Teenager solves stubborn riddle about prime number look-alikes(quantamagazine.org)
quantamagazine.org
Teenager solves stubborn riddle about prime number look-alikes
https://www.quantamagazine.org/teenager-solves-stubborn-riddle-about-prime-number-look-alikes-20221013/
263 comments
Someone once told me, “comparison is the thief of joy.”
I’m confident this is a famous quote. But once I heard it, it kept resonating.
Then years later I watched Bluey with my kids and mom says, “just run your own race” and it all clicked. I’m so thankful that it clicked because I feel liberated from this self-imposed sense that I need to absolutely maximize my time here, which is an impossible task.
I’m confident this is a famous quote. But once I heard it, it kept resonating.
Then years later I watched Bluey with my kids and mom says, “just run your own race” and it all clicked. I’m so thankful that it clicked because I feel liberated from this self-imposed sense that I need to absolutely maximize my time here, which is an impossible task.
Learning a new and at the time somewhat obscure (in my setting) foreign language when I was around 30 taught me this lesson at the time. I did well in the small class I attended with others, but not as well as some of the far younger students with both more free time and perhaps better age-given baseline performance at language acquisition (though this is not entirely clear-cut scientifically).
I realized it didn't matter. What mattered is that I progressed toward my goal at whatever pace I could muster, not making it a race. Interestingly, this mindset has made me willing to take on much larger and more complicated challenges in general, one day at a time, without feeling overwhelmed by the mountain to climb ahead.
I realized it didn't matter. What mattered is that I progressed toward my goal at whatever pace I could muster, not making it a race. Interestingly, this mindset has made me willing to take on much larger and more complicated challenges in general, one day at a time, without feeling overwhelmed by the mountain to climb ahead.
Yes! I love that. Thank you for sharing.
This is the mindset that got me to do so many projects where a younger me thought, “someone smarter has already done a better job at that…”
I now write software for me. Sometimes I publish it.
This is the mindset that got me to do so many projects where a younger me thought, “someone smarter has already done a better job at that…”
I now write software for me. Sometimes I publish it.
Thank you! I feel like this step-by-step mentality is a good replacement for the naivety of the youth when it comes to attacking complex problems.
Reminds me of Seneca:
"However much you possess there's someone else who has more, and you'll be fancying yourself to be short of things you need to exact extent to which you lag behind him."
Lucius Annaeus Seneca, Letters from a Stoic
"However much you possess there's someone else who has more, and you'll be fancying yourself to be short of things you need to exact extent to which you lag behind him."
Lucius Annaeus Seneca, Letters from a Stoic
Though this tends to be true, once you become aware of it and begin to recalibrate your perception of what it means to have ability, possessions, fortune, any of it – it stops mattering nearly as much, and can often even go the other way.
There are many people who are fortunate in material or circumstantial ways, but you see ways in which they’re enslaved by it or it somehow dictates their life. The classic “owned by what your own” paradigm which is eerily common in western culture.
This can have the effect of making you feel fortunate for having less, rather than envious.
I love Seneca. What I haven’t got a good grip on is time. On the Shortness of Life is a bit of a gut punch for me.
There are many people who are fortunate in material or circumstantial ways, but you see ways in which they’re enslaved by it or it somehow dictates their life. The classic “owned by what your own” paradigm which is eerily common in western culture.
This can have the effect of making you feel fortunate for having less, rather than envious.
I love Seneca. What I haven’t got a good grip on is time. On the Shortness of Life is a bit of a gut punch for me.
I don't know about mathematics, but in orgies this is absolutely true.
Ok "run your own race" sounds very different all of a sudden
[deleted]
https://quoteinvestigator.com/2021/02/06/thief-of-joy/
(Seems like a good site for verifying quotes, it's crazy how many commonly-believed-to-have-been-said-by-X quotes simply never were...)
If you want some advice on how to avoid having your "joy" stolen by comparison, you could always go with
https://quoteinvestigator.com/2017/10/03/life-success/
Though I suppose "Z" these days would logically include "avoid replying to online forum comments..."!
(Seems like a good site for verifying quotes, it's crazy how many commonly-believed-to-have-been-said-by-X quotes simply never were...)
If you want some advice on how to avoid having your "joy" stolen by comparison, you could always go with
https://quoteinvestigator.com/2017/10/03/life-success/
Though I suppose "Z" these days would logically include "avoid replying to online forum comments..."!
Very cool article. So it is a famous quote!
> Someone once told me, “comparison is the thief of joy.”
I've seen it phrased better elsewhere, but otherwise yes.
I've seen it phrased better elsewhere, but otherwise yes.
Feel free to share the better phrasing!
"It doesn't matter what they do. It matters what you do."
eh, I'll let my comment stand on its own ;-)
It definitely does stand on its own, haha. I appreciated it anyway
Thanks. Glad at least one person got it :-) I took a risk with the subtlety!
Haha now at last I love it
Sartre perhaps? ..hell is other people.
Consciously knowing that comparing yourself too much to others is one thing, actually stopping it is a whole different story at least for me. I didn‘t succeed yet. But I noticed that the idea of having 20% personal growth each year is really strong
I like to use the phrase "I don't need to be better than someone else; just better today than I was yesterday".
Bluey is a gem.
Have you seen the one where he tries to teach his children chess? Hilarity ensues.
I wasn't as cool as this kid, but I wrote a binomial expansion program in TI BASIC back in high school that I was pretty proud of. Teacher said it was neat, but then banned calculators on our tests/ quizzes after I demoed it.
I had a maths teacher who accused me of cheating because we had coursework to solve a particular problem, I solved the problem for that case and then the general case using math she hadn’t taught me even had a pascal program that you could enter parameters and it’d give you the answer.
She literally couldn’t get her head around a student going to the library, taking out a book on maths and teaching themselves because they where curious.
The UK has a very strange educational system, curiosity isn’t encouraged, you learn like a good little peg in a round hole.
Left a very bad taste in my mouth but at least she did eventually apologise.
She literally couldn’t get her head around a student going to the library, taking out a book on maths and teaching themselves because they where curious.
The UK has a very strange educational system, curiosity isn’t encouraged, you learn like a good little peg in a round hole.
Left a very bad taste in my mouth but at least she did eventually apologise.
"Never let school get in the way of your education" is roughly the philosophy my parents raised me on. Precisely to encourage what you had done vs. what school expected.
Mine too, well mostly. My Grandmother worked through my mother to get that done without invoking other fairly painful, in the consuming too much time sense, family dynamics (Father was a mess and mostly out of this part of the picture.), but I digress save to say me too.
They want to encourage self directed learning and I am glad they did. If nothing else, boredom becomes a good thing more often. The seeking begins and with that can often come some real learning and insight as it may be applied.
They want to encourage self directed learning and I am glad they did. If nothing else, boredom becomes a good thing more often. The seeking begins and with that can often come some real learning and insight as it may be applied.
Wow, my parents were willing to sacrifice my entire education if it meant I participated in school in the ways they cared about.
Your way sounds better.
Your way sounds better.
Not unique to the UK. This is common all over the world. It's the most efficient way to educate the masses cheaply.
Among college-educated professionals, primary and secondary school teachers are near the bottom for intelligence, as measured by SAT scores. https://www.businessinsider.com/heres-the-average-sat-score-...
The pay is shit, so intelligent people decide to pursue other passions instead of teaching.
Also, the profession seems to attract the kind of people who like to exercise arbitrary authority over other people. Not a majority of teachers, but a substantial minority.
Once I realized these things, a lot of negative experiences I had with teachers made more sense. For what it's worth, I also had some great teachers along the way.
The pay is shit, so intelligent people decide to pursue other passions instead of teaching.
Also, the profession seems to attract the kind of people who like to exercise arbitrary authority over other people. Not a majority of teachers, but a substantial minority.
Once I realized these things, a lot of negative experiences I had with teachers made more sense. For what it's worth, I also had some great teachers along the way.
Isn’t the biggest or one of the biggest indicator of SAT scores is how wealthy/well off the test taker is?
Similarly, the IQ test is flawed too. Neither are representative of intelligence or what is required to be a good teacher
Similarly, the IQ test is flawed too. Neither are representative of intelligence or what is required to be a good teacher
No. SAT is in fact fairly resistant to preparation. Not as much as some other tests, and not as much as the older versions of the SAT, but it's still largely an intelligence test.
Families making $200K+ do 400 points higher than families making $40K or $20K or less. I forget the exact figure. This study is widely available. 400 point difference is so massive it shows it’s not largely an intelligence test nor is it resistant to prep.
> The UK has a very strange educational system, curiosity isn’t encouraged, you learn like a good little peg in a round hole.
That was the feeling I had after one week in a UK school through an exchange program. Because my original exchange partner quit on me after seeing my picture, I got assigned a different one who was 2 grades higher. I was pretty average in math back in Germany, but in that class, 2 years above mine (11th while I was 9th), their math riddle that was supposed to take them the whole week was solved by me in one lesson.
This was around 20 years ago, though.
That was the feeling I had after one week in a UK school through an exchange program. Because my original exchange partner quit on me after seeing my picture, I got assigned a different one who was 2 grades higher. I was pretty average in math back in Germany, but in that class, 2 years above mine (11th while I was 9th), their math riddle that was supposed to take them the whole week was solved by me in one lesson.
This was around 20 years ago, though.
25ish for me.
I have a step-son who is (nearly) 13 and about to start seriously in on his GCSE's - the situation hasn't improved, the non-higher tier papers are just embarrassing, the foundational tier he could have done well by the time he was 10 - he likes maths and I always wished I'd pushed it harder when I was young but a spectacularly shitty home-life ensured while I did decent in my GCSE's I didn't do what I could have done - I make sure he doesn't have the same problem.
I have a step-son who is (nearly) 13 and about to start seriously in on his GCSE's - the situation hasn't improved, the non-higher tier papers are just embarrassing, the foundational tier he could have done well by the time he was 10 - he likes maths and I always wished I'd pushed it harder when I was young but a spectacularly shitty home-life ensured while I did decent in my GCSE's I didn't do what I could have done - I make sure he doesn't have the same problem.
I remember my teacher banned the use of software on calculators (not the calculators themselves) but left an exception for me because she knew I was writing all the software I was using and didn't want to discourage me using that to learn about both algorithms and the underlying math.
It was a while ago, and my memory sucks for those kind of things, so maybe it was you! Anyways, programming anything in the space of 5 minutes is hard enough for me, so I'm still amazed at that binomial expansion (which I had to look up, myself).
I wrote programs too. Even had them print out the work.
Checked with the teacher at the beginning of the year and she said no problem. After rampant cheating, mostly people just writing notes in the programs section, she instituted a policy of deleting everyone’s calculator before any test. but let me keep mine too, I had asked to not delete it and I’d take a class calculator if she insisted.
Checked with the teacher at the beginning of the year and she said no problem. After rampant cheating, mostly people just writing notes in the programs section, she instituted a policy of deleting everyone’s calculator before any test. but let me keep mine too, I had asked to not delete it and I’d take a class calculator if she insisted.
Waaaay back at school in the 90s, that was the policy in our public exams, we were allowed the calculator but it would be wiped on the way in to the hall.
So I memorised my quadratic-solver-with-intermediate-steps and tapped it back in while we were waiting for the test to start :)
The calculation was not hard, but it was still faster just to run my program and scribble down the steps it spat out than to do the solving manually each time it came up. IIRC I had one for simultaneous equations too.
So I memorised my quadratic-solver-with-intermediate-steps and tapped it back in while we were waiting for the test to start :)
The calculation was not hard, but it was still faster just to run my program and scribble down the steps it spat out than to do the solving manually each time it came up. IIRC I had one for simultaneous equations too.
> A computer program to factor any polynomial equation string input to it:
Do you mean a program to solve for f(x) = 0 using numerical approximation? Factorization has a specific meaning and is not necessarily possible for quintic and higher-degree polynomials / there is no closed form solution like the quadratic formula for n>=5.
To clarify, there do exist closed form solutions for quintic functions; they just cannot be expressed using rationals and a limited set of operations of additions, subtractions, multiplications, divisions, and root extractions. However, if you do not limit yourself to those operations, there do exist closed form solutions to quintics. See [1] for a few examples.
[1] https://mathworld.wolfram.com/QuinticEquation.html
[1] https://mathworld.wolfram.com/QuinticEquation.html
Yes, polynomial factorization does have a specific meaning. You should learn it.
https://en.wikipedia.org/wiki/Factorization_of_polynomials
[deleted]
If they’re talking about the quadratic equation, this is likely 8th grade to 10th grade in the US education system. Factoring would be y^2 = 3(ab)^2 + c^2, get y all alone so it’s y = _____
[deleted]
And, to add to this, we need to find and nurture these minds so they don't get bored and can grow at their natural speed.
>Throughout high school, Kaczynski was ahead of his classmates academically. Placed in a more advanced mathematics class, he soon mastered the material. He skipped the eleventh grade, and by attending summer school he graduated at age 15. Kaczynski was one of his school's five National Merit finalists and was encouraged to apply to Harvard.[17] While still at age 15, he was accepted to Harvard and entered the university on a scholarship in 1958 at age 16.[19] A classmate later said Kaczynski was emotionally unprepared: "They packed him up and sent him to Harvard before he was ready ... He didn't even have a driver's license."[9]
But we also need to make sure that we respect the humans who possess those brains too...
But we also need to make sure that we respect the humans who possess those brains too...
Letting them get ahead is usually the best way to respect an intelligent student. I wouldn't put a lot of stock in one classmate's opinion about someone's emotional state, and a driving license is hardly the be all and end all (I didn't have one when I went to university, and never missed it). "Don't put students of any age through interrogation-style experiments that are designed to stress them" is probably a better lesson to take from Kaczynski's case.
Sure if you want the student to never be around same aged peers and struggle to develop a social life so they instead focus on their work, I agree.
Being at the same ability level matters way more than being at the same age. Speaking from personal experience, the whole "let's try to pretend that this kid is normal and that with enough time he will fit in" has had terrible consequences on my mental health, and even on my social ability with the rare people that I feel comfortable with.
I think that in many case, trying to pretend that people are the same when they are not is causing a lot of pain for everyone involved. Holding a conversation with a normal person requires me to expand a lot of energy to not be myself, because if I am I will quickly be hurt and/or rejected. And just like I often hurt people by being myself, other people hurt me by being themselves. Trying to explain it to them has been pointless in all situation, and most of the time makes the situation worse.
Why would any of us have to endure that, when we could instead be with groups of like-minded people and enjoy life more? I don't feel superior to those people, and spending more time with them leads me to develop more negative sentiments towards them. I think the sane option is to avoid them as much as possible, while making sure that I treat them with respect. I can't change fundamentally who I am, they can't change who they are. I can't understand why you advocate for a situation that creates so much suffering for everyone involved.
I think that in many case, trying to pretend that people are the same when they are not is causing a lot of pain for everyone involved. Holding a conversation with a normal person requires me to expand a lot of energy to not be myself, because if I am I will quickly be hurt and/or rejected. And just like I often hurt people by being myself, other people hurt me by being themselves. Trying to explain it to them has been pointless in all situation, and most of the time makes the situation worse.
Why would any of us have to endure that, when we could instead be with groups of like-minded people and enjoy life more? I don't feel superior to those people, and spending more time with them leads me to develop more negative sentiments towards them. I think the sane option is to avoid them as much as possible, while making sure that I treat them with respect. I can't change fundamentally who I am, they can't change who they are. I can't understand why you advocate for a situation that creates so much suffering for everyone involved.
You realize there are plenty of places like Exeter, Andover, Stuyvesant, Boston Latin, Bronx Science, etc. full of genius children, right? I agree those places are much better than a given average High School.
The problem is putting a child in an environment where they are surrounded by adults all the time. Like being 17 and having your 'peers' be 30 year old post-docs who are getting married and having children is a complete mind-fuck.
The problem is putting a child in an environment where they are surrounded by adults all the time. Like being 17 and having your 'peers' be 30 year old post-docs who are getting married and having children is a complete mind-fuck.
Why are you so focused on the NYC high-end high school segment? It is utterly unrelatable to 95% of all students and conservatively 85% of users here.
Being surrounded by adults all the time is only a problem if you never spend time around actual peers. Nobody should be forced to isolate themselves like that.
Being surrounded by adults all the time is only a problem if you never spend time around actual peers. Nobody should be forced to isolate themselves like that.
The opposite causes the teacher as well as the peers to resent the more intelligent child, so which is better?
I could read since I was 4. When others at school were reading one word per minute spelling letter by letter, I finished the whole article and then continued to another and another. Result? I got a teacher's note (a big deal where I live) almost every lesson, and bad grades. And the children hated me, probably mostly because of the teacher mocking me for not paying attention. I would give anything to be boosted to similarly skilled children. I was friends with them anyways, never really liked the kids from my grade (but that could've been avoided if I wasn't forced to "learn" with them).
I've never seen a boosted child say it wasn't worth it or that they had much problems being a year/two younger than their classmates. Actually, smaller schools here usually combine 2-3 grades into one anyways and the children are friends alright, so boosting the taught curriculum is really not a big deal (unfortunately the system didn't allow it when I went to school).
I could read since I was 4. When others at school were reading one word per minute spelling letter by letter, I finished the whole article and then continued to another and another. Result? I got a teacher's note (a big deal where I live) almost every lesson, and bad grades. And the children hated me, probably mostly because of the teacher mocking me for not paying attention. I would give anything to be boosted to similarly skilled children. I was friends with them anyways, never really liked the kids from my grade (but that could've been avoided if I wasn't forced to "learn" with them).
I've never seen a boosted child say it wasn't worth it or that they had much problems being a year/two younger than their classmates. Actually, smaller schools here usually combine 2-3 grades into one anyways and the children are friends alright, so boosting the taught curriculum is really not a big deal (unfortunately the system didn't allow it when I went to school).
Students can tell who's smarter and who's not, and someone at the same level of learning will always be far more of a peer than someone who happens to be the same age. All the available evidence is that the best thing for children's social life is to let them interact with people at the same learning level rather than the same age. (To say nothing of the fact that segregating students by age is unnatural in the first place, and traditional cultures where kids grow up around people of a wide range of ages are far better for their social development than the western school model...).
Yes. It is a brutal existence. Social isolation can continue long past when you would expect it to end, after graduation from high school or college, because of the exploitive behaviors of employers and the self-exploitive nature of someone who only finds solace in work.
What is the solution? I will not spoil it for you, but it does not involve giving up.
What is the solution? I will not spoil it for you, but it does not involve giving up.
Would they have success at developing a social life if they were surrounded by same-aged peers of average intellect? It'd be a crapshoot.
I mean sure, Hacker News is proof of that, but most hyper successful adults that were prodigies went to places like Exeter, Andover, Stuyvesant, Boston Latin, Bronx Science, etc. where they are around smart people their age.
For those who, like me, were wondering - this excerpt is from the Wikipedia page [1] about the Ted Kaczynski, aka the Unabomber.
[1] https://en.wikipedia.org/wiki/Ted_Kaczynski
[1] https://en.wikipedia.org/wiki/Ted_Kaczynski
Thank you. I couldn’t figure out who they were talking about.
There should be some sort of university made for people like that - where they’re learning high level stuff but with people their age.
Bards college at Simon's Rock did something like this. My friend went when he was 15ish. I used to visit during the summers and while the education was relatively accelerated, there are a host of other issues that don't often get brought up with this model.
Added bonus is they come out of it being able to cast 4 first level and 2 second level spells each day
It’s a Bard college. More likely they learned to play the lyre and sing ballads.
Bard is also a character class in many RPGs (nethack, Bards Tale); GPs comment makes sense in that context.
I think you’d still have a problem mixing 7, 12 and 15 year old kids. Perhaps not as much as 15 with 20, but still.
This growing up thing is hard.
This growing up thing is hard.
You are so far from wrong, it hurts me to think about it.
Due to having a reading level 10 years ahead of my peers, and already having learned math concepts at least 4 years ahead of them... at age 8... I was put 3 years ahead of my peers in school until the teachers realized the other kids were having none of it.
They put me back in the 'proper' grade, because 'I' somehow wasn't going to develop social skills well enough in such an environment.
They might not have been wrong, but they would have been more correct to tune in the little buggers causing me grief. Or at least let me tune them in...
Due to having a reading level 10 years ahead of my peers, and already having learned math concepts at least 4 years ahead of them... at age 8... I was put 3 years ahead of my peers in school until the teachers realized the other kids were having none of it.
They put me back in the 'proper' grade, because 'I' somehow wasn't going to develop social skills well enough in such an environment.
They might not have been wrong, but they would have been more correct to tune in the little buggers causing me grief. Or at least let me tune them in...
Sorry Manu40, I’m just saying this whole growing up thing isn’t easy and with few correct answers.
Most are still struggling to get over it decades later, myself included.
Most are still struggling to get over it decades later, myself included.
I mean, who can blame any of them, ourselves included. Instead of doing the right thing and setting the little snots straight, they instead hampered our progress all to make things easier for themselves while placating a bunch of children who needed to be taught the reality of the situation, not coddled into thinking they were 'just as capable' (when clearly they weren't).
Quite frankly, I think we all should be banding together to sue our old schools over this stuff.
Quite frankly, I think we all should be banding together to sue our old schools over this stuff.
Surely there’s more to life for the top 1% of learners than trying to get retribution against public institutions. One might ask if you’re really as brilliant as you say if you can’t see that - or maybe you were just early as a child, and that’s all.
I was horribly bullied for being more advanced than my same-age classmates AND older children at high-school, placing me with even older kids (with whom presumably I would have even less in common) doesn't seem like it would have helped my situation.
Yeah, no... it probably wouldn't have helped; very much so for the reasons I have stated in reply to Tambourine.
That said, I should add that 2 of those students ended up running into me again years later. They apologized, which was nice; but then offered me the chance to go do lines of coke with them...
I declined, and went on with my life knowing doubly that my instinct about them was correct the whole time. Wastes of skin.
That said, I should add that 2 of those students ended up running into me again years later. They apologized, which was nice; but then offered me the chance to go do lines of coke with them...
I declined, and went on with my life knowing doubly that my instinct about them was correct the whole time. Wastes of skin.
> Wastes of skin.
I would flag your comment if I had the authority. This is not acceptable language for this forum.
I would flag your comment if I had the authority. This is not acceptable language for this forum.
As discussed in Good Will Hunting: https://www.youtube.com/watch?v=-XCsE5NffMA
[deleted]
I don't hear this viewpoint acknowledged enough.
Society wasting and discarding undiscovered talent is one of my deepest fears.
Society wasting and discarding undiscovered talent is one of my deepest fears.
"I am, somehow, less interested in the weight and convolutions of Einstein's brain than in the near certainty that people of equal talent have lived and died in cotton fields and sweatshops." - Stephen Jay Gould
Mismeasure Of Man?
SJG is underrated.
SJG is underrated.
[deleted]
> Since then, I don't feel bad if someone achieves more than me, because clearly there are some people out there that are born to solve certain classes of problems (maybe their brain structure is better for those, or something, who knows).
As a math teacher, I very much like every part of your comment except for this. There are way too many people who decide never to try mathematics because they weren't born to it. I don't believe (and it seems rather insusceptible to proof) that math is something that you are born with a talent for, or not; instead, it's all about how much passion you have for it. Genius is any field is not born of effortless gift; it is born of tireless effort. The real difference is between the people who are willing to put in that tireless effort, and those who do not wish to do so (which is a reasonable choice for something you're not passionate about!).
As a math teacher, I very much like every part of your comment except for this. There are way too many people who decide never to try mathematics because they weren't born to it. I don't believe (and it seems rather insusceptible to proof) that math is something that you are born with a talent for, or not; instead, it's all about how much passion you have for it. Genius is any field is not born of effortless gift; it is born of tireless effort. The real difference is between the people who are willing to put in that tireless effort, and those who do not wish to do so (which is a reasonable choice for something you're not passionate about!).
In school I essentially hated all math beyond Algebra, because it started to become something less tangible and more rote. Which is a sensible way to teach it as a pure subject, but it completely undercuts it's utility and connection to real world problems and the intrigues that go along with them.
Once I started to really want to understand and write audio processing code, I suddenly had a framework to not only care about Trigonometry, but to actually apply it and to have a tool that would validate my understanding of it. That short feedback loop got me through the subject incredibly quickly.
I was completely blown away when I started reading some Linear Algebra material because I realized that I had essentially been doing it for years without even strictly realizing it. The subject came to me very quickly because I already had a solid "mechanical" understanding of how to implement systems utilizing a particular subset of it.
I think passion for the deeper maths is something most people just have to discover through application, and there just happen to be a rare few who discover a pure love of it very early in life.
Once I started to really want to understand and write audio processing code, I suddenly had a framework to not only care about Trigonometry, but to actually apply it and to have a tool that would validate my understanding of it. That short feedback loop got me through the subject incredibly quickly.
I was completely blown away when I started reading some Linear Algebra material because I realized that I had essentially been doing it for years without even strictly realizing it. The subject came to me very quickly because I already had a solid "mechanical" understanding of how to implement systems utilizing a particular subset of it.
I think passion for the deeper maths is something most people just have to discover through application, and there just happen to be a rare few who discover a pure love of it very early in life.
It's interesting how opposite people's interests in math can be. I hated elementary arithmetic and its focus on memorization and speed. Once the focus switched to solving equations and became less about concrete numbers, that's when I shot from "severely underperforming his potential" to "a natural at math"
What's funny is I was that guy (not literally the same guy, but did similar things) as GP describes who could type in a program implementing the math lesson in the time it took the teacher to describe it.
And. I. Struggled. So. Much. with math classes.
Could never get homework done because I couldn't focus due to undiagnosed (and thus unmedicated) ADHD. On tests it'd take me the whole hour to do three problems out of twenty because I had to re-derive every equation and lemma from first principles before I convinced myself I was doing it right. And then (probably again due to the undiagnosed ADHD) I'd transpose symbols while working the problem or make other working-memory errors.
The best math class experience I had (later in life) was with a professor at a community college who was a military veteran (mentioned because it influenced his attitude toward teaching). He said, "I'm going to spend hardly any time explaining things; we're just going to work the problems." And that's what we did; we just worked problems.
You'd think I'd be bored with that, but it was the opposite. As I think about it now to try to explain why, I realize that other math classes I'd been in the expectation was lecture was for explaining the concepts, homework was for self-directed working the problems, and tests were for demonstrating that. But if you're bright and have ADHD you're quickly bored with the explanation and completely UNable to self-direct working the problems. So the class time was wasted and homework sessions were hell. So this professor using class time to work the problems was just what I needed.
And. I. Struggled. So. Much. with math classes.
Could never get homework done because I couldn't focus due to undiagnosed (and thus unmedicated) ADHD. On tests it'd take me the whole hour to do three problems out of twenty because I had to re-derive every equation and lemma from first principles before I convinced myself I was doing it right. And then (probably again due to the undiagnosed ADHD) I'd transpose symbols while working the problem or make other working-memory errors.
The best math class experience I had (later in life) was with a professor at a community college who was a military veteran (mentioned because it influenced his attitude toward teaching). He said, "I'm going to spend hardly any time explaining things; we're just going to work the problems." And that's what we did; we just worked problems.
You'd think I'd be bored with that, but it was the opposite. As I think about it now to try to explain why, I realize that other math classes I'd been in the expectation was lecture was for explaining the concepts, homework was for self-directed working the problems, and tests were for demonstrating that. But if you're bright and have ADHD you're quickly bored with the explanation and completely UNable to self-direct working the problems. So the class time was wasted and homework sessions were hell. So this professor using class time to work the problems was just what I needed.
I struggled a lot in math too. Got a D in calculus, nearly dropped out of high school. I'd absorbed my dad's excuse, "I'm bad at math." What a crock of shit. Turns out, I was bad at doing work in absence of motivation. But my parents didn't believe in ADHD, so I couldn't have that.
Went back to community college after some time as a web dev, and had a teacher with a booming voice and a gentle attitude, who explained that we were going to be doing a lot of homework. Like your instructor, he'd spend class time working problems, and then I'd go home and do an hour or two of homework every night while it was still fresh. That kicked off a trajectory that resulted in a PhD and a very fulfilling job as a mathematician.
Went back to community college after some time as a web dev, and had a teacher with a booming voice and a gentle attitude, who explained that we were going to be doing a lot of homework. Like your instructor, he'd spend class time working problems, and then I'd go home and do an hour or two of homework every night while it was still fresh. That kicked off a trajectory that resulted in a PhD and a very fulfilling job as a mathematician.
I was on my high school's math team. For three years I was in a special class where all we did was look at math problems selected from competitions and solve them. On the one hand, it was great and fun. On the other hand, trig problems were in short supply, and I came out with a less than excellent grasp of trig, especially identities.
I mean, I agree anyone can do anything with enough effort, but some people have an easier time. It’s really easy for me to program computers. I routinely am perplexed that people I know can’t understand or see the algorithms and structures I see. That doesn’t mean I’m the best or that others cannot learn it, just that somehow my brain is structured, whether through nature or nurture, in a way that helps me do logic and math required for programming. On the other hand, I cannot learn other languages. I just cannot figure them out. I still worked hard to learn that skill, but I definitely worked way less hard to become experienced at this than so many programmers I know.
In my view, not everyone needs to be good at everything, but I do get that society sets up maths and science as way harder than it has to be. Part of that is how we teach math, too. I hated math until I went to college and roundabout earned a maths degree.
In my view, not everyone needs to be good at everything, but I do get that society sets up maths and science as way harder than it has to be. Part of that is how we teach math, too. I hated math until I went to college and roundabout earned a maths degree.
Maryam Mirzakhani is a lovely 3xamoke of somebody who didn't find mathematics effortlessly easy as a child.
On the other hand, there definitely are people for whom mathematics is much easier than others. I found mathematics effortless at school and even to an extent at university in a way that other people clearly didn't. I did work at the Olympiad type stuff at school I guess but that felt like fun.
I didn't become a Fields Medallist though :-)
Most of the international-calibre mathematicians I have crossed paths with did find a certain level of maths effortless in the same way I did I think, but they also worked hard, although not necessarily relentlessly. Most certainly had other interests outside mathematics. They all really really loved mathematics.
On the other hand, there definitely are people for whom mathematics is much easier than others. I found mathematics effortless at school and even to an extent at university in a way that other people clearly didn't. I did work at the Olympiad type stuff at school I guess but that felt like fun.
I didn't become a Fields Medallist though :-)
Most of the international-calibre mathematicians I have crossed paths with did find a certain level of maths effortless in the same way I did I think, but they also worked hard, although not necessarily relentlessly. Most certainly had other interests outside mathematics. They all really really loved mathematics.
> Maryam Mirzakhani is a lovely 3xamoke of somebody who didn't find mathematics effortlessly easy as a child.
"3xamoke" is "example", right? If I seemed to claim that good mathematicians found mathematics effortless, I definitely didn't mean to! Rather I meant to claim the opposite: that doing truly deep mathematics (or anything) is deeply effortful even for those who wind up excelling most at it; those people are just the ones who feel such a passion that they can't stop putting in that effort.
> On the other hand, there definitely are people for whom mathematics is much easier than others. I found mathematics effortless at school and even to an extent at university in a way that other people clearly didn't. I did work at the Olympiad type stuff at school I guess but that felt like fun.
I don't claim anything as absurd as that there is no difference in mathematical ability between different people, only that (1) I think the relevance of ability compared to effort is overstated, and (2) even to the extent to which ability is relevant, looking at certain people and declaring them as just good at math (a) undervalues the effort that they put in and (b) discourages people who struggle with it, and who think that means that they cannot do well at it.
"3xamoke" is "example", right? If I seemed to claim that good mathematicians found mathematics effortless, I definitely didn't mean to! Rather I meant to claim the opposite: that doing truly deep mathematics (or anything) is deeply effortful even for those who wind up excelling most at it; those people are just the ones who feel such a passion that they can't stop putting in that effort.
> On the other hand, there definitely are people for whom mathematics is much easier than others. I found mathematics effortless at school and even to an extent at university in a way that other people clearly didn't. I did work at the Olympiad type stuff at school I guess but that felt like fun.
I don't claim anything as absurd as that there is no difference in mathematical ability between different people, only that (1) I think the relevance of ability compared to effort is overstated, and (2) even to the extent to which ability is relevant, looking at certain people and declaring them as just good at math (a) undervalues the effort that they put in and (b) discourages people who struggle with it, and who think that means that they cannot do well at it.
> This kid walked straight up to the board and explained how you can design a computer program to factor any polynomial equation string input to it, and in fact had implemented a polynomial equation factoring program while the teacher explained how to factor simple quadratic equations.
I find this difficult to square with the well-known theorem that there is no closed-form solution to polynomials of degree five or more. (Where a "solution" and a "factoring" are, for polynomials, the same thing.)
I find this difficult to square with the well-known theorem that there is no closed-form solution to polynomials of degree five or more. (Where a "solution" and a "factoring" are, for polynomials, the same thing.)
And how does "no closed form solution" imply "no solution"?
No closed form solution means you cannot write a program that will, given an arbitrary polynomial, tell you the solution. Which is what was claimed above. Obviously the solution exists, but there is no single algorithm that can tell you what it is.
No closed form solution means we can't write out the solution cleanly using standard algebraic operations. You can still write iterative algorithms that converge on the solution, such as by solving for the eigenvalues of a matrix.
Those won't tell you the solution; they'll tell you a number that is near the solution.
When the solution is irrational, as is common for polynomials, it might be difficult to guess what it is based on knowing an approximation of it.
When the solution is irrational, as is common for polynomials, it might be difficult to guess what it is based on knowing an approximation of it.
When using 64 bit floating point numbers with 54 bits of precision, (or likely in this case, complex values consisting of two 64 bit floats) and you use an algorithm which you can guarantee converges to within 55 bits of precision in X steps, you have found the closest representable solution. Yes, you don't know whether it's 1 + sqrt(17) or 0.9 + Sqrt[17.84...], but you know to a degree of precision that 95% of use cases need what the value is. I do understand what you are saying, but the parent comment was referring to a student factoring polynomials. You doubted what he did based on the fact that he couldn't exactly specify what the roots were. I just assumed that he likely found the roots to a reasonable accuracy and then factored with these approximated values. The parent never specified, but given doubting the authenticity of the comment or celebrating youthful ingenuity, I choose the latter. I understand what you are saying. Yes, you can not find the EXACT values of the roots, but within the context of this comment chain, I don't think that matters.
On the contrary, when the problem posed is "find the roots of this polynomial", it's very common to accept approximate values.
When the problem posed is "factor this polynomial", it's unheard of. This is the conceptual difference between "needing a number to work with" and "studying math".
When the problem posed is "factor this polynomial", it's unheard of. This is the conceptual difference between "needing a number to work with" and "studying math".
Must have been for degree 4 or below.
Or maybe it assumes rational roots.
Which would be useless in context because even for quadratic polynomial problems in middle school, very often the solutions are irrational.
In which case it could say "no rational solutions found"
Yeah, I could also write a program that checks if all solutions are 0 and output the factorization, otherwise say "there are non-zero solutions". That's far from "a program that factors polynomials".
If you brag about how "you can design a computer program to factor any polynomial equation string input to it" in a class about quadratic equations and your code can't factor x^2 - 2, that's just not very impressive, regardless of your age.
If you brag about how "you can design a computer program to factor any polynomial equation string input to it" in a class about quadratic equations and your code can't factor x^2 - 2, that's just not very impressive, regardless of your age.
I figure out the formula of the number of diagonals of n-sided polygon within 15 minutes when I was 10. I thought I would be the greatest mathematician ever. Gauss figured out the formula of 1+2+3+...+n at a similar age and the difficulty was similar, right?
I have never been so wrong haha
I have never been so wrong haha
It's easy to be a genius. All you have to do is give 100% of your attention to your subject, indefinitely.
i mean achievement is at the intersection of luck, talent, and hard work. Knew several people with the raw iq to be great but these people would much rather spend their time doing mental puzzles.
I was by no means this kid. But I remember many times I’d passively listened to the lesson of the day, then get called on because I seemed (was) distracted, and offered a simpler solution. Usually got a “we’ll cover that in future chapters” with very little attempt to hide the frustration.
I’m sorry if I ruined anyone else’s math classes! If it’s any consolation, I ruined them for me too. I lost interest in the whole subject when I recognized the pattern.
I’m sorry if I ruined anyone else’s math classes! If it’s any consolation, I ruined them for me too. I lost interest in the whole subject when I recognized the pattern.
I'm going to be a little obnoxious here. Did he really implement a factoring program, or did he approximate the solution to a high precision numerically? Both would be impressive under the circumstances of course. But if he did it numerically, he still might want to pay attention to how to derive the quadratic equation using partial fractions. It's actually pretty clever.
>there are some people out there that are born to solve certain classes of problems
exact same experience... there is literally a genetic factor but people are not willing to admit it
exact same experience... there is literally a genetic factor but people are not willing to admit it
I wrote a similar program in those years about when we were first learning of quadratics. It's not really that I'm very smart, my parents were just math teachers and they'd force me to go through more advanced math books where I learned stuff, and I'd get bored in math class.
My algorithm was slow garbage though, so I'm sure your friend's was much better.
My algorithm was slow garbage though, so I'm sure your friend's was much better.
Summary:
1. Fermat's Little Theorem: if p is prime, then b^p = b (mod p) for all integers b. i.e. b^p - b is always a multiple of p. 8^3-8 = 512-8 = 504 = 168 x 3.
2. Is the inverse true? Does b^n - b = 0 (mod n) mean that n is prime? No. Sometimes n is non-prime (like n=561, divisible by 3). We call these n, Carmichael numbers.
3. Okay, so these numbers exist. How common are they? For primes we know they're common. Bertrand postulated (Chebyshev proved) that for any n>1, there is a prime p between n and 2n. That's cool!
4. Is it true that there is such a bound for these pretend-primes? Well, we have an interesting fact that there are x^(1/3) of them below any x, once we pass a certain point (i.e. there exists an X such that there are x^(1/3) of them below any x > X) so that makes us think it could be true! Worth seeing!
5. But what about this common-ness measure like the B-C result for primes? Well, it turns out that it exists. It ain't as pretty as just between straight integer multiples, but the fact that it exists in some shape at all is cool! That's what this kid proved. Absolutely rollicking fact. https://arxiv.org/abs/2111.06963
1. Fermat's Little Theorem: if p is prime, then b^p = b (mod p) for all integers b. i.e. b^p - b is always a multiple of p. 8^3-8 = 512-8 = 504 = 168 x 3.
2. Is the inverse true? Does b^n - b = 0 (mod n) mean that n is prime? No. Sometimes n is non-prime (like n=561, divisible by 3). We call these n, Carmichael numbers.
3. Okay, so these numbers exist. How common are they? For primes we know they're common. Bertrand postulated (Chebyshev proved) that for any n>1, there is a prime p between n and 2n. That's cool!
4. Is it true that there is such a bound for these pretend-primes? Well, we have an interesting fact that there are x^(1/3) of them below any x, once we pass a certain point (i.e. there exists an X such that there are x^(1/3) of them below any x > X) so that makes us think it could be true! Worth seeing!
5. But what about this common-ness measure like the B-C result for primes? Well, it turns out that it exists. It ain't as pretty as just between straight integer multiples, but the fact that it exists in some shape at all is cool! That's what this kid proved. Absolutely rollicking fact. https://arxiv.org/abs/2111.06963
By the way, if you don't like reading bulky proprietary PDFs, there is a trick: substitute the x in arxiv.org by the digit 5, and you will see the paper rendered in HTML5, e.g.:
https://ar5iv.labs.arxiv.org/html/1910.06709
(great work by FAU Erlangen's Michael Kohlhase and team).
https://ar5iv.labs.arxiv.org/html/1910.06709
(great work by FAU Erlangen's Michael Kohlhase and team).
Good tip. I should have just linked the arxiv.org page for the article and not the PDF directly https://arxiv.org/abs/2111.06963
Interesting. Entering [ https://ar5iv.org/abs/2111.06963 ] also gives me the HTML render of the paper. Saves one step, since I don't have to fetch the PDF first. :-)
Thanks for this. I find articles like this super hard to read due to the mixing of the topic and all the "back ground".
It does sometimes feel like it's only there to bulk out the article.
It does sometimes feel like it's only there to bulk out the article.
One can hope that future AI autosummarizers can be aware of our personal level of knowledge!
Lazy writers and/or who get paid per word.
The writer can fix the article, or let every reader fix it in his mind.
The writer can fix the article, or let every reader fix it in his mind.
Maybe just consider that the audience of the article is not you?
The article is not for technically minded people who would like a lucid explanation of the math behind the result? Because they're pretty clearly trying to hit that target, and frequent interruption of that kind of exposition works against it...
It could've easily been split up in different articles covering different aspects. It's not only lazy, but it doesn't account for what's going on in the world, where attention is super scarce.
> It ain't as pretty as just between straight integer multiples, but the fact that it exists in some shape at all is cool!
Actually: the form that Larsen proved is stronger than "for sufficiently large X, there is at least one Carmichael number between X and 2X".
So with the more specific proof, he also proved the simpler statement that's easier to express.
Actually: the form that Larsen proved is stronger than "for sufficiently large X, there is at least one Carmichael number between X and 2X".
So with the more specific proof, he also proved the simpler statement that's easier to express.
What is the stronger version?
Bertrand's postulate is that there is at least one prime between x and 2x for x>=1
Daniel Larsen's result here is that there are e^((log x)/((log log x)^(2+d))) Carmichael numbers between x and (x + x/((log x)^(1/(2+d)))) for x>=X (depends on d)
e^((log x)/((log log x)^(2+d))) is >= 1 for all x >= 1.
(x + x/((log x)^(1/(2+d)))) <= 2x
Stronger by being tighter than the (x,2x) bound and being more specific about the >= 1 number of Carmichael numbers
Daniel Larsen's result here is that there are e^((log x)/((log log x)^(2+d))) Carmichael numbers between x and (x + x/((log x)^(1/(2+d)))) for x>=X (depends on d)
e^((log x)/((log log x)^(2+d))) is >= 1 for all x >= 1.
(x + x/((log x)^(1/(2+d)))) <= 2x
Stronger by being tighter than the (x,2x) bound and being more specific about the >= 1 number of Carmichael numbers
Thank you!
Haha very good!
But from the article:
>> In fact, Larsen’s argument didn’t just allow him to show that a Carmichael number must always appear between X and 2X.
And yet the Wikipedia page says 2821 and 6601 are the 5th and 6th Carmichael numbers, which means there are not between 3000 and 6000 (X and 2X). So is his result actually that one must always exist between X and 2.5X or some other small multiple? If so, what multiple did he prove?
>> In fact, Larsen’s argument didn’t just allow him to show that a Carmichael number must always appear between X and 2X.
And yet the Wikipedia page says 2821 and 6601 are the 5th and 6th Carmichael numbers, which means there are not between 3000 and 6000 (X and 2X). So is his result actually that one must always exist between X and 2.5X or some other small multiple? If so, what multiple did he prove?
It’s true for x sufficiently large. So there will be some small counterexamples.
I only read the abstract and the result is in the same spirit but doesn't say exactky between X and 2X. It's between some more complicated expressions (using logs like usual in number theory)
The bounds are tighter than n and 2n. So he actually proved a stronger result.
Ah I didn't mean to mislead that it was x and 2x. Corrected to be slightly clearer.
The bound is easiest seen on the arxiv link above, in the abstract. The HN forum software doesn't do math very well.
The bound is easiest seen on the arxiv link above, in the abstract. The HN forum software doesn't do math very well.
thanks for the really good and concise summary
I've known quite a few families where both parents are scientists or software engineers or quantitative engineers.
Very frequently, their children will have a scary intuitive understanding of concepts that took my many years to understand (I'm a slow learner; didn't really understand hash tables until my 30s) and then apply their abilities to be in the higher echelons at science in a very young age. I see a similar thing in the children of world-class athletes.
Very frequently, their children will have a scary intuitive understanding of concepts that took my many years to understand (I'm a slow learner; didn't really understand hash tables until my 30s) and then apply their abilities to be in the higher echelons at science in a very young age. I see a similar thing in the children of world-class athletes.
Agreed, but this brings up the topic of nature vs. nurture, and perhaps even the previously (~90 years ago now?) popular concept of eugenics.
I think society settled on a response of "yeah, but we shouldn't actively do anything about it as a group." It seems to be best left to individuals to seek out parental partners and enjoy the outcomes on a personal level.
I think society settled on a response of "yeah, but we shouldn't actively do anything about it as a group." It seems to be best left to individuals to seek out parental partners and enjoy the outcomes on a personal level.
> I think society settled on a response of "yeah, but we shouldn't actively do anything about it as a group."
I hope there comes a day when humanity is responsible enough to handle precise gene-tailoring of humans without it devolving into a cliche Sci-Fi dystopia. Until then, I'm glad that humanity has settled on "Eugenics could theoretically work but we know we're not responsible enough to do it right, so let's not". It feels like a very enlightened stance: better than pretending selective breeding would literally not work, and certainly better than doing it badly. Why could we come to such a sensible consensus on this topic but fail to do so on so many others?
All that being said, I still want to see what would happen if Michael Phelps and Katie Ledecky had a child together. You know. For science.
I hope there comes a day when humanity is responsible enough to handle precise gene-tailoring of humans without it devolving into a cliche Sci-Fi dystopia. Until then, I'm glad that humanity has settled on "Eugenics could theoretically work but we know we're not responsible enough to do it right, so let's not". It feels like a very enlightened stance: better than pretending selective breeding would literally not work, and certainly better than doing it badly. Why could we come to such a sensible consensus on this topic but fail to do so on so many others?
All that being said, I still want to see what would happen if Michael Phelps and Katie Ledecky had a child together. You know. For science.
> Why could we come to such a sensible consensus on this topic but fail to do so on so many others?
I would argue that it took thousands of years of history and one of the darkest episodes of humanity in the 20th century to truly realize this.
I would argue that it took thousands of years of history and one of the darkest episodes of humanity in the 20th century to truly realize this.
> I hope there comes a day when humanity is responsible enough to handle precise gene-tailoring of humans without it devolving into a cliche Sci-Fi dystopia.
Power itself is a bit of a paradox - having some is essential to a free society, but too much ends up corrupting its holders and results in human suffering. The power you describe falls in the latter category, IMHO.
Power itself is a bit of a paradox - having some is essential to a free society, but too much ends up corrupting its holders and results in human suffering. The power you describe falls in the latter category, IMHO.
IMO, the reason it feels enlightened is that we don't clearly perceive the negative consequences of not doing it. If societal IQ was as pervasively tracked as global temperature, "let's not" might feel as enlightened as "let's not mess with climate engineering."
> If societal IQ was as pervasively tracked as global temperature
Part of the reason we're not ready for it is that any notion of "societal IQ" is simply not as mature, well-understood, proven, or even useful as our understanding of climate science. Scientists who studied the topic understood the greenhouse effect and the potential for CO2 emissions to result in global warming over 100 years ago. Meanwhile, nobody can agree on whether the Myers-Briggs personality test means anything at all.
We just don't have a way to "rate" humans in a general sense, nor is it obvious that increasing their "rating" would even be a good thing. Selecting for humans with the highest IQ would be like selecting for crops that grow the tallest. Sure, sometimes that's a good measure of general health, but it's so blunt compared to the intricate nuances of how DNA works and the huge variety of outcomes we'd like from our crops. Besides, it's just a weird stance to want to create a "superior human". Why do we want to do that? So they can solve humanity's problems (no pressure!)? So we can have sex with them? So we can "purify our gene pool" (shudder)? So they can live a happier life and avoid disease? Some of those reasons are much better than others.
Yes, messing with the climate has risks, but global climate is a lot less complicated than the DNA of a single human. And we know our climate is headed for disaster if we do nothing, so the risk of not messing with it is much higher. Similarly, we know people with certain genes are doomed to a die young or doomed to have certain major disabilities. Those will (and should) be our first targets of human gene editing, because the risks of not doing anything are higher.
And of course we already are practicing a light form of selective breeding at IVF clinics during embryo selection. Mostly they look for chromosomal abnormalities, but I'd guess there are clinics out there that offer more advanced gene selection (if not now, then soon).
Part of the reason we're not ready for it is that any notion of "societal IQ" is simply not as mature, well-understood, proven, or even useful as our understanding of climate science. Scientists who studied the topic understood the greenhouse effect and the potential for CO2 emissions to result in global warming over 100 years ago. Meanwhile, nobody can agree on whether the Myers-Briggs personality test means anything at all.
We just don't have a way to "rate" humans in a general sense, nor is it obvious that increasing their "rating" would even be a good thing. Selecting for humans with the highest IQ would be like selecting for crops that grow the tallest. Sure, sometimes that's a good measure of general health, but it's so blunt compared to the intricate nuances of how DNA works and the huge variety of outcomes we'd like from our crops. Besides, it's just a weird stance to want to create a "superior human". Why do we want to do that? So they can solve humanity's problems (no pressure!)? So we can have sex with them? So we can "purify our gene pool" (shudder)? So they can live a happier life and avoid disease? Some of those reasons are much better than others.
Yes, messing with the climate has risks, but global climate is a lot less complicated than the DNA of a single human. And we know our climate is headed for disaster if we do nothing, so the risk of not messing with it is much higher. Similarly, we know people with certain genes are doomed to a die young or doomed to have certain major disabilities. Those will (and should) be our first targets of human gene editing, because the risks of not doing anything are higher.
And of course we already are practicing a light form of selective breeding at IVF clinics during embryo selection. Mostly they look for chromosomal abnormalities, but I'd guess there are clinics out there that offer more advanced gene selection (if not now, then soon).
This isn't eugenics. People are allowed to choose their mates for their phenotypes, and we don't sterilize (except in very rare, extreme circumstances, with legal approvals) people who don't have specific phenotypes.
It's a gedanken experiment, but I would expect that people who measure low on IQ tests but are raised immersed in an technical or artistic environment have as much potential to become masters as people with higher IQ, or very nearly so. Further, I think that by increasing the quality of the education system in a country, you can massively increase the intellectual output. There are probably enormous amounts of untapped potential never matched to an equivalent supply of knowledge and skill.
It's a gedanken experiment, but I would expect that people who measure low on IQ tests but are raised immersed in an technical or artistic environment have as much potential to become masters as people with higher IQ, or very nearly so. Further, I think that by increasing the quality of the education system in a country, you can massively increase the intellectual output. There are probably enormous amounts of untapped potential never matched to an equivalent supply of knowledge and skill.
but I would expect that people who measure low on IQ tests but are raised immersed in an technical or artistic environment have as much potential to become masters as people with higher IQ, or very nearly so.
I would strongly expect the opposite. But I agree it would be beneficial to figure out which of us is right.
I would strongly expect the opposite. But I agree it would be beneficial to figure out which of us is right.
According to the latest studies, IQ is largely determined by early exposure to abstract thought. This is the main reason IQ is still rising in first-world nations where malnutrition hasn't been an issue for generations. If you rob a grade school kid of playground time and learning to share toys, and instead force them to grind out math lessons and computer science problems, their adult IQ will be significantly higher.
They might end up tremendously unhappy, of course. It just goes to show how raw IQ is a silly measure to value, let alone optimize for.
They might end up tremendously unhappy, of course. It just goes to show how raw IQ is a silly measure to value, let alone optimize for.
Do you have any links to these latest studies that I can take a look at?
If you're just talking about the Flynn effect, then your conclusion is overreaching.
IQ is still very highly heritable and adjusted for the Flynn effect has very little to do with early exposure to abstract reasoning AFAIK (but would love to see those studies).
If you're just talking about the Flynn effect, then your conclusion is overreaching.
IQ is still very highly heritable and adjusted for the Flynn effect has very little to do with early exposure to abstract reasoning AFAIK (but would love to see those studies).
> This isn't eugenics. People are allowed to choose their mates for their phenotypes
The fact that something is allowed doesn't make it not eugenics. Choosing your mate for their phenotype is eugenics.
The fact that something is allowed doesn't make it not eugenics. Choosing your mate for their phenotype is eugenics.
No - eugenics is by definition arranged. If a government nudged you to pair up with someone, that would be eugenics, but if you choose a mate of your own free will, it's not.
How much nudging do you need before it's eugenics? Would it be eugenics if you were to take young men and women, separate them roughly by area of interest and level of intellectual achievement, remove them from the chaperoning influence of their families, and put them in an institution for four years at the perfect time of their life to form pair bonds? What if you additionally supplied them with easy access to alcohol?
Interesting new take: "College: the hidden eugenics program."
This sort of makes sense to me.
This sort of makes sense to me.
> if you choose a mate of your own free will, it's not
If you're choosing mates at all, it means you're judging the value of other human beings. You're accepting some and rejecting others based on what you think or feel is best. It's just eugenics practiced by the general population according to highly variable personal criteria.
This is not inherently superior to some explicit eugenics program. It's about the principle of individual autonomy.
If you're choosing mates at all, it means you're judging the value of other human beings. You're accepting some and rejecting others based on what you think or feel is best. It's just eugenics practiced by the general population according to highly variable personal criteria.
This is not inherently superior to some explicit eugenics program. It's about the principle of individual autonomy.
its not eugenics because it's not population scale, it's individual. Your interpretation does not correspond to any mainstream understanding.
I wish this could be discussed more openly. I feel the same way.
I don't think that's entirely Fair. Anything you learned at a young age you're going to have more potential in. If you were introduced to X as a kid, you have years of doing or practicing X that someone else will never have, and those years come when your brain is still developing.
In my area the schools are providing computers at a young age and spending time teaching STEAM topics, for example. It's not going to make everybody into Nikola Tesla, but the effort to do something about it does exist.
In my area the schools are providing computers at a young age and spending time teaching STEAM topics, for example. It's not going to make everybody into Nikola Tesla, but the effort to do something about it does exist.
Fundamentally, even ignoring the ethical problems, eugenics doesn't work. The interactions between nature and nurture, genetics, epigenetics, etc. are waaaay to complex (and even multi-generational) to boil down to a simple decision to allow/deny birth.
i mean, we should be able to get to a future when everyone can enjoy in the bounty produced by technology, it is arrogant to think humans with their limited wisdom can determine who gets to breed and who doesn't
Couple of years back there was an article that talked about higher incidences of autism in kids with such parents.
In another area - I knew a guy and his parents were very good with finance and business and he was way ahead of everyone else right out of high school. He did go on to become very successful financially. And me, well I should have done better! (Oh yeah, don't compare yourself. Run your own race. hmmm)
part of this is they are probably wired for it, and part of it is they are immersed in it from a young age. Probably don't need to worry though, a lot of geniuses come from left field where they grow their creativity in isolation.
That's because intelligence is ~80% heritable! Or rather, ~80% of the variance in intelligence is explained by things you inherit.
(Note: "inheritable" here is not exclusive to inherited genetics; it also refers to upbringing.)
Generally, heritability refers to genetics, not upbringing. Specifically: "amount of phenotypic (observable) variation in a population that is attributable to individual genetic differences"
It wouldn't make much sense to report things that were related to upbringing together with genotype, as those are specifically the two things you want to decompose.
It wouldn't make much sense to report things that were related to upbringing together with genotype, as those are specifically the two things you want to decompose.
Yeah, dekhn you're correct. There are mechanisms of inheritance that aren't genetics (like heritable epigenetics, maternal effects, etc.), but when we're talking about "heritable" the vast majority is genes.
success and progress is about favorable contexts and resources
Nobody is talking about the proof, but about the fact that the person who produced it was younger than them and they are trying to explain the achievement by innate abilities or parent influence. The fact is just that most of people don't want to do something like that. They just want to be someone who do things like that.
Of course that's just the first (coarse) filter. Once you've selected for those who actually want to solve the problem (and not just "be" the one who solves it) there is still the the incredibly fine filter of who actually has the ability- that confluence of nature/nurture that is itself such a subject of debate.
The article talks about the proof and how he came up with it as much as it talks about his age.
Congratulations to him! That's an amazing accomplishment. There's a lesson in this line
> For more than a year and a half, Larsen couldn’t stop thinking about a certain math problem.
It's rare to be able to focus on something for that long without giving up.
> For more than a year and a half, Larsen couldn’t stop thinking about a certain math problem.
It's rare to be able to focus on something for that long without giving up.
To your point, one of the more fascinating details about John Nash from Nassar’s book ‘A Beautiful Mind’ was that this very trait set him apart from a number of his contemporaries. He was capable of focusing on problems for what appeared to be quite a bit longer than most, systematically breaking down the barriers that would’ve turned another person toward some other pursuit.
My daughter is in 3rd grade and is bored with her math lessons, so I started to do my own nightly math lessons. Long division was a big hit, but what I really want is to find some accessable number theory material, preferably about interesting patterns. Any recommendations?
Probably something in the 10-15 age range, or 8-10 and originally in Russian
Probably something in the 10-15 age range, or 8-10 and originally in Russian
I remember something a math teacher gave me in middle school as extra work. It was a packet of patterns and it gave you the first few items in the sequence and you needed to write a formula that expresses the value for the nth term. Something like this: https://corbettmaths.com/wp-content/uploads/2018/12/nth-term...
My best friend was a child prodigy at math and his parents started him on "The Number Devil" which I believe is number theory
very impressive to have such fundamental contributions at such an early age. To even know it's applicability to modern day cryptography is also really impressive. All the best to Daniel Larsen!
Quanta is doing really solid science journalism these days. They have a number of fantastic podcasts as well.
It's bankrolled 100% by this generous billionaire https://en.wikipedia.org/wiki/Jim_Simons_(mathematician) and I hope he continues to fund it
It's bankrolled 100% by this generous billionaire https://en.wikipedia.org/wiki/Jim_Simons_(mathematician) and I hope he continues to fund it
Weird how this came out on the day I bomb my probability test. T_T
This guy has Sitzfleisch, a very useful thing for a mathematician to have.
Do Germans really say that people with grit have a lot of "butt" because that's what its looking like here!
Yes, that's the other meaning of the word. :)
Here, it means the mental stamina to sit down and focus on a single problem for a long time.
Disclaimer: I'm not German (although some ancestors in the 1800s were.)
Here, it means the mental stamina to sit down and focus on a single problem for a long time.
Disclaimer: I'm not German (although some ancestors in the 1800s were.)
Getting older, sometimes it can be so tough to accept the fact that people a fraction of your age achieve things you never will.
Given the extreme connectivity of the present, we are also exposed to brilliant minds with incredible capabilities, making us (me at least) feel even more incapable..
I guess it is a lesson for humility.
Good job Daniel, you show us !
Given the extreme connectivity of the present, we are also exposed to brilliant minds with incredible capabilities, making us (me at least) feel even more incapable..
I guess it is a lesson for humility.
Good job Daniel, you show us !
Tom Lehrer once said: "It is a sobering thought that when Mozart was my age, he had been dead for two years."
More Lehrerisms here:
https://en.wikipedia.org/wiki/That_Was_the_Year_That_Was
Not to downplay any of Daniel’s accomplishment but sometimes it isn’t a “fair” comparison when others started younger with more resources. His father is a distinguished professor of mathematics and his mother is a professor of mathematics. When you have that sort of resources available at a young age and advanced training you’ll probably accomplish more sooner than someone of similar IQ without those resources who started later
There are thousands of mathematicians in the US. I am sure many have kids. How many of those kids do even a fraction of what Daniel did even when having every possible advantage? Today, young people have assess to more resources than ever, yet talent is one of those things that resists this trend of egalitarianism seem elsewhere. More resources means that the super-talented will pull way ahead of the untalented or only moderately talented.
Resources also include parental encouragement, not being bullied, not having to do stuff to get by that isn’t delving into deep work, not trying to fight
boring school lessons and exams in subjects not of interest,
no pressure to shape your studies to get a job. These are not universal.
To be honest I care more about the high iq kids interested in math who come from poor or working class families that either won’t be identified or will be identified but not much can be done for them given the lack of resources. Plus this doesn’t negate that someone of a similar IQ may accomplish less / seem less impressive at an early age due to such disadvantages.
I wonder how many more Srinivasa Ramanujan‘s died in poverty in a rural village?
Maybe there’s a path now for these kids.
Maybe there’s a path now for these kids.
I see lots of users here hammering on about IQ - but that's just one factor.
The kid grew up with two mathematicians, so I'm sure they knew how to get him in the right direction. Kids also tend to value whatever their parents are doing.
And not to mention that most kids have all the time in the world to do what they enjoy - and rarely have other concerns.
I'm a musician, and grew up playing my instrument from 4 to 8 hours a day, every day, almost all year round. It was pretty much the only thing I did between age 10 to 18. I had very supportive family and mentors, and got lots of chances others at my age did not.
Naturally, I met a lot of other talented kid. Some child prodigies.
One thing I have noticed afterwards: Being at such a high level as a kid - does not necessarily mean you'll go on to do great things later in life. Some kids just grow out of it, lose interest and motivation, and - well - life often just happens. Even among the most talented, only a small % of them end up doing remarkable stuff later on.
The kid grew up with two mathematicians, so I'm sure they knew how to get him in the right direction. Kids also tend to value whatever their parents are doing.
And not to mention that most kids have all the time in the world to do what they enjoy - and rarely have other concerns.
I'm a musician, and grew up playing my instrument from 4 to 8 hours a day, every day, almost all year round. It was pretty much the only thing I did between age 10 to 18. I had very supportive family and mentors, and got lots of chances others at my age did not.
Naturally, I met a lot of other talented kid. Some child prodigies.
One thing I have noticed afterwards: Being at such a high level as a kid - does not necessarily mean you'll go on to do great things later in life. Some kids just grow out of it, lose interest and motivation, and - well - life often just happens. Even among the most talented, only a small % of them end up doing remarkable stuff later on.
The set of things you will never achieve is infinite compared to the set of things you will, best to stay focused on the latter and enjoy your life. Unless of course you prefer drinking from an infinite well-spring of misery.
I celebrate people who achieve memorable milestones like this or build something that revolutionizes the world - but there are a plethora of different ways to live life. For me, as long as you're happy and you're putting more happiness out into the world than you're consuming then you're achieving a pretty damn good life. We're not all responsible for the entire world - as long as you're leaving your little corner of it better than you found it then good on you.
It's not enough to make the world better or be happier, a lot of people also want acclaim and recognition. Why do so many people apply to Ivy League schools when 50-100 ranking schools can also provide a good education? Status is necessarily scarce.
This may just be the odd musing of a mid-thirty year old but acclaim and recognition feel pretty worthless to me. Chasing individuality through others' approval will always be more difficult than just accepting yourself as yourself and being happy with that.
[deleted]
it's not like everyone can win the IQ lottery. Superior ability is unequally distributed.
people should measure themselves by how best they utilized their gifts and circumstances to live a life that they are happy with. No point in comparing yourself to some kid who probably inherited an amazing mathematical brain through no conscious effort, moral superiority etc.. of his own.
Ability to maximize your own talent is a talent in itself.
Why is it tough?
This may be worth reflecting on [0].
[0] https://www.newadvent.org/summa/3036.htm
This may be worth reflecting on [0].
[0] https://www.newadvent.org/summa/3036.htm
Yeah, I have no problem admitting that there are people much smarter, more talented, better looking, etc than I am. The day that I assume I'm the top of whatever category is a sad sad day, as I must be the only one left.
I think what really bugs me is that for some hobby I chose, there is a always someone who already achieved everything I could dream to achieve in my life.
Then I figured some hobbies are better suited for my use case. For example science is better suited for me. I somehow was never bothered by the fact that some geniuses in 19th century already knew enough math/physics that I could digest in my life. Science is infinite so however much knowledge someone gained, it's zero comparing to the total amount. I guess I need that kind of comfort to commit to something. Fossil collection is also a good one because every piece I collected is MINE and unique so I don't need to compare with someone else.
Then I figured some hobbies are better suited for my use case. For example science is better suited for me. I somehow was never bothered by the fact that some geniuses in 19th century already knew enough math/physics that I could digest in my life. Science is infinite so however much knowledge someone gained, it's zero comparing to the total amount. I guess I need that kind of comfort to commit to something. Fossil collection is also a good one because every piece I collected is MINE and unique so I don't need to compare with someone else.
> I somehow was never bothered by the fact that some geniuses in 19th century already knew enough math/physics that I could digest in my life. Science is infinite so however much knowledge someone gained, it's zero comparing to the total amount.
I think this is important to consider. Anyone even cursorily interested in science can know (just via public information, libraries, etc.) so much more about "everything-except-very-specific-thing" than any of the 19th century geniuses who knew and understood so much about "very-specific-thing".
Ofc, the geniuses are ultimately the ones responsible for the advancement of the state of the art, but... there's also a lot to appreciate for the rest of us.
I think this is important to consider. Anyone even cursorily interested in science can know (just via public information, libraries, etc.) so much more about "everything-except-very-specific-thing" than any of the 19th century geniuses who knew and understood so much about "very-specific-thing".
Ofc, the geniuses are ultimately the ones responsible for the advancement of the state of the art, but... there's also a lot to appreciate for the rest of us.
Yeah. I kinda feel like a medieval monk: we are all equal under God (in the case he is the Cosmos/Natural)
citizenpaul(5)
This kid walked straight up to the board and explained how you can design a computer program to factor any polynomial equation string input to it, and in fact had implemented a polynomial equation factoring program while the teacher explained how to factor simple quadratic equations.
Since then, I don't feel bad if someone achieves more than me, because clearly there are some people out there that are born to solve certain classes of problems (maybe their brain structure is better for those, or something, who knows).