J.S. Mill's autobiography is a very fascinating read. He spends quite a lot of it discussing his early childhood, explaining that in his opinion he was not particularly special, rather, it was his father who pushed him to all those accomplishments. His father sheltered him from other kids so he was not aware that his accomplishments were unusual!
As a father of an 8 years old, this is very moving.
While Terence is -without a doubt- born with prodigious abilities, I think credit should also be given to his parents Billy and Grace who seem to have managed to simultaneously nurture these special abilities while still letting Terence have a happy (?) childhood. This is not easy to do.
Can't find the reference but from an interview with his parents there apparently there wasn't much "nurturing" other than simply making available the necessary materials which he gobbled up. Its not like they put a made him practice for an hour a day.
A boy in my high school class made IMO and got a gold medal (and later on won the Putnam one year). They interviewed his parents and it was a similar story.
I think you might be underrating the value of even that enabling work. Some parents would not have the financial resources to provide those learning materials. And some parents would take a normative stance on how an 8 year old ought to behave.
More importantly, it's not as though individuals like Clements or Erdos was corresponding with Terrence directly to arrange a meeting. His parents clearly played an important role in facilitating and allowing these encounters. That deserves a lot of credit!
Incredible. Knowing about Abelian groups, being able to graph y = x^3 — 2x^2 + x in one minute, and performing integration at age 7. Chomping up university-level math textbooks by 8. A classical math prodigy.
I definitely empathize with "his preference for using an analytic, highly logical problem-solving strategy" (I'm not a genius ofc). It's often more immediately clear for me than visual/spatial manipulation.
This really reminded me of the first part Flowers for Algernon. The main character undergoes a treatment which improves is intelligence and the story is narrated via a series of diary entries which become successively more fluent and sophisticated.
At 8 years I recycled filesystem directories. I didn't know you can create new folders, so when I needed one I grabbed a random one from C:\Windows, moved it to my desktop and deleted its contents.
I know it must be obvious but this proves to me that biological intelligence hasn't nearly reached its peak. If we select for pure intelligence, biological brains can get much smarter. Imagine if we had 5 million geniuses as smart or smarter than Tao doing quantum physics. But life doesn't select for pure intelligence, it selects for survival.
In the Dune books, they banned computers so they bred super mentally capable humans.
Not sure it works like that, I think his biggest superpower was intrinsic motivation. Any child who read maths textbooks with enthusiasm for 3-4 hours a day for years could in theory at least get close to doing what he did, but what kid had that level of motivation?
No .. not really. Not even close.
Just like even if I practiced music 8 hours a day I wouldn't be able to come up with the music Kurt Cobain has or Mozart. There are plenty of musicians who try really hard but lack the innate talent - at best they can learn to play other people's music but never can come up with good original music, at least not something other people want to hear.
As someone wrote here innate ability is a real thing
Any child who read maths textbooks with enthusiasm for 3-4 hours a day for years could in theory at least get close to doing what he did, but what kid had that level of motivation?
There is no way this is true. I've met and worked with enough people to know that not everyone has the same mental ability. There are some exceptionally sharp people and many dim witted ones too.
I think it has to be both. You need some ability to understand and thus find happiness in the thing that you are reading which leads to the motivation.
Another requirement is the emotional capacity at 8 years old to focus, feel confident, and feel safe.
I think that is the main obstacle to most people doing highly effective work and putting in long hours. You hear some call people who don't 'work hard' lazy, but my impression is that it's emotional capacity, and a lot of that comes from family.
I wonder if there is a correlation between prodigies and emotionally stable, healthy, present parents. It's hard to imagine children under a lot of stress - e.g., from abusive parents, highly unreliable parents (e.g., overwhelmed by addictions to drugs), emotionally unstable parents (e.g., narcissists), highly neglectful parents (e.g., who abandon their kids) ... - it's hard to imagine those kids doing what Tao did, regardless of their talent.
I like that test where some of the questions are wrong and wonder whether we should have that kind of thing in maths textbooks.
I think people need to be trained to be more confident in what they know, and if we gave them that kind of thing we could maybe train them to become so.
I read this earlier today and was thinking: how many such mathematically gifted individuals exist I. The world at one time? Assuming there are probably 20-30 Tao-caliber people in the US and an adversarial multiplier of 0.1 (only 1 in 10 such kids are nurtured), we reach 300 for this generation, about 1 in a million.
That means in a generation there are ~ 10k such people in the world. Think about connecting them or nurturing them with AI companions.
Genuine curiosity: if you are gifted with a certain “wiring” (genes, brain chemistry etc) why is that considered an accomplishment? Also - We, as a society, tend to celebrate people with “natural didn’t really need to work for” type gifts quite inconsistently - eg A supermodel who is gifted with the gift of looks, beauty etc is also in the same category of “natural” talent but sure doesn’t get the same celebration as a prodigy in maths or science. In both cases the people are fundamentally bestowed with abilities they didn’t really have to work extremely hard to acquire but are perhaps looked at differently. What’s kind of psychology is at play here? Would love to understand how we tend to interpret such things and then form beliefs.
I realize and acknowledge both sets had talents and the spent thier time doing something with it to produce something extraordinary but we seem to tend to overlook the massive head start they also had. Why so?
(Totally understandable if you feel like downvoting but I would ask you to articulate and share the cord it struck with you if you down vote)
> Genuine curiosity: if you are gifted with a certain “wiring” (genes, brain chemistry etc) why is that considered an accomplishment?
It's complex; first of all society has an interest for exceptional people to be respected and well compensated; if there was absolutely no prestige or compensation in being a math genius it's quite possible Terrence Tao would have become a schoolteacher. So a well functioning capitalist society has both monetary and prestige tools to incentivize extreme accomplishment.
Second, I think it's human nature to like and want hierarchy. Admiring figures for their looks, charisma or intellectual accomplishments could very will be in our wiring - 20 thousand years ago we would admire the shaman, the great hunter or the storyteller.
But ultimately I totally agree with you - not only were these people born into the unique genetic and envrionmental circumstances that made the accomplishment possible , I also don't believe they had any say after being born in becoming what they had become; e.g I don't believe there's a "free will" and that Terrence Tao "chose" to become a math genius. He was born into that reality in a fluke.
There are people wired like Tao (or superstar athletes, supermodels, or other remarkable people) that don't achieve the same results.
Even among the people who have similar "luck" in that respect, some still stand out. The people we think of as elite performers aren't just elite relative to the 99% of us. They're also elite within the top 1% that makes up their field: they're dominant even among the people who should be their peers.
There are very very few people wired like Tao; how many child prodigies like that are there ? He seems to be one in a million but its pretty much impossible to assess IQ at those levels.
Sure, it's not enough. YOu need the obsession for math, but lets not trivialize his intellectual ability - he's definitely not only top 1% that would just put him in the smartest 2-3 kids in his class. No, he was probably among the smartest 10-20 kids of his age group in the whole United States.
> A supermodel who is gifted with the gift of looks, beauty etc is also in the same category of “natural” talent but sure doesn’t get the same celebration as a prodigy in maths or science
We living on the same planet?
Pretty sure the supermodel gets infinitely more attention and certainly makes orders of magnitudes more money than some math prodigy, at least on mine.
There is an inequality between the sexes here. A female model does indeed get more attention and money based purely on the genes they didn't have to work for. It's not the case for men, though. Men also have to actually deliver something, whether it's being a performer like an actor, singer, footballer etc, or winning the Field's medal which you don't just get for being quite good at maths when you're 8. Trying to think of men who are famous just for genetics is quite hard. I guess like Orlando Bloom or the members of K-pop bands and whatnot, but they still have to perform and can't just prance around in fancy clothes and call it a day. In the case of Tao, if he had just decided to do something else or not accomplished anything you'd never have heard of him. Men always have to work for it. Women often don't, and if they try it doesn't work. It's the source of a lot of disgruntlement between the sexes, but probably a "grass is always greener" thing.
All I can say is before you assess the inequality of outcomes across the sexes, perhaps consider the differences in their inherent qualities to begin with.
The two types of talents can be judged by the impact they have. A scientific gifted individual can produce value while a good looking individual has mostly entertainment value.
That being said, supermodels are more famous, have a much larger following and earn much more money than math geniuses. That says we, humans, care more about entertainment than value.
I guess that makes sense in the light of her previous post and work on making a new archiving solution for being able to host singlefile archives more efficiently.
This brings to mind the childhood of John Stuart Mill:
- Learned Greek starting age three.
- Was studying Plato at age six.
- Studied Latin starting at age eight.
And more: https://en.wikipedia.org/wiki/John_Stuart_Mill#Biography
I guess it helps that he had Jeremy Bentham hanging around his house from an early age.
Learning three languages at an early age is completely unremarkable for millions of people around the world. It's just notable which ones his were.
It's notable if he learned Greek and Latin from books. Being classical languages, it sounds that way.
Most people who learn three languages as a kid are surrounded by other speakers, not books.
J.S. Mill's autobiography is a very fascinating read. He spends quite a lot of it discussing his early childhood, explaining that in his opinion he was not particularly special, rather, it was his father who pushed him to all those accomplishments. His father sheltered him from other kids so he was not aware that his accomplishments were unusual!
And imagine what he could have done if he had done something useful at such a young age!
As a father of an 8 years old, this is very moving.
While Terence is -without a doubt- born with prodigious abilities, I think credit should also be given to his parents Billy and Grace who seem to have managed to simultaneously nurture these special abilities while still letting Terence have a happy (?) childhood. This is not easy to do.
Can't find the reference but from an interview with his parents there apparently there wasn't much "nurturing" other than simply making available the necessary materials which he gobbled up. Its not like they put a made him practice for an hour a day.
A boy in my high school class made IMO and got a gold medal (and later on won the Putnam one year). They interviewed his parents and it was a similar story.
I think you might be underrating the value of even that enabling work. Some parents would not have the financial resources to provide those learning materials. And some parents would take a normative stance on how an 8 year old ought to behave.
More importantly, it's not as though individuals like Clements or Erdos was corresponding with Terrence directly to arrange a meeting. His parents clearly played an important role in facilitating and allowing these encounters. That deserves a lot of credit!
Incredible. Knowing about Abelian groups, being able to graph y = x^3 — 2x^2 + x in one minute, and performing integration at age 7. Chomping up university-level math textbooks by 8. A classical math prodigy.
I definitely empathize with "his preference for using an analytic, highly logical problem-solving strategy" (I'm not a genius ofc). It's often more immediately clear for me than visual/spatial manipulation.
This really reminded me of the first part Flowers for Algernon. The main character undergoes a treatment which improves is intelligence and the story is narrated via a series of diary entries which become successively more fluent and sophisticated.
We had to read it in middle school and man did it have me in tears at the end.
Had me in tears by the end. One of my favorite books. So glad a friend recommended it to me.
At 8 years old I was able to expertly dismantle many radios.
Was still a few years away from reassembly.
At 8 years I recycled filesystem directories. I didn't know you can create new folders, so when I needed one I grabbed a random one from C:\Windows, moved it to my desktop and deleted its contents.
Worked ok til it was a system dir and the system wouldn’t boot anymore? :)
Makes total sense, it used to be called "Recycle Bin" after all!
Don't miss the program he wrote after teaching himself BASIC from a book at age six (Fig 5 / book page 222 / PDF page 10):
> 320 print "(brmmmm-brmmmm-putt-putt-vraow-chatter-chatter bye mr. fibonacci!)"
This does feel like something a super smart alien pretending to be an 8 year old would write.
I know it must be obvious but this proves to me that biological intelligence hasn't nearly reached its peak. If we select for pure intelligence, biological brains can get much smarter. Imagine if we had 5 million geniuses as smart or smarter than Tao doing quantum physics. But life doesn't select for pure intelligence, it selects for survival.
In the Dune books, they banned computers so they bred super mentally capable humans.
Not sure it works like that, I think his biggest superpower was intrinsic motivation. Any child who read maths textbooks with enthusiasm for 3-4 hours a day for years could in theory at least get close to doing what he did, but what kid had that level of motivation?
No .. not really. Not even close. Just like even if I practiced music 8 hours a day I wouldn't be able to come up with the music Kurt Cobain has or Mozart. There are plenty of musicians who try really hard but lack the innate talent - at best they can learn to play other people's music but never can come up with good original music, at least not something other people want to hear.
As someone wrote here innate ability is a real thing
whatever helps you sleep at night, brother.
> Any child who read maths textbooks with enthusiasm for 3-4 hours a day for years could in theory at least get close to doing what he did
No, they couldn't. And neither could most adults, for that matter.
Innate ability is real.
I think it has to be both. You need some ability to understand and thus find happiness in the thing that you are reading which leads to the motivation.
Another requirement is the emotional capacity at 8 years old to focus, feel confident, and feel safe.
I think that is the main obstacle to most people doing highly effective work and putting in long hours. You hear some call people who don't 'work hard' lazy, but my impression is that it's emotional capacity, and a lot of that comes from family.
I wonder if there is a correlation between prodigies and emotionally stable, healthy, present parents. It's hard to imagine children under a lot of stress - e.g., from abusive parents, highly unreliable parents (e.g., overwhelmed by addictions to drugs), emotionally unstable parents (e.g., narcissists), highly neglectful parents (e.g., who abandon their kids) ... - it's hard to imagine those kids doing what Tao did, regardless of their talent.
I take it you've never met another human before
I like that test where some of the questions are wrong and wonder whether we should have that kind of thing in maths textbooks.
I think people need to be trained to be more confident in what they know, and if we gave them that kind of thing we could maybe train them to become so.
He’s on Star Talk this week. https://overcast.fm/+AAzXlUoaiV0
My brain initially parsed the title as an obituary title and I was really sad for a moment.
I read this earlier today and was thinking: how many such mathematically gifted individuals exist I. The world at one time? Assuming there are probably 20-30 Tao-caliber people in the US and an adversarial multiplier of 0.1 (only 1 in 10 such kids are nurtured), we reach 300 for this generation, about 1 in a million.
That means in a generation there are ~ 10k such people in the world. Think about connecting them or nurturing them with AI companions.
I am interested in his new book, "Six Math Essentials", but I doubt it will be on my very low level of math understanding..
Genuine curiosity: if you are gifted with a certain “wiring” (genes, brain chemistry etc) why is that considered an accomplishment? Also - We, as a society, tend to celebrate people with “natural didn’t really need to work for” type gifts quite inconsistently - eg A supermodel who is gifted with the gift of looks, beauty etc is also in the same category of “natural” talent but sure doesn’t get the same celebration as a prodigy in maths or science. In both cases the people are fundamentally bestowed with abilities they didn’t really have to work extremely hard to acquire but are perhaps looked at differently. What’s kind of psychology is at play here? Would love to understand how we tend to interpret such things and then form beliefs.
I realize and acknowledge both sets had talents and the spent thier time doing something with it to produce something extraordinary but we seem to tend to overlook the massive head start they also had. Why so?
(Totally understandable if you feel like downvoting but I would ask you to articulate and share the cord it struck with you if you down vote)
> Genuine curiosity: if you are gifted with a certain “wiring” (genes, brain chemistry etc) why is that considered an accomplishment?
It's complex; first of all society has an interest for exceptional people to be respected and well compensated; if there was absolutely no prestige or compensation in being a math genius it's quite possible Terrence Tao would have become a schoolteacher. So a well functioning capitalist society has both monetary and prestige tools to incentivize extreme accomplishment.
Second, I think it's human nature to like and want hierarchy. Admiring figures for their looks, charisma or intellectual accomplishments could very will be in our wiring - 20 thousand years ago we would admire the shaman, the great hunter or the storyteller.
But ultimately I totally agree with you - not only were these people born into the unique genetic and envrionmental circumstances that made the accomplishment possible , I also don't believe they had any say after being born in becoming what they had become; e.g I don't believe there's a "free will" and that Terrence Tao "chose" to become a math genius. He was born into that reality in a fluke.
There are people wired like Tao (or superstar athletes, supermodels, or other remarkable people) that don't achieve the same results.
Even among the people who have similar "luck" in that respect, some still stand out. The people we think of as elite performers aren't just elite relative to the 99% of us. They're also elite within the top 1% that makes up their field: they're dominant even among the people who should be their peers.
There are very very few people wired like Tao; how many child prodigies like that are there ? He seems to be one in a million but its pretty much impossible to assess IQ at those levels. Sure, it's not enough. YOu need the obsession for math, but lets not trivialize his intellectual ability - he's definitely not only top 1% that would just put him in the smartest 2-3 kids in his class. No, he was probably among the smartest 10-20 kids of his age group in the whole United States.
Why for you is innate grit any more commendable than innate intelligence?
> A supermodel who is gifted with the gift of looks, beauty etc is also in the same category of “natural” talent but sure doesn’t get the same celebration as a prodigy in maths or science
We living on the same planet?
Pretty sure the supermodel gets infinitely more attention and certainly makes orders of magnitudes more money than some math prodigy, at least on mine.
There is an inequality between the sexes here. A female model does indeed get more attention and money based purely on the genes they didn't have to work for. It's not the case for men, though. Men also have to actually deliver something, whether it's being a performer like an actor, singer, footballer etc, or winning the Field's medal which you don't just get for being quite good at maths when you're 8. Trying to think of men who are famous just for genetics is quite hard. I guess like Orlando Bloom or the members of K-pop bands and whatnot, but they still have to perform and can't just prance around in fancy clothes and call it a day. In the case of Tao, if he had just decided to do something else or not accomplished anything you'd never have heard of him. Men always have to work for it. Women often don't, and if they try it doesn't work. It's the source of a lot of disgruntlement between the sexes, but probably a "grass is always greener" thing.
All I can say is before you assess the inequality of outcomes across the sexes, perhaps consider the differences in their inherent qualities to begin with.
> based purely on the genes they didn't have to work for
Modeling is notorious for its negative impact on models' health.
They absolutely work for it, and in one of the most toxic work environments.
I wonder how Tao - or a supermodel - might feel about the idea that they don't have to work for their "gifts"
Not a mystery, Tao has written about how, child prodigy aside, he has to work at math on a regular basis with grit and perseverance.
The two types of talents can be judged by the impact they have. A scientific gifted individual can produce value while a good looking individual has mostly entertainment value.
That being said, supermodels are more famous, have a much larger following and earn much more money than math geniuses. That says we, humans, care more about entertainment than value.
It depends on how much value their talents can bring to humankind, I guess.
Wow, incredible read! Amazing what motivated peple (and children!) can achieve.
Interesting it's hosted on gwern...
Gwern hosts a lot of PDFs -- see https://gwern.net/archiving
I guess that makes sense in the light of her previous post and work on making a new archiving solution for being able to host singlefile archives more efficiently.
Thank you
https://www.google.com/search?q=Gwern+Branwen
her?
Humbling.
Indeed. He definitely knows more Math than I do.
I could have been just like him if I tried hard enough.