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...
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...
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 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...
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).
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 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
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...
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 :-)
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!
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...
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.
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...
But in reality this runaway process doesn't get off the ground, and peters out at something called "collegiality and tact."
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, ...
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...
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.)
I don't actually think that wanting to get treated as equals by Jonah even means being a conceited and contentious bunch.
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:
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
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.
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.
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, wh...
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.
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.
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:
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...
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...
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...
First of all, commenters don't owe you to be anyone and they would rightly complain.
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 w...
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.
...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."
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?
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...
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 :)
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...
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...
...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
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 ...
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...
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?
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.
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.
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.
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.
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.