I think there are a lot of toxic default views about how mathematical activity is done floating around. For example, even otherwise very smart folks feel frustrated if they cannot solve something in a day.
I agree with most of your diagnosis in this section—but I still disagree with your prescription. When I was working through my graduate electrodynamics course that used Jackson, I found that, empirically, it took about 10 hours for me to complete a set of homework problems, and each day I could do about 2 hours of work before I ran out of motivation/energy (which would only be replaced by sleep and time). Basically every other class I’ve taken I could solve the homework problems in one session (i.e. I don’t recall systematically running out of energy midway). So I said “alright, if I want to get this weekly homework done on time, I need to work on it for five days and only take two off,” and that’s what I did.
(Incidentally, this is how I know I’m not as clever at solving physics problems as Stephen Hawking—as I recall it, his classmates told the story of how they were sitting down in the common room to work Jackson problems for the first time, and not being able to do them after a few hours of focused effort was a rude shock, and then Hawking came down the stairs from his room. They asked how he was doing on them, and he said “well, I got the first six, but the seventh one is giving me a bit of trouble.”)
It seems to me that viewing my ability to do Jackson problems as a measurable quantity that I can discover and use in planning is helpful. It also seems that trying to nudge the numbers involved (could I solve all of them in only eight hours? How about working 2.5 hours a day, rather than just 2?) is way less useful that arranging other things around those numbers (my old homework schedule won’t work; let’s design a new one that will, and if I can’t find a homework schedule that works, maybe I shouldn’t expect to pass electrodynamics).
It also seems to me that being able to measure my ability and put it as a number on the real line (or vector in R^n) helps break out of discrete categories—rather than just “good at physics” and “bad at physics,” I can observe that I’m about a 10 on the Jackson scale, and a 10 is way more useful and precise than “good” or “bad,” especially if I’m prone to falling into the trap of using myself as the boundary between “good” and “bad.” Among other things, I can schedule my week around a 10; I can’t schedule my week around a “not as good as Stephen Hawking.”
Imagine really trying to explain to your artist friends about correlation and causation, or about why Bayes theorem implies a positive test result for a nasty disease probably means a false positive if the disease is rare enough, or (God forbid..) the Monty Hall problem, etc.
So, some of this is doable with other modules in the brain; the Wason Selection Task is the famous example. A lot of the instrumental rationality things, like WOOP or self-reflection about emotions, seem possible to teach and useful. Monty Hall seems really hard—not to get it right if they’ve got pencil and paper and know how to turn the crank, but to look at the problem and say “this looks easy but is in fact a hard problem! I should get out my pencil and paper.”
I actually think we don’t disagree on anything important (e.g. goals, or easy to verify statements), just on hard empirical matters. Which is a good place for a disagreement to be!
I’m not clear if we’re disagreeing on whether or not (1) the elasticity model is accurate, (2) it’s useful to believe it, or (3) it’s useful to promote it. (Obviously, those aren’t exclusive options.) I’m interpreting eli_sennesh as claiming that even if the elasticity model is true, believing in it is demoralizing and thus will reduce one’s elasticity, and I’m interpreting the first paragraph of this comment of yours in the same way, but I’m also interpreting your comment here as mostly agreeing with (1).
I think how math elasticity is distributed in the general population is an open empirical question. It could be most people will get poor returns on effort wrt math, or it could be we are very bad at teaching math, and making math non-scary to attempt.
Regardless of what the distribution of elasticity is, if you are interested in rationality, you need to be able to push yourself a bit on certain math topics if you want real engagement. I don’t think there is a way around it. So, e.g. I disagree w/ Cyan above where he claims Scott is not being mathy when he’s engaging with rationality. I think Scott absolutely is being mathematic, it just does not look like it because there are no outward signs of “wizarding stuff being done” e.g. scary notation.
[ Random fantasy aside, if you read Scott Bakker, compare to how his Gnostic magic works, there is the audible part, the utteral, and the inaudible part, the inutteral. The inutteral is hard to explain, it is the correct habit of thought to make the magic work. Without the inutteral the spell always fails. My view of math is like this: thinking “in the right way” is the inutteral, the notation/formalization, etc. is the utteral. ]
I worry that all this talk about “top 200 mathematicians” is thinly disguised status talk.
For example, it was taken as given that technique is what’s important (this is common in pure math, and also in theoretical CS). But often conceptual insight is important. Or actually doing the technical proof using a mostly understood approach. There is a lot of heterogeneity in (a) how mathematicians think and (b) what various mathematicians are good at (see e.g. hedgehogs vs foxes). Thinking hard about what sort of contribution is really the most important just feels like status anxiety to me. Let a thousand flowers bloom, I say!
Thinking hard about what sort of contribution is really the most important just feels like status anxiety to me.
I agree that this is mostly status games, and “importance” is not a useful measure relative to, say, marginal value. (That is, I assume most people think of “importance” in terms of, say, “average value,” but the average value of a thing does not tell you whether your current level should increase, decrease, or stay the same.) Find the best niche for you in the market / ecosystem, and make profits/contributions, and only worry about the rest to the extent that it helps you find a better niche.
Re: “value” I am not sure how to think about mathematics in consequentialist terms (and I am not a huge fan of consequentialism in general). The worry is that we should all stop doing math and start working on online ads or something.
I agree with the niche comment as a practical hack given our inability to predict the future.
I’m interpreting eli_sennesh as claiming that even if the elasticity model is true, believing in it is demoralizing and thus will reduce one’s elasticity,
Actually, I think both personal capacity/elasticity and morale/drive are more like real numbers than like booleans, and that since they’re both factors in your actually doing/learning math, there seems to me (just based on observation) to be a very large range of values for the two variables where you can leverage one to make up for a lack of the other in order to get more math done.
I also think that the elasticity/ability variable does not dictate a hard limit on how much math you can learn, but instead on how quickly you learn it.
To pin myself down to concrete predictions, I don’t think, for instance, that most people are incapable of learning, say, beginning multivariable calculus (without any serious analysis, the engineers’ version). I do think that many people have so little innate ability that they cannot learn it quickly enough to pass a math, science, or engineering degree in four years. I just think they can probably learn the material if they repeat certain courses twice over and wind up taking seven years. What we normally consider being Bad At Math simply means being so slow to learn math that it would take decades to learn what more talented students can digest in mere years.
Neither innate ability nor drive, in my view, draws a hard “line in the sand” that dictates an impassable limit. They simply dictate where the “price curve” of pedagogical resource trade-offs will fall; we can always educate more people further if we have the resources to invest and can expect a positive return.
(To further pin myself to concrete experience, I have a dyslexic friend from undergrad who faced exactly this trouble with programming. Because the university and his family knew he was dyslexic, he was allowed to take five years to finish his undergraduate degree in Computer Science, and he used quite a lot of personal grit and drive to ensure he studied enough to pass in those five years. If I recall, he came out with just about a 3.10/4.00 GPA, or somewhere thereabouts—not excellent but respectable. Today he works as a software engineer for Cisco and earns a healthy salary, because the university and his parents decided the extra resources were worth allowing/investing to let him learn the fundamentals at the pace his studying efforts could carry him.)
I agree with most of your diagnosis in this section—but I still disagree with your prescription. When I was working through my graduate electrodynamics course that used Jackson, I found that, empirically, it took about 10 hours for me to complete a set of homework problems, and each day I could do about 2 hours of work before I ran out of motivation/energy (which would only be replaced by sleep and time). Basically every other class I’ve taken I could solve the homework problems in one session (i.e. I don’t recall systematically running out of energy midway). So I said “alright, if I want to get this weekly homework done on time, I need to work on it for five days and only take two off,” and that’s what I did.
(Incidentally, this is how I know I’m not as clever at solving physics problems as Stephen Hawking—as I recall it, his classmates told the story of how they were sitting down in the common room to work Jackson problems for the first time, and not being able to do them after a few hours of focused effort was a rude shock, and then Hawking came down the stairs from his room. They asked how he was doing on them, and he said “well, I got the first six, but the seventh one is giving me a bit of trouble.”)
It seems to me that viewing my ability to do Jackson problems as a measurable quantity that I can discover and use in planning is helpful. It also seems that trying to nudge the numbers involved (could I solve all of them in only eight hours? How about working 2.5 hours a day, rather than just 2?) is way less useful that arranging other things around those numbers (my old homework schedule won’t work; let’s design a new one that will, and if I can’t find a homework schedule that works, maybe I shouldn’t expect to pass electrodynamics).
It also seems to me that being able to measure my ability and put it as a number on the real line (or vector in R^n) helps break out of discrete categories—rather than just “good at physics” and “bad at physics,” I can observe that I’m about a 10 on the Jackson scale, and a 10 is way more useful and precise than “good” or “bad,” especially if I’m prone to falling into the trap of using myself as the boundary between “good” and “bad.” Among other things, I can schedule my week around a 10; I can’t schedule my week around a “not as good as Stephen Hawking.”
So, some of this is doable with other modules in the brain; the Wason Selection Task is the famous example. A lot of the instrumental rationality things, like WOOP or self-reflection about emotions, seem possible to teach and useful. Monty Hall seems really hard—not to get it right if they’ve got pencil and paper and know how to turn the crank, but to look at the problem and say “this looks easy but is in fact a hard problem! I should get out my pencil and paper.”
I’m not clear if we’re disagreeing on whether or not (1) the elasticity model is accurate, (2) it’s useful to believe it, or (3) it’s useful to promote it. (Obviously, those aren’t exclusive options.) I’m interpreting eli_sennesh as claiming that even if the elasticity model is true, believing in it is demoralizing and thus will reduce one’s elasticity, and I’m interpreting the first paragraph of this comment of yours in the same way, but I’m also interpreting your comment here as mostly agreeing with (1).
I think how math elasticity is distributed in the general population is an open empirical question. It could be most people will get poor returns on effort wrt math, or it could be we are very bad at teaching math, and making math non-scary to attempt.
Regardless of what the distribution of elasticity is, if you are interested in rationality, you need to be able to push yourself a bit on certain math topics if you want real engagement. I don’t think there is a way around it. So, e.g. I disagree w/ Cyan above where he claims Scott is not being mathy when he’s engaging with rationality. I think Scott absolutely is being mathematic, it just does not look like it because there are no outward signs of “wizarding stuff being done” e.g. scary notation.
[ Random fantasy aside, if you read Scott Bakker, compare to how his Gnostic magic works, there is the audible part, the utteral, and the inaudible part, the inutteral. The inutteral is hard to explain, it is the correct habit of thought to make the magic work. Without the inutteral the spell always fails. My view of math is like this: thinking “in the right way” is the inutteral, the notation/formalization, etc. is the utteral. ]
I worry that all this talk about “top 200 mathematicians” is thinly disguised status talk.
For example, it was taken as given that technique is what’s important (this is common in pure math, and also in theoretical CS). But often conceptual insight is important. Or actually doing the technical proof using a mostly understood approach. There is a lot of heterogeneity in (a) how mathematicians think and (b) what various mathematicians are good at (see e.g. hedgehogs vs foxes). Thinking hard about what sort of contribution is really the most important just feels like status anxiety to me. Let a thousand flowers bloom, I say!
I agree that this is mostly status games, and “importance” is not a useful measure relative to, say, marginal value. (That is, I assume most people think of “importance” in terms of, say, “average value,” but the average value of a thing does not tell you whether your current level should increase, decrease, or stay the same.) Find the best niche for you in the market / ecosystem, and make profits/contributions, and only worry about the rest to the extent that it helps you find a better niche.
Re: “value” I am not sure how to think about mathematics in consequentialist terms (and I am not a huge fan of consequentialism in general). The worry is that we should all stop doing math and start working on online ads or something.
I agree with the niche comment as a practical hack given our inability to predict the future.
Actually, I think both personal capacity/elasticity and morale/drive are more like real numbers than like booleans, and that since they’re both factors in your actually doing/learning math, there seems to me (just based on observation) to be a very large range of values for the two variables where you can leverage one to make up for a lack of the other in order to get more math done.
I also think that the elasticity/ability variable does not dictate a hard limit on how much math you can learn, but instead on how quickly you learn it.
To pin myself down to concrete predictions, I don’t think, for instance, that most people are incapable of learning, say, beginning multivariable calculus (without any serious analysis, the engineers’ version). I do think that many people have so little innate ability that they cannot learn it quickly enough to pass a math, science, or engineering degree in four years. I just think they can probably learn the material if they repeat certain courses twice over and wind up taking seven years. What we normally consider being Bad At Math simply means being so slow to learn math that it would take decades to learn what more talented students can digest in mere years.
Neither innate ability nor drive, in my view, draws a hard “line in the sand” that dictates an impassable limit. They simply dictate where the “price curve” of pedagogical resource trade-offs will fall; we can always educate more people further if we have the resources to invest and can expect a positive return.
(To further pin myself to concrete experience, I have a dyslexic friend from undergrad who faced exactly this trouble with programming. Because the university and his family knew he was dyslexic, he was allowed to take five years to finish his undergraduate degree in Computer Science, and he used quite a lot of personal grit and drive to ensure he studied enough to pass in those five years. If I recall, he came out with just about a 3.10/4.00 GPA, or somewhere thereabouts—not excellent but respectable. Today he works as a software engineer for Cisco and earns a healthy salary, because the university and his parents decided the extra resources were worth allowing/investing to let him learn the fundamentals at the pace his studying efforts could carry him.)