consider the fact that even I read this stuff, and I have the mathematical ability of a random rock
Seriously, you are too smart to have any trouble in acquiring understanding of mathematics if you made a serious effort. Just read the textbooks starting at an appropriate level. Given that you characterize your skill the way you do, there’s probably some low-hanging fruit there for you. (But it’s possible that you won’t be able to enjoy the process. I know I would know less math if I didn’t have something to protect.)
A fact I didn’t appreciate before encountering this whole AI-related craze followed by overcomingbias followed by lesswrong is that it’s possible to master an arbitrary field of expertise by systematically teaching yourself its skills, even if it’s completely dissimilar to all you’ve ever known.
“Learn math” is kind of a broad imperative. I know the math that’s common to many different applications, like arithmetic and algebra and a bit of calculus, but after that it becomes so fractured that even when I learn how to solve one specific problem in a specific field, I never encounter that problem or field again.
If there were a specific cause for which I needed math, I would force myself to learn the math relevant to that cause, but just “learn topology, who knows when you might need it?” has never been very convincing to me.
I’ve studied a little decision theory, since that seems to be the form of math most relevant to Friendly AI, but so far I’ve found it frustrating and hard to treat with suitable rigor. If anyone wants to recommend an unusually good textbook, preferably online, I suppose I’d take suggestions.
What I would say is that it’s okay not to bother learning math if you don’t need or want to, but for heaven’s sake just don’t go around saying you couldn’t learn it if you tried because you lack some specific cognitive module. (When people say this, it almost always means simply that they weren’t socialized into mastering it in childhood, and they haven’t bothered to update this aspect of their identity since then.)
By the way, I totally agree on the Sequences: I remember when they were being written, I used to look forward to the next post the way a kid looks forward to the next episode of their favorite TV show.
When people say this, it almost always means simply that they weren’t socialized into mastering it in childhood, and they haven’t bothered to update this aspect of their identity since then.
I don’t think you have more evidence for this hypothesis than I do for “some people just don’t have a head for figures.”
I have a generalized sense of “head for figures” in mind, if you mean that you’re good at math but not at calculation. Some people are bad at both, and it’s pretty optimistic to say that what’s holding them back (almost always!) are only their childhood experiences.
I don’t consider myself “good at math” despite having credentials in the subject. As far as I can tell, in order to do math, I have to use thought processes that are quite different from those used by people who are stereotypically “good at math”.
What holds people back is not their childhood experiences per se but the general lesson learned in childhood that having certain abilities and lacking others is an integral part of one’s tribal uniform.
I’ve known many people who are good at math but what they have in common isn’t substantial enough to gel into a “type” for me, so when you say “stereotypically good at math” I draw a blank.
Ahab and Billy are two 14-year old kids in an algebra class this year, and at the end of it they’re both going to get an ‘A’. Ahab will achieve this effortlessly, spending less than an hour per week doing his homework, and Billy will really struggle, spending more than an hour every night on homework and supplementary studying. And it’s always been like this for Ahab and Billy. Maybe you’ll object but I think I’m describing something very plausible and common.
It sounds like you would reject any explanation for the difference between Ahab and Billy of the kind “Billy has less native math ability than Ahab,” and favor an alternative explanation about childhood socialization. But can you spell out what this explanation is, or what some of its consequences are?
To get a sense of what I mean by “stereotypically good at math”, think about the abilities involved in solving tricky puzzles or competition-style problems. Or, consider the comments section of Eliezer’s Drawing Two Aces post, full of people who got the right answer (I didn’t, which resulted in this post). The idea isn’t exactly well-defined, but seems to involve some combination of powerful short-term memory and an ability to quickly identify the particular abstraction that the poser of a concrete problem is attempting to refer to.
The implication of the contrast between Ahab and Billy, on my account, isn’t what you perhaps think. I don’t necessarily deny that some kind of “native” difference could be responsible for Billy’s greater difficulties relative to Ahab. The fact that Billy manages to get an “A”, however, means that anyone with Billy’s level of “native math ability” can’t invoke that to explain why they didn’t get an “A”. Billy may have other native abilities that such an individual may lack, but they won’t be specifically math-related, and instead will be general things like “the ability to overcome akrasia”, etc.
Notwithstanding the above, “lack of native math ability” is still a fake explanation. Whatever “math ability” is, it is reducible. I want to know in detail what goes through Billy’s mind as he attempts to solve an algebra problem, and how it differs from what goes on in Ahab’s mind. Once we know this, we can try to determine what causes this difference: is Billy’s IQ just lower than Ahab’s (which would be a general problem, not a math-specific one), does he lack certain pieces of information that Ahab has (easily fixable), or is he executing particular cognitive habits that prevent him from processing the same information as efficiently as Ahab (fixable via training)?
Teasing this out a little more, bullet points of my own:
If B learns math at a slower pace than A, then it can literally be the case that B will never understand math as well as A. At suitable slow (but common) learning paces, it can be impractical and unrewarding for B to study math. And I think there might even be large numbers of mentally normal human beings walking around for whom this pace is so slow that it’s misleading to call it a “learning pace” at all, e.g. too slow for progress they make one day to stick the next.
I’m sure that “math ability,” like anything else, is reducible, but in these kinds of brain-and-behavior cases it might “reduce” to thousands of different factors that don’t have much to do with each other. In that case it wouldn’t be very easy to give advice on how to be better at math, beyond “arrange each of those thousands of factors in a way favorable to math ability.”
Even if “math ability” has a more satisfying explanation than the kind in 2., so that it’s possible to give good advice on how to improve it, I think that this is not a solved problem. Specifically I still think that your proposed advice (“you simply haven’t bothered to update some aspect of your identity since childhood”) is no good.
In the meantime “lack of math ability” seems to me to be a perfectly good label for a real phenomenon, though I guess I agree with you that it is not an explanation for that phenomenon.
1. I disagree that a slower learning pace is less rewarding. On the contrary, learning is most rewarding when there is time to do it properly, and the frustration many people experience in school settings results from the pressure to (appear-to-) learn things more quickly than their natural pace.
(I owe to Michael Vassar the observation that there is something inherently contradictory and unrealistic about expecting people to learn calculus in a semester when they required five years to learn arithmetic.)
2. It might seem like it could be that complicated, but it turns out not to be. In practice (as revealed by teaching experience), “lacking math ability” usually reduces to something like “I flinch and run away when I realize that I will have to carry out more than two or three steps (especially if there is recursion involved), instead of just gritting my teeth and carrying them out.”
3. Most people don’t even try updating their identities; while on the other hand, I myself have updated my skill-related identities on a number of occasions, and it worked. (Math happens to be an example.)
4. It would be best to have a label that conveys more information about the cause(s) of the phenomenon.
I found that revisiting formal logic/set theory forced more careful intuitions about decision-making, and learning category theory made it less scary to work with more complicated ideas. Learning topology helped with studying set theory and gave some insight into the process of coming up with new mathematical concepts. You’ve probably seen my reading list (all the stuff on it can be downloaded from Kad).
I can’t make a proper explicit argument for studying math being on direct track to contributing to FAI research (particularly since UDT/ADT now look potentially less relevant than I thought before), but it looks like the best available option, giving general enough reasoning skills that could conceivably help, where I’m not aware of other kinds of knowledge that looks potentially useful to a similar extent.
(On the other hand, I probably don’t pay enough attention to the skills I already had two years ago, which include good background in programming and basic background in machine learning.)
I see no easy/convincing way of doing so right now. I’ll write up my ideas when/if they sufficiently mature, or, as is often the case, I’ll move on to a different line of investigation. Basically, morality is seen through a collection of many diverse heuristics, and while a few well-understood heuristics can form the backbone of a tool for boosting the power of an agent, they won’t have foundational significance, and so selection of the heuristics that need to be explicitly understood should be based on the leverage they give, even where they are allowed to have some blind spots.
A fact I didn’t appreciate before encountering this whole AI-related craze followed by overcomingbias followed by lesswrong is that it’s possible to master an arbitrary field of expertise by systematically teaching yourself its skills, even if it’s completely dissimilar to all you’ve ever known.
(If you are sufficiently intelligent and have a certain set of personality traits. This is not something everyone can realistically be considered capable of.)
When I have to learn math, I can learn it, but math is unpleasant to me, and no matter how much I learn it keeps exactly the same level of unpleasantness. I realize the importance of math and I try to know enough to understand all the fields that are important or interesting to me, but there are a lot of equations in the Sequences that I skimmed over because I trusted that Eliezer had derived them correctly and they meant what he said they meant, and while I totally admire people who love math and will work through every step of those equations, I’d recommend anyone else who’s holding off on reading the Sequences just because they’re math-y not to worry about skimming.
Seriously, you are too smart to have any trouble in acquiring understanding of mathematics if you made a serious effort. Just read the textbooks starting at an appropriate level. Given that you characterize your skill the way you do, there’s probably some low-hanging fruit there for you. (But it’s possible that you won’t be able to enjoy the process. I know I would know less math if I didn’t have something to protect.)
A fact I didn’t appreciate before encountering this whole AI-related craze followed by overcomingbias followed by lesswrong is that it’s possible to master an arbitrary field of expertise by systematically teaching yourself its skills, even if it’s completely dissimilar to all you’ve ever known.
“Learn math” is kind of a broad imperative. I know the math that’s common to many different applications, like arithmetic and algebra and a bit of calculus, but after that it becomes so fractured that even when I learn how to solve one specific problem in a specific field, I never encounter that problem or field again.
If there were a specific cause for which I needed math, I would force myself to learn the math relevant to that cause, but just “learn topology, who knows when you might need it?” has never been very convincing to me.
I’ve studied a little decision theory, since that seems to be the form of math most relevant to Friendly AI, but so far I’ve found it frustrating and hard to treat with suitable rigor. If anyone wants to recommend an unusually good textbook, preferably online, I suppose I’d take suggestions.
What I would say is that it’s okay not to bother learning math if you don’t need or want to, but for heaven’s sake just don’t go around saying you couldn’t learn it if you tried because you lack some specific cognitive module. (When people say this, it almost always means simply that they weren’t socialized into mastering it in childhood, and they haven’t bothered to update this aspect of their identity since then.)
By the way, I totally agree on the Sequences: I remember when they were being written, I used to look forward to the next post the way a kid looks forward to the next episode of their favorite TV show.
I don’t think you have more evidence for this hypothesis than I do for “some people just don’t have a head for figures.”
Lacking a “head for figures” myself, I am personally a counterexample to your hypothesis.
Although what I really dislike about it is not even that it’s false but that it’s a curiosity-stopping fake explanation.
I have a generalized sense of “head for figures” in mind, if you mean that you’re good at math but not at calculation. Some people are bad at both, and it’s pretty optimistic to say that what’s holding them back (almost always!) are only their childhood experiences.
I don’t consider myself “good at math” despite having credentials in the subject. As far as I can tell, in order to do math, I have to use thought processes that are quite different from those used by people who are stereotypically “good at math”.
What holds people back is not their childhood experiences per se but the general lesson learned in childhood that having certain abilities and lacking others is an integral part of one’s tribal uniform.
I’ve known many people who are good at math but what they have in common isn’t substantial enough to gel into a “type” for me, so when you say “stereotypically good at math” I draw a blank.
Ahab and Billy are two 14-year old kids in an algebra class this year, and at the end of it they’re both going to get an ‘A’. Ahab will achieve this effortlessly, spending less than an hour per week doing his homework, and Billy will really struggle, spending more than an hour every night on homework and supplementary studying. And it’s always been like this for Ahab and Billy. Maybe you’ll object but I think I’m describing something very plausible and common.
It sounds like you would reject any explanation for the difference between Ahab and Billy of the kind “Billy has less native math ability than Ahab,” and favor an alternative explanation about childhood socialization. But can you spell out what this explanation is, or what some of its consequences are?
Several points to make in reply:
To get a sense of what I mean by “stereotypically good at math”, think about the abilities involved in solving tricky puzzles or competition-style problems. Or, consider the comments section of Eliezer’s Drawing Two Aces post, full of people who got the right answer (I didn’t, which resulted in this post). The idea isn’t exactly well-defined, but seems to involve some combination of powerful short-term memory and an ability to quickly identify the particular abstraction that the poser of a concrete problem is attempting to refer to.
The implication of the contrast between Ahab and Billy, on my account, isn’t what you perhaps think. I don’t necessarily deny that some kind of “native” difference could be responsible for Billy’s greater difficulties relative to Ahab. The fact that Billy manages to get an “A”, however, means that anyone with Billy’s level of “native math ability” can’t invoke that to explain why they didn’t get an “A”. Billy may have other native abilities that such an individual may lack, but they won’t be specifically math-related, and instead will be general things like “the ability to overcome akrasia”, etc.
Notwithstanding the above, “lack of native math ability” is still a fake explanation. Whatever “math ability” is, it is reducible. I want to know in detail what goes through Billy’s mind as he attempts to solve an algebra problem, and how it differs from what goes on in Ahab’s mind. Once we know this, we can try to determine what causes this difference: is Billy’s IQ just lower than Ahab’s (which would be a general problem, not a math-specific one), does he lack certain pieces of information that Ahab has (easily fixable), or is he executing particular cognitive habits that prevent him from processing the same information as efficiently as Ahab (fixable via training)?
Teasing this out a little more, bullet points of my own:
If B learns math at a slower pace than A, then it can literally be the case that B will never understand math as well as A. At suitable slow (but common) learning paces, it can be impractical and unrewarding for B to study math. And I think there might even be large numbers of mentally normal human beings walking around for whom this pace is so slow that it’s misleading to call it a “learning pace” at all, e.g. too slow for progress they make one day to stick the next.
I’m sure that “math ability,” like anything else, is reducible, but in these kinds of brain-and-behavior cases it might “reduce” to thousands of different factors that don’t have much to do with each other. In that case it wouldn’t be very easy to give advice on how to be better at math, beyond “arrange each of those thousands of factors in a way favorable to math ability.”
Even if “math ability” has a more satisfying explanation than the kind in 2., so that it’s possible to give good advice on how to improve it, I think that this is not a solved problem. Specifically I still think that your proposed advice (“you simply haven’t bothered to update some aspect of your identity since childhood”) is no good.
In the meantime “lack of math ability” seems to me to be a perfectly good label for a real phenomenon, though I guess I agree with you that it is not an explanation for that phenomenon.
1. I disagree that a slower learning pace is less rewarding. On the contrary, learning is most rewarding when there is time to do it properly, and the frustration many people experience in school settings results from the pressure to (appear-to-) learn things more quickly than their natural pace.
(I owe to Michael Vassar the observation that there is something inherently contradictory and unrealistic about expecting people to learn calculus in a semester when they required five years to learn arithmetic.)
2. It might seem like it could be that complicated, but it turns out not to be. In practice (as revealed by teaching experience), “lacking math ability” usually reduces to something like “I flinch and run away when I realize that I will have to carry out more than two or three steps (especially if there is recursion involved), instead of just gritting my teeth and carrying them out.”
3. Most people don’t even try updating their identities; while on the other hand, I myself have updated my skill-related identities on a number of occasions, and it worked. (Math happens to be an example.)
4. It would be best to have a label that conveys more information about the cause(s) of the phenomenon.
.
.
I found that revisiting formal logic/set theory forced more careful intuitions about decision-making, and learning category theory made it less scary to work with more complicated ideas. Learning topology helped with studying set theory and gave some insight into the process of coming up with new mathematical concepts. You’ve probably seen my reading list (all the stuff on it can be downloaded from Kad).
I can’t make a proper explicit argument for studying math being on direct track to contributing to FAI research (particularly since UDT/ADT now look potentially less relevant than I thought before), but it looks like the best available option, giving general enough reasoning skills that could conceivably help, where I’m not aware of other kinds of knowledge that looks potentially useful to a similar extent.
(On the other hand, I probably don’t pay enough attention to the skills I already had two years ago, which include good background in programming and basic background in machine learning.)
Expand?
I see no easy/convincing way of doing so right now. I’ll write up my ideas when/if they sufficiently mature, or, as is often the case, I’ll move on to a different line of investigation. Basically, morality is seen through a collection of many diverse heuristics, and while a few well-understood heuristics can form the backbone of a tool for boosting the power of an agent, they won’t have foundational significance, and so selection of the heuristics that need to be explicitly understood should be based on the leverage they give, even where they are allowed to have some blind spots.
(If you are sufficiently intelligent and have a certain set of personality traits. This is not something everyone can realistically be considered capable of.)
When I have to learn math, I can learn it, but math is unpleasant to me, and no matter how much I learn it keeps exactly the same level of unpleasantness. I realize the importance of math and I try to know enough to understand all the fields that are important or interesting to me, but there are a lot of equations in the Sequences that I skimmed over because I trusted that Eliezer had derived them correctly and they meant what he said they meant, and while I totally admire people who love math and will work through every step of those equations, I’d recommend anyone else who’s holding off on reading the Sequences just because they’re math-y not to worry about skimming.