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.