[tgv]

I cannot agree. It's just "feel-good thinking." "Everybody can do everything." Well, that's simply not true. I'm fairly sure you (yes, you in particular) can't run the 100m in less than 10s, no matter how hard you trained. And the biological underpinning of our capabilities doesn't magically stop at the brain-blood barrier. We all do have different brains.

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

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

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

[nestes]

Not the original poster, but I want to push back on one thing -- being capable of something and being one of the best in the world at something are hugely different. Forgive me if I'm putting words in your math -- you mentioned "placing the bar for mathematical skill pretty" low but also mentioned running a sub-10s 100m. If, correspondingly, your notion of mathematical success is being Terence Tao, then I envy your ambition.

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

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

[chipdart]

> cannot agree. It's just "feel-good thinking."

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

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

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

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

[tgv]

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

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

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

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

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

[davidbessis]

I'm the author of what you've just described as clickbait.

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

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

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

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

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

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

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

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

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

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

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

[calf]

This power law argument immediately reminds me of education studies literature that (contrary to the math teachers in this thread) emphasize that mathematical ability is learned cumulatively, that a student's success feeds on itself and advances their ability to grasp more difficult material.

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

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

[hyperthesis]

This beats TFA. Interesting relation between cumulativeness and distribution ("Yule process"). But how does this explain variation is how quickly children pick up maths - would you argue it's due to prior exposure e.g. parental tutoring?

Any comments on the "10x programmer"?

[llm_trw]

There is math the abstract field and math the concrete example you're working on.

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

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

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

[cutemonster]

> document the secret "oral tradition" of top mathematician

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

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

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

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

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

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

> Pareto-like distribution like what we see in math

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

[davidbessis]

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

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

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

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

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

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

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

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

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

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

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

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

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

[cutemonster]

> they do know something about math ... that it requires a certain "attitude"

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

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

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

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

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

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

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

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

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

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

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

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

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

[margorczynski]

> In other words: the people who run the mathematical 100m in under a second don't think it's because of their genes.

Sure they don't. Most extremely successful people (want to) think that the main reason of their success is their commitment and hard work. It runs completely contrary to the findings of modern biology and psychology, most of our intellectual potential at adulthood is genetic.

The floor and ceiling you will operate on in your life is decided the moment of chromosomal crossover.

[cutemonster]

> Most extremely successful people (want to) think that the main reason of their success is their commitment and hard work

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

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

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

[margorczynski]

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

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

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

[tgv]

So, for starters: you don't have any evidence, if I understood it properly. None whatsoever. That's really not the basis for arguing "become 1000x better." If only because your operationalization is missing. If you can't measure someone math's skills, how can you say they can become 1000x better? I think the whole article manages not to even speak about what "math" actually is supposed to be. Symbol manipulation according to axioms?

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

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

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

> What can someone gain by improving their mathematical thinking?

> Joy, clarity and self-confidence.

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

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

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

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

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

[sn9]

You'll want to get your daughter to start using Math Academy for at least 30 minutes per day [0].

New solutions exist for old problems.

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

[Barrin92]

You don't need to be able to run 100m in less than 10 seconds. But almost everyone probably could run a marathon in three and a half hours. How many people do you think have actualized their physical potential, or how far is the average person removed from it?

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

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

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

[llm_trw]

There's a difference between being able to memorize what a square root is and being able to do math - which to mathematicians means being able to organize a proof.

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

The two are rather different.

[Tainnor]

Square roots are not some "mathematical trivia", they're amongst the most fundamental operations in mathematics.

[llm_trw]

In arithmetic. There is a lot more to math than arithmetic.

[Tainnor]

Square roots are fundamental to (real and complex) analysis and to algebra (in the study of polynomials), so the two major branches of modern mathematics.

[llm_trw]

They are fundamental to the arithmetic used in analysis and algebra.

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

[Tainnor]

Just come on. The square root of 2 is the easiest example of an irrational number, this has been known since Ancient Greece. You can't compute distances in Euclidean spaces without the square root. "Solving equations by roots" is the bread and butter of algebra. Adjoining roots to a field is how you get Galois Theory. Several algorithms related to number theory have complexity O(sqrt(n)). And so on.

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

[anticensor]

Also fundamental to geometry (pythagorean equation and others).

[tgv]

Strangely enough, you'd be hard pressed to find a mathematician who doesn't know what a square root is.

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

[llm_trw]

You'd also be hard pressed to find one who doesn't know how to flush a toilet.

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

[tgv]

Are you an LLM? You brought up the point of mathematicians not knowing what a square root is yourself. Anyway, the square root is is so many levels below maths as done by mathematicians, it's laughable.

[blharr]

Pretty sure you followed the thread wrong. They only mentioned the square root in response to you?

[wholinator2]

It certainly is suspicious that their name starts, "llm"

[bloqs]

This is mostly correct. Working memory plays a huge component in grokking more complicated mathematical components, and IQ itself is separated into performance and verbal IQ (which together constitute your IQ score) and its demonstrably robust. Some people find this easier than others and that is OK.

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

[sourcepluck]

Imagine our world was extremely similar to how it is now in any way you'd care to imagine, except two things were different.

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

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

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

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

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

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

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

[hilbert42]

"Kids are told week in week out that maths is stupid, that they are stupid. …."

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

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

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

[wholinator2]

I see it a lot on the internet, mostly I'm the young people places. People talk shit on math all the time, just like they do in person. Just slights and jabs that they've never needed the pythagorean theorem or to do an integral, and thus it was all wasted time and effort. You might just not hang around young people much, or at least their online congregations (i don't blame you). The idea that math education is dumb and useless is very much alive in the young adult to preteen tiktokified spaces online

[sn9]

In math classrooms filled with 30 students on average for 12 years? With usually all 30 of those students having mathematically illiterate parents who can't help with their homework?

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

[tgv]

> how often are kids exposed to such stupid talk? I suspect very infrequently

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

[vacuity]

I think most people can become fairly skilled in useful fields if educated properly, and the people who can't are a small group that can be cared for. I agree that even in a better education system, people aren't all going to be equally skilled in the same fields, just that most people can contribute something of value.

[theclansman]

Anybody can do everything if we restrict everything to things everyone can do.

[antegamisou]

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

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

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

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

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

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

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

[tgv]

> the vast majority of people in the Soviet Union were very math literate

I doubt that.

> although indeed most became engineers

And that is demonstrably false.

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

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

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

[antegamisou]

> I doubt that.

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

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

which is also why the following statement

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

is strictly US-centric.

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

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

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

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

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

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