This post is a third installment to the sequence that I started with The Truth About Mathematical Ability and Innate Mathematical Ability. I begin to discuss the role of aesthetics in math. 

There was strong interest in the first two posts in my sequence, and I apologize for the long delay. The reason for it is that I've accumulated hundreds of pages of relevant material in draft form, and have struggled with how to organize such a large body of material. I still don't know what's best, but since people have been asking, I decided to continue posting on the subject, even if I don't have my thoughts as organized as I'd like. I'd greatly welcome and appreciate any comments, but I won't have time to respond to them individually, because I already have my hands full with putting my hundreds of pages of writing in public form.

Where I come from

My father is a remarkable creature, and I'm grateful for the opportunity to have grown up around him. Amongst other things, we share a love of music. There's a fair amount of overlap in our musical tastes. But there's an important difference between us.

When a piece of music is complex, like a piano sonata or a symphony, I often need to listen to it repeatedly before I figure out what I like about it. When I share the piece with him that he's never heard before, he'll often highlight the parts that I like most in real time, on first listening, without my having said anything.

In the past, people would have attributed this to magic, or other supernatural constructs like telepathy. We now know that these explanations don't suffice. 

You might hypothesize that the difference comes from him having greater abstract pattern recognition ability than my own. In fact, this is the case, but it doesn't suffice to account for the phenomenon. Some people with greater pattern recognition ability than me don't appreciate music at all. More significantly, my father doesn't figure out what I like by thinking about it – his reactions are instead emotionally rooted, for example, he broke into tears upon hearing the repetition of the original theme in the final movement of Beethoven's piano sonata Op. 109.

For whatever reason, my father's initial emotional responses are surprisingly often closely aligned with my eventual emotional responses than my own initial emotional responses are. They also seem to be more closely aligned with the average person's eventual emotional responses than my own initial emotional responses are. The phenomenon extends beyond music, into the visual arts and even math. It plays a role in his work as Art Director for the Wells Fargo website.

People are often surprised to learn that my IQ is about average for the Less Wrong community: they think that it you need to be a lot smarter to be as good at math as I am. They're not the only ones: a leading researcher in the field of exceptional intellectual talent expressed surprise that I was able to become a mathematician given that I have a nonverbal learning disability. 

When I hear people say these things I smile inwardly.

Math is an art

You see, there are broad misconceptions that math is about intelligence. No, math is an art. This isn't just true of pure math, it's also true of applied math, statistics, physics and computer science. Sufficiently high quality mathematical thinking of any kind has a large aesthetic component. My unusually high mathematical ability doesn't come me having higher intelligence than my conversation partners. It comes from me having unusually high aesthetic discernment, something that I acquired from my father, both out of virtue of inheriting his genes, and out of virtue of having him as a strong environmental influence in my life.

That's how I was able to go from failing geometry in 9th grade to being the best calculus student in my high school class of ~650 people. I was far from being the sharpest of my classmates, but my aesthetic sense drove me in the direction of rediscovering how to do mathematical research, and at that point it became easy for me to reconstruct any part of the high school math curriculum. I transcended the paradigm of "memorizing without understanding very well" to gain a deep conceptual understanding of the material, without needing outside assistance.

Just as levels of innate intelligence vary greatly, levels of innate aesthetic discernment vary greatly, and this has profound ramifications. Even if I were as smart as John von Neumann, I still wouldn't be able to discover the fast Fourier transform in the early 1800's like Gauss did: I don't have enough aesthetic discernment. This shouldn't be surprising – even though I have some musical talent, there's no way that I could write music as great as Beethoven's late string quartets

But if you're reading this post with interest, you've already distinguished yourself as somebody who can probably understand and appreciate math much more deeply than you would have imagined possible.

I understand that you may doubt me. The great mathematician Alexander Grothendieck understood too. He wrote to people in your position:

It's to that being inside of you who knows how to be alone, it is to this infant that I wish to speak, and no-one else. I'm well aware that this infant has been considerably estranged. It's been through some hard times, and more than once over a long period. It's been dropped off Lord knows where, and it can be very difficult to reach. One swears that it died ages ago, or that it never existed - and yet I am certain it's always there, and very much alive.

Is Scott Alexander "bad at math"?

In The Parable of The Talents Scott Alexander discusses his mathematical ability:

In Math, I just barely by the skin of my teeth scraped together a pass in Calculus with a C-. [...] Meanwhile, there were some students who did better than I did in Math with seemingly zero effort. I didn’t begrudge those students. But if they’d started trying to say they had exactly the same level of innate ability as I did, and the only difference was they were trying while I was slacking off, then I sure as hell would have begrudged them. Especially if I knew they were lazing around on the beach while I was poring over a textbook.

I don't doubt that Scott Alexander struggled to get a C- in calculus, and worked much harder than some other students. But almost surely, what he was seeing wasn't math in a meaningful sense. What he was seeing was more akin a course that teaches scales and chords to piano students. It's just not true that if someone has substantially more trouble learning scales and chords than his or her classmates, he or she is "worse than them at music." 

The signals of Scott's mathematical ability coming outside of formal math classes are much stronger. Some of these are fairly obvious — as Ilya Shpitser wrote:

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."

But these don't even constitute the main evidence that Scott Alexander is good at math. 

When a friend pointed out a couple of his blog posts back in early 2010, I did a double take, and thought "wow, this guy has something really special." I'm not alone: there's a broad consensus that he's a great writer, both within and outside of the Less Wrong community. Ezra Klein has been named one of the 50 most powerful people in Washington DC and he responded to one of Scott's blog posts.

A large part of what makes Scott's posts a pleasure to read is his storytelling ability, which overlaps strongly with the ability to write narrative fiction. There are hints that come across in the cultural references that he makes that he has a strong appreciation for art in general.

When I mentioned the unsolvability of quintic to Scott in passing, it grabbed his attention, and he was visibly very curious as to how it could be possible to show that a general quintic polynomial has no solutions in terms of radicals. It's the exact same reaction that my father has had to some of the deep math that I've showed him. There aren't very many mathematicians who have such a strong level of interest in the unsolvability of the quintic when they first encounter it.

What accounts for the difference? Like my father, Scott has exceptional aesthetic discernment. If most mathematicians had as much as he did, they would rightly find what I mentioned as striking as Scott did: the problem of showing that the quintic isn't solvable in radicals is what led to Galois Theory, one of the pinnacles of mathematical achievement, and the backdrop for the study of the Absolute Galois Group, one of the deepest areas of contemporary mathematical research.

People don't believe me when I tell them they're good at math!

When I try to convince people like Scott that they're actually very good at math, they often say "No, you don't understand, I'm really bad at math, you're overestimating my mathematical ability because of my writing ability." To which my response is "I know you think that, I've seen many people in your rough direction who think that they're really bad at math, and say that I don't understand how bad they are, and they're almost always wrong: they almost never know that what they were having trouble with wasn't representative of math."

I taught myself how to do mathematical research in order to understand calculus deeply. I've been thinking deeply about mathematical education for 12 years. I spent hundreds of hours tutoring students in calculus in high school and college. I taught calculus for 6 semesters at University of Illinois. I completed a PhD in math. Scott's exposure to calculus seems to consist of a single year in calculus. Your Bayesian prior should be that I know more about Scott's mathematical potential than Scott does. :-)

But so often I've seen people in Scott's position not believe me. By the time people have reached their mid-20's, they generally have such strong negative perceptions of their mathematical ability that I can't get through to them: their confirmation bias is too strong, there's nothing that I can do about the situation. So it may be that Scott will incorrectly think that he's bad at math forever, and that there's nothing that I can do about it. But maybe this article will influence at least someone's thinking.

I'll substantiate my claim that aesthetic sense drives a large fraction of mathematical accomplishment in future posts.

New Comment
223 comments, sorted by Click to highlight new comments since: Today at 7:32 AM
Some comments are truncated due to high volume. (⌘F to expand all)Change truncation settings

My terse summary of this post's argument is "I have good aesthetic sense and unremarkable calculation ability; I was able to translate my aesthetic sense into mathematical ability. Scott Alexander has great aesthetic sense, thus he should be able to translate that into mathematical ability, even in the presence of poor calculation ability."

I think this only works if you keep 'mathematics' a broad and vague category. Yes, Scott could probably do well in a group theory class--when I took it, I was surprised at how much of it would have been intelligible to a much younger me, whereas other math classes I took at around that time really did require linear algebra and calculus and so on as a foundation.

But being good at group theory is different from being good at "math" in general! If Scott's dream was to become an actuary or accountant, aesthetic sense would be irrelevant in the face of calculation ability. If you start with the goal (say, mastery of Bayesian probabilistic reasoning, or causality discovery, or so on) and then try to learn the math necessary to achieve that goal, it seems obvious to me that someone could be poorly matched to their goal, and possibly... (read more)

I think this only works if you keep 'mathematics' a broad and vague category.

I agree. But group theory isn't less "math" than actuarial math is!! I'd be happy with just dropping "math" as a term and renaming second semester calculus "computing integrals" or something. Then Scott could say that he's bad at computing integrals, rather than thinking that because he's bad at computing integrals, group theory must be way beyond him.

But since people call both calculus class and group theory "math," I need to respond to that.

So, I'll defer to Scott if he disagrees, but my impression is that he has substantially more trouble learning scales and chords than his brother, and that he is "worse than him at music." It might not be logically necessary, but we can certainly notice that it is probabilistically likely.

Yes, but I don't think that Scott's innately worse at music than most people who can easily pick up on scales and chords, the countervailing forces cutting in his favor are too strong.


My terse summary of this post's argument is "I have good aesthetic sense and unremarkable calculation ability; I was able to translate my aesthetic sense into mathematical ability. Scott Alexander has great aesthetic sense, thus he should be able to translate that into mathematical ability, even in the presence of poor calculation ability."

Well, actually, if the class Scott got a C- in was the famed Calculus 2: Sequences and Series and Integral Calculus, then I have to mention that I've heard from many people that they did terribly in that class, even when they went on to do quite well in other math courses. I myself got a C+ in that class, despite getting an A in Calculus 1, an A- in Multivariable Calculus, another A- in Linear Algebra, and generally somewhere from B to A in most math or theoretical CS classes I've ever taken, and even better marks in most programming-based CS courses I've ever taken.

That's before we get into JonahSinick's actual theory, which is that "verbal" general intelligence can be traded off with strictly calculative ability to get better at math even when one is mediocre (or "merely above average", a rather awful term if I... (read more)

Scott: I am bad at math.

Jonah: You are good at math.

Scott: No, I really am bad at math.

Jonah: No, you really are good at math.

Nisan: Esteemed colleagues, it is no use! If you continue this exchange, Scott will continue to believe they are bad at math, and Jonah will continue to disagree — forever!

Scott: Thank you for the information, but I still believe I am bad at math.

Jonah: And I still believe Scott is good at math.

Scott: And I still believe I am bad at math.

Nisan: Esteemed colleagues, give it up! Even if you persist in this exchange, neither of you will change your stated beliefs. In fact, I could truthfully repeat my previous sentence a hundred times (including the first time), and Scott would still believe they are bad at math, and Jonah would still disagree.

Scott: That's good to know, but for better or for worse, I still believe I am bad at math.

Jonah: And I still believe Scott is good at math.

Scott: Ah, but now I realize I am good at math after all!

Jonah: I agree, and what's more, I now know exactly how good at math Scott is!

Scott: And now I know that as well.

I totally wasn't expecting this and it made me chuckle aloud. Aaaand, peak-end rule, I now close out LW.

I am not sure for how many people it is true, but my own bad-at-mathness is largely about being bad at reading really terse, dense, succint text, because my mind is used to verbose text and thus filtering out half of it or not really paying close attention.

I hate the living guts out of notation, Greek variables or single-letter variables. Even the Bayes theorem is too terse, succint, too information-dense for me. I find it painful that in something like P(B|A) all three bloody letters mean a different thing. It is just too zipped. I would far more prefer something more natural langauge like Probability( If-True (Event1), Event2) (this looks like a software code - and for a reason).

This is actually a virtue when writing programs, I am never the guy who uses single letter variables, my programs are always like MarginPercentage = DivideWODivZeroError((SalesAmount-CostAmount), SalesAmount) * 100. So never too succint, clearly readable.

Let's stick to the Bayer Theorem. My brain is screaming don't give me P, A, B. Give me "proper words" like Probability, Event1, and Event2. So that my mind can read "Pro...", then zone out and rest while reading "bability" a... (read more)


Math notation is optimized for doing math, not learning math. Once you've internalized what P(A|B) is, you know what it means at a glance, and when you look at a large equation, you're more interested in the structure of the whole thing than the identities of it's constituents (Because abstracting the details away getting results based only on structure is what algebra is).

Perhaps math really needs multiple "programming languages". One for teaching, one for higher level stuff...
People that do novel math often invent their own notation -- it's sort of like domain-specific languages written on top of LISP. Teaching programming languages haven't proven to be a popular idea, e.g. MIT and Berkeley moved their famous introductory CS class from scheme to python (because python is actually used, even though it is a much less elegant language).
I think historically math started with longer variables, but it wasn't so convenient. Compare: 3x^2 + 4x + 1 = 0 with three times the square of a value, and four times the value, and one, equals nothing The latter may be easier to read, but the former is easier to divide by x+1.
0[anonymous]9y Also, being very frugal with token length seems to be a thing into the 1960's, see Unix e.g. "ls -l" instead of the far more human eye friendly "list -long" I don't exactly understand why but apparently this wasn't really a priority until about, say, 1995 when more and more programmers said fsck Perl with its unreadably frugal letter soup and use stuff like Python, where things are expressed in actual words. I guess there are good reasons behind it. I still don't have to like it.
To get the link use the following code in your comment: [\_(mathematics)#Genesis\_and\_evolution\_of\_the\_concept](\)#Genesis_and_evolution_of_the_concept) See Comment formatting/Escaping special symbols on the wiki for more details (I've backslash-escaped underscores _ in the text part of the link to avoid their turning surrounding texts into italics, and the closing round bracket in the URL part of the link to avoid its interpretation as the end of the URL).

Hm, interesting, I have an aversion to what I see as fluffy and low-info-density content, and I have a hard time pushing my brain in to "high gear" so I can just skim through it. I do think I can shift gears but it seems to take a few months to change my preferred reading mode.

It's interesting to speculate what math notation would be like if there were competing math notation schemes the same way there are competing programming languages; arguably math notation is terrible when judged by the standards of programming languages. reddit thread

I think this gear thing may be a strong difference between STEM and humanities orientation. (I work more or less STEM, but just to pay bills, I am more of a hobby historian and suchlike at heart.) Taking everything literally, and liking high-density content, while skimming fluff and focusing on intended meaning instead of literal meaning is more of a humanities thing. This is why I tend to insist that programming should not be called software engineering. Programs are written primarily for people to read, and only secondarily for computers to execute, and thus it floats somewhere in between STEM-type precision and humanities type good readable writing. In my experience programmers don't exactly have the highly precise, literalist minds of e.g. mechanical engineers. Nor the "there are multiple viewpoints" type of overly-fluffy humanities angle, but somewhere in between - something more like a craft than engineering or philosophy. BTW a sad reminder of how good Reddit used to be. Thanks. (This is entirely offtopic, but is there a way to stop this fluctuation of subcultures? Large websites count as subcultures the same way as musical styles count as one. Usually a subculture is started by more high-brow people and as it gets popularized and more low-brow people move in, the starters move on to the next one. I was strongly suspecting that the idea of subreddits may be the killer feature that stops it. Alas, not. even /r/insightfulquestions are insightful only on a high school debating club level. From another angle, it is not merely just an IQ based in and out migration, it is also age based.)
I agree strongly. IMO, this is a significant part of why tutoring works. The tutor "translates" the terse explanations into easier natural language ones. Edditionally, I think this is partly why the mathematics programs I've been exposed to tend to involve a lot of people hanging around each other socially, moreso than is common for other majors. I think even the mathematics majors mainly rely on translations from older students, at least until they progress past a certain point.
Most of the notation was developed for people doing math rather than learning it. If you're doing a lengthy calculation, you probably want to cut down on the amount of unnecessary symbols. With something like conditional probability, "P(A|B)" makes perfect sense if you're going to be writing some variant of it many times over the course of many pages. Or let's take the Einstein summation convention. Mathematics already had a compact notation for writing sums, but even this proved to be too cumbersome for the tastes of people doing differential geometry in coordinates. This led Einstein to introduce a convention wherein a repeated index denoted summation. As you pointed out, this tendency toward brevity leads to some very dense writing that can be difficult to interpret if you're not already used to it. It's good for working, bad for pedagogy. Compare shorthand writing. It's great for writing faster, but I can't imagine reading a novel written in shorthand.
That resembles what happens to me when I have to 'read some math'. I hated integrals and multivariate equations because solving them always took so long, and somewhere halfway through I invariably began thinking 'okay, we did this and then broken off this piece to pretend it's actually like that, though we'll have to weld it back on someday... is there really no meaning to what we've already done, except that in some distant future we'll write the final "="?..' In highschool physics, when they gave you an equation, you sure could use it as kind of a lever to turn the model Earth.
I'm sympathetic, something very similar is true of me. Thanks for making your thoughts explicit - it'll be helpful for me when I try to explain the situation to others.
I have ADHD and read a lot (and read more as a kid), so this definitely is interesting to me. Then again, I also like compression at the aesthetic level, but also find it quite difficult to learn/use. I too have to translate Bayes' Theorem! (I found the version I like best is where the letters are "O" "H" and "E" for observation, hypothesis, evidence.)
People do think of conditional probabilities that way, one just has to be careful not to conflate with p(IF event1 THEN event2), which is a probability of a disjunction, and quite different from p(event2 | event1). Conditioning is kind of weird, Simpsons paradox is a paradox because we are really bad at processing what conditioning actually means.
I find that what helps for me is re-writing maths as I'm learning it. When I glance at an equation or formula (especially an unfamiliar one), I usually can't take it in because my mind is trying to glance it all at once. I have to force myself to scan it slowly, either by re-writing it, writing out its definition, or by (holding a ruler under it) and scanning one symbol at a time. Then again, I'm currently studying a postgraduate degree in maths and I'm not someone who's ever considered themselves 'bad at math'.
I myself consider that a large degree of why I find myself to be bad at math is because I have spent very little time really trying to do math as a result of it being actually mentally painful to do due to this effect. This makes me sad, because even without accomplishment, I feel as if my reasoning ability is "merely above average" and I have no apparent way of leveraging that besides hacking that into making me seem more verbally intelligent (a lame result in my opinion). However I'm not a programmer either, yet.
I almost immediately start glazing over when I see some math in a text and it takes real effort to make myself pull it apart. I find that I almost always understand an algorithm when implemented in a programming language with much less effort than if I'm reading it in some mathematical notation.

I say everything I'm about to say as a person who is more certain than not that you have something valuable to contribute through this sequence, and who eagerly awaits more.

All of your posts in this sequence have purportedly been written to motivate your main thesis, but it's not clear to me what that is. I think you should stop motivating and very clearly reveal your Big Secret. What can I do right now to improve my mathematical ability? That's what I want to know.

Consider these points:

  • Eliezer tried to explain his metaethics the first time and failed, but it was okay because he was able to write his epistemology sequence after that and clarify. His posts pretty much always motivate people to continue reading because they usually have such high insight density and people know that they do; but your posts in this sequence, as far as I can tell, have just been anecdotes, quotes, nonstandard definitions, references detailing your nonstandard definitions, and promises of elucidation in future posts, and in your case, there isn't common knowledge of high insight density to make people trust you even when they don't understand where you're going. Your second post had less karma than yo

... (read more)

Let me take a stab at my (not OPs) views on math:

(a) A single number model of intelligence is toxic and silly. IQ is a single number proxy for a complex multidimensional space.

(b) Effective test taking has very little to do with math ability. Many excellent mathematicians are bad test takers (e.g. do not think quickly on their feet): this means basically nothing.

(c) Brains are complicated, and there is a huge amount of heterogeneity in how people process information and think about mathematics (and indeed all topics, but it is clearer in mathematics perhaps). Some are very visual, some are big on calculation.

(d) There is no separate magisterium called "math," there is a gently sloping continuum from common sense to "novel math work." When someone says "I am bad at math," I am not sure if they mean "I can't think carefully at all," "math notation scares me," "I can't think abstractly," [something else].

(e) If you haven't engaged with math beyond high school, you probably don't have enough information to evaluate the counterfactual "would a hypothetical me that pursued a math education make a good mathematician?&quo... (read more)

A data point for you: I am not particularly good at math. What this means is that at certain levels going forwards suddenly becomes much more difficult. I can continue, but slowly and only with a lot of effort. It's a slog. By comparison, I'm much better at logic/patterns and going deeper there is just easier. I do NOT mean that I can't think carefully or abstractly or that notation scares me. Note that I'm using a fairly narrow definition of math here. In particular, I distinguish math and statistics and believe that they require two different propensities. People good at math are rarely good at statistics; people good at statistics are rarely good at math.
I am not sure what exactly going deeper at logic/patterns means if not getting into mathematical logic. It is incredibly easy to read mathematics you know and incredibly difficult to read mathematics that you don't due to how dense it is. It might be the case that your impression is due to comparing these two. I am training to become a mathematician and I do not know of a single person for whom learning mathematics is not slowly and with a lot of effort, I do not think you are particularly exceptional in that but I know very little about your particular scenario.
I am not sure how to properly express the difference, but it has to do with math being more, um, hard-edged and rigid (in the sense of a rigid mechanical construction as opposed to a reed bending with the wind). To use a quote from Conan Doyle, a logic/patterns ability would allow one to "from a drop of water ... infer the possibility of an Atlantic or a Niagara without having seen or heard of one or the other." Mathematical logic cannot do that.
So something more like intuitive pattern-recognition/completion than rigorous formalization?
Definitely less rigorous formalization and more gestalt pattern recognition. In general, I think of math as dealing with well-defined "things" -- you may not know the shape/properties/characteristics at the moment, but they exist, they are precisely defined, and they are not going anywhere. In contrast to math, statistics deals with fuzzy amorphous "things" that you will likely never know in precise detail, that mutate as more data becomes available, and that usually require interpretation and/or some guessing to fill in the gaps.
Cutting edge math is actually mostly about converting fuzzy stuff, at least the parts of math I am interested in(Algebraic Geometry - Grothendieck/Weil for example). Both the mentioned mathematicians worked in a field where people had some stuff that worked but no foundations. Also, the foundations of math have been changing for quite a long time and continue to do so. I think your reaction to mathematics might be to badly taught mathematics rather than mathematics as practiced. However, I don't see an easy way to fix it. To teach mathematics well would require a high amount of mastery and we don't have enough people like that around.
I doubt it -- I generally teach myself things and just ignore bad instruction. The underlying cause is likely to be the curse of the gifted -- I'm lazy and when I run into walls I usually go around instead of starting a wall disassembly project. And I was never attracted to math sufficiently to apply a lot of effort.
Can you give an example of the level where things suddenly become more difficult? As I said in another post, I struggled quite a bit with early calculus classes, but breezed through later "more difficult" classes that built on them. Also, I disagree with the math and stats thing. Many of the best statisticians I know have strong grounding in mathematics, as do many of the best data scientists I know.
I hit a wall in my string theory course, after having to apply a lot more effort than expected in a QFT course the year before. Didn't have that issue with GR at all. Well, maybe with some finer points involving algebraic topology, but nothing insurmountable.
And importantly, brains are more heterogeneous at the extremes than at the means. A person of average intelligence might not have much difference between their verbal and calculation abilities; a person of great verbal or calculative intelligence may have a large gap between their stronger and (comparatively) weaker cognitive abilities.
I very much appreciate your interest. I'm sympathetic to the points that you raise. The trouble is that it's hard to even state my main thesis without presenting a lot of background information (!!). It's as though someone wanted to know about monstrous moonshine, and I started explaining what a modular function is, and the person said "ok, rather than giving so much motivation, I'd prefer it if you just told me what monstrous moonshine is." Then I state the theorem, and the person says "wait, what's the modular j-function?" But it's much easier to speak to an individual's situation than it is to speak to the general question of how people can get better at math. What's your background and what are your goals? I may not be able to respond at length individually, but I'll try to offer quick thoughts at least, and your comments will inform what I write about subsequently.
I did consider the possibility when I wrote that that I was grossly underestimating the amount of background information necessary to understand your main thesis. After all, you've been tutoring for, 15 years, I believe you said? Presumably you've collected a ton of anecdata, besides the information you've gathered from actual research. That's a lot to distill. But do you really think it's comparable to having to understand advanced abstract algebra/group theory? If you're really confident that stating your main thesis in your next post will only discredit you, then I don't think you should do it. But I do think you should be open to the possibility that that's not the case, and I think there's a non-negligible risk that you'll alienate the portion of your audience that strongly believes that mathematical ability is almost completely, if not completely, innate, if you take too much time to motivate. The rest of this comment is going to require considerable personal disclosure, so others should move on if they don't like that or care. I hope that my comment doesn't decrease my credibility with you or others; I think I've made valid points. My background is weak, far below what I expect to be the LessWrong norm. I'm going to explain my background, innate ability, interests, and the provisional goals that I've set based on my preliminary research. I understand that your time is limited, so I should say that any links are for elaboration and do not necessarily need to be read. I do apologize if this is too long, and I won't be offended if you don't offer a response. I only have a high school diploma and I'm not a college student. I was trapped in the bowels of anti-epistemology until the end of last year. I have strong ugh fields around the traditional math curriculum because I always memorized passwords and my math classes took place very early in the newest part of my school, in which the floors were ceramic, the walls concrete, and the air conditioning diabolically
Completely unrelated: long ago, I took a lower division math class from Borcherds. I don't think anything rubbed off, though :(.
Yeah, I know people who have seen his calculus lectures, and if I understand correctly he sticks to the book when he teaches undergraduate courses and doesn't share his special insights. You would have had to go to office hours :P.
I am a bachelor's in mathematics and estimate my current knowledge to be around a second year graduate student's if my mathematical knowledge is useful. I am interested in getting better at doing math as well as teaching it. Note: I am not the person you replied to.
Well, I certainly find it plausible that most reasonably intelligent (ie: at the mean to one standard deviation above) people can learn math to the level of, say, a first-year undergrad, or an advanced high-schooler. For one thing, some countries have more advanced and rigorous math curricula for high-schoolers than others ("engineering-major calculus" and linear algebra are reasonably common in high school curricula), and yet their class-failure rate, to my very limited knowledge, seems to vary with the quality of the pedagogy rather than being uniformly higher. Could most people of such an intelligence level also learn an entire undergraduate math major? I don't know: nobody is trying that experiment. I do think most of the people who already complete engineering, natural science, or computer-science degrees could probably complete undergraduate math -- but nobody is trying the experiment of controlling the "double major" variable, either. Instead we discourage physics, engineering, and comp sci majors who don't voluntarily double-major from doing additional theoretical math courses in favor of the applied math they need for their own domain (ie: algebra for the quantum physicist, differential equations and optimization for the engineer, logic and computability for the computer scientist). And that's when they take a theoretical track, instead of just cutting out to the applications ASAP! Could most people who do PhD-level research work in other natural and formal sciences do work in mathematics? At the research level, the distinction has collapsed: they partly already do! I've actually heard it said that you're not really well-prepared for PhD-level CS or physics if you didn't double-major in math, or for PhD-level biology if you didn't take an undergrad major or minor in statistics, anyway. You certainly can't work in type theory or machine learning (to bang on my own interests) these days without using and doing research-level work in "math" as an inherent pa
I'm finding this discussion very interesting because of my personal background. The general population would describe me as "good at maths". I would describe myself (because of context) as "bad at maths". I was one of the best all the way through high school and then started an undergraduate maths course known for being challenging. After a few weeks I completely hit a wall and couldn't progress any further with it. (I changed course to music.) My sister, father and brother-in-law all completed a whole undergraduate course in maths - I couldn't finish the first year. So I think I am bad at maths. Following on: I think I have a much deeper aesthetic understanding of music than my father and sister. They, the "better" mathematicians, are excellent musicians, but in a functional sense. I'd say that I, the "worse" mathematician, have a much more profound insight into music than they do.
And I failed my first go at Machine Learning, and nearly failed my first go at Intro to Statistics! (Because I hadn't taken continuous probability first, and was depressed.) What was it like from the inside, hitting that wall? Math is a particularly easy subject to hit a wall in, because if you're lacking a prerequisite of some sort, all of a sudden everything you're seeing turns to apparent nonsense.
Actually I found it very difficult to understand how it had happened. At school I was one of the best, I enjoyed maths, I understood the concepts and mostly found it easy. At University that all reversed: I was one of the worst, I couldn't do the assignments, I found the lectures boring, and I thoroughly disliked it. I found it very hard to comprehend how such a complete reversal had happened. And more than 15 years later, I still don't really get it... It's rather destabilising when you can't do the thing you expected to devote three years of your life to!
This is a gentle reminder for everyone reading these comments that if you enjoyed this post, please go back up top and upvote it real quick. I myself forgot at first. I think when I see a post that looks especially interesting to me, I tear right into it and don't look back.

I've always been pretty good at math, so I can't empathize very well with people who are "bad at math". Sure, it's easy to imagine some specific difficulties they might be having, but they could also be having other difficulties which I've never had.

In a sense, calculus isn't "representative of math". But in another sense, it is. If you approach it as a typical student, it requires you to focus on abstract ideas without seeing the payoff, which makes many people uncomfortable.

Now, of course, people who are "good at math" do actually see the payoff. We get bored by pointless things just like everyone else, but math doesn't feel pointless to us, because we feel that it's going somewhere specific. Maybe all the subfields of math could do a better job at explaining the kinds of questions they want to answer, and why.

With that in mind, IMO the perfect kind of math for Scott to study would be basic game theory. He already has a deep understanding of the motivations behind it, why it's important and interesting, and it has almost no prerequisites besides arithmetic. I'd be really curious to see him try. If the intuition behind the Prisoner's Dilemma led him to write the Moloch post, I can't wait to see what he will do with things like imperfect information and mechanism design :-)

Remember his sequence here :-).

Wow. And I even commented on that. And then forgot about it. Sorry :-p So, what kind of math does Scott feel that he's bad at?
I also would like to know what the evidence is exactly that Scott is bad at math.
Mostly his self-description, like here.
Thanks. A great quote from the link: I would guess myself that someone able to write the game theory sequences he wrote is someone able to fit all of the relevant concepts together like LEGOs.
Wow! How did I miss that? And why does he keep saying that he's bad at math?

Since much of this sequence has focused on case studies (Grothendiek, Scott Alexander), I'd be curious as to what you think of Douglas Hofstadter. How does he fit into this whole picture? He's obviously a man of incredible talent in something - I don't know whether to call it math or philosophy (or both). Either way it's clear that he has the aesthetic sense you're talking about here in spades. But I distinctly remember him writing something along the lines of how, upon reaching graduate mathematics he hit a "wall of abstraction" and couldn't progress any further. Does your picture of mathematical ability leave room for something like that to happen? I mean, this is Douglas freakin' Hofstadter we're talking about - it's hard to picture someone being more of a mathematical aesthete than he is. And even he ran into a wall!

Excuse me, I have to don a flame-proof suit now. Just a question: what useful results for predicting and modelling a preexisting reality has Douglas Hofstadter produced? I mean, yes, GEB is... well, it's GEB. I find it quite dated and think that it skates by on having fun with patterns rather than explaining observed phenomena. I'm also a little aggravated that GEB includes no discussions of model theory, ordinal logic, and w-incompleteness, nor of algorithmic randomness and halting problems, nor of the Curry-Howard Isomorphism and how it matches computational systems to logical systems. It goes on and on about recursion and formal systems for a very long time without actually addressing the formal sciences that handle the various phenomena arising from talking recursively in logic! Whereas something more recent like Universal Artificial Intelligence by Hutter succeeds on mathematical rigor and Probabilistic Models of Cognition on beauty of compression and presentation.
I'm sure GEB says at least a little bit about omega-incompleteness. Is my memory defective?
I think it does, and I know it at least alludes to ordinal logic and model theory.
Yes, though I think it's fair to say it says little enough about those that Eli's complaint could be reasonable.
Maybe I just didn't reach that part yet.
It's not a modeling handbook. It just fucks with your mind. Most people's minds are much too unfucked-with.
So am I fucked-up enough yet if I find it kinda boring and wish it would skip to the part where we model real phenomena, especially since, if I just want to make stuff up, I can make up rather crazier things than this?
Mirror, mirror on the wall... X-D
Perhaps, but that doesn't mean randomly fucking with them will improve them.
That's OK: GEB fucks with minds nonrandomly.
GEB makes a very strong case for the idea that intelligent systems might not be formal systems. That idea is then developed in his subsequent writings, and has lead to a quiet branch of AI that still flourishes.
Depending on how you define "preexisting reality", most professional mathematics can be said not to achieve this. In any case, the terms under which people usually praise Douglas Hofstadter do not include this sort of achievement. And if you really want to know what Hofstadter has done, there's this.
Is it being too specific to say that what Hofstadter has is a talent for putting useful labels on recursive phenomena?

Your Bayesian prior should be that I know more about Scott's mathematical potential than Scott does. :-)

I don't think so. Your priors aren't worth much until you have been on both sides of the fence. There are people who are bad at musics. There are people who are bad at language. There are people who are bad at sports. Some people are bad at programming. And Scott is indeed bad at math.

He can certainly internalize some math he finds relevant, but if you take him and someone of his age but with aptitude for math and try to teach them, to the best of your abilities, some math they have never been exposed to and have no intuitive frame of reference for, you will see the difference in the uptake rate immediately. Maybe elements of abstract algebra, or something.

This is an experimental fact that you must have come across many times in your tutoring, and I don't understand why you seem to be denying that. Some people learn faster, retain better and can learn more about certain subjects than other people. Some people can use their aptitude elsewhere as crutches. The aesthetical discernment you mentioned is one of those crutches. Scott is certainly multi-talented enough to be able to... (read more)

Part of what I'll be arguing is that the whole conceptual framework that people are using is wrong. :-) As far as I can tell, it's empirically true that Scott's emotional reaction to the unsolvability of quintic is unusual amongst mathematicians (while being almost uniform amongst elite mathematicians). If true, then on that dimension, he's better at math than the average mathematician, even without having any technical knowledge, even not knowing calculus well enough to have gotten a grade higher than a C-. I don't doubt that his struggling to get a C- in calculus reflects some sort of relative lack ability on his part, but I don't think that it carves reality at its joints to call that "mathematical ability." Separately, I think that his calculus experience would have been very different if it had been immersive: I don't think that he would have gotten a grade below C in calculus if he had spent all waking hours talking about calculus with me for 6 months. Of the ~200 calculus students who I taught at University of Illinois, I don't think that there are any students for whom this is true.

I don't think that he would have gotten a grade below C in calculus if he had spent all waking hours talking about calculus with me for 6 months.

Of course he would have gotten an A. The difference between being good and bad at math is whether you need to "spent all waking hours talking about calculus" to get an A.

Extrapolating from 1 course is silly. I worked like a demon to do mediocre (low Bs) in both calc 1 and physics 1, but somewhere towards the end of my freshman year of college something fell into place for me. By my first year of grad school I was breezing through quantum field theory and math methods for string theory courses with minimal effort.
Fascinating... do you have any idea what might have "fallen in to place"? (I'm always eager to learn from people who became good, as opposed to people who were always good or always bad, because I figure the people who became good have the most to tell us about whatever components of being good are non-innate. For example, Elon Musk thinks he's been highly driven ever since he was a kid, which suggests that he doesn't have much to teach others about motivation.)
Well, one thing was definitely changed was my approach to the coursework. I started taking a lot of notes as a memory aid, but then when I worked through problems I relied on what I remembered and refused to look things up in the text book or my notes. This forced me to figure out ways to solve problems in ways that made sense to me- it was really slow going at first but I slowly built up my own bag of tricks.
Interesting. I think I've read research suggesting that answering questions is significantly better for learning than just reading material (similar to how Anki asks you questions instead of just telling you things). Val at CFAR likes to make the point that if you look at what students in a typical math class are actually practicing during class, they are practicing copying off the blackboard. In the same way maybe what most people are "actually practicing" when they do math homework is flipping though the textbook until they find an example problem that looks analogous to the one they're working on and imitating the structure of the example problem solution in order to do their homework.
Consider that you're given a magic formula (the derivative) to determine the vertex of a quadratic equation when learning how to graph equations. That nonsense is how mathematics is -taught-. It shouldn't surprise us when students adopt the "magic pattern" approach to problem-solving. (And my own experience is that most of the teachers are following magic patterns they don't understand themselves, anyways.)
That would explain why story problems seem to be perceived as hard by average students at the high school level. I remember being confused by that, since mathematically they were usually the easiest problems in a set -- but they wouldn't be trivially pattern-matched to sample problems.
That's true. I've seen this go both ways, too. Though the priors are against a turn of events like yours, it does happen.
You seem to be nitpicking over a semantic issue at this point. With considerable due respect, you have better things to do.
OK, sorry, didn't mean to upset you. Disengaging.
I wasn't offended :-).
Sounds more like a lack of enthusiasm. Allow me to illustrate. There's a story of Thomas Hobbes finding a copy of Euclid's Elements on the table at a friend's house. He opened it up, found a proposition, and disbelieved it at first. Then he started reading the proof. Whenever a previous result was referenced, he looked up that proposition, went over its proof, and so on. Eventually, he made his way back to the beginning of the book and became amazed at the whole structure--of a seemingly far out result being carefully built on an edifice founded on statements so obvious that no one could dispute. He gained a great deal of respect for geometry and you can see some echo of this sort of thinking in his Leviathan. Without that kind of spark (and the discipline to back it up), study just becomes an exercise in drudgery. If your motivation to learn a subject is to get an A so you can go to law school and please your father, then class performance turns into a game of Guessing the Teacher's Password. Some have a lot of trouble forcing themselves to play. I certainly hope so. I very much doubt the average C student spends more than 10 hours a week (including classroom time) for one semester doing calculus problems. Retreating to a monastery and meditating upon the calculus for six months should work for any student. Scott might just have the problem where he has trouble proceeding to a new step without understanding the old ones, and the class went too fast to keep up. Some students can manage by taking everything on faith or mimicking what the professor did on the blackboard, but this causes other students much distress. Unsolvability? Bah. It just takes a radical approach (well, figuratively, not actually using radicals).
Seconding shminux. While there technically exist edge cases who wouldn't recall anything you said the previous day, I think nearly all humans could learn calculus this way. We don't do it because we don't have the teachers/money, nor the time, nor the interest on the part of prospective students (nor any clear reason they should value calculus that highly). This lowers my expectation of you getting around to a sensible recommendation.
This sort of thing is common on Less Wrong, and I don't mean to single you out (you're behaving within a cultural context that you didn't create). But what you're saying here, connotatively, doesn't make any sense. You're implicitly adopting the premise that it's my responsibility to convince you of something, as though you were a judge and I was a lawyer. The actual situation is that I have many more orders of magnitude of knowledge about the subject than you do, there are millions of people who I could be helping, and if helping you specifically understand isn't yielding high marginal returns, then I shouldn't waste my time interacting with you, because it takes away time from other people who would benefit. It's as though I were a selective college and you were sending me an application essay about why my offerings aren't valuable. The reaction that you should anticipate is me discarding your application and moving onto the next one. Again, I don't intend to be harsh, but you should seriously reconsider your conceptual framework on this point. This is why it's taken me so long to write about these things on LW: because I found the prospect of dealing with ever more nitpicking and straw-manning to be exhausting. So often people don't meet me half way and put in the work of steelmanning my arguments that would be necessary to make it cost-effective for me to engage.

This is why it's taken me so long to write about these things on LW: because I found the prospect of dealing with ever more nitpicking and straw-manning to be exhausting. So often people don't meet me half way and put in the work of steelmanning my arguments that would be necessary to make it cost-effective for me to engage.

Two points:

First, I think the recurring advice to state your main thesis, and then motivate it, applies. Among other reasons, it makes it easier for people to not make leaps in the wrong direction. If you show me some bizarre theorem, and then explain the pieces that make up that theorem, I can keep returning to the bizarre theorem, adjusting my concept of it with the new explanation until it clicks. If you just show me the pieces that make up the theorem, the part of me that's trying to model your motives in the conversation has to search many possibilities for why you might be introducing any particular piece. Unless I can independently discover the theorem you want to talk about, I'm probably going to get it wrong even once I have all the pieces! While I have only a subset of the pieces, how do I have any hope?

Remember, a steelman is when one takes an argu... (read more)

This is helpful feedback. I do recognize that I have a lot of room for improvement in these regards. But making comments like should be against community norms, not for my sake, but for the sake of the commenters – this is not a good mode of operation for overcoming bias and becoming less wrong (!!). Commenters should be inquisitive and open-minded rather than combative and dismissive.
I dislike the trend to cuddlify everything, to make approving noises no matter what, then framing criticisms as merely some avenue for potential further advances, or somesuch. On the one hand, I do recognize that works better for the social animals that we are. On the other hand, aren't we (mostly) adults here, do we really need our hand held constantly? It's similar to the constant stream of "I LOVE YOU SO MUCH" in everday interactions, it's a race to the bottom in terms of deteriorating signal/noise ratios. How are we supposed to convey actual approval, shout it from the rooftops? Until that is the new de facto standard of neutral acknowledgment? A Fisherian runaway, in which a simple truth is disregarded: When "You did a really good job with that, it was very well said, and I thank you for your interest" is a mandatory preamble to most any feedback, it loses all informational content. A neutral element of speech. I do wish for a reset towards more sensible (= information-driven) communication. Less social-affirmation posturing. But, given the sensitive nature of topics here, this may be the wrong avenue to effect such a reset, invoking Crocker's Rules or no. Actually skipping the empty phraseology should be one of the later biases to overcome.
8IlyaShpitser9y This is a thing because we have complex brains, with only a part devoted to processing information of the kind you mean, and others worried about contingent social facts: dominance/submission/status/etc. I think the broadly right response is to make peace-via-compromise between those parts, and that involves speaking on multiple bandwidths, as it were. This, to me, is a type of instrumental rationality in interpersonal communication.
Citing phatic expressions is not really enough. The issue is what creates the signal: presence or absence of something. If the default is "Thanks" then saying nothing is the negative signal and saying "Thank you, you did such a great job!" is a positive signal. But if the default is "Thank you, you did such a great job!" then just "Thanks" becomes a negative signal and for a positive signal you have to escalate to "Oh my God this was the greatest thing ever I thank you so much how could I ever..." It's easy to see how this could get to be very inefficient and, frankly, ridiculous.

But in reality this runaway process doesn't get off the ground, and peters out at something called "collegiality and tact."

The default amount of "gratitude" expressed on LW seems to be considerably less than that expressed by even "thanks". Actually, most of the time it seems that the default response is to find some flaw of wording to nitpick, and usually such a flaw is only tangentially related to the thrust of the argument. That's not what we should be encouraging.
Congratulatory comments, even of the empty sort like "Great job!", serve as positive Pavlovian reinforcement, which helps to motivate/encourage people to post. In addition, they signal appreciation and gratefulness at the fact that someone was willing to make a top-level post in the first place. The fact that the people on LessWrong are at times so damn unfriendly is in my opinion a non-trivial part of the cause of LW's too often insular atmosphere. Furthermore, studies consistently show that humans respond better to positive reinforcement than to negative reinforcement, regardless of age. This isn't about whether we're "adults who don't need our hands held". It's about how to motivate people to post more. If Jonah gets a torrent of criticisms every time he posts something, that's going to create an ugh field around the idea of posting. If he then points this out in a comment, and people respond by saying what effectively amounts to "Well, it's your own fault for not being clear enough," well, you can imagine how it might feel. This is an issue entirely separate from that of whether the criticisms are right. The bottom line is that transmission of useful information isn't the only kind of transmission that occurs in human communication. "This post is so messy and obfuscated as to be nearly unreadable" and "I think your point may benefit from some clarification" are denotationally similar, but connotationally they are very different. If you insist on ignoring this distinction or dismissing it as unimportant (as it seems so many LWers are wont to do), you run the risk of generating an unpleasant social atmosphere. Seriously. This isn't rocket science. (See what I did there?)
I don't want approval, I want to help people. If people think that they can offer helpful feedback (as Vaniver did), they should do so. Empty praise is just as useless as empty criticism. Vaniver's feedback had substantive information value – that's why I'm glad that he made his comment. If I fail to help people because I'm not receptive enough to critical feedback, it's my own fault. I accept responsibility for the consequences of my actions.
Here too, it's unclear to me what your intent is in engaging with me. You seem upset with me, and I don't have an intuitive understanding of why. How could I interact with you with you and others in a way that wouldn't rub you the wrong way? I'm happy to seriously consider any suggestions. I don't want to rub anyone the wrong way. But I'm not going to apologize for who I am. I'm someone who's deeply devoted to helping people, and who's spent thousands of hours of hard work developing very deep understanding of substantive intellectual material. I should be able to tell people who I am without facing hostility, in the same way that a gay person should be able to say that he or she is gay without facing hostility.

Disclosing one's sexual orientation won't be (mis)construed as a status grab in the same way as disclosing one's (real or imagined) intellectual superiority. Perceived arguments from authority must be handled with supreme care, otherwise they invariably set the stage for a primate hierarchy contest. Minute details in phrasing can make all the difference: "I could engage with people much smarter than you, yet I choose to help you, since you probably need my help and my advice" versus "I made the following experiences, hopefully someone [impersonal, not triggering status comparisons] can benefit from them". sigh, hoo-mans ... I could laugh at them all day if I wasn't one of them.

I'm happy to read your posts, but then I may be less picky about my cognitive diet than others. I mean, the alternative would be watching Hell's Kitchen. You do beat Gordon Ramsay on the relevant metrics, by a large amount.

Then again, maybe I'm just a bit jealous of your idealism.

That's the thing – it's not zero sum! Other LWers can become thousands of times more intellectually sophisticated than they are. Some of them may have substantially more potential in principle than I have. Similarly for idealism. We should have a culture of positive sum cooperation, where people are happy to have someone more who's much more knowledgable around because they can benefit from it, rather than thinking in terms of "if they're around than they have more status, so I have lower status, so it's bad for me if they signal intellectual superiority." If people had consistently adopted such attitudes throughout history, we would still be in the dark ages.

I feel like we could have a more productive discussion on this in another format (maybe a Hangout sometime this weekend?), but for now a short comment (that might take years to unpack):

I should be able to tell people who I am without facing hostility, in the same way that a gay person should be able to say that he or she is gay without facing hostility.

I have found that the word "should" is dangerous, and that any time one uses it, one could benefit from contemplation on the underlying belief.

I am intentionally speaking high handedly here. I spent years being suicidal because people pathologized me when I was doing what I was doing to help people. I received so many accusations of disingenuousness and arrogance that I involuntarily internalized them, and it caused me unthinkable psychological damage. I'm not going to give weight such accusations anymore. I can take the perspective that the people who accuse me of disingenuousness or arrogance are evil, or I can take the perspective that I'm morally sophisticated than they are out of virtue of being privileged. I've chosen the latter. In exchange, I'm committed to striving for moral purity.

Ok, look, I get that you are trying hard to be a good person, and that's great, but you're not doing such a great job of it right now. And I think that's kind of the crux here: You've somehow gotten the idea that being a Good Person automatically makes you good at it, or should, whatever that means.

You say that you like helping people. I identify with that. I like helping people too. But all that really tells you is how I get my jollies, you know? Other people are not obliged to give me said jollies by being helped, and they may have good reasons not to. Here are some possible reasons:

  • They don't think they need my help.
  • They don't think I am competent to help them, and perhaps are worried that I may make things worse.
  • They suspect that I am optimizing for fuzzies rather than for actually helping, which may cause conflict or poor outcomes from their point of view.
  • They feel disrespected by the implication that I am in a position to help them, and fear loss of status.

Now, you may think some of these reasons are mistaken or irrational (I think any of them might be perfectly sane, myself), but the fact remains that people are quite possibly going to have these concerns, and if I c... (read more)

Things have changed. I finally got over it over the past 24 months, and feel so much better now. I'm just offering explanation for where my apparently aloof tone is coming from – it may seem disrespectful, but it's actually what I need to be mentally healthy: I need to be able to be open about who I am and totally discount people's reactions when they're angry and hostile in response. For so many years, the egalitarian pressures were suffocating. The irony is that my experience probably actually has a great deal in common with the experience of many LWers on account of being smarter than others earlier on in life – mine is in the same direction, only more extreme. And I couldn't find kindred spirits even here: people who were ok with me being smart and thoughtful even when it signaled superiority in a way that would result in social backlash from mainstream society. To the extent that I'm dismissive, that's why. A sense of the type "these people aren't on my side, they're in the same reference class as the women who construed me giving them interesting math books as an attempt to coercively obtain sexual favors, when the actual situation was that I was starved for intellectual companionship, and mistakenly thought that they were the same as me and would be happy to have someone to talk to."

I find it funny that I'm finally getting the feedback that I needed 25 years ago, from so many people at once. See here and here: over the past ~6 months, I finally started to get it.

Thanks very much for your comment, I appreciate the time that you put into it. The points that you make have largely been made already by other commenters, and I feel a little bit sheepish that you went through so much effort, but I might find your framing of things to be helpful at the margin, even on reflection.

YES YES YES. I LOVE when things like this happen.
If you find that something is suddenly happening a lot, probably it was always happening and you never noticed. Particularly if it is something that is easy to misinterpret, like advice.

A general principle that I think is sufficient for this case (there are alternative reasons also sufficient on their own) is that in most situations, you should only assert things when you expect justified agreement from nontrivial portion of your target audience. So when you say "I'm not going to apologize for who I am", this assumes the context of your assertions about who you are, and I don't think you've given good arguments about that.

Environmental conditions don't reliably determine the outcome, so even though you might correctly have private knowledge about that, pointing out environmental conditions doesn't communicate sufficient evidence for your audience to accept the conclusion (whose meaning/application also wasn't very clear, but that seems secondary in this case). There are many high-status geniuses trained in excellent environments who are both confident and confused in particular domains outside of their areas of brilliance, such as reasons for their success or correctness of some non-mainstream theory.

Without establishing agreement on such details, you can't rely on their influence on social norms that you'd expect in situations where they can be communicated. The acting social norms are implied by what was successfully communicated, not by what you privately know. If you follow the norms implied by your private knowledge, you break the acting social norms.

I'm knowingly breaking social norms. I reject the social norms that are in place as maladaptive, in the same way that Martin Luther King rejected social norms around segregation as maladaptive. And no, I'm not going to apologize for analogizing myself to Martin Luther King on account of it coming across as a status grab: even if I'm totally inconsequential, I still identify with him strongly, and whatever other people think, it's not a status grab.

I reject the social norms that are in place as maladaptive

Do you expect the social norms to accept your arguments, and should they, given the evidence (i.e. what is the role of addressing them in this context, expressing disapproval of certain responses)? That's the frustration of hard-to-communicate facts: you can (1) give up, (2) turn to the dark side and cut through your audience's epistemology with a machete, insisting that they accept the conclusion based on insufficient evidence and appeals to on-reflection irrelevant things, or (3) put in so much work that the result isn't worth the trouble.

(I personally dislike the machete more than the breaking of social norms, but that might be unusual.)

Sometimes you can make subtle changes to your wording to communicate the same facts with different status modifiers. I'll give it a shot: Let me know if you thought I failed in my objective to communicate the same facts while appearing humbler :P
This sounds like a deontological argument. You're talking about responsibilities (or lack thereof) and rights. It's possible that Jonah is operating within a consequentialist framework... that is, he believes that a culture of not griping about well written and informative posts (like his) will lead to the valuable consequence of more such posts being written. (By delivering social punishment, in the form of complaints, to people who make quality LW contributions, one works against the public good of quality LW contributions.) It's also possible that if Jonah adjusts his tone, it will could lead to positive consequences like people not getting annoyed by the feeling that he is lording it over them or whatever. It doesn't annoy me much when Jonah lets me know that he's an expert; I shrug and await the next post in his sequence; but I recognize that others may feel differently. Personally, I think both these hypotheses are true.
"I am the pearl-caster and you're swine" arguments tend to go badly. It's "cost-effective" for you to engage only if people put in the work of steelmanning your arguments?? Excuse me if I feel underwhelmed by the benevolent wisdom that you condescend to bestow on us unworthy ingrates.
Domain expertise is a thing, and society possesses a general social norm in favor of being charitable to domain experts. He also doesn't come across to me as particularly hostile.

Domain expertise is a thing, and society possesses a general social norm in favor of being charitable to domain experts.

I don't think the norm is as general as this implies. Western society expects a great deal of charity toward the mentor in a mentor/student relationship, but that relationship is usually a consensual one -- it can be assumed in some situations, such as between adults and children or within certain business relationships, but it isn't automatically in effect in a casual context even if one person has very much more subject matter expertise than the other. It's usually considered very rude to assume the mentor role without a willing student, unless you're well-known as a public intellectual, which no one here is.

And the pattern's weaker still online, where credentials are harder to verify and more egalitarian norms tend to prevail. Except in a venue specifically set up to foster such relationships (like a Reddit AMA), they're quite rare -- even people known as intellectual heavyweights in a certain context, like Scott or Eliezer around here, can usually expect to relate to people more in a first-among-equals kind of way. In fact it's not uncommon for them to receive more criticism.

Well the issue is, JonahSinick doesn't come across to me as arrogant, hostile, or assuming any kind of relationship of superiority in the first place. He's sharing his domain knowledge with us for the sheer pleasure of doing so, and wants to be helpful to people who've gotten discouraged about learning mathematics. Given his motivations, his actions, and the context for all of them, I just don't see the rudeness. It looks to me like some very conceited LW regulars are reading a preaching into this article and JonahSinick's comments that just isn't there, by action or intention. I usually don't see that much vehement criticism of Scott; it helps that he behaves in a very egalitarian fashion. Eliezer tends to take somewhat heavy criticism, including sometimes from me, precisely because he adheres to the LW community norm of "We here at LW are smarter and know better than everyone else, and we don't need your stinking domain knowledge." Oh, and also because Eliezer is phenomenally bad at explaining his thoughts and intentions to people outside Bay Area techno-futurist circles, which probably comes of training himself to be good at explaining his thoughts and intentions to an incredibly narrow, self-selected, and psychologically unusual circle of people. Once you've been reading him for long enough to have a clear idea what he's trying to say, even he's really not that bad. It's funny: when I got here, I thought Eliezer's Sequences were basically nothing special, just explaining some science and machine-learning stuff to people who apparently can't be arsed to read the primary sources. But the longer I'm here, the more I sometimes want to exasperatedly say to some or another "aspiring rationalist" who thinks they're being ever-so-clever, no, you are actually being a Straw Vulcan, read the fucking Sequences.
Hostile, no. Arrogant -- a bit, but quite within the LW norm. But asserting superiority? Very much so. Here is a direct quote: And the problems arose not because of claims about superior domain knowledge, but rather claims about superior "crystallized intelligence" and "intellectual caliber" which are much wider than "I'm really good at math".
Eliezer's Sequences contain a lot of science and machine-learning stuff as you describe, and a few core bits that... aren't. Going by volume, most of them are good. But the actual objections to them, of course, will be disproportionately around those few core bits. And sometimes not agreeing with something that is phrased like a scientific lecture can look an awful lot like refusal to listen, even when it's not.
Well yeah, but hey: take what's useful and chuck out the rest. Besides which, by bothering to try to come up with a wholly naturalistic worldview that never resorts to mysticism in the first place, Eliezer is massively ahead of the overwhelming majority of, for example, laypeople and philosophers. Practicing scientists are better at science, but often resort to mysticism themselves when confronted with questions outside their own domain (ie: Roger Penrose and his quantum-woo on consciousness). I do dislike the degree of mathematical Platonism and Tegmarkism I occasionally see around here, but that's just my personal extreme distaste for mysticism coming out. Basically, it's really nice to have a community where words like "irreducible" will get you lynched, and if I have to put up with a few old blog entries being kinda bad at conveying their intended point, or just plain being wrong, so be it.
That means that if I just say "space aliens have replaced the President", I'm saying something bad, but if I copy a math textbook, and add a footnote "also, space aliens have replaced the President", I'm saying something good, because the sum total of what I am saying (a lot of good math + one bad thing about aliens) is good. In one sense that's correct; people could certainly learn lots of math from my footnoted math textbook. But we don't generally add these kinds of things together.
Did you write the rest of the math textbook?
I really don't care what people think of me. The point that I'm making is a general one that doesn't have to do with me specifically at all: people are responsible for their own learning. I welcome constructive criticism, but it doesn't help anyone for people to read what I write without approaching with an inquisitive and open mindset. I'm not being paid to write these articles, I'm doing it for the benefit of others. Doing it requires large time commitments from me. If people don't think that they can learn from me, they don't have to read my articles. [Edit:] If you aren't benefiting from reading my writing and don't believe that you can offer constructive criticism that would lead me to make my writing more beneficial to you, don't waste your time arguing with me – do something more productive. Do something that makes you happy. I don't have delusions of grandeur: I don't think that reading my articles is an optimal use of time for everyone – there are lots of other very fulfilling ways to spend time .

people are responsible for their own learning

Yes, they are. One of the consequences of that is that they don't owe anything to you -- not to steelman your arguments and not even to not nitpick or spindle, fold, and mutilate them.

I'm doing it for the benefit of others

That's the problem. If you feel you're doing a charitable act, a mitzvah, shut up and do it. Why are you expecting gratitude and bitching about the lack of it?

You take the position of someone from above bestowing wisdom upon those below. LW has always been sensitive to status and you are assuming the role of a lord to whom lowly peasants should show obeisance wherever he throws them scraps from his table. That will not and does not play well.

People are responsible for themselves -- you, too. It's your own responsibility to figure out what's cost-efficient for you and whether it's a good use of your time to post things on LW. Complaining about ingratitude and threatening to pick up your toys and go home is unlikely to get you much.

You take the position of someone from above bestowing wisdom upon those below. LW has always been sensitive to status and you are assuming the role of a lord to whom lowly peasants should show obeisance wherever he throws them scraps from his table. That will not and does not play well.

This actually is helpful feedback. Can you elaborate on your thoughts on the sensitivity of LWers to status? I'm not sure that I have a clear understanding of the situation here.

My comments above were not intended as a slight toward you or anyone else. I was relating factual information: I know much more about what I'm writing about than most LWers, and have high opportunity cost of time, but I don't feel smug about it.

Presumably I'm missing something really important. I'd welcome the opportunity to better understand it.

People are responsible for themselves -- you, too. It's your own responsibility to figure out what's cost-efficient for you and whether it's a good use of your time to post things on LW. Complaining about ingratitude and threatening to pick up your toys and go home is unlikely to get you much.

This was not my intention. I don't care about whether I get gratitude, I care about people learning from me. I value constructive criticism and explanation of why people aren't finding my posts more useful. As a factual matter, my efforts to help people throughout my life have been largely fruitless. I take responsibility for that.

Let me offer another angle of view.

As I understand you spent a lot of time teaching and tutoring math. This means you are used to being the master in the master-disciple relationship. This relationship has a few relevant characteristics. The disciple voluntarily enters it and agrees to accept the authority of the master with the understanding that it's going to be for his own benefit. The master accepts the responsibility of guiding the disciple and correcting him when he strays away from the path. Such a relationship can be very useful and productive, especially for the disciple.

This is NOT the relationship between you and your LW readers.

You are accustomed to not only teaching the subject matter, but also telling the student how best to understand and absorb it. That involves telling the student what not do (e.g. not to nitpick the details but rather pay attention to the general thrust of the argument). The student accepts this because he has agreed to let you guide him. The problem is, LW people did no such thing.

LW regulars are a conceited and contentious bunch. Even if you may feel that it will be quite good for them to accept you as a master and learn useful things from you, ... (read more)

LW regulars are a conceited and contentious bunch.

What do you think is going on here? Why are LW regulars a conceited and contentious bunch? I've been wondering this since I started posting under a pseudonym back in 2010, and I still don't understand.

Even if you may feel that it will be quite good for them to accept you as a master and learn useful things from you, you don't get to decide that. If you want to offer learning, you can only offer it.

This is absolutely correct, and a lesson that it's taken me decades to start to appreciate deeply.

I'm still learning. This is actually the main reason that I started this subthread – because I had (before starting this sequence of posts) been just not taking the time to post to LW anymore out of exasperation (without voicing my frustration), and I'm breaking from that behavior by initiating a conversation around it.

And speaking of gratitude, while you may not care whether you get gratitude, you do seem to care when you get pushback and criticism (gratitude with the flipped sign) -- this is why this whole sub-thread exists.

Until several months ago, I had been finding it insulting to receive responses along the lines "I don... (read more)

Btw, I find it slightly uncomfortable that we are discussing Scott's personal life, and he might too (yes I realize he shared this stuff. Still.)

Ok, I'll take note of this. I was using him as an example because people in the community are familiar with him and because the information is public – it's hard to talk about these things without being able to get into concrete specifics. Feel free to PM me if you have specific concerns in mind.
LW is basically a high-IQ club. People with abnormally high IQ get used to being smarter than most around them -- and specifically get used to winning arguments, if not by superior knowledge than by superior logic and rhetoric. If you're, say, in the top 1% of the population (by IQ), 99% of the people are not as smart as you. That's enough to make you conceited and contentious :-) Yes, I have a similar attitude, though originating slightly differently. I treat the ability to insult me as a right that no one has by default and one that I give out via respect. It's not really a super-high status pose, it's more of a "you're outside of my circle of concern, so you don't get to affect me". However I'm not sure adopting this will help with e.g. being ignored by cute girls. Defanging insults is essentially self-defence while getting others to like you is active reaching out. And self-worth/self-confidence issues are generally more complex than just having been insulted too many times.

I don't actually think that wanting to get treated as equals by Jonah even means being a conceited and contentious bunch.

Nah, more like disagreeing for seemingly no benefit.
Yes, ok, this is a good point (and an explanation that I had considered, but you saying it is an update in the direction of that being the driver). The trouble is that then one falls into a pattern of spending a lot of time bickering, while simultaneously feeling resentful about not being recognized by the world. The sense of superiority coming from being right ends up being wireheading that distracts from just optimizing for achieving one's goals. And when people who have even greater genetic advantages, or unusual environmental advantages, observe the behavior, they often look down on the people who are engaging in it. They think "These people think that they're smart, but they're actually really stupid and uneducated! It's hilarious!" I myself have no such contempt, but it's the generic thing, so in practice, people who are like this end up facing a glass ceiling that prevents them to crack into the upper echelons of society, without having a clear sense for what's going on. My posts are in large part an attempt to help LWers crack through that glass ceiling, but a lot of LWers don't get it, instead they just hate me because I come across as thinking that I'm superior. Even though the main difference between me and other people who think themselves to be superior is that I actually care about helping LWers and so talk about it, when others are too contemptuous to even consider engaging. And they wonder "why am I in a dead end job when I'm so smart?" And it's frustrating, because I can't do anything about it. No, once you don't feel insulted anymore, you become more confident, and that makes you feel more prosocial feeling, which is conducive to reaching out.
I don't think it's a charitable assumption that some large proportion of the people here are in dead-end jobs, or consider themselves unsuccessful at achieving their goals in general. This is one of the more accomplished social clubs I've ever found, actually, and that's been an immense boon for helping me to personally up my game by getting better at more things! Now I've got other people to meet up with and talk to who also try to get good at many related things, and can talk about that experience.
Did it ever occur to you that perhaps you can dish it out but you can't take it? That phrase is often used to refer to insults, but it also applies to "helpfulness". You need to be willing to be helped by others in the same way that you want to help them. And you don't seem to be. When someone disagrees with you, take it as a learning opportunity for yourself just like you expect others to take learning opportunities from you.
Oh no, I'm very grateful to people for having helped me. Lumifer's elaboration and Vaniver's comments were great. I haven't found your comments useful yet, but I can easily imagine that I might if you wrote more than a few lines.
In the context of you "teaching" others, it means that others are trying to "teach" you as well. This is a discussion forum. That means that it has discussions, which are two-way. The people whom you describe as "nitpicking" and "strawmanning" are people on the other end of the discussion. We're permitted to nitpick here--even to nitpick you--because you're discussing, you're not teaching. And while strawmanning is bad everywhere, what may look like strawmanning can actually be a result of you failing to communicate.
I don't claim that I've been communicating well :-). It's clear that I haven't been. I feel as though I'm out of touch with the goals of LW readers. When I read a post, it's usually because I'm eager to learn something from the author. I almost never respond to posts that I disagree with: it's only when I have high regard for the author that I go out of my way to engage. So I've been very puzzled as to why when I post to LW, it's not uncommon for people to respond in confrontational / standoffish ways, implicitly or explicitly expressing skepticism as to the value of what I have to offer. It's not that I'm never skeptical of the value of an author's writing: there are just things that I'd rather be doing than talking about it! Can you help me understand what's going on here?

So I've been very puzzled as to why when I post to LW, it's not uncommon for people to respond in confrontational / standoffish ways, implicitly or explicitly expressing skepticism as to the value of what I have to offer.

Possibilities that hinge on the way you post are worth extra attention if you notice that people are responding that way to you but not to others. I don't have a fully formed opinion on that, though, and so will ignore it in favor of generic possibilities. The first three that come to mind:

  1. People are busy, and collaborate to conserve attention. Suppose A posts 5k words; B reads it and responds with "I think this is low quality for reason X," then C can see the comment first and avoid spending time on the post. B can't recover their lost time by writing the comment, but they can save C's time, and by creating a culture of quality / calling out bad quality, they can have their time saved in the future. (This is more typically a role for karma, but comments also have a function here. Comments often remind people to vote, one way or another--one of my early posts was hovering at a very low score until someone commented that they thought the post was surp

... (read more)
Perhaps it's just the cynic in me talking, but of the reasons you posted, I find 3 the most compelling one by far.
Thanks, this is helpful. I wasn't talking about people's responses to my posts specifically: I've had the same reaction to people's comments on other people's as well – I don't see a difference on that front. My intuitive response has been "these people are just belligerent nitpickers who care more about arguing than about overcoming bias!" and so it's useful to have more charitable possible explanations in mind.
I liked your post but didn't really have anything substantive to add to it. In general, it's harder to think of good constructive ideas than to think of decent flaws in an idea. Combine that with a tendency for status seeking, and you get a big threat to productive group conversations.
Because that's what it means for a discussion to be two way. People criticize you. That's how it works. I doubt it, because that would imply that even if you're trying to teach someone, you never try to dispel any misconceptions, correct errors, etc. You probably don't think of those as "being skeptical of the value of an author's writing", but in fact, that's what it is. Well, in a two way discussion, this is going to be happening in both directions, and just like you do it to other people, other people will do it to you.
Has this been your experience in real life interactions? LW is virtually the only context in which I've seen this dynamic as a community norm. :-) Some reasons why I might engage somebody: 1. I have a lot to learn from the person. 2. The person is high potential enough so that if I communicate relevant information him or her, that'll enable him or her to use it to powerful effect. 3. The person is in a position of influence such that it's actually really important that misconceptions get corrected, and the person seems open-minded enough so that the chance of influencing the person's thinking is reasonable. 4. I find the person pleasant to be around. All of these things are signals of respect. What I'm puzzled by is the fact that some LWers who engage with me don't seem to respect me in the way that I respect them (as shown by my taking time to communicate with them when the time could be spent in other ways!). I feel like "If you don't respect me, why are you talking to me at all? Why don't you instead spend time talking with people who you do respect?" Can you help me understand what's going on here?
The ultimate problem is that you seem to have a double standard, and this is an example of it. If you taking time to communicate with them counts as a sign of you respecting them, then them taking the time to communicate with you should count as a sign of them respecting you. Just like the double standard where someone who criticizes you is "skeptical of the value of an author's writing" but when you do the same thing to other people, you're just correcting misconceptions and influencing the person's thinking. You're nobody special here, just like everyone else is nobody special. [1] [1] There are some people, like Eliezer, who sometimes get treated as special. I don't agree at all with this.
Your comments have been giving me the sense that you don't respect me – have I been misreading you?
The thing is, most people on LW don't disagree for any specific reason like respect or wanting to correct misconceptions or whatever. Disagreement just happens to be the status quo here. I haven't worked out why people here like to disagree so much, even when there seems to be no benefit from doing so--but then again, humans aren't perfectly instrumentally rational, and perhaps LWers are less instrumentally rational than most. (We certainly do seem to have more people suffering from akrasia around here than most places, after all.) It's also possible that Lumifer and Vaniver are right, and that LW users are a conceited, contentious bunch that like to disagree to signal intelligence and/or gain karma. I know I've fallen victim to the urge to nitpick before, and so have many others here, including even prominent users like shminux. (Jiro, too, from my previous experience talking with him/her.) It's just nitpicking. Respect, for better or for worse, doesn't even really enter the equation. For the record, however, I do respect your mathematical ability, and while I'm less confident about your metacognitive abilities, I think that if you've spent as much time pondering this subject as you claim--certainly a lot more time than most people around here have, probably--you have a reasonably good grasp on what you're talking about. Just don't expect most LWers to feel the same way.
I think you are too quick to take disagreement personally. Interpreting disagreement as lack of respect is an example of this.
I think it's more issues like tone, personally. (Particularly egregious are examples of "you are" statements, instead of "I feel" statements, which I've noticed many LWers seem really prone to making. It's a lot less pretentious-sounding when you prefix your statements with an "I feel" or "I think" or "in my opnion".)
You didn't answer my question: do you respect me? :-) You're not giving any positive feedback whatsoever. I don't care what you think of me, but it's reasonable for me to assume that you don't have any positive feelings toward me if you're not saying anything positive.
I respect you by my standards, but apparently not by your standards.
Based on JonahSinick's prior comments, his motivation for asking this question is pretty clear. You have already critiqued the thought process that made him think this question is necessary, to attack it again is almost double-counting. I think if you had answered the question directly the discussion would have a better chance of bootstrapping out of mutual unintelligibility. Then again, I mostly lurk and only rarely participate in internet debates so I don't feel I really understand how any given discussion strategy would actually play out. Also, I cheated, since Jonah already expressed a desire for a direct answer.
That's not an uncommon failure mode, but I don't think it's limited to high-IQ people. Plus the usual argument applies: if you're smart, reflection is easier for you so you have a better chance of realizing you're stuck in a pit but can climb out. What do you mean by that? At first glance, acquiring the respect of a Princeton department, getting invited to Rihanna parties, and being able to afford a $50,000 plate at a Hillary fundraiser all qualify... Well, is it a correct evaluation? :-D Regardless of your desire to help?
I agree. No, I meant by the standards that I imagine LWers to have – e.g. Luke Muehlhauser, Holden Karnofsky, Scott Alexander, etc. [Note: I'm not attributing contempt of the sort that I described to these people – the point is that they need to be respected by people who would be contemptuous of the average LWer in order to be where they are.] I've been trying to figure out how to communicate the situation without causing offense: it's really hard, because people are so sensitive to perceived slights. I had major environmental advantages growing up that most LWers didn't. Perhaps the greatest advantage was growing up around my father, as I described above. But beyond that: I grew up in San Francisco and so was able to attend an academic magnet high school with 650 students per grade, where my first year (at age 14) I met Dario Amodei, a Hertz Fellow who now works with Baidu's AI group. I went to Swarthmore, one of the top 3 ranked liberal arts colleges in the country, where my first year I met Andy Drucker, who did a PhD under the direction of Scott Aaronson and will be starting as a theoretical computer science professor at University of Chicago next year. The advantages of early interactions with these people compounded (e.g. they recommended books to read that upon reading led me to other books, etc.). By way of contrast, a lot of LWers grew up without knowing basically anyone similar to themselves who they might have been able to learn from. The end effect of this was that it resulted in me developing such much more crystallized intelligence that I outstrip all but a small handful of LWers in intellectual caliber by a very large margin. And I feel an impulse to help the people who I would have been without having had such decisive environmental advantages. But it's very difficult, because LWers have developed very strong priors that they're probably right when they disagree with someone. My reaction is "that's only because unlike me, you weren't fortunate
With due respect to those involved, this is not "upper echelons of society", this is a set of people highly respected in a small and isolated bubble. It all depends on the baseline, but these advantages don't sound huge to me. Going to a magnet school and to Swarthmore is nothing extraordinary. And what evidence do you have to support this view?
This is a semantic distinction. They're much higher status than most people in mainstream society, the same is not true of most LWers. That's what I meant. The more significant thing was growing up around my father: that gave me a large advantage over the people who I went to school with as well. But even putting that aside, what fraction of LW commenters do you think had better environmental conditions than I did? In particular, what about yourself? There are surface indicators, e.g. I have a PhD in math, which isn't true of almost any LWers. But even stronger than that, I've met with a number of elite mathematicians (advisors of multiple Fields medalists, etc., professors at the Institute for Advanced Studies, where Einstein, Von Neumann and Godel were, etc.) who have expressed high regard for me as a thinker.
I'd like to point out that the 2014 survey found 7.0% of LWers to have PhDs and 2.9% to have other professional degrees. These objective measures are considered by society at large to be of roughly equal intellectual caliber. You probably don't outstrip this roughly 1 in 10 lesswrongers by a such a large margin. Of course, the survey results may not be accurate. Furthermore while most of those degrees are in sciences, only a handful are in math or a close field. Thus if you consider math to require higher intellectual caliber (as I'm sure we both do) then you are still probably right about being of at least "higher" intellectual caliber. I guess you think the expressions of high regard from elite mathematicians are pretty big indicators though.
"Most people in mainstream society" is, within this context, a very low bar. So let's say I go to my doctor for a check-up. She is a licensed MD with her own practice which puts her higher on the mainstream-society status ladder than Scott Alexander, for example. Is that the upper echelon of the society I should be trying to break into? Luke, by mainstream-society standards, runs a small non-profit and the guy who owns a large car dealership nearby is more successful than him. Should I aspire to be like the car dealer? I don't know about LW commenters. From my personal perspective your upbringing is pretty normal and I think my "environmental conditions" were comparable. IQ is much more genetic than environmental, in any case. First, that's your side of the equation (errr, not equation, inequality :-D). What about the other side? It's not like Ph.Ds (or people in Ph.D. programs) are rare here. Second, your arguments are that of a child. To put it crudely, "I jumped through the hoops necessary to get a degree and important people patted me on the back". The proper criterion is achievement in real life. What have you done that demonstrates your sky-high crystallized intelligence?
Well, I mean... it's certainly higher than the baseline, but I wouldn't exactly call 7.0% common.
Still down on that ridiculous and inefficient phatic stuff? :) That stuff is a "load-bearing poster," to quote Bart Simpson.
You're in the dangerous position of suffering from confirmation bias on account of having so little exposure to people who are highly accomplished. The people who I'm talking about have mathematical productivity of order ~100,000x that of the average mathematician. Most mathematicians are terrified of talking to them on account of the expectation that they'd come across as really stupid. These are not people who pat people on the back for jumping through hoops. On an object level: in the course of working on my speed dating project, I rediscovered logistic regression, collaborative filtering, and hierarchical modeling. I rediscovered cross validation and how it can be combined with stepwise regression to identify robustly generalizable patterns in data. This led me to the discovery that principal component analysis greatly reduces concerns about multiple hypothesis testing, and greatly clarifies what's going on. The trouble is that I can't credibly signal that this represents unusually high quality work, because you don't have the subject matter knowledge that you would need to make an assessment. This is my point above: it's not clear to me that there's anything that I can say to change your mind. The question that you should ask yourself is: if you're so rational and intelligent, why aren't you more successful? It's convenient to attribute it to luck of the draw, but the fact is that you're actually roughly 1 million times lower in intellectual caliber than the highest intellectual caliber people in the world. Returns to IQ and aesthetic discernment aren't linear in expectation, they're exponential. And you have no way of knowing this. Which is why I'm taking the time to explain this to you. You're totally misreading the situation: I could be talking to people who are thousands of times more sophisticated than you, and it would be much more interesting to me, and instead I'm talking to you because I care about you. It should make you feel much higher status th
Whether what you write in the above comment is true or not (and by the way, I should mention that I believe you), it's an empirical fact about human psychology that taking a "holier than thou" attitude never helps if you want the other person to actually listen. And maybe it doesn't feel to you like you're taking a "holier than thou" attitude--or even any attitude at all. Maybe to you, you're just stating the facts. That's fine. But you've got to take into the account how the other person feels--and speaking for myself, I perceived a lot of condescension from your comment. (And then there's also the fact that the average person on LW is much less likely to take authority as an argument, anyway.) I'm not quite sure how to signal greater knowledge without also issuing a status challenge, and I somewhat doubt that there is a way. But you could do a lot better simply by cleaning up your tone a bit. For example, this could have been phrased as EDIT: I'm not saying Lumifer's been doing any better. In particular, "your arguments are that of a child" was really poorly phrased, IMO.
Thanks for the feedback, I do really appreciate it – I think that you're absolutely right. I was showing an empathy deficit there. Consistently showing empathy is difficult, and I'm working on it. But I shouldn't face social punishment for spending thousands of hours developing deep subject matter knowledge. I shouldn't face social punishment for having a deep desire to help people. It shouldn't be that people who have the stated objectives of being less wrong and overcoming bias are hostile to me for speaking the truth. That's not a good incentive structure for our culture (whether on LW or in the world) to adopt. For many years I felt like I couldn't be open about who I am, even amongst Less Wrong people or mathematicians. I'm not going to hide who I am just so that people don't have to feel uncomfortable about someone being more sophisticated and empathetic than they are. The sin of underconfidence is just as dangerous as the sin of overconfidence. If people can't handle knowing the facts about me, it's because they have psychological issues to work out rather than because there's something wrong with me. Edit: I may appear to be exhibiting an empathy deficit here as well – it's sort of inevitable, I shouldn't be internalizing perspectives that are fundamentally misguided at the cost of my own mental health.
So, I certainly cringed empathetically when I read that--but on reflection I agreed with the assessment, and the issue I saw was that it was said in public, and by someone who doesn't seem to have established rapport beforehand. So I'm not sure I agree that it's a phrasing issue.
You rediscovered? You didn't know logistic regression existed? What exactly did you rediscover? I suspect you're wrong about that. Rotating a matrix (which is what PCA does) doesn't actually reduce concerns about "hidden" degrees of freedom which you use up by trying multiple hypotheses. I actually think that the usefulness of PCA is often overstated -- all you're doing is selecting linear combinations with the highest variance which is not always the right thing to pay attention to. All in all that just sounds like pretty standard statistics. Well, yes, you can't. Speaking of "unusually high quality", your Github code contains things like which should be mildly embarassing. Along with values hardcoded as numbers in the body of the function, etc. I can read (and write) R code just fine -- what is it that you consider to be "unusually high quality"? I don't have much of a mind to change. I am doubtful of your assertions of great superiority, but that's a doubt, not a conviction that you're just an average math geek. Yeah: if you're so smart how come you ain't rich? :-D Why aren't I more successful than what? Which metric are you using? IQ values are ranks and I just don't know what "1 million times lower in intellectual caliber" even means. Evidence, please. Not to mention that for particular parameters exponential can be pretty close to linear :-) I'm sorry, did I stumble into some Christian revival meeting? What is this shit about trying to guilt me into agreement because you sacrifice so much of your highly valuable utils and hedons only because you care? I think your ego is in dire need of some deflation.
I was just learning R at the time and in a rush to get things to work. The code itself is not high quality. I see that you don't know what you're missing, I know it's because you didn't have the environmental advantages that I did, I know that I could have been in your position if not for the luck of the draw, and so I have pangs of sympathy for you, because your situation is in some sense very close to my own. It would make me feel so good if I could help you. That's why it's worth it to me in expectation, even if it's unpleasant in real time. But you can't love someone who doesn't want you to love him/her. I spent ~15 years on that and it helped no one and came at great cost to my myself. So I'll withdraw from this conversation.
LOL. I don't know if you're imitating a Christian missionary or a Jewish mother, but you're doing it badly. You can, but the relevant thing is that yes, I have no particular desire for you to love me. I suspect the same is true for the great majority of the LW population. And if I ever go looking for unconditional love, Jesus has a much better spiel that you do -- and He, at least, died for my sins :-P
To be blunt, I think that what's going on here is that you have an empathy deficit and so don't experience the warm fuzzy feelings around helping people that some people do. There are some people who would immediately understand where I'm coming from, and say "that totally makes sense." You seem to be Generalizing From One Example. I don't know whether it's genetic or environmental, mutable or immutable, but it's sad. Yes, so I'll stop and withdraw.
I'm not generalizing -- I'm pointing out that you, singular, you personally are doing it badly. And you are putting a lot of effort into not hearing this message. By the way, have you noticed how your last few comments started to focus on me and my shortcoming and deficiencies?
Not exclusively, I also mentioned my many years of failed efforts along these lines. I'm not claiming that my efforts have been useful. It's possible that you've actually helped people more than I have. My comments about you were made with a view toward giving a comprehensive explanation of why I'm bowing out of the conversation. The general pattern is "I tried to help people and they misconstrued it because they didn't have enough empathy to have a visceral understanding of the fact that I wanted to help them, so rather than being touched, they just found it irritating, and I made sacrifices when it should have been a priori clear that they were doomed to failure." I finally get it now.
The problem is that you believe that your internal motivation justifies your expectations of other people. Because your intentions are virtuous you expect that other people be "touched", be grateful, help you by steelmanning your arguments, etc. And it's not a matter of empathy, it's a matter of whether your state of mind imposes obligations on other people. It looks reasonable to you because from your point of view you only want to teach and it's reasonable that other people help you teach them. But try taking an external view (and try being more consequentialist, too). Christian missionaries appeared in this subthread not by accident -- they also care and also want to help and also make sacrifices to teach what they teach.
We're not in disagreement! :-) What I'm saying is that after many years, I finally came around to understanding what you're telling me right now. Your remarks are a useful update further in the same direction. I've genuinely benefitted from this interaction.
Uhhhh but the whole point of LW is that "argument-winning power" is a very different thing from "entangling-yourself-with-reality power", which is precisely why you can have a very high IQ and still need to learn all kinds of domains, like rationality, or scuba diving, or mathematics. Yes, but that's called being an arrogant asshole, and I personally prefer to do as little of it as possible, especially because I know it's the easiest bad habit for me to fall into and one of the worst for my ability to get along with others, which is very much something I care about. An Arrogant People's Club is a very bad thing to consider having.
I was being descriptive, not normative. Do you think the description is incorrect?
I think the description is correct for high-IQ clubs, by virtue of the norms those groups inform. Many high-IQ people who don't belong to those groups learn different social norms, and thus act differently.
I'm not even making a claim about the opportunity cost of my time relative to your own. For all I know, you have higher opportunity cost of time. Note that the amount of time that I put into writing the article is far greater than the amount of time that you've spent writing comments. If you were to write an article of comparable length engaging with me in detail, I would read it with great interest. My point is just that people should have a strong prior on me actually having something useful to say, and that if it's not coming across, and that to the extent that people have time, the focus should be on helping me understand how I could be more clear rather than on expressing skepticism that I have valuable information to share.
Well you should, but now you're having to make stand-offish statements because people are being bizarrely hostile to the notion that you possess domain expertise and direct experience, and are doing the rest of us the favor of trying to convey it.
:D I meant that I don't feel an emotional desire to be respected by the LW community. It should be obvious – I'm not interacting with people in a way that's optimal for getting respect ;-). Can you help me understand why people are being hostile to my claim that I have orders of magnitude more subject matter knowledge than they do? The most obvious explanation is ugly – that it makes them feel inferior by comparison, independently of whether or not I have any smugness about it (which I don't). If true, that's their problem, not my problem: the costs of not being explicit about the situation are prohibitively high to me. Is there something that I'm missing? If someone wants to give a detailed explanation for why he or she doubts my subject matter knowledge, I'll read it with great interest. It shouldn't be offensive that I don't have time to carefully optimize to not come across as thinking that I'm higher status than other community members. I'm putting much more time into my posts than they are into their comments! They're implicitly saying that I'm not worth their time in offering detailed thoughtful responses. This is fine: everybody has limited time, but the situation of people pressing me to justify the value of my posts seems so bizarre to me, given that they're not putting nearly as much effort in as I am. If someone wants to signal that he's intellectually serious, he can write a full length article carefully responding to mine, going into details about where he disagrees and where he agree, and why. That's all it would take for me to take him seriously.
Well, let me tell you: academics often come across as somewhat smug to everyone who's not an academic. But, you missed an even more abundantly obvious explanation: you're an outsider, so anything you say, other than blatant gestures of joining-the-ingroup, comes across as more hostile than it should.
Jonah has something like 91 posts and has been posting since May 2013.
Ah, really? Interesting! I thought everyone was reacting to his being an outsider. Huh, turns out he's an insider. Dafuck?
... and we're down to definitional quibbles, which are rarely worth the effort, other than simply stating "I define x as such and such, in contrast to your defining x as such as such". Reality has no intrinsic, objective dictionary with an entry for "mathematical ability", so such discussions can mostly be solved by using some derivative terms x1 and x2, instead of an overloaded concept x. Of course, the discussion often reduces to who has the primacy on the original wording of x, which is why I'd suggest that neither get it / taboo x. I agree that a more complex, nuanced framework would better correspond to different aspects of cognitive processing, but then that's the case for most subject matters. Bonus for not being as generally demotivating as "you lack that general quality called math ability", malus points because of a complexity penalty.
How do you know Scott is bad at math?
See his quote in the OP.

I mostly agree with this post.

I am actually not even sure there is a single predictor for being a strong mathematician (such as strong aesthetic discernment as you put it, although I agree it is a strong predictor) -- I think people's math thinking is just that heterogeneous.

For example, I am very visual, so it is easier for me to think about graphs than about logic. Some people have very strong calculation abilities, but can't rotate shapes in their head, etc. Maybe math is less about aesthetics to them, or about a very different kind of aesthetic, who knows!

I do think professional mathematicians need to combat the panic reaction people have to mathematics. Really, mathematics is just a gentle slope from common sense into the infinite.

My own impression (coming from thousands of hours of conversation and reflection) is that there's less genetic variability than superficially appears to be the case: a lot of the variance in styles of mathematical thinking is environmental / circumstantial. There is more than a single intelligence variable and a single aesthetic discernment variable, but I think that there's a collection of 4-6 genetic variables that explain basically all genetic variance amongst mathematicians. I agree that thinking in terms of mathematical ability being captured by a few numbers is connotatively misleading: it's like characterizing all human behavior as being about survival and reproduction, those were the influences that shaped who we are, but that's not what goes on on a day-to-day level. Yes :-).

To be great at anything creative, you must have both skill and taste. Painting, music, programming -- every art I've ever studied, or even heard of, has worked this way. You need the technical skill to create, and the eye that decides what's worth trying, and worth keeping.

You've made a good case that math, like music, requires taste for true greatness. And you've persuaded me that Scott Alexander has it. But you also seem to be saying that math doesn't have a skill component, in the sense I mean here, and I do not find that part of your argument persuasive.

There's an enormous skill component: it matters roughly as much as the aesthetic component. Even if I were as aesthetically discerning as Beethovn, I still wouldn't be able to invent the fast Fourier transform in the early 1800's like Gauss did. You need both for achievement at the highest levels. I'm counterbalancing the standard attitude of the type "huh? Aesthetic component? What's that?"
I think that pretty much everyone who knows any number of mathematicians and has talked to them at any length about their work has received exactly this sort of counterbalancing. As someone in a similar position to Scott, I've heard it more times than I can count, and I've honestly come to resent it somewhat. I've been told no end of times about how the beauty and elegance of "real" math, and how unrepresentative the sort of calculating work done at lower levels is of that sort of mathematics, but this is pretty much always being expressed by people who didn't have certain difficulties with the work at lower levels that the people they're expressing it to did. I've been on the other end of this a lot, trying to teach stuff to people which seems to me to be so intuitively, even beautifully obvious once you look at it from the right perspective, that it seems impossible for a person of any intellectual capacity not to grasp it, only to find that it takes a herculean effort on both our parts for them to make any sense of it at all. It's forced me to accept that there's a lot more human variability than I once thought in the capacity to be really bad at things. Like Scott, there are some kinds of "real math" which I have a reasonable amount of familiarity with and fluency in. And I have a fair amount of curiosity about and enthusiasm for mathematical curiosities of a certain sort. But I've never been able to muster the slightest bit of enthusiasm for doing math except to the extent that it lets me work out non-math things I'm interested in the answers to. I would love to like math more for its own sake, because there are times when figuring things out which I'm interested in the answers to requires learning more math which is a lot easier if I can appreciate it for its own sake throughout the steps I have to make it through. But lacking that immediate motive, I find much of the necessary learning incredibly dull and frustrating.

I think Scott Alexander is a terrible example.

He wrote a rebuttal.

There are several messages that would be good to send to lots of calculus students. One is that math is diverse. Another is that classes are twisted by the focus on evaluation and grading. Calculus is particularly bad: no one computes closed form integrals -- not pure mathematicians, not engineers.

But Scott is not a calculus student. He has encountered math in many contexts outside of math class and outside of school. He needs statistics. Telling him that the statistics that he grapples with... (read more)

It's hard to call a post written two years before the OP a rebuttal. :-)
??? Not sure how you got this from the OP. Scott's relationship with stats is, as far as I am concerned, evidence he's not nearly as bad at math as he says.
I think you have the chronology wrong: JonahSinick wrote this article in response to that one (among other things).

Would you agree with Scott if he said "I am bad at math classes"?

I think that, to most people, "math" brings to mind "the skills taught in math classes, which are used by people such as engineers and actuaries in their job, and by ordinary people when adding up grocery bills" much more than it does "what professional mathematicians do when they invent and prove new theorems"...
(Do you mind if I play Socrates for a while?) If Scott said "I am bad at the kind of math used by engineers", would you agree with him?
Do you think you could teach Scott to be better at math classes?
I heard that as "I am bad at math classes)".

I remember my mom, who was a math teacher, telling me for the first time that e^(i*pi) = -1. My immediate reaction was incredulity - I literally said "What??!" and grabbed a piece of paper to try to work out how that could be true. Of course I had none of the required tools to grapple with that kind of thing, so I got precisely nowhere with it. But that's the closest I've come to having a reaction like you describe with Scott and quintics. I consider the quintic thing far more impressive of course - the weirdness of Euler's identity isn't exactly... (read more)

But... it's just rotation! I think the thing that's weird about Euler's identity is that the symbology looks odd (especially if you're more used to degrees than radians), not that the underlying reality is odd. (Maybe I've just dealt with exponentials of complex numbers for so long that I can't be surprised by them anymore, but I don't remember being surprised by it before.)
Sure, I understand the identity now of course (or at least I have more of an understanding of it). All I meant was that if you're introduced to Euler's identity at a time when exponentiation just means "multiply this number by itself some number of times", then it's probably going to seem really odd to you. How exactly does one multiply 2.718 by itself sqrt(-1)*3.14 times?
You simply measure out a length such that, if you drew a square that many meters on a side, and also drew a square 3.1415 meters on a side, they would enclose no area between the two of them. Then evenly divide this length into meters, and for each meter write down 2.7183. Now multiply those numbers together, and you'll find they make -1. Easy!

The short example (from somebody who went to college with Scott and took Calc II in the same class with him) is yes. But that's an answer relative to the students of an elite college and only based on the fact that he asked me for to work on math homework with him.

There's an argument, which I find somewhat persuasive, that the usual belief that one is "not a math person" stems from learned helplessness, from many years of being forced to attempt difficult mathematical tasks in school without the required grounding. Mathematics, or at least the parts that are taught in standard curricula, is a very linear subject. Failure to grasp eg. fractions in the semester they are introduced could conceivably haunt a student for the rest of their school career, as it makes it difficult to understand essentially everything that follows.

If this theory of learned helplessness is correct, then perhaps if Scott could be convinced to complete the Khan Academy math courses he could be cured :)

I hope the next post in this series is about how to cultivate aesthetic discernment.

When I try to convince people like Scott that they're actually very good at math, they often say "No, you don't understand, I'm really bad at math, you're overestimating my mathematical ability because of my writing ability." To which my response is "I know you think that, I've seen many people in your rough direction who think that they're really bad at math, and say that I don't understand how bad they are, and they're almost always wrong: they almost never know that what they were having trouble with wasn't representative of math."

... (read more)
I would recommend trying these books (at high school level or earlier, depending on when it becomes possible to follow them): * H. Rademacher & O. Toeplitz (1967). The Enjoyment of Math. * J. R. Weeks (2001). The Shape of Space. * R. Courant & H. Robbins (1996). What Is Mathematics?
What Is Mathematics? was the only one I was able to find from a local library. I've put a request in for it and I should be getting it soon. Thanks for the recommendation; if it helps me to not hate math then I might be able to do something actually useful for existential risk reduction.
These are available on Library Genesis. Also, "What is Mathematics?" is more serious than the other two. "The Shape of Space" is probably the easiest and most fun, and "The Enjoyment of Math" is a collection of almost completely independent small pieces that don't assume any background, but some of them are a bit involved for something that doesn't assume any background.
Thanks, these are great!

Sorry for being dense but what is 'high aesthetic discernment' precisely differing from 'precise pattern matching'? Maybe I can't pattern match that to anything I posses - despite being quite good at pattern matching and other IQ-measured tasks (except spatial). I can appreciate elegance and beauty in mathematical proofs. I also hugely enjoyed Hofstadters GEB. But apparently something is amiss here. What?

just not true that if someone has substantially more trouble learning scales and chords than his or her classmates, he or she is "worse than them at music."

Following this analogy, is it also true that almost anyone can learn scales and chords then, then? Is it true that being bad at scales and chords will not forever be a major limitation? Is this because scales and chords are not fundamental to music, or is it because anyone can learn scales and chords given enough time and effort, after which innate musicality decides the rest?

(basically, s... (read more)

I agree that mathematical ability builds on more than some one dimension of IQ. Same as IQ has many dimensions. I quite clearly see that in the different ways my sons are smart: The oldest (11) has an enormous episodic and procedural memory - and uses it to solve complex tasks by combining methods. The second enjoys operating with complex algebraic expressions mentally - he also has a very good memory for facts. The third takes very long to deeply observe and then ultimately deeply grasp vague concepts. He also has a very good motor ability and spatial co... (read more)

There was strong interest in the first two posts in my sequence, and I apologize for the long delay. The reason for it is that I've accumulated hundreds of pages of relevant material in draft form, and have struggled with how to organize such a large body of material. I still don't know what's best, but since people have been asking, I decided to continue posting on the subject, even if I don't have my thoughts as organized as I'd like. I'd greatly welcome and appreciate any comments, but I won't have time to respond to them individually, because I alread

... (read more)
0Swimmer963 (Miranda Dixon-Luinenburg) 9y
Ooh! I think 'clerical intelligence' is the thing that my husband and I have taken to calling 'attention to detail' amongst ourselves. It's also been at least occasionally studied under that name – when applying for an admin job, they gave me a test of 'attention to detail' that consisted of several hundred timed questions comparing a block of six numbers to another block, having to answer whether they were the same or not, with around 5-10 seconds to spend per question. I don't think I'm outright bad at this, but it's not effortless for me. (Luckily, I had math teachers who gave points for the work getting to the solution, not just the solution, so I could get 7/8 points on a complicated problem even if I substituted a + for a - somewhere and got the wrong answer). My husband tends to use 'attentional to detail' to some degree also to mean what Paul Graham would call 'taste' or what Jonah would call 'aesthetic discernment'. I think the causal relationship is probably that in order to develop 'taste'–intuitions for what's good that correspond to what's generally agreed to be good – you need to be paying close attention to its details for a few years. Thus I have 'taste' for music, writing, and to some degree math, but not for fashion, since I never looked at what people were wearing.

Aesthetic ability as such hasn't been extracted as a cognitive ability factor. My guess would be that it's mainly explained by g and the temperamental factor of openness to experience. (I don't know what the empirical data is on this subject, but I think some immersion in the factor-analytic data would prove rewarding.)

[Added.] On aesthetic sense: the late R.B. Cattell (psychologist) devised an IQ test based on which jokes were preferred.

[Added.2] I'm wondering if you're not misinterpreting your personal experience. You say your IQ is only LW-average. You ... (read more)


Perhaps different people are also aestetically predisposed to favor continuous or discrete and rational or irr. variables. I aml much more comfortable with discrete rational ones, in part, perhaps, due to my father trying to install math appreciation in me since I was learning to count. He said '6 oranges, 6 sheep and 6 hours have a wonderful property in common. Nobody knows just what it is, but it objectively exists...'

Other commenters have said similar things, but I want to express this with my own words. To do mathematics requires multiple skills, and an aesthetic sense may be an underappreciated one of them. You argue that Scott has a good aesthetic sense. I also think that Scott probably has good abilities in some of the skills necessary for doing mathematics. But from Scott's account he appears to be lacking in other skills. Why do you think that what Scott has is sufficient? You mention that early college courses are not representative of real math, but even at hig... (read more)

Interesting! I have a BA in Mathematics, I was always 'good at math', and I'm currently a programmer.

I've never before considered that an "aesthetic sense" could related to mathematical ability, but it makes a lot of sense. I'm an extremely visual thinker – when I'm confused I feel blind – but I realize now that I spend a lot of time 'reasoning' by translating and transforming claims and statements and beliefs into different forms. The maximally insightful forms are almost always also the most pleasing to my eye – and I think you're right that that's not a coincidence.

I'm sure not only "elite" mathematicians intuit the interest of problems like the unsolvability of the quintic. That one can prove a construction impossible, the very concept of an invariant, is startling to the uninitiated. So many classic problems of this nature are held up as paradigms of beauty--the Konigsberg bridge problem, ruler and compass constructions of cube roots, the irrationality of sqrt(2),..

When I mentioned the unsolvability of quintic to Scott in passing, it grabbed his attention, and he was visibly very curious as to how it could be possible to show that a general quintic polynomial has no solutions in terms of radicals.

Why should this be so surprising to us? I guess I think it's a bit interesting that it starts at the 5th degree rather than elsewhere, but I'm sort of used to seeing such discontinuities. Naive induction doesn't really get much of my faith anymore. Is there anything else to the problem here that I'm not seeing?

The first time I heard of the unsolvability of general quintic equations, my response was "Wait, there's unsolvable polynomials?" followed immediately by "Why five? It seems so arbitrary!". Then I looked it up, and realized that it would take a lot more background knowledge before I could come even close to being able to understand the proof. Even so, my initial reaction was very similar to the one Jonah mentioned from Scott in the post, so I think he's on the right track with what he's saying.
Not the fact that it's true, the fact that it's possible to prove that it's true.