I think this is a largely overestimated concept, especially on LW. I doubt most people here are “smarter” than average Joe. A lot of it is due to education, a difference of interest, and a little more ease when it comes to symbol manipulation. Surely there are many more factors, like the ability to concentrate, not getting bored too quickly, being told as a child that one can learn anything if one tries hard enough etc., but little has to do with insurmountable hardware limitations.
You know how there are people who, even though you could train them to carry out the steps of a Universal Turing Machine, you can’t manage to teach them linear algebra...
I haven’t heard of any evidence that would suggest that there are human beings who can’t understand linear algebra. I myself have not yet arrived at linear algebra, because I didn’t bother to learn any math when I was a teenager, but I doubt that it is something only superhuman beings can understand. I would go as far as to bet that you could teach it to someone with down syndrome.
Take for example the number 3^^^^3. Can I hold a model of 3^^^^3 objects in my memory? No. Can I visualize 3^^^^3? No. Does that mean that I am unable to fathom some of its important properties, e.g. its scope? No.
Someone who has no legs can’t run faster than you. Similar differences are true about different brains, but we don’t know enough about brains, or what it means to understand linear algebra, to indiscriminately claim that someone is “smarter”...
I’m not convinced anybody could teach me to understand linear algebra. Or maybe what I mean by that is that I’m not convinced of my own ability to understand linear algebra, which may be a different thing.
I have trouble with maths. More specifically, I have trouble with numbers. What I experience when faced with lots of numbers is akin to how people with dyslexia often describe trying to parse lots of written text—they swim and shift beneath my eyes, and dissolve into a mass of meaningless gobbledegook that I can’t pick any sense from. And then after a while, even if I’ve ploughed through some of this, I start to get what I can only describe as “number fatigue” and things that previously I’d almost started to comprehend seem to slip out from my grasp.
And, when asked to do simple maths, I panic and fly into what is pretty much an anxiety attack. Which, of course, means that I’m not thinking clearly enough to untangle it all and try to start making sense of it.
Maths feels utterly, utterly impenetrable to me. Half the time I can’t even work out what the necessary sum is—recent examples include my having no notion of the calculations required for aspect ratio or 10% of a weight in stones and pounds, but this also applies to much simpler things, like the time I couldn’t figure out how to calculate the potential eventual fundraising total from the time elapsed, the time remaining and the money so far achieved.
I realise that in a community like this I’m going to stick out like a sore thumb, mind you ;-)
Certainly some of it is. The anxiety and fluster and horrible panic feeling is certainly emotional, and the “number blindness” thing is probably related too. It’s much, much worse if there’s anyone else around—the only thing more embarrassing than knowing I’ve failed simple arithmetic is failing simple arithmetic when other people who might assume I’m moronically stupid can see me doing it.
And of course that makes me a nightmare to teach, because I’m horribly resistant to learning maths because I know I’ll fail and look stupid and whoever it is will think I’m thick. You of all people have encountered that in me!
Struggling to parse strings of numbers, though, can happen no matter how calm and unpressured and private I am. I’ve emailed myself things like my debit card number so that I can just cut and paste them when I buy things, because I can’t always reliably type them in by looking at the card.
That’s certainly entirely plausible, and something my mother (a primary school teacher of a quarter-century’s experience, who’s known a lot of children well) has always suspected. I’ve never had it checked out, though. Maybe I should.
ETA—particularly as I’ve just had a look at the wikipedia article and every single thing in the symptoms list applies to me to some degree. I’m even a pretty good writer. Good grief.
I realise that in a community like this I’m going to stick out like a sore thumb, mind you ;-)
Check the following links, here is my homepage from when I was 18 and this is another page from that year. Looks more like something made by a 14 year old, doesn’t it? And since Desrtopa mentioneddiscalculia, your case might be stronger, but it took me 3 attempts to figure out how old I was in 2002 :-)
There do exist neurological deficits that prevent people from acquiring certain skills, understand some concepts and reach certain performance levels, but I wouldn’t jump to a conclusion in your case. A lot of it might very well has to do with what you believe to be the case, rather than what is actual. I haven’t hit any barrier yet. And I only learnt to read analog clocks when I was around 14.
I admit that I can’t put myself in your position, so maybe I am wrong and you should stop worrying about mathematics. I am only saying that you might as well stop caring, but not give up. In other words, do not panic, just try it and don’t expect to succeed. Start small, think about the most basic problem for as long as necessary, without feeling coerced to understand it. Use objects and drawings to approach the problem. Read up on various different sources explaining the same problem. Do not stop reading, or listening to explanations when you feel that you can’t follow anymore, just read it over and over again. Then stop for a few hours or days and think about it again. And remember not to push yourself to understand it, you just do it in your spare-time, for fun. If you feel overwhelmed, just forget about it and get back to it later. Write it down, print it out and plaster the walls in your bedroom with it so that you don’t need any willpower to approach the problem, the problem will approach you. You have all the time you need, even if it takes decades to understand that one simple problem.
It also helps to remember that almost everyone knows someone who is much better at something. Many people learn to play a musical instrument and never expect to become a professional musician. People play golf or soccer, just for fun or because of the challenge. Almost nobody turns out to be good at what they are doing. Personally I like to play an online racing game called Trackmania. I play it since 2007 and only managed to reach the world rank 34795. And I still play it, even though I almost never win. If you have trouble doing basic arithmetic, well then, try to enjoy the challenge, don’t worry, don’t panic!
A lot of it is due to education, a difference of interest, and a little more ease when it comes to symbol manipulation [...] but little has to do with insurmountable hardware limitations.
I wonder if that makes a difference in practical terms. There’s all sorts of potential in one’s genes, but one has the body, brain and personal history one ends up with.
What I mean is no longer feeling like the smartest person in the room and quite definitely having to put in effort to keep up.
I haven’t heard of any evidence that would suggest that there are human beings who can’t understand linear algebra.
I first encountered humans who couldn’t understand basic arithmetic at university, in the bit of first-year psychology where they try to bludgeon basic statistics into people’s heads. People who were clearly intelligent in other regards and not failures at life, who nevertheless literally had trouble adding two numbers with a result in the thirties. I’m still boggling 25 years later, but I was there and saw it …
first encountered humans who couldn’t understand basic arithmetic at university
When I first saw a fraction, e.g. 1⁄4, I had real trouble to accept that it equals .25. I was like, “Uhm, why?”...when other people are like, “Okay, then by induction 2/4=.5″...it’s not that I don’t understand, but do not accept. Only when I learnt that .25 is a base-10 place-value notation, which really is an implicit fraction, with the denominator being a power of ten, I was beginning to accept that it works (it took a lot more actually, like understanding the concept of prime factorization etc.). Which might be a kind of stupidity, but not something that would prevent me from ever understanding mathematics.
The concept of a function is another example:
f:X->Y (Uhm, what?)
f(x) : X → Y (Uhm, what?)
f(x) = x+1 (Hmm.)
f(1) = 1+1 (Okay.)
y = f(x) (Hmm.)
(x, y)
(x, f(x))
(1,2) (Aha, okay.)
(x,y) is an element of R (Hmm.)
R is a binary relation (Uhm, what?)
x is R-related to y (Oh.)
xRy
R(x,y) (Aha...)
R = (X, Y, G)
G is a subset of the Cartesian product X × Y (Uhm, what?)
...so it goes. My guess is that many people appear stupid because their psyche can’t handle apparent self-evidence very well.
I wonder if that makes a difference in practical terms.
If only by its effect on yourself and other people. If you taboo “smarter” and replace it with “more knowledgeable” or “large inferential distance”, you do not claim that one can’t reach a higher level:
“That person is smarter than you.” = Just give up trying to understand, you can’t reach that level by any amount of effort.
vs.
“That person is more knowledgeable than you.” = Try to reduce the inferential distance by studying hard.
I first encountered humans who couldn’t understand basic arithmetic at university...
I believe that to be the case with literally every new math problem I encounter. Until now I have been wrong each time.
Basic arithmetic can be much harder for some people than others because some just do the logic of symbol manipulation while others go deeper by questioning axiomatic approaches. There are many reasons for why people apparently fail to understand something simple, how often can you pinpoint it to be something that can’t be overcome?
I first encountered humans who couldn’t understand basic arithmetic at university, in
the bit of first-year psychology where they try to bludgeon basic statistics into people’s > heads. People who were clearly intelligent in other regards and not failures at life, who > nevertheless literally had trouble adding two numbers with a result in the thirties. I’m
still boggling 25 years later, but I was there and saw it …
See above, but I am basically one of those people. My own intelligence lies in other areas ;-)
I first encountered humans who couldn’t understand basic arithmetic at university
Thinking about this a bit longer, I think mathematical logic is a good example that shows that their problem is unlikely to be that they are fundamentally unable to understand basic arithmetic. Logic is a “system of inference rules for mechanically discovering new true statements using known true statements.” Here the emphasis is on mechanical. Is there some sort of understanding that transcends the knowledge of logical symbols and their truth values? Is arithmetic particularly more demanding in this respect?
BTW, you should pop along to a London meetup, even if only to boggle slightly. A nice bunch.
But I suspect you’re all disturbingly humanoid! I know you are!
You’ll see me sipping water in a real ale pub. A deeply disturbing sight.
...who are you and what have you done with, you know, /you/?!?
Someone attempting to keep up with a room full of people smarter than me :-)
I point you at the welcome thread!
I think this is a largely overestimated concept, especially on LW. I doubt most people here are “smarter” than average Joe. A lot of it is due to education, a difference of interest, and a little more ease when it comes to symbol manipulation. Surely there are many more factors, like the ability to concentrate, not getting bored too quickly, being told as a child that one can learn anything if one tries hard enough etc., but little has to do with insurmountable hardware limitations.
Eliezer Yudkowsky recently wrote:
I haven’t heard of any evidence that would suggest that there are human beings who can’t understand linear algebra. I myself have not yet arrived at linear algebra, because I didn’t bother to learn any math when I was a teenager, but I doubt that it is something only superhuman beings can understand. I would go as far as to bet that you could teach it to someone with down syndrome.
Take for example the number 3^^^^3. Can I hold a model of 3^^^^3 objects in my memory? No. Can I visualize 3^^^^3? No. Does that mean that I am unable to fathom some of its important properties, e.g. its scope? No.
Someone who has no legs can’t run faster than you. Similar differences are true about different brains, but we don’t know enough about brains, or what it means to understand linear algebra, to indiscriminately claim that someone is “smarter”...
I’m not convinced anybody could teach me to understand linear algebra. Or maybe what I mean by that is that I’m not convinced of my own ability to understand linear algebra, which may be a different thing.
I have trouble with maths. More specifically, I have trouble with numbers. What I experience when faced with lots of numbers is akin to how people with dyslexia often describe trying to parse lots of written text—they swim and shift beneath my eyes, and dissolve into a mass of meaningless gobbledegook that I can’t pick any sense from. And then after a while, even if I’ve ploughed through some of this, I start to get what I can only describe as “number fatigue” and things that previously I’d almost started to comprehend seem to slip out from my grasp.
And, when asked to do simple maths, I panic and fly into what is pretty much an anxiety attack. Which, of course, means that I’m not thinking clearly enough to untangle it all and try to start making sense of it.
Maths feels utterly, utterly impenetrable to me. Half the time I can’t even work out what the necessary sum is—recent examples include my having no notion of the calculations required for aspect ratio or 10% of a weight in stones and pounds, but this also applies to much simpler things, like the time I couldn’t figure out how to calculate the potential eventual fundraising total from the time elapsed, the time remaining and the money so far achieved.
I realise that in a community like this I’m going to stick out like a sore thumb, mind you ;-)
This all sounds less like a lack of innate ability and more like a barrier of fear. Not to say that can’t be just as disabling.
Certainly some of it is. The anxiety and fluster and horrible panic feeling is certainly emotional, and the “number blindness” thing is probably related too. It’s much, much worse if there’s anyone else around—the only thing more embarrassing than knowing I’ve failed simple arithmetic is failing simple arithmetic when other people who might assume I’m moronically stupid can see me doing it.
And of course that makes me a nightmare to teach, because I’m horribly resistant to learning maths because I know I’ll fail and look stupid and whoever it is will think I’m thick. You of all people have encountered that in me!
Struggling to parse strings of numbers, though, can happen no matter how calm and unpressured and private I am. I’ve emailed myself things like my debit card number so that I can just cut and paste them when I buy things, because I can’t always reliably type them in by looking at the card.
It could be a case of discalculia.
That’s certainly entirely plausible, and something my mother (a primary school teacher of a quarter-century’s experience, who’s known a lot of children well) has always suspected. I’ve never had it checked out, though. Maybe I should.
ETA—particularly as I’ve just had a look at the wikipedia article and every single thing in the symptoms list applies to me to some degree. I’m even a pretty good writer. Good grief.
Check the following links, here is my homepage from when I was 18 and this is another page from that year. Looks more like something made by a 14 year old, doesn’t it? And since Desrtopa mentioned discalculia, your case might be stronger, but it took me 3 attempts to figure out how old I was in 2002 :-)
There do exist neurological deficits that prevent people from acquiring certain skills, understand some concepts and reach certain performance levels, but I wouldn’t jump to a conclusion in your case. A lot of it might very well has to do with what you believe to be the case, rather than what is actual. I haven’t hit any barrier yet. And I only learnt to read analog clocks when I was around 14.
I admit that I can’t put myself in your position, so maybe I am wrong and you should stop worrying about mathematics. I am only saying that you might as well stop caring, but not give up. In other words, do not panic, just try it and don’t expect to succeed. Start small, think about the most basic problem for as long as necessary, without feeling coerced to understand it. Use objects and drawings to approach the problem. Read up on various different sources explaining the same problem. Do not stop reading, or listening to explanations when you feel that you can’t follow anymore, just read it over and over again. Then stop for a few hours or days and think about it again. And remember not to push yourself to understand it, you just do it in your spare-time, for fun. If you feel overwhelmed, just forget about it and get back to it later. Write it down, print it out and plaster the walls in your bedroom with it so that you don’t need any willpower to approach the problem, the problem will approach you. You have all the time you need, even if it takes decades to understand that one simple problem.
It also helps to remember that almost everyone knows someone who is much better at something. Many people learn to play a musical instrument and never expect to become a professional musician. People play golf or soccer, just for fun or because of the challenge. Almost nobody turns out to be good at what they are doing. Personally I like to play an online racing game called Trackmania. I play it since 2007 and only managed to reach the world rank 34795. And I still play it, even though I almost never win. If you have trouble doing basic arithmetic, well then, try to enjoy the challenge, don’t worry, don’t panic!
I wonder if that makes a difference in practical terms. There’s all sorts of potential in one’s genes, but one has the body, brain and personal history one ends up with.
What I mean is no longer feeling like the smartest person in the room and quite definitely having to put in effort to keep up.
I first encountered humans who couldn’t understand basic arithmetic at university, in the bit of first-year psychology where they try to bludgeon basic statistics into people’s heads. People who were clearly intelligent in other regards and not failures at life, who nevertheless literally had trouble adding two numbers with a result in the thirties. I’m still boggling 25 years later, but I was there and saw it …
When I first saw a fraction, e.g. 1⁄4, I had real trouble to accept that it equals .25. I was like, “Uhm, why?”...when other people are like, “Okay, then by induction 2/4=.5″...it’s not that I don’t understand, but do not accept. Only when I learnt that .25 is a base-10 place-value notation, which really is an implicit fraction, with the denominator being a power of ten, I was beginning to accept that it works (it took a lot more actually, like understanding the concept of prime factorization etc.). Which might be a kind of stupidity, but not something that would prevent me from ever understanding mathematics.
The concept of a function is another example:
f:X->Y (Uhm, what?)
f(x) : X → Y (Uhm, what?)
f(x) = x+1 (Hmm.)
f(1) = 1+1 (Okay.)
y = f(x) (Hmm.)
(x, y)
(x, f(x))
(1,2) (Aha, okay.)
(x,y) is an element of R (Hmm.)
R is a binary relation (Uhm, what?)
x is R-related to y (Oh.)
xRy
R(x,y) (Aha...)
R = (X, Y, G)
G is a subset of the Cartesian product X × Y (Uhm, what?)
...so it goes. My guess is that many people appear stupid because their psyche can’t handle apparent self-evidence very well.
If only by its effect on yourself and other people. If you taboo “smarter” and replace it with “more knowledgeable” or “large inferential distance”, you do not claim that one can’t reach a higher level:
“That person is smarter than you.” = Just give up trying to understand, you can’t reach that level by any amount of effort.
vs.
“That person is more knowledgeable than you.” = Try to reduce the inferential distance by studying hard.
I believe that to be the case with literally every new math problem I encounter. Until now I have been wrong each time.
Basic arithmetic can be much harder for some people than others because some just do the logic of symbol manipulation while others go deeper by questioning axiomatic approaches. There are many reasons for why people apparently fail to understand something simple, how often can you pinpoint it to be something that can’t be overcome?
See above, but I am basically one of those people. My own intelligence lies in other areas ;-)
Thinking about this a bit longer, I think mathematical logic is a good example that shows that their problem is unlikely to be that they are fundamentally unable to understand basic arithmetic. Logic is a “system of inference rules for mechanically discovering new true statements using known true statements.” Here the emphasis is on mechanical. Is there some sort of understanding that transcends the knowledge of logical symbols and their truth values? Is arithmetic particularly more demanding in this respect?