I have no insight to offer here but I would just like to say a very very interesting post.
Thanks!
I had no idea this was the case. Again, nor am I in the Grothendieck category, but I am very uneven in abilities too. I thought I was an exception in that regard.
Yvain / Scott Alexander is another example. See section 2 of his post The Parable Of Talents. I agree with most of what he says and find his post is quite insightful. But I think that his assessment of his mathematical ability is probably wrong, even though him struggling to get a C- in calculus probably reflects some sort of innate difference between his classmates. In fact, observing this was one of the proximate causes for me writing on the nature of mathematical ability.
Thanks for writing this post, and specifically for trying to change Scott’s mind. Scott’s complaints about his math abilities often go like this:
“Man, I wish I wasn’t so terrible at math. Now if you will excuse me, I am going to tear the statistical methodology in this paper to pieces.”
Put me in as yet another “clearly not in the genius category” person in a somewhat mathy area awaiting the rest of this series. I think a lot about what “mathematical sophistication” is, I am curious what your conclusions are.
I think mathematical sophistication gets you a lot of what is called “rationality skills” here for free, basically.
Scott’s technique for shredding papers’ conclusions seem to me to consist mostly of finding alternative stories that account for the data and that the authors have overlooked or downplayed. That’s not really a math thing, and it plays right to his strengths.
I actually disagree that having a good intuitive grasp of “stories” of this type is not a math thing, or a part of the descriptive statistics magisterium (unless you think graphical models are descriptive statistics). “Oh but maybe there is confounder X” quickly becomes a maze of twisty passages where it is easy to get lost.
“Math things” is thinking carefully.
I think equating lots of derivation mistakes or whatever with poor math ability is: (a) toxic and (b) wrong. I think the innate ability/genius model of successful mathematicians is (a) toxic and (b) wrong. I further think that a better model for a successful mathematician is someone who is past a certain innate ability threshold who has the drive to keep going and the morale to not give up. To reiterate, I believe for most folks who post here the dominating term is drive and morale, not ability (of course drive and morale are also partly hereditary).
I have the sort of math skills that Scott claims to lack. I lack his skill at writing, and I stand in awe (and envy) at how far Scott’s variety of intelligence takes him down the path of rationality. I currently believe that the sort of reasoning he does (which does require careful thinking) does not cluster with mathy things in intelligence-space.
Look at his latest post: “hey wait a second, there is bias by censoring!” The “hard/conceptual part” is structuring the problem in the right way to notice something is wrong, the “bookkeeping” part is e.g. Kaplan-Meier / censoring-adjustment-via-truncation.
I think equating lots of derivation mistakes or whatever with poor math ability is: (a) toxic and (b) wrong. I think the innate ability/genius model of successful mathematicians is (a) toxic and (b) wrong.
Obviously, everyone will do best if they work as hard as they can, and innate talent models can prevent people from working harder. But innate talent models are really useful at avoiding wasted effort and frustration, and so I think it’s easier to make the case for the toxicity and wrongness of the inverse of the models you’re describing.
In particular, I think Scott’s easy response is to say “what happens when you put drive and morale into the ability bucket along with calculation ability?” (You bring up that possibility a bit with the hereditary section.) Given that Scott doesn’t seem to have the drive or the morale, then you either need to claim to him that he’s misdiagnosed his drive to do mathematics, which seems toxic and wrong,* or to say “yep, not a mathematician.”
Also, I do work both as a research mathematician and as a industrial mathematician, and while you might be right about research mathematics, I think that equating a lot of derivation mistakes with poor industrial math ability is healthy and right. I don’t want to hand off an analysis problem to someone who makes derivation mistakes, because the results can’t be trusted, and I wouldn’t want my accountant to be bad at calculation, and so on.
*Scott’s made the comparison to sexual orientation before, and I think it fits; just like a gay guy would get fed up with people saying “you just haven’t met the right girl yet!”, someone who does not have the drive or talent for mathematics will get fed up with people saying “you just haven’t met the right theorem yet!”
But innate talent models are really useful at avoiding wasted effort and frustration.
I think that equating a lot of derivation mistakes with poor industrial math ability is healthy and right.
I think this is how this conversation is playing out:
“I think there are wizards and muggles (from birth). Wizards can do magic because they are born with magic-sensitivity, muggles cannot because they are not born magic-sensitive.”
“I think there is some amount of magic sensitivity you need, but a lot of it is just general psych factors which are partly hereditary but which you can partly control. In particular, miscasting spells sometimes isn’t evidence you are a bad wizard. WIzards can be good at lots of other types of magic stuff.”
“Ok maybe for certain types of wizards that’s true, but you know, in industrial settings your wand-work needs to be bulletproof. Also, you don’t want to give muggles false hope lest they waste a lot of time at Hogwarts.”
It seems to me you just like the muggle/wizard worldview, and I don’t. I view the “raising the sanity waterline” project as related to the “raising average math proficiency” project. I think on net, people should be less scared of math, not more, and shouldn’t be afraid to dip their toes in this or that even if they are not going to do cutting edge work, necessarily. Math is infinite and eventually everyone gets confused and lost, even the best among us.
The big difference between the wizard/muggle model and the drive/morale model is that with the latter you can decide your tradeoffs regarding efforts in a magic-wardly direction (magic is pretty useful!). You might not want to spend 10 years in a wizarding school, but you may take the time to familiarize yourself with classes of spells, be able to read magical literature of some types, etc. With the former you just write yourself off as a muggle.
It seems to me you just like the muggle/wizard worldview, and I don’t.
I feel like “just like” is about half right; I think it’s much more plausible than alternatives, and I get that you don’t think it’s much more plausible than alternatives. But I also think that whether or not we should promote a view publicly depends a lot on the impacts of that view. In particular:
I view the “raising the sanity waterline” project as related to the “raising average math proficiency” project. I think on net, people should be less scared of math, not more, and shouldn’t be afraid to dip their toes in this or that even if they are not going to do cutting edge work, necessarily.
I think the main empirical difference between the two models is the increase in ability we expect to see from a marginal increase in expended effort. (I’ll call this ‘elasticity.’) The wizard/muggle model thinks that elasticity varies heavily from person to person, and that some people have very high elasticities and other people have very low elasticities. The drive/morale model thinks that people have comparable elasticities, and the input is the main thing that matters.
Let’s move from Scott to an example I feel a bit more strongly about: I have artist friends who sometimes have difficulty adding single-digit numbers. I wish they were less scared of math, and I suspect a large part of their difficulty with arithmetic is trauma from subpar math education being inflicted on them before they were ready for it—but my goal is to get them up to the point where they are comfortable with the sorts of numbers they deal with in their professional work, which is basically just arithmetic. Maybe they could go further, with heroic effort, but that doesn’t seem worth trying.
I agree with Scott that you are making the exact wrong prediction about the impacts of the wizard/muggle model vs. the drive/morale model on motivation and self-worth (and then eventual ability). I see the wizard/muggle model as allowing radical self-acceptance: it is okay for my friends to have difficulty adding single-digit numbers because that’s where their competence level is, and it is okay for it to be a major project for them to erase that difficulty, because that’s where their elasticity is. And if they move at their pace with their needs taken care of, then they might be willing to dip their toes in more and more, and regain the sense of play that makes learning fun. How can I go to them with the same drive/morale model that they’ve been cudgeled with for decades and say “but if you just tried harder, you too could be where I was when I was five!”, when it is neither kind nor true nor conducive to play or seeking?
I see the wizard/muggle model as allowing radical self-acceptance.
How can I go to them with the same drive/morale model that they’ve been cudgeled with for decades …
I agree that a set of folks just isn’t great at math. However, among the LW-posting crowd in particular (the set I am concerned with), while there are indeed some folks who aren’t great at math, I suspect more often than not there are morale (partly due to shitty math edu) and akrasia (partly due to untrained human’s issues maintaining a sustained line of effort in any direction) issues instead.
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. But doing non-trivial math, in my experience, takes months, possibly years, of work. If getting a non-trivial result out takes that long, you need to be able to: (a) not write yourself off as dimwitted and slow, even in the face of there being people who might be able to deal with what you are working on faster, perhaps much faster, and (b) maintain the motivation to keep banging on the problem.
These are non-trivial “psych skills” that I think are needed for doing serious math for people who aren’t Terry Tao. Note: I am not implying that Terry Tao magically spits out results in a day, there is every indication that he works very very hard (it is just that he is so smart he moves up to the frontier of what is possible for him). I just think that the existence of people like Tao doesn’t imply other folks cannot make meaningful contributions, they just need to learn to work in an environment that has Taos of the world running around.
Leaving “being a mathematician” aside, I think if you really aren’t good at math, “rationality” probably isn’t for you, it all sits on math in the end. If you really cannot grasp certain basics, you cannot have more than a religious engagement with rationality. Note: I don’t mean you must be able to do novel work in mathematics, I mean you have serious issues thinking mathematically at all.
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.
More of this sort of engagement (engagement on the margin for folks who might be capable of a bit more math thinking but for akrasia/morale issues) is what I feel the goal of mathematician outreach should be.
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 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.)
what happens when you put drive and morale into the ability bucket along with calculation ability?
I must protest: absolutely nothing happens. Real limitations on your drive and morale don’t feel from the inside like “running out of hit points”, they feel like akrasia. So as long as you can choose, from the inside, to keep going, you haven’t run out of drive yet.
Real limitations on your drive and morale don’t feel from the inside like “running out of hit points”, they feel like akrasia.
I am reluctant to generalize to the internal experience of others, especially if the difference between my and someone else’s internal experience are the cause of those externally observable drive and morale differences.
So as long as you can choose, from the inside, to keep going, you haven’t run out of drive yet.
I feel like this is contentless. Suppose you and I are observing a third person, and they do not keep going. Can either of us state whether or not they could choose to keep going from the inside?
Now, if you are struggling, and say to yourself “this sucks, but I will keep going,” and you do keep going for longer having said that to yourself, then great! As you point out, that’s part of your overall drive, and so is irrelevant once we step outside how you get the drive you have and are instead quantifying how much drive you have.
Now, if you are struggling, and say to yourself “this sucks, but I will keep going,” and you do keep going for longer having said that to yourself, then great! As you point out, that’s part of your overall drive, and so is irrelevant once we step outside how you get the drive you have and are instead quantifying how much drive you have.
Unfortunately, the key word there is outside: reasoning about some other system than one’s self encounters no Loebian obstacles. Reasoning about one’s self usually tends to involve reasoning about one’s self-model, which is actually a necessarily less accurate description of your state of being than the raw data of how things feel-from-the-inside to Objectively Be.
When you are going “UUUUGGGGGH, THIS IS FUCKING IMPOSSIBLE GODDAMNIT!” you may have hit your limit for the time being. When you are frustrated and thinking, “I probably just have limited ability at maths”, that’s just anxiety.
When you are frustrated and thinking, “I probably just have limited ability at maths”, that’s just anxiety.
I’m not sure I buy this, though. If you view the ability as learning ability, or what I call elasticity over here, then it seems like I can say “I find it more difficult than I expect to learn a foreign language; I’ll downgrade my expected elasticity for foreign languages” and that might switch the EV of spending more effort on learning foreign languages from positive to negative. If there are multiple things I could do, and which thing is wisest depends on the relative elasticities, then trying to estimate those elasticities seems useful and doable without hitting the hard limit.
“Thinking carefully” is necessary but not sufficient for “math things”.
a better model for a successful mathematician is someone who is past a certain innate ability threshold who has the drive to keep going and the morale to not give up
I don’t know about that—there are opportunity costs. Let’s say you’re smart, and conscientious, and have good analytical skills, etc., but not particularly good at math. Yes, you can probably make a passable mathematician if you persevere and sink a lot of time and effort into learning math. But since math is not your strong point, you probably would have made a better X (social scientist, hedge manager, biologist, etc.) with a lot less effort and frustration. Thus going for math would be a losing move.
But I think that his assessment of his mathematical ability is probably wrong, even though him struggling to get a C- in calculus probably reflects some sort of innate difference between his classmates.
Anecdata: I got an A in Calculus 1, a C+ in Calculus 2, and an A- in Calculus 3. Of them all, Calculus 2 seemed to be the most focused on “memorize this bunch of unjustified heuristics”, and Calculus 3 was one of the first and only times I really experienced the Wonders of Math in an actual course.
Oh, and for further anecdata, without being able to convert to letter grades, I got a 75% in Statistics 1 and failed (post-grad level) Intro to Machine Learning last year due to taking the courses without the continuous probability-theory prereq, and then retook Machine Learning this year to get an 86%.
It seems to me that a lot of variation in math grades can be very easily explained by differences in previous preparation.
As someone whose day-job largely consists of teaching Calculus 1, 2, and 3, I heartily with you about what they are like! If I could redesign the curriculum from scratch, Calculus 3 would definitely come before Calculus 2 (for the most part), and far fewer people would be required to ever take Calculus 2 at all.
ETA: I’m talking about the curriculum in most colleges in the U.S., so I hope that you are too; other countries’ curricula can vary a lot.
Actually, yeah, requiring more people to take Multivariate Calculus and fewer people to take Assorted Sequence/Series and Integration Heuristics sounds like a fine idea.
Thanks!
Yvain / Scott Alexander is another example. See section 2 of his post The Parable Of Talents. I agree with most of what he says and find his post is quite insightful. But I think that his assessment of his mathematical ability is probably wrong, even though him struggling to get a C- in calculus probably reflects some sort of innate difference between his classmates. In fact, observing this was one of the proximate causes for me writing on the nature of mathematical ability.
Thanks for writing this post, and specifically for trying to change Scott’s mind. Scott’s complaints about his math abilities often go like this:
“Man, I wish I wasn’t so terrible at math. Now if you will excuse me, I am going to tear the statistical methodology in this paper to pieces.”
Put me in as yet another “clearly not in the genius category” person in a somewhat mathy area awaiting the rest of this series. I think a lot about what “mathematical sophistication” is, I am curious what your conclusions are.
I think mathematical sophistication gets you a lot of what is called “rationality skills” here for free, basically.
Scott’s technique for shredding papers’ conclusions seem to me to consist mostly of finding alternative stories that account for the data and that the authors have overlooked or downplayed. That’s not really a math thing, and it plays right to his strengths.
Causal stories in particular.
I actually disagree that having a good intuitive grasp of “stories” of this type is not a math thing, or a part of the descriptive statistics magisterium (unless you think graphical models are descriptive statistics). “Oh but maybe there is confounder X” quickly becomes a maze of twisty passages where it is easy to get lost.
“Math things” is thinking carefully.
I think equating lots of derivation mistakes or whatever with poor math ability is: (a) toxic and (b) wrong. I think the innate ability/genius model of successful mathematicians is (a) toxic and (b) wrong. I further think that a better model for a successful mathematician is someone who is past a certain innate ability threshold who has the drive to keep going and the morale to not give up. To reiterate, I believe for most folks who post here the dominating term is drive and morale, not ability (of course drive and morale are also partly hereditary).
I have the sort of math skills that Scott claims to lack. I lack his skill at writing, and I stand in awe (and envy) at how far Scott’s variety of intelligence takes him down the path of rationality. I currently believe that the sort of reasoning he does (which does require careful thinking) does not cluster with mathy things in intelligence-space.
Look at his latest post: “hey wait a second, there is bias by censoring!” The “hard/conceptual part” is structuring the problem in the right way to notice something is wrong, the “bookkeeping” part is e.g. Kaplan-Meier / censoring-adjustment-via-truncation.
I don’t disagree with this. A lot of the kind of math Scott lacks is just rather complicated bookkeeping.
(Apropos of nothing, the work “bookkeeping” has the unusual property of containing three consecutive sets of doubled letters: oo,kk,ee.)
Obviously, everyone will do best if they work as hard as they can, and innate talent models can prevent people from working harder. But innate talent models are really useful at avoiding wasted effort and frustration, and so I think it’s easier to make the case for the toxicity and wrongness of the inverse of the models you’re describing.
In particular, I think Scott’s easy response is to say “what happens when you put drive and morale into the ability bucket along with calculation ability?” (You bring up that possibility a bit with the hereditary section.) Given that Scott doesn’t seem to have the drive or the morale, then you either need to claim to him that he’s misdiagnosed his drive to do mathematics, which seems toxic and wrong,* or to say “yep, not a mathematician.”
Also, I do work both as a research mathematician and as a industrial mathematician, and while you might be right about research mathematics, I think that equating a lot of derivation mistakes with poor industrial math ability is healthy and right. I don’t want to hand off an analysis problem to someone who makes derivation mistakes, because the results can’t be trusted, and I wouldn’t want my accountant to be bad at calculation, and so on.
*Scott’s made the comparison to sexual orientation before, and I think it fits; just like a gay guy would get fed up with people saying “you just haven’t met the right girl yet!”, someone who does not have the drive or talent for mathematics will get fed up with people saying “you just haven’t met the right theorem yet!”
I think this is how this conversation is playing out:
“I think there are wizards and muggles (from birth). Wizards can do magic because they are born with magic-sensitivity, muggles cannot because they are not born magic-sensitive.”
“I think there is some amount of magic sensitivity you need, but a lot of it is just general psych factors which are partly hereditary but which you can partly control. In particular, miscasting spells sometimes isn’t evidence you are a bad wizard. WIzards can be good at lots of other types of magic stuff.”
“Ok maybe for certain types of wizards that’s true, but you know, in industrial settings your wand-work needs to be bulletproof. Also, you don’t want to give muggles false hope lest they waste a lot of time at Hogwarts.”
It seems to me you just like the muggle/wizard worldview, and I don’t. I view the “raising the sanity waterline” project as related to the “raising average math proficiency” project. I think on net, people should be less scared of math, not more, and shouldn’t be afraid to dip their toes in this or that even if they are not going to do cutting edge work, necessarily. Math is infinite and eventually everyone gets confused and lost, even the best among us.
The big difference between the wizard/muggle model and the drive/morale model is that with the latter you can decide your tradeoffs regarding efforts in a magic-wardly direction (magic is pretty useful!). You might not want to spend 10 years in a wizarding school, but you may take the time to familiarize yourself with classes of spells, be able to read magical literature of some types, etc. With the former you just write yourself off as a muggle.
I feel like “just like” is about half right; I think it’s much more plausible than alternatives, and I get that you don’t think it’s much more plausible than alternatives. But I also think that whether or not we should promote a view publicly depends a lot on the impacts of that view. In particular:
I think the main empirical difference between the two models is the increase in ability we expect to see from a marginal increase in expended effort. (I’ll call this ‘elasticity.’) The wizard/muggle model thinks that elasticity varies heavily from person to person, and that some people have very high elasticities and other people have very low elasticities. The drive/morale model thinks that people have comparable elasticities, and the input is the main thing that matters.
Let’s move from Scott to an example I feel a bit more strongly about: I have artist friends who sometimes have difficulty adding single-digit numbers. I wish they were less scared of math, and I suspect a large part of their difficulty with arithmetic is trauma from subpar math education being inflicted on them before they were ready for it—but my goal is to get them up to the point where they are comfortable with the sorts of numbers they deal with in their professional work, which is basically just arithmetic. Maybe they could go further, with heroic effort, but that doesn’t seem worth trying.
I agree with Scott that you are making the exact wrong prediction about the impacts of the wizard/muggle model vs. the drive/morale model on motivation and self-worth (and then eventual ability). I see the wizard/muggle model as allowing radical self-acceptance: it is okay for my friends to have difficulty adding single-digit numbers because that’s where their competence level is, and it is okay for it to be a major project for them to erase that difficulty, because that’s where their elasticity is. And if they move at their pace with their needs taken care of, then they might be willing to dip their toes in more and more, and regain the sense of play that makes learning fun. How can I go to them with the same drive/morale model that they’ve been cudgeled with for decades and say “but if you just tried harder, you too could be where I was when I was five!”, when it is neither kind nor true nor conducive to play or seeking?
I agree that a set of folks just isn’t great at math. However, among the LW-posting crowd in particular (the set I am concerned with), while there are indeed some folks who aren’t great at math, I suspect more often than not there are morale (partly due to shitty math edu) and akrasia (partly due to untrained human’s issues maintaining a sustained line of effort in any direction) issues instead.
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. But doing non-trivial math, in my experience, takes months, possibly years, of work. If getting a non-trivial result out takes that long, you need to be able to: (a) not write yourself off as dimwitted and slow, even in the face of there being people who might be able to deal with what you are working on faster, perhaps much faster, and (b) maintain the motivation to keep banging on the problem.
These are non-trivial “psych skills” that I think are needed for doing serious math for people who aren’t Terry Tao. Note: I am not implying that Terry Tao magically spits out results in a day, there is every indication that he works very very hard (it is just that he is so smart he moves up to the frontier of what is possible for him). I just think that the existence of people like Tao doesn’t imply other folks cannot make meaningful contributions, they just need to learn to work in an environment that has Taos of the world running around.
Leaving “being a mathematician” aside, I think if you really aren’t good at math, “rationality” probably isn’t for you, it all sits on math in the end. If you really cannot grasp certain basics, you cannot have more than a religious engagement with rationality. Note: I don’t mean you must be able to do novel work in mathematics, I mean you have serious issues thinking mathematically at all.
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.
More of this sort of engagement (engagement on the margin for folks who might be capable of a bit more math thinking but for akrasia/morale issues) is what I feel the goal of mathematician outreach should be.
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 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.)
I must protest: absolutely nothing happens. Real limitations on your drive and morale don’t feel from the inside like “running out of hit points”, they feel like akrasia. So as long as you can choose, from the inside, to keep going, you haven’t run out of drive yet.
I am reluctant to generalize to the internal experience of others, especially if the difference between my and someone else’s internal experience are the cause of those externally observable drive and morale differences.
I feel like this is contentless. Suppose you and I are observing a third person, and they do not keep going. Can either of us state whether or not they could choose to keep going from the inside?
Now, if you are struggling, and say to yourself “this sucks, but I will keep going,” and you do keep going for longer having said that to yourself, then great! As you point out, that’s part of your overall drive, and so is irrelevant once we step outside how you get the drive you have and are instead quantifying how much drive you have.
Unfortunately, the key word there is outside: reasoning about some other system than one’s self encounters no Loebian obstacles. Reasoning about one’s self usually tends to involve reasoning about one’s self-model, which is actually a necessarily less accurate description of your state of being than the raw data of how things feel-from-the-inside to Objectively Be.
When you are going “UUUUGGGGGH, THIS IS FUCKING IMPOSSIBLE GODDAMNIT!” you may have hit your limit for the time being. When you are frustrated and thinking, “I probably just have limited ability at maths”, that’s just anxiety.
I’m not sure I buy this, though. If you view the ability as learning ability, or what I call elasticity over here, then it seems like I can say “I find it more difficult than I expect to learn a foreign language; I’ll downgrade my expected elasticity for foreign languages” and that might switch the EV of spending more effort on learning foreign languages from positive to negative. If there are multiple things I could do, and which thing is wisest depends on the relative elasticities, then trying to estimate those elasticities seems useful and doable without hitting the hard limit.
“Thinking carefully” is necessary but not sufficient for “math things”.
I don’t know about that—there are opportunity costs. Let’s say you’re smart, and conscientious, and have good analytical skills, etc., but not particularly good at math. Yes, you can probably make a passable mathematician if you persevere and sink a lot of time and effort into learning math. But since math is not your strong point, you probably would have made a better X (social scientist, hedge manager, biologist, etc.) with a lot less effort and frustration. Thus going for math would be a losing move.
And, of course, this.
Anecdata: I got an A in Calculus 1, a C+ in Calculus 2, and an A- in Calculus 3. Of them all, Calculus 2 seemed to be the most focused on “memorize this bunch of unjustified heuristics”, and Calculus 3 was one of the first and only times I really experienced the Wonders of Math in an actual course.
Oh, and for further anecdata, without being able to convert to letter grades, I got a 75% in Statistics 1 and failed (post-grad level) Intro to Machine Learning last year due to taking the courses without the continuous probability-theory prereq, and then retook Machine Learning this year to get an 86%.
It seems to me that a lot of variation in math grades can be very easily explained by differences in previous preparation.
As someone whose day-job largely consists of teaching Calculus 1, 2, and 3, I heartily with you about what they are like! If I could redesign the curriculum from scratch, Calculus 3 would definitely come before Calculus 2 (for the most part), and far fewer people would be required to ever take Calculus 2 at all.
ETA: I’m talking about the curriculum in most colleges in the U.S., so I hope that you are too; other countries’ curricula can vary a lot.
Calc 3 for me was Multivariate Calculus.
Actually, yeah, requiring more people to take Multivariate Calculus and fewer people to take Assorted Sequence/Series and Integration Heuristics sounds like a fine idea.
Yep, sounds like we’re talking about the same curriculum.