Is Scott Alexander bad at math?

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.