Monday, February 2, 2015

Yitang Zhang Article in the New Yorker


The lead article in this week's New Yorker (Feb-2, 2015) is on Yitang Zhang, the UNH professor who appeared from obscurity last year to prove the first real result in the direction of the twin-primes conjecture (specifically, a concrete repeating bounded gap between primes; full article here).

To some extent I feel an unwarranted amount of closeness to this story. First, I grew up very close to UNH (just over the border in Maine), I would use the library there all through high school to do research for papers, and I worked at the dairy facility there for a few summers while I was in college. Secondly, the "twin primes conjecture" is basically the only real math research problem that I ever even had any intuition about -- along about senior year in my math program I think wrote a paper in abstract algebra where after some investigation on a computer I wrote "it's interesting to note that primes separated by only two repeat infinitely", to which the professor wrote back in red pen, "unproven conjecture!". I sort of have a running debate with a colleague at school that it's sort of intuitively obvious if you look at it, while course there's no rigorous proof. Yet.

When I saw this New Yorker in our apartment around midnight Friday after I got back from school, I first noticed that there was some article on math (the writer makes it pretty opaque initially about exactly who or what the subject is). My partner Isabelle immediately said, "For god's sake, don't read it tonight and go to bed angry!", to which I said, "Mmmm-hmmm, probably a good idea." But I did so anyway. Frankly, I got less angered by it than you might expect, because while it's big pile of dumb fucking shit, it's dumb in a way that so stupidly predictable it almost turns around and becomes comedy if you know what's going on. It's dumb in exactly the carbon-copy way, almost word-for-word that all of these articles are dumb -- so it's at least unsurprisingly stupid. Let's check off some boxes...

The "Beautiful Math" Trope

Sure enough, the title of the article is "The Pursuit of Beauty". Paragraph #2 of the article is a string of predictable quotes by some dead white guys about "proofs can be beautiful" (G.H. Hardy, Bertrand Russell). The writer managed to find one living professor who he got to use that word one time (Edward Frankel, UC Berkeley, the proof having "a renaissance beauty", sounding like the author pressed him on the question and was grudgingly humored). And then he gets a hail-Mary sentence on neuroscientists connecting math to art in some lobe of the brain. But Zhang never says that. Nor anyone else in the article from that point.

This is so goddamn predictable that, yes, it's the raison d'etre of this very six-year blog, to respond to that exact piece of nonsense in pop math writing (see tagline above; and the "Manifesto" in the first post). It's bullshit, it's not part of the real work of math. Sure, shorter is better, and it's far more convenient to get at a proof quickly with some heavy-caliber technique or clever trick, and I'd argue this is all that's meant by the "beautiful" trope. Someone gets careless and uses "beautiful" as a metaphor, in the way that Einstein or someone likes to pitch "God" as a metaphor -- when they secretly have some nonstandard definition like, "scientific research reducing superstition" (see: letter to Herbert Goldstein) -- and then it gets repeated by a thousand propagandists for their personal crusades. In the case of a pop media writer, they can latch onto the "beautiful" tag line and feel that they've got a hook on the story, and approach the rest of it like it's an article on Jeff Koons or some other high-society, celebrity scam artist.

But at any rate, the "beautiful math" pitch is entirely isolated to the article title and a single paragraph, it has zero connection to the rest of the story, it's basically just clickbait, so let's move on.


Journalist-Mathematician Antimatter

The broader issue that makes article count as downright comedy is the completely predictable acid-and-water interaction between the journalist and the mathematician. The writer here, Alec Wilkinson, is an exemplar of his industry -- scammy, full of bullshit, and just downright really fucking stupid. We've all met these folks at this point, have we not? Doesn't really know about anything. Has a single journalistic move up his sleeve for every article: "put a human face on the story", make it personal, make it about the people, "how did X make you feel?". (Elsewhere in the magazine, another writer waxes nostalgic for the classic traditions of New Yorker staffers: "all the editors dressed up and out every night for dinner and a show... a shrine of exotic booze...", Talk of the Town).

But here Wilkinson confronts a person who is ultimately patient, disciplined, humble, hard-working, and truth-seeking. And he doesn't know what the hell to do with that. No other professional mathematician had known what Zhang was doing for over 10 years. He received no accolades nor enemies. He doesn't seem aggrieved or jealous that other people's careers advanced ahead of his own. He speaks softly at awards ceremonies and talks. There's no "personal face" meat here.

So here's how Wilkinson responds; he makes the article about himself. Specifically about how he's a stupid damn bullshit artist. The opening paragraph is specifically about how apparently proud he is to know nothing about math, to be unqualified to write this story, and about how he's a fucking lying cheater:

I don’t see what difference it can make now to reveal that I passed high-school math only because I cheated. I could add and subtract and multiply and divide, but I entered the wilderness when words became equations and x’s and y’s. On test days, I sat next to Bob Isner or Bruce Gelfand or Ted Chapman or Donny Chamberlain—smart boys whose handwriting I could read—and divided my attention between his desk and the teacher’s eyes.


Later, here's a summary of his interactions with Zhang:
Zhang is deeply reticent, and his manner is formal and elaborately polite. Recently, when we were walking, he said, “May I use these?” He meant a pair of clip-on shades, which he held toward me as if I might want to examine them first. His enthusiasm for answering questions about himself and his work is slight. About half an hour after I had met him for the first time, he said, “I have a question.” We had been talking about his childhood. He said, “How many more questions you going to have?” He depends heavily on three responses: “Maybe,” “Not so much,” and “Maybe not so much.” From diffidence, he often says “we” instead of “I,” as in, “We may not think this approach is so important.”... Peter Sarnak, a member of the Institute for Advanced Study, says that one day he ran into Zhang and said hello, and Zhang said hello, then Zhang said that it was the first word he’d spoken to anyone in ten days. 

This is not the kind of thing that a drinky, likely coke-blowing, social butterfly bullshit artist has any way of processing. And come on, that's pretty fucking funny; in that regard you almost couldn't make this stuff up. But on the downside it argues that these articles are always thricefold doomed; no journalist will ever write about the practice or results of math in any intelligible or useful way, because they're constitutionally, commercially, and philosophically opposed to it.

This is how predictable it all is: I give a mini-rant to Isabelle and she says, "Oh, he probably just went after something a family member mentioned to him once", and that is in fact exactly what motivated the article (see end of the first paragraph). So frankly I could read about three sentences and map out in advance the progression of all the rest of the article. Hard to get usefully enraged by that; just standard-stupid is all.

The Community of Math

That said, the article does brush up against a real essential issue that I've been wrestling with for a few years now. Many of the math blogs that I've been reading in the last half-decade make a powerful and sustained case for the "community of math", that math cannot be done in isolation, that it only exists in the context of communicating with colleagues. Even that the writing of papers is inherently a peripheral and transient distraction, that the "true" productive activity of math is done verbally face-to-face and via body language with other experts -- writing being a faint shadow of that true work. (Hit me up for references on any of these points if you want them.)

Unfortunately, this hits me something like an attack right through my own person. This is exactly the way that I personally failed in graduate school, and did not proceed on to the doctorate -- by continuing to work furiously in isolation while the rest of the classes basically passed me by. I've recently seen this called "John Henry Disease" in the work of Claude Steele (although there he holds it out as uniquely a phenomenon for black students). When I bring this up to colleagues nowadays, I can tell this story lightly enough that I get a laugh out of them, "Obviously you had to know better than that", or some-such. But a combination of personality and cultural upbringing literally left me completely unaware of the idea that you'd go get someone else's help on a math problem. So in that regard the "math community" thesis is a strong one.

But on the other hand, the whole prime-directive that I've established for my math classes in the last few years is: Learn how to read and write math properly. It's literally the first thing on my syllabi now; the idea that math (algebraic) language is inherently a written language and not primarily verbal, and that this is the hard thing to master if you're a standard poorly-prepared city public high school graduate. That learning to read a math book was the key that got me through calculus and all the rest of a math program (through the undergraduate level, anyway). That the software that runs our world are fundamentally products of writing (see a prior post here). And once I commit to this goal in class, and get most of the students to buy in to it, I've been getting what I think are wonderful and satisfying results with it, incredibly encouraging, in the last few years. Ken Bain's book "What the Best College Teachers Do" hits on this as an even more universal theme: "We found among the most effective teachers a strong desire to help students learn to read in the discipline." (Chapter 3, item #8).

And now here we have, in the very recent past, multiple cases of major mathematical breakthroughs by people working entirely in isolation, effectively in secret, for one to two decades, interacting only with the published literature in the field and their own brainpower. This is what's held out as Zhang's experience. And the same could be said for Perelman with the Poincaré conjecture, right? And also Andrew Wiles with Fermat's Last Theorem. And maybe Shinichi Mochizuki with the abc conjecture? (Here's where the argument rages.) A common theme recently is that with ever-more stringent publishing requirements for tenure, people on the standard academic track must publish every year or two, not meditate on the deepest problems for a decade. And so does the institution actually force isolation on the people tackling these giant problems? Or is it merely the nature of the beast itself?
When Zhang wasn’t working [at a Subway sandwich shop], he would go to the library at the University of Kentucky and read journals in algebraic geometry and number theory. “For years, I didn’t really keep up my dream in mathematics,” he said...

When we reached Zhang’s office, I asked how he had found the door into the problem. On a whiteboard, he wrote, “Goldston-Pintz-Yıldırım”and “Bombieri-Friedlander-Iwaniec.” He said, “The first paper is on bound gaps, and the second is on the distribution of primes in arithmetic progressions. I compare these two together, plus my own innovations, based on the years of reading in the library.”


An aside: I have this exact same issue in terms of my gaming work with Dungeons & Dragons (see that blog here). The conventional wisdom is "obviously we all know that no one could learn D&D on their own, we all had some older mentor(s) who inducted us into the game". And I am in the very rare situation for whom that is absolutely false. Growing up in a rural part of Maine, the only reason I ever heard about the game was through magazines; I was the first person to get the rulebooks and read them; and the catalyst in my town and school, among anyone I ever knew, to introduce and run the game for them. Purely from the written text of the rulebooks. To the extent that there were any other conventions or understandings about the game that didn't get into the books, I never knew about them. Which in retrospect has been both a great strength and in some fewer cases a weakness for me. In short: I learned purely from the book and most people don't believe that's possible.

But back to the article by Wilkinson: he expresses further dismay and incredulity at Zhang's solitary existence, his disinterest in social gatherings, and his preference for taking a bus to school so that he can get more thinking time in. All things which I could say pretty much identically for myself; and all things which our standard-template journalist is going to find alien and utterly bewildering:
Zhang’s memory is abnormally retentive. A friend of his named Jacob Chi said, “I take him to a party sometimes. He doesn’t talk, he’s absorbing everybody. I say, ‘There’s a human decency; you must talk to people, please.’ He says, ‘I enjoy your conversation.’ Six months later, he can say who sat where and who started a conversation, and he can repeat what they said.”

“I may think socializing is a way to waste time,” Zhang says. “Also, maybe I’m a little shy.”

A few years ago, Zhang sold his car, because he didn’t really use it. He rents an apartment about four miles from campus and rides to and from his office with students on a school shuttle. He says that he sits on the bus and thinks. Seven days a week, he arrives at his office around eight or nine and stays until six or seven. The longest he has taken off from thinking is two weeks. Sometimes he wakes in the morning thinking of a math problem he had been considering when he fell asleep. Outside his office is a long corridor that he likes to walk up and down. Otherwise, he walks outside.


Conclusion

There are more things I could criticize -- For example, in the absence of anything useful to say, the author has to hang onto any dumb or tentative attempt at an analogy that anyone throws at him, and is really helpless to double-check or confirm any assessment with anyone else; he literally can't understand anything anyone says about the math, even when interviews multiple professors on the same subject. He refers to Terry Tao like he's just "some professor", not one of the brightest and clearest thinkers on the planet. He has a paragraph on pp. 27-27, running 40 lines on the page, simply listing every variety of "prime number" he could find defined on Wikipedia (probably) -- the most blatant attempt at bloating up the word count of an article I think I've ever seen. Of course, it's intended to make your eyes cross and seem opaque. The exact opposite of a mathematical discipline dedicated to clear and transparent explanations.

But those are just nit-picky details, and we've probably already given the article writer more attention than he deserves. Let me finish by addressing the elephantine angel in the room. Are the true, greatest breakthroughs really made by loners, working in isolation with just the written text, over decades of time? Or is that just another journalistic illusion?


6 comments:

  1. I found this blog via Delta's D&D Hotspot. Nice post. I had previously read the New Yorker article and didn't pick up on any of the points you raise here.

    Apologies for an irrelevant query, but not sure where else to post this: I want to learn statistics. Is there a method you recommend for self-study? I'm asking you in particular since it's clear you've given a lot of thought to teaching and learning statistics.

    I teach philosophy at a university and am super busy when classes are in session, but I could manage to work on studying statistics over the summer. Are there any textbooks which you recommend? I recently purchased my friend Joe Blitzstein's _Introduction to Probability_, but discovered after cracking it open that it presumes familiarity with calculus (which I haven't studied for 20 years). Not sure where to go from here. Was considering starting with some Khan Academy or YouTube videos, but I recall you getting "angry" about those on this blog before.

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    1. Hey, really glad you wrote I love your blog! I had to resist the temptation to blow through and leave a comment on all your recent posts.

      As you can tell (in the above blog, say), I've got a pretty strong anti-video and pro-text bias; it's always seemed most efficient for me. I use Neil Weiss', "Introductory Statistics" in my classes; it's my favorite book to teach out of and I can't recommend it highly enough. It does not require calculus (although in the first pages of Chapter 6 you steal some broad ideas from calculus, without proving them rigorously). I think it's very clear and readable; any prior edition on Amazon or whatever will be great and inexpensive. Follow the table of contents and skip the optional sections (marked with asterisks [*], incl. all of Ch. 5); consider reading section 7.3 prior to reading the rest of Ch. 6-7 (it's the reason why normal curves are so central). Note that this is a course in classical frequentist statistics; you'll find Bayes' Rule in the optional part of the probability chapter, but not the broader subject of Bayesian Statistics.

      If you want, send me an email at dcollins at-symbol superdan dot net and I can send you my brief lecture notes (one page per chapter) as an overview of the course.

      And here's another thing to try. I was looking at the Carnegie Foundation Statway course last week, and while I'm not broadly in favor (it's mostly a political dodge to waive students' algebra requirement) I must say that I adore the first lecture. It actually does a surprisingly good job of laying out the whole basic thought-process of statistical inference in a 1.5 hour class (namely: specify a research hypothesis, determine the likely results if that was not the case, and then require that an experiment come out radically different to support a new discovery; basically Weiss Ch. 1-9). Consider reading that one sample lecture as a capsule case study; following link, bottom of page, click on "Statway" for link to a ZIP file:

      Statway Info

      What courses do you teach at U. Texas? I think I get a PHI-heavy readership on my blog here (and have a degree in it myself).

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  2. Thanks. I will check out Weiss and also the Statway lecture.

    I currently teach intro to philosophy, intro to ethics, and assorted upper division philosophy courses (history of modern, Asian philosophy, etc.).

    The program at my university is very small but we are looking to grow. In the fall I will start teaching a logic class. Part of my interest in statistics is how it relates to logic and critical thinking generally. Traditionally, logic courses focus on deductive logic, but it would be nice to be able to cover a few topics related to statistical reasoning as well, howsoever briefly.

    In general, philosophers don't get much training in statistics or in experimental design, even though both would be quite useful for a wide variety of philosophical specializations. Most of us don't run experiments or analyze data, but it would still be helpful to be able to read and evaluate scientific literature relevant to our domain of inquiry--and to avoid falling into fallacies which could be corrected by a knowledge of statistics!

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    1. Great. If there was any magic wand I could wave, I would require every high school or college student to have a primer on basic logic before their college curriculum. It's crazy frustrating to be midstream in a math class and have students totally unable to parse an "if/then" or "or" statement. Like even a 1 credit course as freshmen would help a lot.

      One of the things our school promotes are interdisciplinary linked courses (called "learning communities"). I'm kind of not thrilled at the prospect in general, but if forced then the one thing I could actually be excited and find value in would be linked logic/math courses.

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  3. By the way, where did you study philosophy? I assume that's a bachelor's degree? I got my PhD at Bowling Green State University in Ohio. (They traditionally specialize in applied ethics and political philosophy.)

    I also have a D&D question but that will go on your other blog. Cheers.

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    1. You're right, my B.A. in philosophy was at the University of Maine (Orono), at the same time as my B.A. in Math (followed by an M.A. in math & statistics at the same place). Very small department, but intimate and delightful in a lot of ways. My favorite professor publishes in Foucault and feminism, now at Williams College.

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