Can anyone learn advanced maths? (Researches)

[Drakkith]

i think you need to reread posts 1 and 6.

Hmmm...

On a high level, I mean the level like when you study maths in university. That you need some kind of talent to get a Field-Medal or even compete it is obvious. That is not what I meant with high level. High level is university level for me.

Well there we go then.

 

[IjustlikeMaths]

Just about what level are we talking about as "advanced maths" here? Are we talking undergraduate? Graduate? Post doc?

I would say Doctor Professional level

 

[IjustlikeMaths]

The OP originally asked about the math that wins the Fields medal, so quite advanced. Definitely beyond undergrad and possibly beyond graduate.

Nono. I never meant Fiel-Medal level. I just used the Field-Medal as a Benchmark where you need to be blessed! You need kind of talent to get there.

I mean the University level like doctor or professor in math.

 

[FactChecker]

I would say Doctor Professional level

At the research level, there certainly are individual limitations that would prevent a large percent of people from achieving that ability.

 

[StatGuy2000]

There is no qualitative difference between an average mathematician and a Fields candidate. The difference is quantitative: One has more talent than the other.
If some mathematicians are not good enough to become Fields medalists, then it follows that some people are not good enough to be mathematicians.
Saying anyone could become a successful mathematician is like saying any mathematician could win the Fields medal, if he only worked hard enough.
Not going to happen.

First of all, I dispute your characterization between the average mathematician and a Fields candidate -- Fields medals (like Nobel prizes) are awarded based on discoveries, and raw talent is not the only ingredient in making discoveries -- there is a considerable element of random chance involved as well.

But setting that aside, your very premise that some mathematicians cannot become Fields medalists => some people are incapable of learning math (your contention, @SchroedingersLion ) is both false, and is extremely arrogant and elitist on your part (I'm being kind, btw, with my choice of words, due to PF rules).

As for that 14 year girl, how much of her difficulty is based on poor instruction in her earlier years? If she had received better resources earlier in her life (and encouraged to both study and persist in studying math), she very well could have learned math effectively.

Again, what is the difference between studying mathematics and, say, studying say, French as a foreign language? Or geography? Or history? Or biology? I've met plenty of students who have trouble in all of these subjects -- do you then conclude that students are incapable of learning all of these subjects?
How absurd does this sound to you?

 

[StatGuy2000]

And since people quoted IQ here, physicist-turned-statistician Cosma Shalizi has given superb critiques of its so-called effectiveness in measuring "intelligence" in these blog posts:

http://bactra.org/weblog/494.html

http://bactra.org/weblog/495.html

http://bactra.org/weblog/520.html

http://bactra.org/weblog/523.html

 

[Drakkith]

But setting that aside, your very premise that some mathematicians cannot become Fields medalists => some people are incapable of learning math (your contention, @SchroedingersLion ) is both false, and is extremely arrogant and elitist on your part (I'm being kind, btw, with my choice of words, due to PF rules).

Per wiki:
Elitism is the belief or attitude that individuals who form an elite — a select group of people with a certain ancestry, intrinsic quality, high intellect, wealth, special skills, or experience — are more likely to be constructive to society as a whole, and therefore deserve influence or authority greater than that of others.

Since no one has said anything about high-skill mathematicians (or anyone else) deserving greater influence or authority, I find your accusation simply wrong.
If you cannot discuss this without making such accusations then please leave the thread.

As for that 14 year girl, how much of her difficulty is based on poor instruction in her earlier years? If she had received better resources earlier in her life (and encouraged to both study and persist in studying math), she very well could have learned math effectively.

That's pure speculation. We can't go back in time and try things differently, so there's no way to know.

Again, what is the difference between studying mathematics and, say, studying say, French as a foreign language? Or geography? Or history? Or biology? I've met plenty of students who have trouble in all of these subjects -- do you then conclude that students are incapable of learning all of these subjects?

Yes, absolutely. I completely agree with the idea that some students will never be able to learn a second language, or geography, or history, etc. Are most students who are having difficulty in these areas incapable of learning them? No, probably not. Are some? In my opinion, yes.

How absurd does this sound to you?

It doesn't sound absurd at all. There are many, many students that have significant difficulty in all of the subjects you listed.

I'd also like to say that I don't like this black and white concept of learning math. Instead of asking whether or not anyone could learn some level of math, it seems much more reasonable to ask will some students find learning math so difficult that it would be unreasonable to expect them to do so? I mean, if someone spent ten years learning Calculus, how long would it take them to learn higher level math? If it would take them longer than their own lifespan to learn all the math necessary to get to a certain level then I would say that that means they will never learn math at that level. It doesn't mean that they can't learn math at all, it just means that the difficulty is so large that they cannot be reasonably expected to learn math to that level.

 

[symbolipoint]

Crazy hot-headed arguing just like would have been expected of this kind of topic. No clear conclusions.

 

[StoneTemplePython]

At the research level, there certainly are individual limitations that would prevent a large percent of people from achieving that ability.

I don't understand how this statement is controversial.

A lot of mathematics is about clear thinking and I'm seeing a large deficit of that on this thread.

 

[gmax137]

If some mathematicians are not good enough to become Fields medalists, then it follows that some people are not good enough to be mathematicians.

That's nonsense. It is the same as this:

"There are runners who will never win a marathon. It follows that some people can't even breathe."

 

[Drakkith]

That's nonsense. It is the same as this:

"There are runners who will never win a marathon. It follows that some people can't even breathe."

Well that's a bit extreme. A more appropriate statement would be: "There are runners who will never win a marathon. It follows that some people will not be good runners." The implied assumption here is that running is a skill and that various people have different skill levels or ability at it.

 

[StatGuy2000]

Yes, absolutely. I completely agree with the idea that some students will never be able to learn a second language, or geography, or history, etc. Are most students who are having difficulty in these areas incapable of learning them? No, probably not. Are some? In my opinion, yes.
It doesn't sound absurd at all. There are many, many students that have significant difficulty in all of the subjects you listed.

I'd also like to say that I don't like this black and white concept of learning math. Instead of asking whether or not anyone could learn some level of math, it seems much more reasonable to ask will some students find learning math so difficult that it would be unreasonable to expect them to do so? I mean, if someone spent ten years learning Calculus, how long would it take them to learn higher level math? If it would take them longer than their own lifespan to learn all the math necessary to get to a certain level then I would say that that means they will never learn math at that level. It doesn't mean that they can't learn math at all, it just means that the difficulty is so large that they cannot be reasonably expected to learn math to that level.

My issue with your stance above is that the logical conclusion you would make is the following:

some people are incapable of learning Subject A (math, history, geography, language) => it's a waste of time for some people to study Subject A => we should identify these people and stop them from even learning Subject A

The problem is that educators (whether at the K-12 level, or in post-secondary level) by and large have no idea why their students are struggling with their subjects. What I fear is that educators may well see a student struggling and automatically conclude that these students are hopeless cases, whereas they may well be suffering from poor preparation in their preceding years (due to poor teaching or poor resources).

The other issue is that people do not always learn subjects in the same pace nor do they necessarily learn material in an orderly, linear path. There have been many documented cases where students who have struggled with a subject like math in the early years end up catching up with the material and excelling in the subject at an older age. However, if an educator (or parent) looks at said student from the earlier years, they may be led to believe (erroneously) that the student will never learn math, and thus actively discourage or prevent the student in further studies. To me this is a tragedy.

As educators, the goal should not be to focus their attention on the high achievers, but to be evangelists in their subjects, to bring their passion to the subject accessible to as broad a swathe of students as possible, and to instill the discipline the students will need to learn the subject matter at hand. Concluding beforehand that certain students can never learn a subject is a betrayal of that goal.

 

[symbolipoint]

That's nonsense. It is the same as this:

"There are runners who will never win a marathon. It follows that some people can't even breathe."

That analogy does not work.

 

[Swamp Thing]

This may be a bit tangential, so dear moderators, please feel free to (re) move it.

My niece has no problem with calculus, linear algebra, etc. She enjoys the math part of her syllabus. On the other hand, she is not too comfortable with her programming language courses. She has trouble figuring out how to approach a programming problem, i.e. what kind of loop structures she would probably have to use, etc. etc. (Once I walk her through the solution, she ultimately understands it, but if she doesn't revisit it for a few days, she might find the same problem nearly as impenetrable as before).

I'd like to know if anyone else knows someone who is OK with math but not so OK with programming. Any thoughts about how to nurture the programmer's way of thinking for such a student?

Coming back to the original thread topic, it is conceivable that my niece may, at some point, break through the barrier and begin to progress much faster. It's also possible that her progress will remain slow compared to other students. Extrapolating from this, it is conceivable that anyone may in principle be able to learn a lot of math or programming if they went at it persistently and diligently. But then, if it took them most of their life to achieve college level competence, then it's practically equivalent to a total inability to progress to doctoral level stuff, IMHO. So it's reasonable to say that in a practical sense, some may be incapable of learning doctoral level math in one lifetime, simply due to the sheer number of years they might need. It's not a huge leap to then say the same thing about college level math (e.g. calculus and linear albegra), except that statistically the fraction of people in that category would be less.

Another thought: Consider some hypothetical person, whose manifested math aptitude is just slightly below average. Imagine that somehow they are fortunate (?) enough to live for 300 years with undiminished mental faculties and reasonable physical condition. Can they manage Fields Medal type achievements just by staying around and working really, really hard for those 300 years?

 

[Bobman]

My niece has no problem with calculus, linear algebra, etc. She enjoys the math part of her syllabus. On the other hand, she is not too comfortable with her programming language courses. She has trouble figuring out how to approach a programming problem, i.e. what kind of loop structures she would probably have to use, etc. etc. (Once I walk her through the solution, she ultimately understands it, but if she doesn't revisit it for a few days, she might find the same problem nearly as impenetrable as before).

I'd like to know if anyone else knows someone who is OK with math but not so OK with programming.

Answering this side note, i am similar to your niece. I enjoyed calculus (at least multivariable, vector and tensor) and linear algebra, but i absolutely loathe programming. I cannot make sense of it and it is the first subject i have come across that i actually cannot bring myself to want to learn even a little. I seem to be near immune to the logic and structure of it as well.

This is actually part of the reason i decided to not go for robotics as i originally planned and decided to switch to physics instead.

I don't feel like the two subjects necessarily are interwined when it comes to the actual understanding, sort of like physics and maths are a bit different in that way.

 

[clope023]

Answering this side note, i am similar to your niece. I enjoyed calculus (at least multivariable, vector and tensor) and linear algebra, but i absolutely loathe programming. I cannot make sense of it and it is the first subject i have come across that i actually cannot bring myself to want to learn even a little. I seem to be near immune to the logic and structure of it as well.

This is actually part of the reason i decided to not go for robotics as i originally planned and decided to switch to physics instead.

I don't feel like the two subjects necessarily are interwined when it comes to the actual understanding, sort of like physics and maths are a bit different in that way.

I know people who are good at math and bad at physics and vice versa.

With respect to your loathing of programming, I think you just haven't been doing the right programming if you really enjoy multivariable, vector, and tensor calc and linear algebra; lots of programming research having to do with those things related to numerical solutions of partial differential equations, image analysis, ray tracing, machine learning, and so forth.

 

[neilparker62]

I say absolutely not - no way you can be Field's Medal material unless you are extraordinarily 'talented'. In terms of neural activity or neural 'hardware' there must be some particular defining factor which we haven't quite figured out yet. To take another example: for me as a chessplayer I just don't have what it takes to become a chess grandmaster. And guys such as Timur Gareyev who took on a world record 48 opponents simultaneously blindfolded are completely mind blowing! I can't even conceive of playing one game without sight of the board! So in conclusion you are undoubtedly born - not made - as far as this type of talent is concerned. That said , there may also be childhood upbringing factors which decisively shape the neural wiring of the adult brain.

Interesting discussion - thanks!

 

[pinball1970]

Good genetics and environmental factors just like every other human trait.

What do you guys think? Can anyone learn maths to a high level?

Only on PF could people get so passionate about maths!

As an outsider (none mathematician or scientist) who was not great at maths at school, If you practice you will get better but as has been pointed out very good singers, runners, body builders are born not made.

If by high level you mean University entry I would guess probably not, A level? Possibly but with a lot more work required if you have questionable innate ability (again from experience)

 

[IjustlikeMaths]

I say absolutely not - no way you can be Field's Medal material unless you are extraordinarily 'talented'. In terms of neural activity or neural 'hardware' there must be some particular defining factor which we haven't quite figured out yet. To take another example: for me as a chessplayer I just don't have what it takes to become a chess grandmaster. And guys such as Timur Gareyev who took on a world record 48 opponents simultaneously blindfolded are completely mind blowing! I can't even conceive of playing one game without sight of the board! So in conclusion you are undoubtedly born - not made - as far as this type of talent is concerned. That said , there may also be childhood upbringing factors which decisively shape the neural wiring of the adult brain.

Interesting discussion - thanks!

Yep I am with you. But the point is I never said Field-Medal Level. I used the Field-Medal as a benchmark where you need to be blessed.

I am talking about doctor or professor in maths level.

 

[pinball1970]

Well that's a bit extreme. A more appropriate statement would be: "There are runners who will never win a marathon. It follows that some people will not be good runners." The implied assumption here is that running is a skill and that various people have different skill levels or ability at it.

I should have read this first, yes I think that is a good description of natural ability.

BTW Are you stuck on 20,003 posts? You have posted a few after your 20,003 but your counter isn't moving forward.

 

[Dragon27]

Again, there is a commonly-held view throughout Western countries (particularly by Americans and the British) that somehow mathematical ability is a "genetic" trait that only a certain people are blessed with the capability to understand. In no other subject that I can think of is such a view held -- not in, say, foreign languages, not in geography, history, art, music, etc.

But it is. People do think that they're bad at art/drawing/music/learning languages just because they don't have some innate talent, so they shouldn't even try. And when someone get good at, say, Japanese (having put in a lot of hard work), people just naturally assume that said person is just very talented at learning languages.

I personally like to think of it this way - everyone can achieve a reasonably high level of <subject>, but not everyone (statistically) will.

"There are runners who will never win a marathon. It follows that some people will not be good runners."

Is it necessary to "win" a marathon to be a good runner?
What does "winning a marathon" even mean? If I'm the only competitor (running a marathon by myself) - does it count as "winning"?

 

[StatGuy2000]

But it is. People do think that they're bad at art/drawing/music/learning languages just because they don't have some innate talent, so they shouldn't even try. And when someone get good at, say, Japanese (having put in a lot of hard work), people just naturally assume that said person is just very talented at learning languages.

I personally like to think of it this way - everyone can achieve a reasonably high level of <subject>, but not everyone (statistically) will.

This just further reinforces my point -- too many people in Western countries put too much stock in talent being "innate", when in fact hard work, persistence, and a solid training/education in the foundations of a subject (especially in the early childhood years) can make far more of an impact. I also feel that in Western countries (particularly in countries like the US and the UK, and even Canada as well) the broader society gives up on people too readily based on their initial struggles on a subject.

I will bring up an anecdote here. As someone who is half-Japanese, I was long brought up to believe that any subject is accessible so long as I worked hard at it. There were some subjects which I had comparatively little interest in (e.g. art), but with all others, I have always been made to feel by my parents that the harder I worked, the better I could become, even in subjects which I felt weren't my strongest areas (e.g. English literature). Many of my classmates in elementary and secondary school, by contrast, never seemed to put in much effort in any class, and the moment they encountered the slightest difficult, concluded that they were "bad" at a subject.

 

[Dragon27]

Coming back to the original thread topic, it is conceivable that my niece may, at some point, break through the barrier and begin to progress much faster. It's also possible that her progress will remain slow compared to other students. Extrapolating from this, it is conceivable that anyone may in principle be able to learn a lot of math or programming if they went at it persistently and diligently. But then, if it took them most of their life to achieve college level competence, then it's practically equivalent to a total inability to progress to doctoral level stuff, IMHO. So it's reasonable to say that in a practical sense, some may be incapable of learning doctoral level math in one lifetime, simply due to the sheer number of years they might need.

There's so many conditions and unanswered assumptions. What if they learn in an inefficient manner and are capable of progressing much faster (and becoming a PhD), if they find a way of learning best suitable for them? How much time do they actually have? There's life, other hobbies, work, family. Is this factored in into the phrase "in a practical sense"? Psychology and attitude may be one of the biggest factors. It may be a hopeless struggle to fight a person's disbelief into their own abilities (even if that person themselves say that they do want to learn the subject, and they do think that they can do it).

 

[Dragon27]

I enjoyed calculus (at least multivariable, vector and tensor) and linear algebra, but i absolutely loathe programming.
...
This is actually part of the reason i decided to not go for robotics as i originally planned and decided to switch to physics instead.
...
I don't feel like the two subjects necessarily are interwined when it comes to the actual understanding

I personally feel many common points between programming and math/physics at some level. Specifically, advanced math/theoretical physics/computer science kinda level. Where abstractions and the ability to abstract come to play. I think that abstraction is the main workhorse of scientific thinking (modelling of reality). It's not a coincidence that category theory (which is a celebration of high-level abstraction) is such a big thing for functional programming.

 

[ZeGato]

Not that it answers your question, but I've met smart people who had an extremely difficult time learning even high-school level mathematics. I guess you need to have an analytical personality so that's it not too hard to learn it, and so that you're able to take pleasure from learning mathematics.

 

[Drakkith]

My issue with your stance above is that the logical conclusion you would make is the following:

some people are incapable of learning Subject A (math, history, geography, language) => it's a waste of time for some people to study Subject A => we should identify these people and stop them from even learning Subject A

If this were a perfect world I would I agree with this. If some people cannot learn subject A at skill level X, then it would be beneficial for them not to be forced to try to learn subject A up to skill level X. I don't see how anyone could argue against this. The problem is that we can't identify these people ahead of time. We just can't know who will be incapable of learning a subject. That's why we have to push people along and make them try as hard as we can get them to try to see just how far they can go in a subject. So in our real, imperfect world that is not the logical conclusion to make.

The problem is that educators (whether at the K-12 level, or in post-secondary level) by and large have no idea why their students are struggling with their subjects. What I fear is that educators may well see a student struggling and automatically conclude that these students are hopeless cases, whereas they may well be suffering from poor preparation in their preceding years (due to poor teaching or poor resources).

The other issue is that people do not always learn subjects in the same pace nor do they necessarily learn material in an orderly, linear path. There have been many documented cases where students who have struggled with a subject like math in the early years end up catching up with the material and excelling in the subject at an older age. However, if an educator (or parent) looks at said student from the earlier years, they may be led to believe (erroneously) that the student will never learn math, and thus actively discourage or prevent the student in further studies. To me this is a tragedy.

I don't think any is, or realistically could, argue against this. That's not what I'm arguing, nor anyone else as far as I can tell. The only thing being argued is that, in addition to the problems you've given, there are at least a small percentage of people who just cannot learn high-level math. If we include absolutely everyone then this is just a given, as we've already talked about people with severe mental handicaps. So the answer to the OP's question, if we consider absolutely everyone, is a firm "No. Not everyone can learn advanced math."

What I have been saying is that there are people out there who are not considered to be mentally handicapped that simply cannot learn advanced math, regardless of how they were raised and educated. I'm not arguing that most people are like this, I'm not even arguing that the percentage is large. Frankly I believe that with enough time and given enough effort, most people could learn math up to the undergrad level.

 

[Pleonasm]

Hello guys,

I often ask myself if anyone can learn maths to an advanced level? And get really good on it.
I think that every healthy person can get very good at maths. The only condition is that the person is interested in math.
Of course, to get on the level of a Field-Medal winner you have to be blessed a little bit. But I think you can reach and understand a lot just by working out hard.

But are there researches which proof the current state of science in relation to how much the genetic predisposition affects the learning of math?
When you are healthy our neural system should work nearly the same as the neural system of a high-level mathematician or nah? What do you guys think? Can anyone learn maths to a high level?
Or is it important to be 'blessed'? Or do you think it is pretty irrelevant and only relevant for the level of Field-Medal member?

I am really sorry for the grammatic issues. I am still improving my English!

Math yes, physics no.

 

[SchroedingersLion]

First of all, I dispute your characterization between the average mathematician and a Fields candidate -- Fields medals (like Nobel prizes) are awarded based on discoveries, and raw talent is not the only ingredient in making discoveries -- there is a considerable element of random chance involved as well.

I never said raw talent is the only ingredient. But it is a necessity.

But setting that aside, your very premise that some mathematicians cannot become Fields medalists => some people are incapable of learning math (your contention, @SchroedingersLion ) is both false, and is extremely arrogant and elitist on your part (I'm being kind, btw, with my choice of words, due to PF rules).

Some people are incapable of learning high level mathematics, because it requires a certain amount of intelligence they don't have. This is neither false, nor is stating it arrogant or elitist in any way. Are you trying to attack me because you realize your naive view of the world doesn't hold up to logic and everyday experience?
"Anyone can learn anything, if he tries hard enough or gets the right tutorship" is a nice political correct statement that might even lead to a better education system. But it does not describe reality since talent is real. You can deny it all you want, it doesn't change the fact. How much raw brainpower a certain individuum has depends on a) genetics, b) early childhood education. Psychology is pretty clear on that. And the amount of brainpower determines a certain level of intellectual ability and knowledge you will not cross in the finite lifetime you have.

As for that 14 year girl, how much of her difficulty is based on poor instruction in her earlier years? If she had received better resources earlier in her life (and encouraged to both study and persist in studying math), she very well could have learned math effectively.

None of this matters. This 14 years old girl I met is SOMEONE. This SOMEONE will probably never be able to learn high level mathematics.
This answers the question of this thread, wether anyone could learn anything at a high level.
If you could go back in time to her earlier years, you could have improved her current condition. Sure.
But even then: Genetics play a role in that as well so even if we changed the thread's question to "Could anyone learn anything to a high level, if he got the perfect learning environment from the very first minute of his life?", the answer "yes" would become much more realistic, but still not certain.

Again, what is the difference between studying mathematics and, say, studying say, French as a foreign language? Or geography? Or history? Or biology? I've met plenty of students who have trouble in all of these subjects -- do you then conclude that students are incapable of learning all of these subjects?
How absurd does this sound to you?

It's not absurd at all. In fact, it is absurd to assume anyone could learn anything to a high level.

 

[StatGuy2000]

I never said raw talent is the only ingredient. But it is a necessity.Some people are incapable of learning high level mathematics, because it requires a certain amount of intelligence they don't have. This is neither false, nor is stating it arrogant or elitist in any way. Are you trying to attack me because you realize your naive view of the world doesn't hold up to logic and everyday experience?
"Anyone can learn anything, if he tries hard enough or gets the right tutorship" is a nice political correct statement that might even lead to a better education system. But it does not describe reality since talent is real. You can deny it all you want, it doesn't change the fact. How much raw brainpower a certain individuum has depends on a) genetics, b) early childhood education. Psychology is pretty clear on that. And the amount of brainpower determines a certain level of intellectual ability and knowledge you will not cross in the finite lifetime you have.

Again, your contention above assumes that "intelligence" (however way you define it) is a fixed quantity that remains impervious to change over the development of an individual. That is neither an obvious and far from trivial assumption you are making, and has been widely criticized and debated by psychologists for years.

As for the statement "talent is real" -- again, if someone works extremely hard on a certain task and becomes sufficiently skilled at that task, how much of this is dependent on genetics and how much is based on the individual effort? You seem to imply that this is primarily a genetic attribute, whereas I'm proposing that environmental influences (including the very actions of the individual in question) could play just as important a factor.

I should further argue that you repeatedly mention genetics here. Can you, or anyone else, actually pinpoint to research that indicate which genes (or which collective interactions of genes) can be definitively linked to mental or intellectual abilities in humans? As far as I'm aware, I cannot think of any geneticists, neuroscientists, psychologists, etc. who can provide even tentative evidence of such. Of course, if you can link me to published research in this area, I would be more than happy to read this.

 

[gmax137]

In addition to "genetics," repeated use of the word "logic" is inaccurate. This is not a question to be settled by "logic." So far we have four pages of "opinion."

Existence of a gradient in "ability" does not allow us to "logically" deduce that some individuals have zero ability, or even that some have an ability below any given threshold. I've seen no evidence for the slope of this gradient. Maybe it is more shallow than some believe. Maybe it isn't. That's the question, right? How steep is the "ability" gradient among individuals?

Opinions vary. Most driven by anecdotal experiences.

 

[mhl47]

It is really astounding how this discussion polarizes. Unfortunately, I was really hoping that someone would have taken a deeper look in the research of this topic since I have been too lazy ever since to dig into this area. But to be honest, even if there would be empirical studies on this, they would have to face immense issues with the setup, since you can not really isolate all the factors here.

So some interesting questions regarding this topic might be:

.) How much in the brain is really "hard-wired" based on genetics (assuming that nothing is really "hard-wired" but that there is some information already there in the biological structures)
.) Can you circumvent this existing structures in the brain or is there a chance they or their function changes in your lifetime (neuroplasticity)
.) Are you really bound to this preexisting structures whilst building a model of your world? (in AI commonly referred to as world model)
.) What does it actually mean to have brainpower?

One would really have to investigate how "easy" it is for a given state of a brain, starting at birth, to be shaped such that it has a model powerful enough to make a conclusion in mathematics (e.g. linking different fields). At this level of complexity, probably neither neuroscience nor research in artificial intelligence is close to answering such question. But I would be happy to be wrong.

Having said that I would also like to throw my personal opinion and experiences in. From my point of view, anyone can learn anything but it will take longer for some people because the information provided to them might not fit well in their current model of the world. Especially motivation seems to be the key to build knowledge in a way that it can be used to solve problems. Why on Earth would you restructure everything you believe, that kept you alive till now, if you do not have a strong motivation to do so (not something superficial like grades)? If a person does not think, that understanding mathematics can be really beneficial for their life, they will have a hard time learning it. Convincing people that they can benefit from understanding mathematics, and do awesome things with that knowledge, is for me the most important part of teaching. Since one tends to judge that also based on the person that teaches (do I really want to be like my teacher?) it does not exactly help that a lot of math teachers are a bit introverted.

In all the years at university or whilst tutoring, I never had the feeling that a persons struggle with a concept was based on their inability to learn it. Never have I experienced a student with a deep and real interest (working hard on the problems, gathering additional information through books etc., trying to apply the knowledge to real-world situations) who hit a line he/she could not cross.

In a way, it is a really comfortable world-view to say that the ability to learn maths is limited. For the people who are good at it, feeling special and secure in their position, and for the people who seem to be bad at math, convincing them self that they can not learn it anyway. I do not see any real evidence supporting that view.

 

[Drakkith]

Again, your contention above assumes that "intelligence" (however way you define it) is a fixed quantity that remains impervious to change over the development of an individual.

I don't agree with your conclusion here. A 'variable intelligence' could still mean that certain people are simply not going to be able to learn something, as the amount that their intelligence can vary may be too low.

I should further argue that you repeatedly mention genetics here. Can you, or anyone else, actually pinpoint to research that indicate which genes (or which collective interactions of genes) can be definitively linked to mental or intellectual abilities in humans?

There are plenty of genetic diseases that negatively affect mental ability, so there is obviously some link between genetics and mental ability. While there is no conclusive proof that certain specific genes give a person above average abilities, it seems reasonable to conclude that it is possible, perhaps likely. But any link is not obvious and it may not result in a particularly large difference.

Existence of a gradient in "ability" does not allow us to "logically" deduce that some individuals have zero ability, or even that some have an ability below any given threshold.

We aren't looking purely at a mathematical gradient. We have real life people we can look at. The spectrum of mental ability (however you might define that or break it down) runs from 0 (brain dead) through those with severe mental handicaps, continuing on up through people with less severe handicaps, average/near-average, above average, and so on.

In all the years at university or whilst tutoring, I never had the feeling that a persons struggle with a concept was based on their inability to learn it. Never have I experienced a student with a deep and real interest (working hard on the problems, gathering additional information through books etc., trying to apply the knowledge to real-world situations) who hit a line he/she could not cross.

Perhaps the reason that you haven't seen anyone with an interest in a subject hit a hard wall is because the people who have severe difficulty with math (or another subject) don't want to do math because of how difficult it is for them. People rarely enjoy things that are immensely difficult and taxing for them. Also realize that the more difficulty someone has with math and other subjects, the less likely they are to go to college or pursue non-required education. So you aren't as likely to encounter them as you might think.

 

[atyy]

What I have been saying is that there are people out there who are not considered to be mentally handicapped that simply cannot learn advanced math, regardless of how they were raised and educated. I'm not arguing that most people are like this, I'm not even arguing that the percentage is large. Frankly I believe that with enough time and given enough effort, most people could learn math up to the undergrad level.

Can anyone (with "normal intelligence") learn to count? There an interesting case about this: https://en.wikipedia.org/wiki/Pirahã_language.

 

[mathwonk]

Here is a quote from Heisuke Hironaka, Fields medalist in mathematics: "I like basic things. Very clever people tend to jump to the new techniques: something is developing very fast, and you want to be on top of it; and if you are smart, you can be a top runner. But I am not so smart, so it is better that I start something where there are no techniques for the problem, and then I can just build step by step."

http://www.ams.org/notices/200509/fea-hironaka.pdf page 1013, line -9.

So even a not so smart person can be a Fields medalist.

Or to quote Adam Sandler: "I am not particularly smart or good looking or talented, and yet I am a multi millionaire".

To include myself, most people think I am as dumb as a bag of rocks, and yet I have a PhD in math, which I obtained when my university said they would fire me if I did not get one, and I was a young father. So I am an advocate of the hard work and serious motivation school of thought.

I rest my case.

 

[Drakkith]

To include myself, most people think I am as dumb as a bag of rocks, and yet I have a PhD in math, which I obtained when my university said they would fire me if I did not get one, and I was a young father. So I am an advocate of the hard work and serious motivation school of thought.

I failed out of college, despite my best effort. So where does that put me? Did I just not try hard enough? Or not put in enough effort?