and apparently he isn't mathematically equipped to fully comprehend the intrinsic limitations of Bouchard's approach. Did you read Bouchard's 1990 paper on twins reared apart? Do you find it convincing? Are you aware that even The Bell Curve's Charles Murray thinks that this approach, abundantly cited by Pinker, is structurally flawed? Are you aware of the fundamental instability of IQ estimates based on twins reared together? Aren't you concerned that even a mild violation of Equal Environment Assumption, plugged into Falconer's equation, would drastically reduce the estimates?

If you don't understand what I'm talking about, if you've never read the authors and the primary research I'm citing, then it's quite likely that you're the one living in a social media bubble.

If you're interesting in learning more about the actual science of IQ heritability, I recommend using Sasha Gusev's Substack as an entry point: https://theinfinitesimal.substack.com/p/comments-on-no-intel...

Feel free to also subscribe to my own Substack, where I plan to cover these topics in the coming months: https://davidbessis.substack.com

All the best, David.

You're wrong about that, but you couldn't have know. I've lived in far more different places with more different people, than most people you've met.

> innate talent predated IQ tests and twin studies by many millenia

That's why I wrote it hadn't been well studied, not that it hadn't been studied at all.

> You're declaring that

Of course not. I'm not the source.

> incredibly smart people were wrong about their own domains, which is a pretty bold claim to make. What do you have in support of this claim? A fake Einstein quote?

That's from a letter Einstein wrote 1926 to Bohr. He wrote in German, that quote is a paraphrase in English.

Look here: https://en.wikipedia.org/wiki/Bohr%E2%80%93Einstein_debates, And, about him being mistaken, quoting that article:

"As mentioned above, Einstein's position underwent significant modifications over the course of the years. In the first stage, Einstein refused to accept quantum indeterminism [...]" -- indicating that, at some points, he had the wrong beliefs, right.

Here's the quote explained further: https://hsm.stackexchange.com/questions/5937/why-did-einstei...

> It's true that a number of people have been fooled by twin studies, most notably Steven Pinker, in Chapter 19 of the Blank Slate (did you read it?)

> You see, Pinker ... Bouchard's 1990 ... The Bell Curve's Charles Murray ... thinks ... structurally flawed

No, didn't read that book. Continuing in another comment.

I tried to find what he might have said that you were referring to, and stumbled upon that phrase, and since it was "your" quote, I didn't double check it.

But it's something else then, or maybe a misunderstanding somehow.

Thank you for taking the time to comment here!

Here's a twin study from 2015, newer than the books (Clean Slate etc) and papers you (David) mentioned.

"Thinking positively: The genetics of high intelligence" https://pmc.ncbi.nlm.nih.gov/articles/PMC4286575/

Figure 3 indicates that intelligence is pretty strongly inherited, and they arrive at 0.44.

Now you're saying that that doesn't matter because of GWAS? Sounds a bit hand-wavy to me.

> > Sasha Gusev's Substack as an entry point: https://theinfinitesimal.substack.com

Blog post looks biased. So there's a controversy: https://en.wikipedia.org/wiki/Missing_heritability_problem

And there's two camps:

https://www.clearerthinking.org/post/the-missing-heritabilit...

(I like that article!)

And the two of you (Davind and Thomas) seem to be in the "The DNA Proponents" camp. The other is "The Twin Study Advocates" camp.

I guess now I'm in "The middle ground" camp, no longer in the "Twin Study Advocates".

Thanks for that. Maybe I'll check back in 10 years later and see what has happened.

A 2015 twin study result is basically a citation to the phlogiston era of polygenic population-wide genetic surveys. Heritability estimates of that vintage basically define away indirect genetic effects, which subsequent work appears to have very clearly established; the work now is on characterizing and bounding it, not asking whether it's real.

"Blog post looks biased" is not a good way to address this unless you actually practice in the space, like the author does, and are in conversation with other practitioners in the space, like the author is. You find lots of --- let's generally call them pop science writers --- knee-jerk responding to the new rounds of heritability numbers, but those same authors often wrote excitedly about how GWAS results would bolster their priors in the years before the results were published. It's worth paying attention to the backgrounds of the people writing about this stuff!

I substantially rewrote this comment, which was sprawling; the original is preserved here: https://gist.github.com/tqbf/b118ec9f9e69e0f3f61003c152d0d44...

Ok, good to know :-)

> the work now is on characterizing and bounding it

Using GWAS I suppose, ok.

> "Blog post looks biased" is not a good way ... but those same authors often wrote excitedly about how GWAS results would bolster their priors

Ok, yes I think I agree. ... Interesting

Thanks for keeping the original comment text. (I had a super quick glance at the blog post it mentioned, this one, right: https://theinfinitesimal.substack.com/p/book-review-eric-tur..., maybe will read at some point. "But Turkheimer sets a trap for GWAS Guys" (in the blog post) made me smile :-))

A few posts ago you were alluding to heritability in the 0.7-0.8 range, as a reason to dismiss the writings of Einstein, Newton, Descartes and Grothendieck.

Now you're at 0.44. If you discount for a mild EEA violation correction, you'd easily get to 0.3 or below — a figure which I personally find believable.

Just FYI, I don't belong to any "camp". These aren't camps but techniques and models. Intra-family GWAS provide underestimated lower bounds, twin studies provide wildly overestimated upper bounds. I don't care about the exact value, as long at it doesn't serve as a distraction from the (much more interesting!) story of how one can develop one's ability for mathematics.

In any case, IQ is a pretty boring construct, especially on the higher end where it's clearly uncalibrated. And it's a deep misunderstanding of mathematics to overestimate the role of "computational ability / short term memory / whatever" vs the particular psychological attitude and mental actions that are key to becoming better at math.

Now that the smoke screen has evaporated, can we please return to the main topic?

> A few posts ago you were alluding to heritability in the 0.7-0.8 range, as a reason to dismiss the writings of Einstein, Newton, Descartes and Grothendieck.

No. This is what I wrote:

"Back when they were active, intelligence, IQ tests and the heritability of intelligence hadn't been well studied. They didn't have enough information, like we do today: ... twin studies ..."

And now that changes to: "like we do today: ... GWAS (and twin studies) ...". The precise numbers were not the point.

> you'd easily get to 0.3 or below — a figure which I personally find believable

That's interesting. I thought you were closer to zero. Well, 0.3 or 0.7 or 0.2 -- it's a little bit all the same to me, as long as it's not 0 or 0.0001.

> I don't care about the exact value

Ok, makes sense :-)

> as long at it doesn't serve as a distraction

Aha, so that's why you didn't like 0.7 or 0.8 and reacted to it. Yes that's maybe a bit depressingly high numbers, in a way.

And I don't like 0 or close to 0 because that'd indicate that this animal species was "stuck".

> ... how one can develop one's ability for mathematics ... psychological attitude and mental actions that are key to becoming better at math

Yes, to becoming better. If you have time, I wonder what's the level of maths you think most people on the planet can reach? If everyone had the right encouragement, time and attitude.

- High school maths in economy and finance programs? (needed for example for accounting and running one's own business)

- The most advanced maths classes in high school if you study natural sciences?

- Technical mathematics or theoretical physics a few years at university?

- General theory of relativity?

I'm wondering if you're saying that just as long as someone starts early enough, they can reach the highest levels?

But then what about today's topic:

California's most neglected group of students: the gifted ones

https://news.ycombinator.com/item?id=42247334

Is it just that they started earlier and have the right attitude?

I suppose that makes sense from their own personal perspectives (but that doesn't make them right), in that they had to put in lots of time and work, but didn't do anything to become bright people.

> The floor and ceiling you will operate on in your life

Interesting that what you wrote got downvoted. Lots of flat-earthers here? (figuratively speaking)

I call it secular creationism - basically humans are special beings to which the rules and laws of biology (evolution and natural selection) do not apply fully.

And people with a liberal disposition who pride themselves as rational thinkers quickly switch off that rationality when it comes to natural differences between humans especially when those differences are in cognitive abilities.

Your starting point is the way elite mathematicians think about themselves. But people don't understand themselves. They don't understand their own motivations, their own capabilities, their own logic. You know who are best at explaining what/how other people think? Average people. Hence the success of mediocrity in certain types of quizzes and politics.

I'm sure you're right about the mixture of logic and intuition. I've had the thought myself, mainly about designing systems, but there is some analogy: you've got to "see through" the way from the top to the bottom, how it connects, and then fill the layers in between. But that intuition is about a very, very specific domain. And it's not given that is a priori equally distributed. More likely than not, it's isn't.

Your whole argument then is based in naive psychology. E.g., this

> What can someone gain by improving their mathematical thinking?

> Joy, clarity and self-confidence.

> Children do this all the time. That’s why they learn so fast.

Are there no other reasons children learn so fast? It's not even given that joy and clarity makes children learn faster. What is known is that children do learn fast under pressure. Have you seen the skills of child soldiers? It's amazing, but it comes of course at great cost. But they did learn. Children pick up languages at a relatively high speed (note: learning a new language is still very well possible at later ages, certainly until middle age), but that's got nothing to do with joy, clarity and self-confidence. They also do it under the dreariest of circumstances.

So I'd say: your argument, or at least the quanta article, is at odds with common sense, and with psychological research, and doesn't provide concrete evidence.

You might have ideas for teaching maths better. But beware there's a long tradition of people who've tried to improve the maths curriculum, and basically all failed.

I'll give you one more point for thought (if you ever read this): intuition can also be a negative. I've practiced with my daughter for her unprepared math exam (she dropped it at one point, and then wanted to have it on her grade list anyway). One thing that I clearly remember, and it's not just her, is that she had very weird ideas about the meaning of e.g. x, even in simple equations. They were nearly magical. It was hard to get her to treat x like she would treat any other term. At one point, she failed to see that e.g. 1/3 = x^-1 is easy to solve, even when she had written down 1/x = x^-1 right next to it. Her intuition blocked her logic. My conclusion is that it's certainly easy to frak up someone's understanding of maths, unless you're really teaching, tutoring and monitoring 1-on-1. There's no solution for maths but good teachers, and a lot of fast feedback. Quite an old lesson.

New solutions exist for old problems.

[0] https://www.justinmath.com/books/#the-math-academy-way

If someone's smart enough to get into a psychology class they are smart enough to be thought basic undergrad math. It wasn't your teaching failure necessarily, but it was someone's teaching failure at some point.

Not everyone can play Rachmaninov like Lugansky or do math like Terence Tao, but there is absolutely no doubt that almost all people are magnitudes away from their latent potential in almost all domains. I'm fairly certain you could teach any average person how to play Rachmaninov decently. You could bring any person to a reasonably high mathematical level. You can get any person to lift a few hundred pounds.

Most people today read at a 7th grade level, can't do basic math, and are out of air after 3 flights of stairs. "Everyone can do everything" is maybe not literally right but directionally right given how utterly far removed we are from developing practically anyone's potential.

I've found that the people who most believe in math being a genetic ability are the ones who do not work in the symbolic world of modern math, but in the semantic world of whatever the field the math describes is.

The two are rather different.

One can get along quite well without them once you replace the arithmetic expressions with more general objects.

You chose an extremely poor example and now you're trying to die on that hill. Please don't die on that hill.

And I didn't mention genetics. Nature is complicated.

Neither trivia has anything to do with being good at mathematics as done by mathematicians.

I dont disagree with the premise that mathematical thinking can benefit anybody, but its absurd notion that everything abstract is teachable and learnable to all is a fantasy of a distinctly left-wing variety, who would have you believe that everything is just social conditioning and human beings dont differ from one-another.

1. Everyone (young, old, poor, rich) thinks that maths is interesting and fun and beautiful and important. Not important "to get a good job" or "to go to a good college" or "to be an impressive person", but rather important because it's fun and interesting. And maybe it also helps you think clearly and get a good job and all these practical things, but they're secondary to the tremendous beauty and wondrousness of the domain.

2. Everyone believes that barring actual brain injuries people can learn mathematics to a pretty high level. Not Ramanujan level, not Terrence Tao, not even a research mathematician at one of the smaller universities, but a level of extreme comfort, let's say a minimum level of being able to confidently ace the typical types of exams 17 and 18 year olds face to finish secondary school in various countries.

Would you claim that in that world - people think maths is great, and that anyone can learn it - we'd see similar levels of ability and enjoyment of mathematics?

My claim is that we don't live in "Math-World", as described above, but "Anti-Math-World". And further, that anyone suggesting things have to be the way they are in Anti-Math-World is not only wrong, but also fundamentally lacking imagination and courage.

Kids are told week in week out that maths is stupid, that they are stupid, that their parents themselves are stupid, that the parents hated maths, that the teachers are stupid, and then when they end up doing poorly, people say: "ahhh, some kids just aren't bright!"

Parents who like things like learning and maths and reading and so on, have kids that tend to like those things. And parents that don't, usually don't. Saying that this somehow tells us something concrete and inalterable about the nature of the human brain is preposterous.

It's a card that's used by grown-ups who are terrified by the idea that our education systems are fundamentally broken.

Come on, how often are kids exposed to such stupid talk? I suspect very infrequently.

My grandmother, who wasn't stupid by any means but who knew only basic arithmetic, would never have uttered such nonsense.

And I'd stress, like many of her generation and background, her knowledge of mathematics was minimal, if she'd been ask what calculus was she'd likely have been perplexed and probably have guessed it to be some kind of growth on one's foot.

Every student hears it enough for it to be a widespread sentiment. It spreads like an infection.

Pretty often, by other kids. And kids listen to kids, not well-meaning, mealy-mouthed adults.

Bad example, it's much more likely to create a musical prodigy by providing early and appropriate guidance. Of course this is not easy as it assumes already ideal teaching methods and adequate motivation to the youngling, but even those with some learning difficulties have the potential to excel. The subtypes of intellect required to play complex music and absord advanced abstract math subjects are quite different, former requiring strong short-term memory (sightreading) the latter fluid intelligence -I think almost everyone is familiar with these terms by now and knows that one can score high/low on certain subtypes of an IQ test affecting the total score-.

BTW IDK if the Rachmaninoff choice was deliberate to imply that even the most capable who lack the hand size won't be able to perform his works well yeah, but there are like 1000s of others composers accessible that the audiences appreciate even more. Attempting to equate music with sports in such manner is heavily Americanized and therefore completely absurd. Tons of great pianists who didn't have the hand size to interpret his most majestic works and of others. Tons of others who could but never bothered. There have been winners of large competitions who barely played any of his works during all stages of audition or generally music requiring immense bodily advantage. Besides, it's almost 100% not a hand size issue when there are 5 year old kids playing La Campanella with remarkable fluidity.

And even in this case this isn't even the point. Most conservatory alumni today are 100x skilled than the pianists of previous generations... yet they all sound the exact same, their playing lacks character/variability, deepness, elegance to the point where the composers ideas end up distorted. And those can be very skilled but just have poor understanding of the art, which is what music is, not the fast trills/runs, clean arpeggios, very strict metronomic pulse.

> So no, unless you've placed the bar for "mathetical skill" pretty low, or can show me proper evidence, I'm not going to believe it. "Everyone is capable of..." reeks of bullshit.

Well the vast majority of people in the Soviet Union were very math literate, regardless of what they ended up working as (although indeed most became engineers) and in quite advanced subjects. This is obviously a product of the extensive focus of primary and secondary education on the sciences back then.

So the point isn't to make everyone have PhD level math background and I heavily dislike the dork undertones/culture that everyone should love doing abstract math on their freetime or have to have some mathematical temperament' . But let's not go the other way and claim that those not coming close to achieving the knowledge those in the top % of the fields possess, they are chumps.

I doubt that.

> although indeed most became engineers

And that is demonstrably false.

Anyway, most of your argument boils down to: there's a bunch of people that can't do a certain task, there's a bunch that has mediocre skills, a few that are good, and a handful that's really good. There's no argument, just observation, not even related to effort, which is what this discussion is about.

And maths is no different from sports or music in that sense. Most people suck at math, and will always suck at it. The things described in the article are personal reflections of elite mathematicians. They have no bearing on development of knowledge and skills of us mortals, if only because those reflections have no truth value. It's all just "feel good" thoughts, no data, nothing provable, etc.

Yes, effort improves skill, but everyone has a limit, and the ones with the high limits we call talented.

Not even the most vicious polemicists of communist states deny that i.e. the superiority of the Soviet-era education system.

https://www.amazon.com/What-Ivan-Knows-Johnny-Doesnt/dp/4871...

which is also why the following statement

> Most people suck at math, and will always suck at it.

is strictly US-centric.

> Anyway, most of your argument boils down to: there's a bunch of people that can't do a certain task, there's a bunch that has mediocre skills, a few that are good, and a handful that's really good. There's no argument, just observation, not even related to effort, which is what this discussion is about.

No, you came up with that cause you have a very poor understanding of what constitutes a good musician which, like any other typical HNer believes, is another LeetCode type of thing where the more problems like a good little monkey you can solve, the smarter you become. And I already stated that there are people striving in the arts, even more than the ones with the supposedly 'better' skills according to absurd and clueless standards you set, i.e. no, those who can't access specific repertoire easily are not people that can't do a certain task or a bunch that has mediocre skills.

> And maths is no different from sports or music in that sense.

Let me guess, you also think that if painters can't draw photorealistically they're not deserving the artist title and lack talent? Or that the opera is all about who can sing the highest note?

Anyone who lumps every discipline within one other like that without realizing they require completely different things to be considered successful at and believe everything boils down to some supposedly 'objective' absurd video-game like character strategy always serves to remind that the US today has nothing to offer other than hi-tech bombing technology and horrific subculture.

Nowhere did I deny the existence of some people having the innate ability to absorb skills faster and better than the others and of course this is an interdisciplinary fact. But it definitely doesn't hold the same weight for every single discipline for one to strive.

I think that's also a trap. Even professional athletes spend a little bit of their time doing simple drills: shooting free throws, fielding fly balls, hitting easy groundstrokes.

Sometimes your daily work keeps up the "easy" skills, but if you haven't used a skill in a while, it's not a bad idea to do some easy reps before you try to combine it with other skills in difficult ways.

One thing I'm anticipating from LLM-based tutoring is an adaptive test that locates someone's frontier of knowledge, and plots an efficient route toward any capability goal through the required intermediate skills.

Trying to find the places where math starts getting difficult by skimming through textbooks takes too long; especially for those of us who were last in school decades ago.

Expert-systems using techniques from the 80s are doing this right now [0].

[0] https://www.mathacademy.com/how-our-ai-works

LLMs currently can't find efficient paths longer than 5 hops when given a simple itinerary. Expecting them to do anything but a tactical explanation of issues they have seen in training is extremely naive with something as high dimensional as math.

I have heard it is the ego protecting itself by rejecting something outright rather than admitting you can't do it. It still happens to me all the time. My favorite technique was one I heard from a college professor. He starts reading while filling a notepad with sloppy notes, once a page is filled he just throws it away. He claimed it was the fastest way to "condition his brain to the problem space". More than the exercise I like the idea that your brain cannot even function in that space until it has been conditioned.

I've tried night classes, tutors, activities. Nothing sticks.

Even the standard 12x tables I struggle at. I want to understand it but my brain just can't understand the non-practicality side of things.

My brother's mom tutors math. One of her insights with a former student was that they were in need of forming some number sense. She started by walking them both out to the street: "how many tires are there on this street of parked cars?" The student, already flummoxed, started panic guessing. So she started with counting.

For times tables, have you developed any intuition around it? For me, times tables are rectangles composed of unit squares and that helps with my intuition. Modern Common Core standards in the US focuses a lot on exposing different mental models to students. And after seeing the same 4x6 enough times your brain will automatically associate that with its solution. Instead of calculating, it is memorized.

My brain doesn't require car tires, geology, or other practical needs: it likes puzzles. I struggle with medical stuff and I can feel my brain switching to meh-mode and hardly anything sticks. I don't know how many times I have been told about the different kinds of sugar and how your body uses that energy and I would still have to look it up.

I've tried different approaches. 4x3 being 3x4

But somehow I end up miscounting and giving the answer for 4x4 or be off a digit every-time.

A good example was that I was at a brewery last night.

They didn't do pint glasses but glasses of: 1/4, 1/2 and 2/3rd's.

I thought 1/2 was more than 2/3rd's so I ordered a 1/2 rather than thinking I was getting more.

However I was unable to visual how 2/3 is more than 1/2 when 1/2 is half a pint, or half a glass.

My visual capabilities are great at others but just couldn't formulate the equation of 2/3 is more than 1/2.

Very simple stuff, but it just doesn't meld.

https://awrestaurants.com/blog/aw-third-pound-burger-fractio...

Maybe visualize splitting a pint with a friend. If you split the pint into 2 equal parts and each of you gets 1 of those 2 parts you each get the same amount.

Then visualize splitting it instead into 3 equal parts. You get 1 of those parts and your buddy gets 2. There's no fractions there so it should be easier to visualize that your buddy got twice as much as you did.

Comparing those two visualizations might make it easier to see that someone who gets 2/3 of a pint gets more than someone who gets 1/2 of a pint.

For me, a progress bar or a pizza is the default. And because I cook rice daily and we were talking volume, my "progress bar" was like a measuring cup. Mentally, I can stand two measuring cups next one another, and filling one of two parts is less than filling two of three parts.

Maybe there are ways to be more aware of or insert ways to force the use of fractions in physical space. More cooking, more building, etc. The more comfortable you get the more intuition you should build

Absolutely. There's also a pernicious idea that only young people can master complex maths or music. This is a self-fulfilling prophecy - why bother try if you're going to fail due to being old? Or perhaps it's an elitist psy-op, giving the children of wealthy parents further advantage because of course no-one can catch up.

While it’s indisputable that terse symbolic formalisms have great utility, one can capture all the same information verbally.

This is perhaps most evident in formal logic. It’s not hard to imagine a restricted formalized subset of natural language that is amenable to mechanical manipulation that is isomorphic to say modal logic.

And finally, for logic at least, there is something of a third way. Diagrammatic logical systems such as Existential Graphs capture the full power of propositional, predicate, and modal logic in a way that is neither verbal nor conventionally symbolic.

Who knows? You might also contribute meaningfully to these fields as you embrace your own unique path.

[1] https://dc.fandom.com/wiki/Batcomputer

Can you talk about the path or resources you are using to improve here?

But when I started focusing on making money I've come to believe it's a bad heuristic for that purpose.