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.
> Not really. There's nothing inherently special about people who dedicated enough time to learn a subject.
"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.
I'm the author of what you've just described as clickbait.
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.
This power law argument immediately reminds me of education studies literature that (contrary to the math teachers in this thread) emphasize that mathematical ability is learned cumulatively, that a student's success feeds on itself and advances their ability to grasp more difficult material.
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.
This beats TFA. Interesting relation between cumulativeness and distribution ("Yule process"). But how does this explain variation is how quickly children pick up maths - would you argue it's due to prior exposure e.g. parental tutoring?
There is math the abstract field and math the concrete example you're working on.
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.
> document the secret "oral tradition" of top mathematician
> 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
> Sounds like people mostly living in different bubbles? What do they know about the world?
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?
> they do know something about math ... that it requires a certain "attitude"
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.
Current estimates of the "heritability" of intelligence are far, far lower than "0.7 or 0.8"; they're probably below 0.1, and that's before digging into what "heritability" means, which is not generally what people think it does.
I'd guess the person you're responding to has thought more carefully about this issue than the median HN commenter has.
They've studied men in Sweden during 40 years. From the abstract:
"We found that high intelligence is familial, heritable, and caused by the same genetic and environmental factors responsible for the normal distribution of intelligence."
"... 360,000 sibling pairs and 9000 twin pairs from 3 million 18-year-old males with cognitive assessments administered as part of conscription to military service in Sweden between 1968 and 2010 ..."
Looking at Figure 3, in that pager, about identical twins and non-identical (two-egg) twins -- I think that settles it for me.
Seems they arrive at a bit above 0.4 as heritability. Yes that's less than 0.7 - 0.8 but I wouldn't say "far far lower", and more than 0.1. Also, they're 18 years old, not adults.
> I'd guess the person you're responding to has thought more carefully about this issue than the median HN commenter has.
Well, in his reply to me, he was sort of name dropping and appealing to (the wrong) authorities, didn't make a good impression on me. Plus writing about himself, but he's a single person. -- I would have preferred links to research on large numbers of people.
> what "heritability" means, which is not generally what people think it does.
That sounds interesting. Can I guess: You mean that people believe that heritability means how likely a trait is to get inherited from parent to child? When in fact it means: (https://en.wikipedia.org/wiki/Heritability)
"What is the proportion of the variation in a given trait within a population that is not explained by the environment or random chance?"
My sibling comment (unsuprisingly) goes into more depth and with more sourcing. The 0.4 result you've cited is from 2015, which is in the phlogiston era of this science given what we've learned since 2018. As he has aptly demonstrated: his authorities are sound, and he has thought carefully about this matter --- respectfully, far more than you seem to have. That's OK! We're just commenting on a message board. I wouldn't even bring it up, except that you've decided to make his grasp on the subject a topic of debate.
> he has thought carefully about this matter --- respectfully, far more than you seem to have
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.
> Did you ever consider the possibility that you might be the one living in a bubble?
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.
"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.
Aha. That phrase was supposed to support your viewpoint, not mine. "99% hard work" -- in contrary to intelligence.
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.
The "warring camps" framing is very overstated. Greenberg, who doesn't practice in this space, believes it to be a vital concern, but giants in the twin-study practitioner field freely cite GWAS results, including the EA studies.
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!
> "warring camps" framing is very overstated ... twin-study practitioner field freely cite GWAS results
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
> In other words: the people who run the mathematical 100m in under a second don't think it's because of their genes.
Sure they don't. Most extremely successful people (want to) think that the main reason of their success is their commitment and hard work. It runs completely contrary to the findings of modern biology and psychology, most of our intellectual potential at adulthood is genetic.
The floor and ceiling you will operate on in your life is decided the moment of chromosomal crossover.
> Most extremely successful people (want to) think that the main reason of their success is their commitment and hard work
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)
> 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.
So, for starters: you don't have any evidence, if I understood it properly. None whatsoever. That's really not the basis for arguing "become 1000x better." If only because your operationalization is missing. If you can't measure someone math's skills, how can you say they can become 1000x better? I think the whole article manages not to even speak about what "math" actually is supposed to be. Symbol manipulation according to axioms?
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.
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.