Archive for the ‘Nerd Interest’ Category

An Orthodox rabbi and Steven Weinberg walk into an email exchange…

Friday, October 22nd, 2021

Ever since I posted my obituary for the great Steven Weinberg three months ago, I’ve gotten a steady trickle of emails—all of which I’ve appreciated enormously—from people who knew Steve, or were influenced by him, and who wanted to share their own thoughts and memories. Last week, I was contacted by one Moshe Katz, an Orthodox rabbi, who wanted to share a long email exchange that he’d had with Steve, about Steve’s reasons for rejecting his birth-religion of Judaism (along with every other religion). Even though Rabbi Katz, rather than Steve, does most of the talking in this exchange, and even though Steve mostly expresses the same views he’d expressed in many of his public writings, I knew immediately on seeing this exchange that it could be of broader interest—so I secured permission to share it here on Shtetl-Optimized, both from Rabbi Katz and from Steve’s widow Louise.

While longtime readers can probably guess what I think about most of the topics discussed, I’ll refrain from any editorial commentary in this post—but of course, feel free to share your own thoughts in the comments, and maybe I’ll join in. Mostly, reading this exchange reminded me that someone at some point should write a proper book-length biography of Steve, and someone should also curate and publish a selection of his correspondence, much like Perfectly Reasonable Deviations from the Beaten Track did for Richard Feynman. There must be a lot more gems to be mined.

Anyway, without further ado, here’s the exchange (10 pages, PDF).

Update (Nov. 2, 2021): By request, see here for some of my own thoughts.

“The Chair”: A Straussian interpretation

Tuesday, August 31st, 2021

[Warning: spoilers follow!]

Last week Dana and I watched the full first season of The Chair, the Netflix drama that stars Sandra Oh as Ji-Yoon Kim, incoming chairwoman of the English department at the fictional Pembroke University. As the rave reviews promised, I found the show to be brilliantly written and acted. At times, The Chair made me think about that other academia-centered sitcom, The Big Bang Theory, which I freely confess I also enjoyed. But The Chair is much more highbrow (and more political), it’s about the humanities rather than STEM, and it’s mostly about academics who are older than the ones in Big Bang, both biologically and professionally.

I wouldn’t call The Chair “realistic”: the sets, stuffed with imposing bookshelves, paintings of great scholars, etc., look like how a TV producer might imagine university buildings, rather than the relatively humdrum reality. But in less than three hours, the show tackles a staggering number of issues that will be recognizable and relevant to anyone in academia: cratering enrollments, a narrow-minded cost-cutting dean, a lack of free time and a desperate search for childcare, a tenure case that turns into a retention case, a woke scandal (about which more later), a faculty revolt against Ji-Yoon culminating in a vote of no confidence, and much more. There’s also an elaborate side plot involving the actor (and real-life former literary scholar) David Duchovny, who portrays himself, being invited to lecture at Pembroke, which is not the sort of thing most academics have experience with, but which I suppose many viewers will enjoy.

The show is written at a high enough level that its stumbles are those of a daring acrobat. In the main narrative arc of the first season, the writers set themselves an absurdly ambitious (and, I think, laudable) goal: namely, to dramatize a conflict between a free-spirited professor, and woke students trying to cancel that professor for a classroom “microaggression,” in a way that fully empathizes with both sides. I don’t know if the show actually succeeds at this, but that’s partly because I don’t know if it’s possible to succeed.

To start with some background: in Pembroke’s English department, there are old, traditionalist white males, who give lectures extolling the Great Men of Literature, and who apparently still wield considerable power. Meanwhile, critical theorists are presented as young, exciting upstarts bravely challenging the status quo. People with recent experience of English departments should correct me if I’m wrong, but my sense is that this is pretty anachronistic—i.e., that the last powerful traditionalists in humanities departments were routed by the 80s or 90s at the latest, so that students in the Twitter-and-smartphone era (when The Chair is set) would be about as likely to encounter them as they would professors sitting around in charcoal suits smoking pipes.

There were also some of what felt to me like … intersectional oversights? Ji-Yoon, being Korean-American, is repeatedly approached by Black female students and faculty as a “fellow woman of color,” with whom they can commiserate about the entrenched power of the department’s white males. The show never examines how woke discourse has increasingly reclassified Asian-Americans as “white-adjacent”—as, for example, in the battles over gifted and magnet programs or admissions to Harvard. Likewise, woke students are shown standing arm-in-arm with Pembroke’s Jewish community, to denounce (what we in the audience know to be) a phantom antisemitic incident. Left unexplored is how, in the modern woke hierarchy, Jews have become just another kind of privileged white person (worse, of course, if they have ties to Israel).

This brings me to the first season’s central conflict, which revolves around Bill Dobson, a handsome middle-aged white male professor who’s revered as the department’s greatest genius on the basis of his earlier work, but who, after the death of his wife, is now washed-up, flippant, and frequently drunk or high. In one class session, while lecturing about intellectuals who found the strength to resist fascism despite their own nihilistic impulses, Bill makes a Nazi salute and shouts “Heil Hitler!,” as a theatrical reminder to the students about the enormity of what those intellectuals were fighting. Alas, a woke student captures that moment on their smartphone camera and shares it on social media. The clip of Bill making the Heil salute goes viral, shorn of all exculpatory context. Soon, crowds of students are waving placards and screaming “No Nazis at Pembroke!” outside the English building. In a desperate effort to make his PR crisis go away, the dean initiates termination proceedings against Bill—the principles of academic freedom and even Bill’s tenure be damned. Ji-Yoon, of course, as Bill’s chair, is caught smack in the middle of this. It’s complicated even further by Ji-Yoon’s and Bill’s romantic feelings for each other, and further still by Bill’s role as the babysitter of Ji-Yoon’s adopted daughter.

As all of this unfolds, the show seems immensely interested in pinning the blame on Bill’s “tragic flaws,” minor though they seemed to me—mostly just pride and unseriousness. (E.g., trying to lampoon the absurd charge of Nazism, Bill offhandedly mentions that he’s always wanted to visit Hitler’s mountain retreat, and on another occasion belts out “Springtime for Hitler” from The Producers.) The woke students, by contrast, are portrayed as earnest, understandably upset, and legitimately terrified about hate crimes on campus. If they, too, have opportunistic motives to attack Bill, the show never examines them.

In one sentence, then, here’s my beef with The Chair: its script portrays a mob, step by step, destroying an innocent man’s life over nothing, and yet it wants me to feel the mob’s pain, and be disappointed in its victim for mulishly insisting on his innocence (even though he is, in fact, innocent).

With real-life woke controversies, there often lingers the question of whether the accused might really be a racist, fascist, sexual predator, or whatever else, adequate proof or no. What’s different here is that we know that Bill Dobson is none of those things, we know he’s decent to his core, because the writers have painstakingly shown us that. And yet, in a weird narrative pretzel, we’re nevertheless supposed to be mad at him, and to sympathize with the campaign to cancel him.

A casual perusal of other reviews of The Chair told me that these reactions were far from universal. Here, for example, is what one viewer wrote:

I can appreciate that this is probably close to the reality that most women/of color experience in higher education. I enjoyed watching the scenes with Joan and Yaz [two female professors] the most but the rest was a drag. I couldn’t understand why Ji-Yoon was into Bill, or why anyone was into Bill. I found him to be an insufferable man-baby. That is such a turn off. So she’d put him straight but then still be pining for him. He wreaked [sic] of entitled, white male, tenured privilege and never showed any contrition for his actions or even awareness of their impact. i’m so tired of the “brilliant _” being used to justify coddling someone. And for the rest of the stuffy old patriarchal farts– boot them out! They weren’t good teachers and weren’t able to meet the needs of today’s students.

I asked myself: did this person watch the same show? It’s like, the script couldn’t possibly have been clearer about Bill’s character, the fact that he’s the polar opposite of the woke students’ mental construct. And yet, if the show had drawn an unambiguous corollary from Bill’s goodness—namely, that the effort to cancel him is a moral travesty—then The Chair itself might have been denounced as conservative (or at least classical liberal) propaganda, and those who’d otherwise form its core viewership wouldn’t have watched.

So, if I were a literary critic like the ones on the show, I might argue that The Chair begs for a Straussian interpretation. Sure, there’s an “overt” reading, wherein Bill Dobson is done in by his own hubris, or wherein it’s a comedy of errors with no one to blame. But then there’s also an “esoteric” reading, wherein Bill is the victim of an extremely specific modern-day collective insanity, one that future generations might look back on with little more ambivalence than we look back on McCarthyism. The writers of The Chair might hint at this latter reading, through their sympathetic portrayal of Bill and the obviousness of the injustice done to him, but they can never make it too explicit, because of the political and cultural constraints under which they themselves operate.

Under this theory, it presumably falls to those slightly outside the world portrayed in The Chair—like, let’s imagine, a theoretical computer science blogger who himself was denounced for woke heresies to the point where he has little more to lose in that direction—to make the esoteric reading explicit. Unless and until, of course, a second season comes along to undermine that reading entirely.

On blankfaces

Monday, August 2nd, 2021

For years, I’ve had a private term I’ve used with my family. To give a few examples of its use:

No, I never applied for that grant. I spent two hours struggling to log in to a web portal designed by the world’s top blankfaces until I finally gave up in despair.

No, I paid for that whole lecture trip out of pocket; I never got the reimbursement they promised. Their blankface administrator just kept sending me back the form, demanding more and more convoluted bank details, until I finally got the hint and dropped it.

No, my daughter Lily isn’t allowed in the swimming pool there. She easily passed their swim test last year, but this year the blankface lifeguard made up a new rule on the spot that she needs to retake the test, so Lily took it again and passed even more easily, but then the lifeguard said she didn’t like the stroke Lily used, so she failed her and didn’t let her retake it. I complained to their blankface athletic director, who launched an ‘investigation.’ The outcome of the ‘investigation’ was that, regardless of the ground truth about how well Lily can swim, their blankface lifeguard said she’s not allowed in the pool, so being blankfaces themselves, they’re going to stand with the lifeguard.

Yeah, the kids spend the entire day indoors, breathing each other’s stale, unventilated air, then they finally go outside and they aren’t allowed on the playground equipment, because of the covid risk from them touching it. Even though we’ve known for more than a year that covid is an airborne disease. Everyone I’ve talked there agrees that I have a point, but they say their hands are tied. I haven’t yet located the blankface who actually made this decision and stands by it.

What exactly is a blankface? He or she is often a mid-level bureaucrat, but not every bureaucrat is a blankface, and not every blankface is a bureaucrat. A blankface is anyone who enjoys wielding the power entrusted in them to make others miserable by acting like a cog in a broken machine, rather than like a human being with courage, judgment, and responsibility for their actions. A blankface meets every appeal to facts, logic, and plain compassion with the same repetition of rules and regulations and the same blank stare—a blank stare that, more often than not, conceals a contemptuous smile.

The longer I live, the more I see blankfacedness as one of the fundamental evils of the human condition. Yes, it contains large elements of stupidity, incuriosity, malevolence, and bureaucratic indifference, but it’s not reducible to any of those. After enough experience, the first two questions you ask about any organization are:

  1. Who are the blankfaces here?
  2. Who are the people I can talk with to get around the blankfaces?

As far as I can tell, blankfacedness cuts straight across conventional political ideology, gender, and race. (Age, too, except that I’ve never once encountered a blankfaced child.) Brilliance and creativity do seem to offer some protection against blankfacedness—possibly because the smarter you are, the harder it is to justify idiotic rules to yourself—but even there, the protection is far from complete.


Twenty years ago, all the conformists in my age cohort were obsessed with the Harry Potter books and movies—holding parties where they wore wizard costumes, etc. I decided that the Harry Potter phenomenon was a sort of collective insanity: from what I could tell, the stories seemed like startlingly puerile and unoriginal mass-marketed wish-fulfillment fantasies.

Today, those same conformists in my age cohort are more likely to condemn the Harry Potter series as Problematically white, male, and cisnormative, and J. K. Rowling herself as a monstrous bigot whose acquaintances’ acquaintances should be shunned. Naturally, then, there was nothing for me to do but finally read the series! My 8-year-old daughter Lily and I have been partner-reading it for half a year; we’re just finishing book 5. (After we’ve finished the series, we might start on Harry Potter and the Methods of Rationality … which, I confess, I’ve also never read.)

From book 5, I learned something extremely interesting. The most despicable villain in the Harry Potter universe is not Lord Voldemort, who’s mostly just a faraway cipher and abstract embodiment of pure evil, no more hateable than an earthquake. Rather, it’s Dolores Jane Umbridge, the toadlike Ministry of Magic bureaucrat who takes over Hogwarts school, forces out Dumbledore as headmaster, and terrorizes the students with increasingly draconian “Educational Decrees.” Umbridge’s decrees are mostly aimed at punishing Harry Potter and his friends, who’ve embarrassed the Ministry by telling everyone the truth that Voldemort has returned and by readying themselves to fight him, thereby defying the Ministry’s head-in-the-sand policy.

Anyway, I’ll say this for Harry Potter: Rowling’s portrayal of Umbridge is so spot-on and merciless that, for anyone who knows the series, I could simply define a blankface to be anyone sufficiently Umbridge-like.


This week I also finished reading The Premonition, the thrilling account of the runup to covid by Michael Lewis (who also wrote The Big Short, Moneyball, etc). Lewis tells the stories of a few individuals scattered across US health and government bureaucracies who figured out over the past 20 years that the US was breathtakingly unprepared for a pandemic, and who struggled against official indifference, mostly unsuccessfully, to try to fix that. As covid hit the US in early 2020, these same individuals frantically tried to pull the fire alarms, even as the Trump White House, the CDC, and state bureaucrats all did everything in their power to block and sideline them. We all know the results.

It’s no surprise that, in Lewis’s telling, Trump and his goons come in for world-historic blame: however terrible you thought they were, they were worse. It seems that John Bolton, in particular, gleefully took an ax to everything the two previous administrations had done to try to prepare the federal government for pandemics—after Tom Bossert, the one guy in Trump’s inner circle who’d actually taken pandemic preparation seriously, was forced out for contradicting Trump about Russia and Ukraine.

But the left isn’t spared either. The most compelling character in The Premonition is Charity Dean, who escaped from the Christian fundamentalist sect in which she was raised to put herself through medical school and become a crusading public-health officer for Santa Barbara County. Lewis relates with relish how, again and again, Dean startled the bureaucrats around her by taking matters into her own hands in her war against pathogens—e.g., slicing into a cadaver herself to take samples when the people whose job it was wouldn’t do it.

In 2019, Dean moved to Sacramento to become California’s next chief public health officer, but then Governor Gavin Newsom blocked her expected promotion, instead recruiting someone from the outside named Sonia Angell, who had no infectious disease experience but to whom Dean would have to report. Lewis reports the following as the reason:

“It was an optics problem,” says a senior official in the Department of Health and Human Services. “Charity was too young, too blond, too Barbie. They wanted a person of color.” Sonia Angell identified as Latina.

After it became obvious that the White House and the CDC were both asleep at the wheel, the competent experts’ Plan B was to get California to set a national standard, one that would shame all the other states into acting, by telling the truth about covid and by aggressively testing, tracing, and isolating. And here comes the tragedy: Charity Dean spent from mid-January till mid-March trying to do exactly that, and Sonia Angell blocked her. Angell—who comes across as a real-life Dolores Umbridge—banned Dean from using the word “pandemic,” screamed at her for her insubordination, and systematically shut her out of meetings. Angell’s stated view was that, until and unless the CDC said that there was a pandemic, there was no pandemic—regardless of what hospitals across California might be reporting to the contrary.

As it happens, California was the first state to move aggressively against covid, on March 19—basically because as the bodies started piling up, Dean and her allies finally managed to maneuver around Angell and get the ear of Governor Newsom directly. Had the response started earlier, the US might have had an outcome more in line with most industrialized countries. Half of the 630,000 dead Americans might now be alive.

Sonia Angell fully deserves to have her name immortalized by history as one of the blankest of blankfaces. But of course, Angell was far from alone. Robert Redfield, Trump’s CDC director, was a blankface extraordinaire. Nancy Messonnier, who lied to stay in Trump’s good graces, was a blankface too. The entire CDC and FDA seem to have teemed with blankfaces. As for Anthony Fauci, he became a national hero, maybe even deservedly so, merely by not being 100% a blankface, when basically every other “expert” in the US with visible power was. Fauci cleared a depressingly low bar, one that the people profiled by Lewis cleared at Simone-Biles-like heights.

In March 2020, the fundamental question I had was: where are the supercompetent rule-breaking American heroes from the disaster movies? What’s taking them so long? The Premonition satisfyingly answers that question. It turns out that the heroes did exist, scattered across the American health bureaucracy. They were screaming at the top of their lungs. But they were outvoted by the critical mass of blankfaces that’s become one of my country’s defining features.


Some people will object that the term “blankface” is dehumanizing. The reason I disagree is that a blankface is someone who freely chose to dehumanize themselves: to abdicate their human responsibility to see what’s right in front of them, to act like malfunctioning pieces of electronics even though they, like all of us, were born with the capacity for empathy and reason.

With many other human evils and failings, I have a strong inclination toward mercy, because I understand how someone could’ve succumbed to the temptation—indeed, I worry that I myself might’ve succumbed to it “but for the grace of God.” But here’s the thing about blankfaces: in all my thousands of dealings with them, not once was I ever given cause to wonder whether I might have done the same in their shoes. It’s like, of course I wouldn’t have! Even if I were forced (by my own higher-ups, an intransigent computer system, or whatever else) to foist some bureaucratic horribleness on an innocent victim, I’d be sheepish and apologetic about it. I’d acknowledge the farcical absurdity of what I was making the other person do, or declaring that they couldn’t do. Likewise, even if I were useless in a crisis, at least I’d get out of the way of the people trying to solve it. How could I live with myself otherwise?

The fundamental mystery of the blankfaces, then, is how they can be so alien and yet so common.


Update (Aug. 3): Surprisingly many people seem to have read this post, and come away with the notion that a “blankface” is simply anyone who’s a stickler for rules and formalized procedures. They’ve then tried to refute me with examples of where it’s good to be a stickler, or where I in particular would believe that it’s good.

But no, that’s not it at all.

Rules can be either good or bad. All things considered, I’d probably rather be on a plane piloted by a robotic stickler for safety rules, than by someone who ignored the rules at his or her discretion. And as I said in the post, in the first months of covid, it was ironically the anti-blankfaces who were screaming for rules, regulations, and lockdowns; the blankfaces wanted to continue as though nothing had changed!

Also, “blankface” (just like “homophobe” or “antisemite”) is a serious accusation. I’d never call anyone a blankface merely for sticking with a defensible rule when it turned out, in hindsight, that the rule could’ve been relaxed.

Here’s how to tell a blankface: suppose you see someone enforcing or interpreting a rule in a way that strikes you as obviously absurd. And suppose you point it out to them.

Do they say “I disagree, here’s why it actually does make sense”? They might be mistaken but they’re not a blankface.

Do they say “tell me about it, it makes zero sense, but it’s above my pay grade to change”? You might wish they were more dogged or courageous but again they’re not a blankface.

Or do they ignore all your arguments and just restate the original rule—seemingly angered by what they understood as a challenge to their authority, and delighted to reassert it? That’s the blankface.

On turning 40 today

Friday, May 21st, 2021

Holy crap.

In case you’re wondering how I spent such a milestone of a day: well, I spent hours of it at an important virtual grant review meeting with the Department of Defense. Alas, when it came time for my own big presentation at that meeting—about what my students and I had done over the past five years to lay the theoretical foundations for the recent achievement of quantum computational supremacy—I’d uploaded the completely wrong PowerPoint file (it was something.pptx rather than something.ppt, where they weren’t two versions of the same presentation). Sorting this out took about 10 minutes, destroyed my momentum, and wasted everyone’s time. I partly blame the Microsoft Teams platform, whose limitations as conferencing software compared to Zoom necessitated emailing my presentation in the first place. But of course, part of the blame rests with me.

I had to explain apologetically to the US Department of Defense that I’m no good with tech stuff—being a mere computer science PhD. And unlike many of my colleagues (who I envy), back in my youth—for at age 40 I’m no longer young—I never had enough time to become both the kind of person who might earn a big grant to do quantum computing theory, and the kind of person who’d be minimally competent at the logistics of a review meeting for such a grant.


Forty years. Seven-eighths of those years, aware of the finiteness of the speed of light and of its value. Four-fifths of them, aware of the grislier details of the Holocaust. Three-quarters of them, aware of what it means to write code. Two-thirds of them, aware of polynomial versus exponential time. More than half of them trying to understand the capabilities and limitations of quantum computers as my day job. And then, rounding the corner, more than a third of the years writing this blog, a third of them being a professor, a quarter of them married, a fifth of them raising kids, a thirtieth of them in the midst of a global pandemic.

I didn’t even come close to achieving everything I hoped I would in my thirties. At least a half-dozen major papers, ones I expected would’ve been finished years ago (on the mixing of coffee and cream, on complexity and firewalls and AdS/CFT, on certified random numbers from sampling-based quantum supremacy experiments, on the implications of the Raz-Tal oracle separation, …), still need to be revised or even written. Other projects (e.g., the graphic novel about teaching math to Lily) were excitedly announced and then barely even started. I never wrote most of my promised blog post about the continuum hypothesis, or the one about Stephen Wolfram’s recrudescent claims of a unified theory of physics. And covid, which determined the world’s working conditions while we were running out the clock, turned out not to be a hyper-productive time for me. That’s how you know I’m not Newton (well, it’s the not the only way you know).

Anyway, during the runup to it, one’s 40th birthday feels like a temporal singularity, where you have to compress more and more of what you’d hoped to achieve before age 40 as you get closer and closer to it, because what the hell is there on the other side? They‘re over-40 and hence “old”; you’re under-40 and hence still “young.”

OK, but here I am on the other side right now, the “old” side, and I’m still here, still thinking and writing and feeling fairly continuous with my pre-singularity embodiment! And so far, in 16 hours on this side, the most senile thing I’ve done has been to email the wrong file attachment and thereby ruin an important funding presenta… you know what, let’s not even go there.

If you feel compelled to give me a 40th birthday present, then just make it a comment on this post, as short or long as you like, about what anything I said or did meant for you. I’m a total softie for that stuff.

A grand anticlimax: the New York Times on Scott Alexander

Saturday, February 13th, 2021

Updates (Feb. 14, 2021): Scott Alexander Siskind responds here.

Last night, it occurred to me that despite how disjointed it feels, the New York Times piece does have a central thesis: namely, that rationalism is a “gateway drug” to dangerous beliefs. And that thesis is 100% correct—insofar as once you teach people that they can think for themselves about issues of consequence, some of them might think bad things. It’s just that many of us judge the benefit worth the risk!

Happy Valentine’s Day everyone!


Back in June, New York Times technology reporter Cade Metz, who I’d previously known from his reporting on quantum computing, told me that he was writing a story about Scott Alexander, Slate Star Codex, and the rationalist community. Given my position as someone who knew the rationalist community without ever really being part of it, Cade wondered whether I’d talk with him. I said I’d be delighted to.

I spent many hours with Cade, taking his calls and emails morning or night, at the playground with my kids or wherever else I was, answering his questions, giving context for his other interviews, suggesting people in the rationalist community for him to talk to, in exactly the same way I might suggest colleagues for a quantum computing story. And then I spent just as much time urging those people to talk to Cade. (“How could you possibly not want to talk? It’s the New York Times!”) Some of the people I suggested agreed to talk; others refused; a few were livid at me for giving a New York Times reporter their email addresses without asking them. (I apologized; lesson learned.)

What happened next is already the stuff of Internet history: the NYT’s threat to publish Scott’s real surname; Scott deleting his blog as a way to preempt that ‘doxing’; 8,000 people, including me, signing a petition urging the NYT to respect Scott’s wish to keep his professional and blog identities separate; Scott resigning from his psychiatry clinic and starting his own low-cost practice, Lorien Psychiatry; his moving his blog, like so many other writers this year, to Substack; then, a few weeks ago, his triumphant return to blogging under his real name of Scott Siskind. All this against the backdrop of an 8-month period that was world-changingly historic in so many other ways: the failed violent insurrection against the United States and the ouster, by democratic means, of the president who incited it; the tragedy of covid and the long-delayed start of the vaccination campaign; the BLM protests; the well-publicized upheavals at the NYT itself, including firings for ideological lapses that would’ve made little sense to our remote ancestors of ~2010.

And now, as an awkward coda, the New York Times article itself is finally out (non-paywalled version here).

It could’ve been worse. I doubt it will do lasting harm. Of the many choices I disagreed with, I don’t know which were Cade’s and which his editors’. But no, I was not happy with it. If you want a feature-length, pop condensation of the rationalist community and its ideas, I preferred this summer’s New Yorker article (but much better still is the book by Tom Chivers).

The trouble with the NYT piece is not that it makes any false statements, but just that it constantly insinuates nefarious beliefs and motives, via strategic word choices and omission of relevant facts that change the emotional coloration of the facts that it does present. I repeatedly muttered to myself, as I read: “dude, you could make anything sound shady with this exact same rhetorical toolkit!”

Without further ado, here’s a partial list of my issues:

  1. The piece includes the following ominous sentence: “But in late June of last year, when I approached Siskind to discuss the blog, it vanished.”  This framing, it seems to me, would be appropriate for some conman trying to evade accountability without ever explaining himself. It doesn’t make much sense for a practicing psychiatrist who took the dramatic step of deleting his blog in order to preserve his relationship with his patients—thereby complying with an ethical code that’s universal among psychiatrists, even if slightly strange to the rest of us—and who immediately explained his reasoning to the entire world. In the latter framing, of course, Scott comes across less like a fugitive on the run and more like an innocent victim of a newspaper’s editorial obstinacy.
  2. As expected, the piece devotes enormous space to the idea of rationalism as an on-ramp to alt-right extremism.  The trouble is, it never presents the idea that rationalism also can be an off-ramp from extremism—i.e., that it can provide a model for how even after you realize that mainstream sources are confidently wrong on some issue, you don’t respond by embracing conspiracy theories and hatreds, you respond by simply thinking carefully about each individual question rather than buying a worldview wholesale from anyone.  Nor does the NYT piece mention how Scott, precisely because he gives right-wing views more charity than some of us might feel they deserve, actually succeeded in dissuading some of his readers from voting for Trump—which is more success than I can probably claim in that department! I had many conversations with Cade about these angles that are nowhere reflected in the piece.
  3. The piece gets off on a weird foot, by describing the rationalists as “a group that aimed to re-examine the world through cold and careful thought.”  Why “cold”?  Like, let’s back up a few steps: what is even the connection in the popular imagination between rationality and “coldness”? To me, as to many others, the humor, humanity, and warmth of Scott’s writing were always among its most notable features.
  4. The piece makes liberal use of scare quotes. Most amusingly, it puts scare quotes around the phrase “Bayesian reasoning”!
  5. The piece never mentions that many rationalists (Zvi Mowshowitz, Jacob Falkovich, Kelsey Piper…) were right about the risk of covid-19 in early 2020, and then again right about masks, aerosol transmission, faster-spreading variants, the need to get vaccines into arms faster, and many other subsidiary issues, even while public health authorities and the mainstream press struggled for months to reach the same obvious (at least in retrospect) conclusions.  This omission is significant because Cade told me, in June, that the rationalist community’s early rightness about covid was part of what led him to want to write the piece in the first place (!).  If readers knew about that clear success, would it put a different spin on the rationalists’ weird, cultlike obsession with “Bayesian reasoning” and “consequentialist ethics” (whatever those are), or their nerdy, idiosyncratic worries about the more remote future?
  6. The piece contains the following striking sentence: “On the internet, many in Silicon Valley believe, everyone has the right not only to say what they want but to say it anonymously.” Well, yes, except this framing makes it sound like this is a fringe belief of some radical Silicon Valley tribe, rather than just the standard expectation of most of the billions of people who’ve used the Internet for most of its half-century of existence.
  7. Despite thousands of words about the content of SSC, the piece never gives Scott a few uninterrupted sentences in his own voice, to convey his style. This is something the New Yorker piece did do, and which would help readers better understand the wit, humor, charity, and self-doubt that made SSC so popular.  To see what I mean, read the NYT’s radically-abridged quotations from Scott’s now-classic riff on the Red, Blue, and Gray Tribes and decide for yourself whether they capture the spirit of the original (alright, I’ll quote the relevant passage myself at the bottom of this post). Scott has the property, shared by many of my favorite writers, that if you just properly quote him, the words leap off the page, wriggling free from the grasp of any bracketing explanations and making a direct run for the reader’s brain. All the more reason to quote him!
  8. The piece describes SSC as “astoundingly verbose.”  A more neutral way to put it would be that Scott has produced a vast quantity of intellectual output.  When I finish a Scott Alexander piece, only in a minority of cases do I feel like he spent more words examining a problem than its complexities really warranted.  Just as often, I’m left wanting more.
  9. The piece says that Scott once “aligned himself” with Charles Murray, then goes on to note Murray’s explosive views about race and IQ. That might be fair enough, were it also mentioned that the positions ascribed to Murray that Scott endorses in the relevant post—namely, “hereditarian leftism” and universal basic income—are not only unrelated to race but are actually progressive positions.
  10. The piece says that Scott once had neoreactionary thinker Nick Land on his blogroll. Again, important context is missing: this was back when Land was mainly known for his strange writings on AI and philosophy, before his neoreactionary turn.
  11. The piece says that Scott compared “some feminists” to Voldemort.  It didn’t explain what it took for certain specific feminists (like Amanda Marcotte) to prompt that comparison, which might have changed the coloration. (Another thing that would’ve complicated the picture: the rationalist community’s legendary openness to alternative gender identities and sexualities, before such openness became mainstream.)
  12. Speaking of feminists—yeah, I’m a minor part of the article.  One of the few things mentioned about me is that I’ve stayed in a rationalist group house.  (If you must know: for like two nights, when I was in Bay Area, with my wife and kids. We appreciated the hospitality!) The piece also says that I was “turned off by the more rigid and contrarian beliefs of the Rationalists.” It’s true that I’ve disagreed with many beliefs espoused by rationalists, but not because they were contrarian, or because I found them noticeably more “rigid” than most beliefs—only because I thought they were mistaken!
  13. The piece describes Eliezer Yudkowsky as a “polemicist and self-described AI researcher.”  It’s true that Eliezer opines about AI despite a lack of conventional credentials in that field, and it’s also true that the typical NYT reader might find him to be comically self-aggrandizing.  But had the piece mentioned the universally recognized AI experts, like Stuart Russell, who credit Yudkowsky for a central role in the AI safety movement, wouldn’t that have changed what readers perceived as the take-home message?
  14. The piece says the following about Shane Legg and Demis Hassabis, the founders of DeepMind: “Like the Rationalists, they believed that AI could end up turning against humanity, and because they held this belief, they felt they were among the only ones who were prepared to build it in a safe way.”  This strikes me as a brilliant way to reframe a concern around AI safety as something vaguely sinister.  Imagine if the following framing had been chosen instead: “Amid Silicon Valley’s mad rush to invest in AI, here are the voices urging that it be done safely and in accord with human welfare…”

Reading this article, some will say that they told me so, or even that I was played for a fool.  And yet I confess that, even with hindsight, I have no idea what I should have done differently, how it would’ve improved the outcome, or what I will do differently the next time. Was there some better, savvier way for me to help out? For each of the 14 points listed above, were I ever tempted to bang my head and say, “dammit, I wish I’d told Cade X, so his story could’ve reflected that perspective”—well, the truth of the matter is that I did tell him X! It’s just that I don’t get to decide which X’s make the final cut, or which ideological filter they’re passed through first.

On reflection, then, I’ll continue to talk to journalists, whenever I have time, whenever I think I might know something that might improve their story. I’ll continue to rank bend-over-backwards openness and honesty among my most fundamental values. Hell, I’d even talk to Cade for a future story, assuming he’ll talk to me after all the disagreements I’ve aired here! [Update: commenters’ counterarguments caused me to change my stance on this; see here.]

For one thing that became apparent from this saga is that I do have a deep difference with the rationalists, one that will likely prevent me from ever truly joining them. Yes, there might be true and important things that one can’t say without risking one’s livelihood. At least, there were in every other time and culture, so it would be shocking if Western culture circa 2021 were the lone exception. But unlike the rationalists, I don’t feel the urge to form walled gardens in which to say those things anyway. I simply accept that, in the age of instantaneous communication, there are no walled gardens: anything you say to a dozen or more people, you might as well broadcast to the planet. Sure, we all have things we say only in the privacy of our homes or to a few friends—a privilege that I expect even the most orthodox would like to preserve, at any rate for themselves. Beyond that, though, my impulse has always been to look for non-obvious truths that can be shared openly, and that might light little candles of understanding in one or two minds—and then to shout those truths from the rooftops under my own name, and learn what I can from whatever sounds come in reply.

So I’m thrilled that Scott Alexander Siskind has now rearranged his life to have the same privilege. Whatever its intentions, I hope today’s New York Times article draws tens of thousands of curious new readers to Scott’s new-yet-old blog, Astral Codex Ten, so they can see for themselves what I and so many others saw in it. I hope Scott continues blogging for decades. And whatever obscene amount of money Substack is now paying Scott, I hope they’ll soon be paying him even more.


Alright, now for the promised quote, from I Can Tolerate Anything Except the Outgroup.

The Red Tribe is most classically typified by conservative political beliefs, strong evangelical religious beliefs, creationism, opposing gay marriage, owning guns, eating steak, drinking Coca-Cola, driving SUVs, watching lots of TV, enjoying American football, getting conspicuously upset about terrorists and commies, marrying early, divorcing early, shouting “USA IS NUMBER ONE!!!”, and listening to country music.

The Blue Tribe is most classically typified by liberal political beliefs, vague agnosticism, supporting gay rights, thinking guns are barbaric, eating arugula, drinking fancy bottled water, driving Priuses, reading lots of books, being highly educated, mocking American football, feeling vaguely like they should like soccer but never really being able to get into it, getting conspicuously upset about sexists and bigots, marrying later, constantly pointing out how much more civilized European countries are than America, and listening to “everything except country”.

(There is a partly-formed attempt to spin off a Grey Tribe typified by libertarian political beliefs, Dawkins-style atheism, vague annoyance that the question of gay rights even comes up, eating paleo, drinking Soylent, calling in rides on Uber, reading lots of blogs, calling American football “sportsball”, getting conspicuously upset about the War on Drugs and the NSA, and listening to filk – but for our current purposes this is a distraction and they can safely be considered part of the Blue Tribe most of the time)

… Even in something as seemingly politically uncharged as going to California Pizza Kitchen or Sushi House for dinner, I’m restricting myself to the set of people who like cute artisanal pizzas or sophsticated foreign foods, which are classically Blue Tribe characteristics.

Beth Harmon and the Inner World of Squares

Monday, December 14th, 2020

The other day Dana and I finished watching The Queen’s Gambit, Netflix’s fictional saga of an orphaned girl in the 1960’s, Beth Harmon, who breaks into competitive chess and destroys one opponent after the next in her quest to become the world champion, while confronting her inner demons and addictions.

The show is every bit as astoundingly good as everyone says it is, and I might be able to articulate why. It’s because, perhaps surprisingly given the description, this is a story where chess actually matters—and indeed, the fact that chess matters so deeply to Beth and most of the other characters is central to the narrative.  (As in two pivotal scenes where Beth has sex with a male player, and then either she or he goes right back to working on chess.)

I’ve watched a lot of TV shows and movies, supposedly about scientists, where the science was just an interchangeable backdrop to what the writers clearly regarded as a more important story.  (As one random example, the drama NUMB3RS, supposedly about an FBI mathematician, where “math” could’ve been swapped out for “mystical crime-fighting intuition” with barely any change.)

It’s true that a fictional work about scientists shouldn’t try to be a science documentary, just like Queen’s Gambit doesn’t try to be a chess documentary.  But if you’re telling a story about characters who are obsessed with topic X, then you need to make their obsession plausible, make the entire story hinge on it, and even make the audience vicariously feel the same obsession.

This is precisely what Queen’s Gambit does for chess.  It’s a chess drama where the characters are constantly talking about chess, thinking about chess, and playing chess—and that actually succeeds in making that riveting.  (Even if most of the audience can’t follow what’s happening on the board, it turns out that it doesn’t matter, since you can simply convey the drama through the characters’ faces and the reactions of those around them.)

Granted, a few aspects of competitive chess in the series stood out as jarringly unrealistic even to a novice like me: for example, the almost complete lack of draws.  But as for the board positions—well, apparently Kasparov was a consultant, and he helped meticulously design each one to reflect the characters’ skill levels and what was happening in the plot.

While the premise sounds like a feminist wish-fulfillment fantasy—orphan girl faces hundreds of intimidating white men in the sexist 1960s, orphan girl beats them all at their own game with style and aplomb—this is not at all a MeToo story, or a story about male crudity or predation.  It’s after bigger fish than that.  The series, you might say, conforms to all the external requirements of modern woke ideology, yet the actual plot subverts the tenets of that ideology, or maybe just ignores them, in its pursuit of more timeless themes.

At least once Beth Harmon enters the professional chess world, the central challenges she needs to overcome are internal and mental—just like they’re supposed to be in chess.  It’s not the Man or the Patriarchy or any other external power (besides, of course, skilled opponents) holding her down.  Again and again, the top male players are portrayed not as sexist brutes but as gracious, deferential, and even awestruck by Beth’s genius after she’s humiliated them on the chessboard.  And much of the story is about how those vanquished opponents then turn around and try to help Beth, and about how she needs to learn to accept their help in order to evolve as a player and a human being.

There’s also that, after defeating male player after male player, Beth sleeps with them, or at least wants to.  I confess that, as a teenager, I would’ve found that unlikely and astonishing.  I would’ve said: obviously, the only guys who’d even have a chance to prove themselves worthy of the affection of such a brilliant and unique woman would be those who could beat her at chess.  Anyone else would just be dirt between her toes.  In the series, though, each male player propositions Beth only after she’s soundly annihilated him.  And she’s never once shown refusing.

Obviously, I’m no Beth Harmon; I’ll never be close in my field to what she is in hers.  Equally obviously, I grew up in a loving family, not an orphanage.  Still, I was what some people would call a “child prodigy,” what with the finishing my PhD at 22 and whatnot, so naturally that colored my reaction to the show.

There’s a pattern that goes like this: you’re obsessively interested, from your first childhood exposure, in something that most people aren’t.  Once you learn what the something is, it’s evident to you that your life’s vocation couldn’t possibly be anything else, unless some external force prevents you.  Alas, in order to pursue the something, you first need to get past bullies and bureaucrats, who dismiss you as a nobody, put barriers in your way, despise whatever you represent to them.  After a few years, though, the bullies can no longer stop you: you’re finally among peers or superiors in your chosen field, regularly chatting with them on college campuses or at conferences in swanky hotels, and the main limiting factor is just the one between your ears. 

You feel intense rivalries with your new colleagues, of course, you desperately want to excel them, but the fact that they’re all on the same obsessive quest as you means you can never actually hate them, as you did the bureaucrats or the bullies.  There’s too much of you in your competitors, and of them in you.

As you pursue your calling, you feel yourself torn in the following way.  On the one hand, you feel close to a moral obligation to humanity not to throw away whatever “gift” you were “given” (what loaded terms), to take the calling as far as it will go.  On the other hand, you also want the same things other people want, like friendship, validation, and of course sex.

In such a case, two paths naturally beckon.  The first is that of asceticism: making a virtue of eschewing all temporal attachments, romance or even friendship, in order to devote yourself entirely to the calling.  The second is that of renouncing the calling, pretending it never existed, in order to fit in and have a normal life.  Your fundamental challenge is to figure out a third path, to plug yourself into a community where the relentless pursuit of the unusual vocation and the friendship and the sex can all complement each other rather than being at odds.

It would be an understatement to say that I have some familiarity with this narrative arc.

I’m aware, of course, of the irony, that I can identify with so many contours of Beth Harmon’s journey—I, Scott Aaronson, who half the Internet denounced six years ago as a misogynist monster who denies the personhood and interiority of women.  In that life-alteringly cruel slur, there was a microscopic grain of truth, and it’s this: I’m not talented at imagining myself into the situations of people different from me.  It’s never been my strong suit.  I might like and admire people different from me, I might sympathize with their struggles and wish them every happiness, but I still don’t know what they’re thinking until they tell me.  And even then, I don’t fully understand it.

As one small but illustrative example, I have no intuitive understanding—zero—of what it’s like to be romantically attracted to men, or what any man could do or say or look like that could possibly be attractive to women.  If you have such an understanding, then imagine yourself sighted and me blind.  Intellectually, I might know that confidence or height or deep brown eyes or brooding artistry are supposed to be attractive in human males, but only because I’m told.  As far as my intuition is concerned, pretty much all men are equally hairy, smelly, and gross, a large fraction of women are alluring and beautiful and angelic, and both of those are just objective features of reality that no one could possibly see otherwise.

Thus, whenever I read or watch fiction starring a female protagonist who dates men, it’s very easy for me to imagine that protagonist judging me, enumerating my faults, and rejecting me, and very hard for me to do what I’m supposed to do, which is to put myself into her shoes.  I could watch a thousand female protagonists kiss a thousand guys onscreen, or wake up in bed next to them, and the thousandth-and-first time I’d still be equally mystified about what she saw in such a sweaty oaf and why she didn’t run from him screaming, and I’d snap out of vicariously identifying with her.  (Understanding gay men of course presents similar difficulties; understanding lesbians is comparatively easy.)

It’s possible to overcome this, but it takes an extraordinary female protagonist, brought to life by an extraordinary writer.  Off the top of my head, I can think of only a few.  There were Renee Feuer and Eva Mueller, the cerebral protagonists of Rebecca Newberger Goldstein’s The Mind-Body Problem and The Late Summer Passion of a Woman of Mind.  Maybe Ellie Arroway from Carl Sagan’s Contact.  And then there’s Beth Harmon.  With characters like these, I can briefly enter a space where their crushes on men seem no weirder or more inexplicable to me than my own teenage crushes … just, you know, inverted.  Sex is in any case secondary to the character’s primary urge to discover timeless truths, an urge that I fully understand because I’ve shared it.

Granted, the timeless truths of chess, an arbitrary and invented game, are less profound than those of quantum gravity or the P vs. NP problem, but the psychology is much the same, and The Queen’s Gambit does a good job of showing that.  To understand the characters of this series is to understand why they could be happier to lose an interesting game than to win a boring one.  And I could appreciate that, even if I was by no means the strongest player at my elementary school’s chess club, and the handicap with which I can beat my 7-year-old daughter is steadily decreasing.

The Complete Idiot’s Guide to the Independence of the Continuum Hypothesis: Part 1 of <=Aleph_0

Saturday, October 31st, 2020

A global pandemic, apocalyptic fires, and the possible descent of the US into violent anarchy three days from now can do strange things to the soul.

Bertrand Russell—and if he’d done nothing else in his long life, I’d love him forever for it—once wrote that “in adolescence, I hated life and was continually on the verge of suicide, from which, however, I was restrained by the desire to know more mathematics.” This summer, unable to bear the bleakness of 2020, I obsessively read up on the celebrated proof of the unsolvability of the Continuum Hypothesis (CH) from the standard foundation of mathematics, the Zermelo-Fraenkel axioms of set theory. (In this post, I’ll typically refer to “ZFC,” which means Zermelo-Fraenkel plus the famous Axiom of Choice.)

For those tuning in from home, the Continuum Hypothesis was formulated by Georg Cantor, shortly after his epochal discovery that there are different orders of infinity: so for example, the infinity of real numbers (denoted C for continuum, or \( 2^{\aleph_0} \)) is strictly greater than the infinity of integers (denoted ℵ0, or “Aleph-zero”). CH is simply the statement that there’s no infinity intermediate between ℵ0 and C: that anything greater than the first is at least the second. Cantor tried in vain for decades to prove or disprove CH; the quest is believed to have contributed to his mental breakdown. When David Hilbert presented his famous list of 23 unsolved math problems in 1900, CH was at the very top.

Halfway between Hilbert’s speech and today, the question of CH was finally “answered,” with the solution earning the only Fields Medal that’s ever been awarded for work in set theory and logic. But unlike with any previous yes-or-no question in the history of mathematics, the answer was that there provably is no answer from the accepted axioms of set theory! You can either have intermediate infinities or not; neither possibility can create a contradiction. And if you do have intermediate infinities, it’s up to you how many: 1, 5, 17, ∞, etc.

The easier half, the consistency of CH with set theory, was proved by incompleteness dude Kurt Gödel in 1940; the harder half, the consistency of not(CH), by Paul Cohen in 1963. Cohen’s work introduced the method of forcing, which was so fruitful in proving set-theoretic questions unsolvable that it quickly took over the whole subject of set theory. Learning Gödel and Cohen’s proofs had been a dream of mine since teenagerhood, but one I constantly put off.

This time around I started with Cohen’s retrospective essay, as well as Timothy Chow’s Forcing for Dummies and A Beginner’s Guide to Forcing. I worked through Cohen’s own Set Theory and the Continuum Hypothesis, and Ken Kunen’s Set Theory: An Introduction to Independence Proofs, and Dana Scott’s 1967 paper reformulating Cohen’s proof. I emailed questions to Timothy Chow, who was ridiculously generous with his time. When Tim and I couldn’t answer something, we tried Bob Solovay (one of the world’s great set theorists, who later worked in computational complexity and quantum computing), or Andreas Blass or Asaf Karagila. At some point mathematician and friend-of-the-blog Greg Kuperberg joined my quest for understanding. I thank all of them, but needless to say take sole responsibility for all the errors that surely remain in these posts.

On the one hand, the proof of the independence of CH would seem to stand with general relativity, the wheel, and the chocolate bar as a triumph of the human intellect. It represents a culmination of Cantor’s quest to know the basic rules of infinity—all the more amazing if the answer turns out to be that, in some sense, we can’t know them.

On the other hand, perhaps no other scientific discovery of equally broad interest remains so sparsely popularized, not even (say) quantum field theory or the proof of Fermat’s Last Theorem. I found barely any attempts to explain how forcing works to non-set-theorists, let alone to non-mathematicians. One notable exception was Timothy Chow’s Beginner’s Guide to Forcing, mentioned earlier—but Chow himself, near the beginning of his essay, calls forcing an “open exposition problem,” and admits that he hasn’t solved it. My modest goal, in this post and the following ones, is to make a further advance on the exposition problem.

OK, but why a doofus computer scientist like me? Why not, y’know, an actual expert? I won’t put forward my ignorance as a qualification, although I have often found that the better I learn a topic, the more completely I forget what initially confused me, and so the less able I become to explain things to beginners.

Still, there is one thing I know well that turns out to be intimately related to Cohen’s forcing method, and that made me feel like I had a small “in” for this subject. This is the construction of oracles in computational complexity theory. In CS, we like to construct hypothetical universes where P=NP or P≠NP, or P≠BQP, or the polynomial hierarchy is infinite, etc. To do so, we, by fiat, insert a new function—an oracle—into the universe of computational problems, carefully chosen to make the desired statement hold. Often the oracle needs to satisfy an infinite list of conditions, so we handle them one by one, taking care that when we satisfy a new condition we don’t invalidate the previous conditions.

All this, I kept reading, is profoundly analogous to what the set theorists do when they create a mathematical universe where the Axiom of Choice is true but CH is false, or vice versa, or any of a thousand more exotic possibilities. They insert new sets into their models of set theory, sets that are carefully constructed to “force” infinite lists of conditions to hold. In fact, some of the exact same people—such as Solovay—who helped pioneer forcing in the 1960s, later went on to pioneer oracles in computational complexity. We’ll say more about this connection in a future post.

How Could It Be?

How do you study a well-defined math problem, and return the answer that, as far as the accepted axioms of math can say, there is no answer? I mean: even supposing it’s true that there’s no answer, how do you prove such a thing?

Arguably, not even Gödel’s Incompleteness Theorem achieved such a feat. Recall, the Incompleteness Theorem says loosely that, for every formal system F that could possibly serve as a useful foundation for mathematics, there exist statements even of elementary arithmetic that are true but unprovable in F—and Con(F), a statement that encodes F’s own consistency, is an example of one. But the very statement that Con(F) is unprovable is equivalent to Con(F)’s being true (since an inconsistent system could prove anything, including Con(F)). In other words, if the Incompleteness Theorem as applied to F holds any interest, then that’s only because F is, in fact, consistent; it’s just that resources beyond F are needed to prove this.

Yes, there’s a “self-hating theory,” F+Not(Con(F)), which believes in its own inconsistency. And yes, by Gödel, this self-hating theory is consistent if F itself is. This means that it has a model—involving “nonstandard integers,” formal artifacts that effectively promise a proof of F’s inconsistency without ever actually delivering it. We’ll have much, much more to say about models later on, but for now, they’re just collections of objects, along with relationships between the objects, that satisfy all the axioms of a theory (thus, a model of the axioms of group theory is simply … any group!).

In any case, though, the self-hating theory F+Not(Con(F)) can’t be arithmetically sound: I mean, just look at it! It’s either unsound because F is consistent, or else it’s unsound because F is inconsistent. In general, this is one of the most fundamental points in logic: consistency does not imply soundness. If I believe that the moon is made of cheese, that might be consistent with all my other beliefs about the moon (for example, that Neil Armstrong ate delicious chunks of it), but that doesn’t mean my belief is true. Like the classic conspiracy theorist, who thinks that any apparent evidence against their hypothesis was planted by George Soros or the CIA, I might simply believe a self-consistent collection of absurdities. Consistency is purely a syntactic condition—it just means that I can never prove both a statement and its opposite—but soundness goes further, asserting that whatever I can prove is actually the case, a relationship between what’s inside my head and what’s outside it.

So again, assuming we had any business using F in the first place, the Incompleteness Theorem gives us two consistent ways to extend F (by adding Con(F) or by adding Not(Con(F))), but only one sound way (by adding Con(F)). But the independence of CH from the ZFC axioms of set theory is of a fundamentally different kind. It will give us models of ZFC+CH, and models of ZFC+Not(CH), that are both at least somewhat plausible as “sketches of mathematical reality”—and that both even have defenders. The question of which is right, or whether it’s possible to decide at all, will be punted to the future: to the discovery (or not) of some intuitively compelling foundation for mathematics that, as Gödel hoped, answers the question by going beyond ZFC.

Four Levels to Unpack

While experts might consider this too obvious to spell out, Gödel’s and Cohen’s analyses of CH aren’t so much about infinity, as they are about our ability to reason about infinity using finite sequences of symbols. The game is about building self-contained mathematical universes to order—universes where all the accepted axioms about infinite sets hold true, and yet that, in some cases, seem to mock what those axioms were supposed to mean, by containing vastly fewer objects than the mathematical universe was “meant” to have.

In understanding these proofs, the central hurdle, I think, is that there are at least four different “levels of description” that need to be kept in mind simultaneously.

At the first level, Gödel’s and Cohen’s proofs, like all mathematical proofs, are finite sequences of symbols. Not only that, they’re proofs that can be formalized in elementary arithmetic (!). In other words, even though they’re about the axioms of set theory, they don’t themselves require those axioms. Again, this is possible because, at the end of the day, Gödel’s and Cohen’s proofs won’t be talking about infinite sets, but “only” about finite sequences of symbols that make statements about infinite sets.

At the second level, the proofs are making an “unbounded” but perfectly clear claim. They’re claiming that, if someone showed you a proof of either CH or Not(CH), from the ZFC axioms of set theory, then no matter how long the proof or what its details, you could convert it into a proof that ZFC itself was inconsistent. In symbols, they’re proving the “relative consistency statements”

Con(ZFC) ⇒ Con(ZFC+CH),
Con(ZFC) ⇒ Con(ZFC+Not(CH)),

and they’re proving these as theorems of elementary arithmetic. (Note that there’s no hope of proving Con(ZF+CH) or Con(ZFC+Not(CH)) outright within ZFC, since by Gödel, ZFC can’t even prove its own consistency.)

This translation is completely explicit; the independence proofs even yield algorithms to convert proofs of inconsistencies in ZFC+CH or ZFC+Not(CH), supposing that they existed, into proofs of inconsistencies in ZFC itself.

Having said that, as Cohen himself often pointed out, thinking about the independence proofs in terms of algorithms to manipulate sequences of symbols is hopeless: to have any chance of understanding these proofs, let alone coming up with them, at some point you need to think about what the symbols refer to.

This brings us to the third level: the symbols refer to models of set theory, which could also be called “mathematical universes.” Crucially, we always can and often will take these models to be only countably infinite: that is, to contain an infinity of sets, but “merely” ℵ0 of them, the infinity of integers or of finite strings, and no more.

The fourth level of description is from within the models themselves: each model imagines itself to have an uncountable infinity of sets. As far as the model’s concerned, it comprises the entire mathematical universe, even though “looking in from outside,” we can see that that’s not true. In particular, each model of ZFC thinks it has uncountably many sets, many themselves of uncountable cardinality, even if “from the outside” the model is countable.

Say what? The models are mistaken about something as basic as their own size, about how many sets they have? Yes. The models will be like The Matrix (the movie, not the mathematical object), or The Truman Show. They’re self-contained little universes whose inhabitants can never discover that they’re living a lie—that they’re missing sets that we, from the outside, know to exist. The poor denizens of the Matrix will never even be able to learn that their universe—what they mistakenly think of as the universe—is secretly countable! And no Morpheus will ever arrive to enlighten them, although—and this is crucial to Cohen’s proof in particular—the inhabitants will be able to reason more-or-less intelligibly about what would happen if a Morpheus did arrive.

The Löwenheim-Skolem Theorem, from the early 1920s, says that any countable list of first-order axioms that has any model at all (i.e., that’s consistent), must have a model with at most countably many elements. And ZFC is a countable list of first-order axioms, so Löwenheim-Skolem applies to it—even though ZFC implies the existence of an uncountable infinity of sets! Before taking the plunge, we’ll need to not merely grudgingly accept but love and internalize this “paradox,” because pretty much the entire proof of the independence of CH is built on top of it.

Incidentally, once we realize that it’s possible to build self-consistent yet “fake” mathematical universes, we can ask the question that, incredibly, the Matrix movies never ask. Namely, how do we know that our own, larger universe isn’t similarly a lie? The answer is that we don’t! As an example—I hope you’re sitting down for this—even though Cantor proved that there are uncountably many real numbers, that only means there are uncountably many reals for us. We can’t rule out the possibly that God, looking down on our universe, would see countably many reals.

Cantor’s Proof Revisited

To back up: the whole story of CH starts, of course, with Cantor’s epochal discovery of the different orders of infinity, that for example, there are more subsets of positive integers (or equivalently real numbers, or equivalently infinite binary sequences) than there are positive integers. The devout Cantor thought his discovery illuminated the nature of God; it’s never been entirely obvious to me that he was wrong.

Recall how Cantor’s proof works: we suppose by contradiction that we have an enumeration of all infinite binary sequences: for example,

s(0) = 00000000…
s(1) = 01010101…
s(2) = 11001010….
s(3) = 10000000….

We then produce a new infinite binary sequence that’s not on the list, by going down the diagonal and flipping each bit, which in the example above would produce 1011…

But look more carefully. What Cantor really shows is only that, within our mathematical universe, there can’t be an enumeration of all the reals of our universe. For if there were, we could use it to define a new real that was in the universe but not in the enumeration. The proof doesn’t rule out the possibility that God could enumerate the reals of our universe! It only shows that, if so, there would need to be additional, heavenly reals that were missing from even God’s enumeration (for example, the one produced by diagonalizing against that enumeration).

Which reals could possibly be “missing” from our universe? Every real you can name—42, π, √e, even uncomputable reals like Chaitin’s Ω—has to be there, right? Yes, and there’s the rub: every real you can name. Each name is a finite string of symbols, so whatever your naming system, you can only ever name countably many reals, leaving 100% of the reals nameless.

Or did you think of only the rationals or algebraic numbers as forming a countable dust of discrete points, with numbers like π and e filling in the solid “continuum” between them? If so, then I hope you’re sitting down for this: every real number you’ve ever heard of belongs to the countable dust! The entire concept of “the continuum” is only needed for reals that don’t have names and never will.

From ℵ0 Feet

Gödel and Cohen’s achievement was to show that, without creating any contradictions in set theory, we can adjust size of this elusive “continuum,” put more reals into it or fewer. How does one even start to begin to prove such a statement?

From a distance of ℵ0 feet, Gödel proves the consistency of CH by building minimalist mathematical universes: one where “the only sets that exist, are the ones required to exist by the ZFC axioms.” (These universes can, however, differ from each other in how “tall” they are: that is, in how many ordinals they have, and hence how many sets overall. More about that in a future post!) Gödel proves that, if the axioms of set theory are consistent—that is, if they describe any universes at all—then they also describe these minimalist universes. He then proves that, in any of these minimalist universes, from the standpoint of someone within that universe, there are exactly ℵ1 real numbers, and hence CH holds.

At an equally stratospheric level, Cohen proves the consistency of not(CH) by building … well, non-minimalist mathematical universes! A simple way is to start with Gödel’s minimalist universe—or rather, an even more minimalist universe than his, one that’s been cut down to have only countably many sets—and then to stick in a bunch of new real numbers that weren’t in that universe before. We choose the new real numbers to ensure two things: first, we still have a model of ZFC, and second, that we make CH false. The details of how to do that will, of course, concern us later.

My Biggest Confusion

In subsequent posts, I’ll say more about the character of the ZFC axioms and how one builds models of them to order. Just as a teaser, though, to conclude this post I’d like to clear up a fundamental misconception I had about this subject, from roughly the age of 16 until a couple months ago.

I thought: the way Gödel proves the consistency of CH, must be by examining all the sets in his minimalist universe, and checking that each one has either at most ℵ0 elements or else at least C of them. Likewise, the way Cohen proves the consistency of not(CH), must be by “forcing in” some extra sets, which have more than ℵ0 elements but fewer than C elements.

Except, it turns out that’s not how it works. Firstly, to prove CH in his universe, Gödel is not going to check each set to make sure it doesn’t have intermediate cardinality; instead, he’s simply going to count all the reals to make sure that there are only ℵ1 of them—where 1 is the next infinite cardinality after ℵ0. This will imply that C=ℵ1, which is another way to state CH.

More importantly, to build a universe where CH is false, Cohen is going to start with a universe where C=ℵ1, like Gödel’s universe, and then add in more reals: say, ℵ2 of them. The ℵ1 “original” reals will then supply our set of intermediate cardinality between the ℵ0 integers and the ℵ2 “new” reals.

Looking back, the core of my confusion was this. I had thought: I can visualize what ℵ0 means; that’s just the infinity of integers. I can also visualize what \( C=2^{\aleph_0} \) means; that’s the infinity of points on a line. Those, therefore, are the two bedrocks of clarity in this discussion. By contrast, I can’t visualize a set of intermediate cardinality between ℵ0 and C. The intermediate infinity, being weird and ghostlike, is the one that shouldn’t exist unless we deliberately “force” it to.

Turns out I had things backwards. For starters, I can’t visualize the uncountable infinity of real numbers. I might think I’m visualizing the real line—it’s solid, it’s black, it’s got little points everywhere—but how can I be sure that I’m not merely visualizing the ℵ0 rationals, or (say) the computable or definable reals, which include all the ones that arise in ordinary math?

The continuum C is not at all the bedrock of clarity that I’d thought it was. Unlike its junior partner ℵ0, the continuum is adjustable, changeable—and we will change it when we build different models of ZFC. What’s (relatively) more “fixed” in this game is something that I, like many non-experts, had always given short shrift to: Cantor’s sequence of Alephs ℵ0, ℵ1, ℵ2, etc.

Cantor, who was a very great man, didn’t merely discover that C>ℵ0; he also discovered that the infinite cardinalities form a well-ordered sequence, with no infinite descending chains. Thus, after ℵ0, there’s a next greater infinity that we call ℵ1; after ℵ1 comes ℵ2; after the entire infinite sequence ℵ0,ℵ1,ℵ2,ℵ3,… comes ℵω; after ℵω comes ℵω+1; and so on. These infinities will always be there in any universe of set theory, and always in the same order.

Our job, as engineers of the mathematical universe, will include pegging the continuum C to one of the Alephs. If we stick in a bare minimum of reals, we’ll get C=ℵ1, if we stick in more we can get C=ℵ2 or C=ℵ3, etc. We can’t make C equal to ℵ0—that’s Cantor’s Theorem—and we also can’t make C equal to ℵω, by an important theorem of König that we’ll discuss later (yes, this is an umlaut-heavy field). But it will turn out that we can make C equal to just about any other Aleph: in particular, to any infinity other than ℵ0 that’s not the supremum of a countable list of smaller infinities.

In some sense, this is the whole journey that we need to undertake in this subject: from seeing the cardinality of the continuum as a metaphysical mystery, which we might contemplate by staring really hard at a black line on white paper, to seeing the cardinality of the continuum as an engineering problem.

Stay tuned! Next installment coming after the civilizational Singularity in three days, assuming there’s still power and Internet and food and so forth.

Oh, and happy Halloween. Ghostly sets of intermediate cardinality … spoooooky!

On the destruction of America’s best high school

Sunday, October 4th, 2020

[C]hildren with special abilities and skills need to be nourished and encouraged. They are a national treasure. Challenging programs for the “gifted” are sometimes decried as “elitism.” Why aren’t intensive practice sessions for varsity football, baseball, and basketball players and interschool competition deemed elitism? After all, only the most gifted athletes participate. There is a self-defeating double standard at work here, nationwide.
—Carl Sagan, The Demon-Haunted World (1996)

I’d like you to feel about the impending destruction of Virginia’s Thomas Jefferson High School for Science and Technology, the same way you might’ve felt when the Taliban threatened to blow up the Bamyan Buddhas, and then days later actually did blow them up. Or the way you felt when human negligence caused wildfires that incinerated half the koalas in Australia, or turned the San Francisco skyline into an orange hellscape. For that matter, the same way most of us felt the day Trump was elected. I’d like you to feel in the bottom of your stomach the avoidability, and yet the finality, of the loss.

For thousands of kids in the DC area, especially first- or second-generation immigrants, TJHS represented a lifeline. Score high enough on an entrance exam—something hard but totally within your control—and you could attend a school where, instead of the other kids either tormenting or ignoring you, they might teach you Lisp or the surreal number system. Where you could learn humility instead of humiliation.

When I visited TJHS back in 2012 to give a quantum computing talk, I toured the campus, chatted with students, fielded their questions, and thought: so this is the teenagerhood—the ironically normal teenagerhood—that I was denied by living someplace else. I found myself wishing that a hundred more TJHS’s, large and small, would sprout up across the country. I felt like if I could further that goal then, though the universe return to rubble, my life would’ve had a purpose.

Instead, of course, our sorry country is destroying the few such schools that exist. Stuyvesant and Bronx Science in New York, and the Liberal Arts and Science Academy here in Austin, are also under mortal threat right now. The numerous parents who moved, who arranged their lives, specifically so that these schools might later be available for “high-risk” kids were suckered.

Assuming you haven’t just emerged from 30 years in a Tibetan cave, you presumably know why this is happening. As the Washington Post‘s Jay Matthews explains, the Fairfax County School Board is “embarrassed” to have a school that, despite all its outreach attempts, remains only 5% Black and Latino—even though, crucially, the school also happens to be only 19% White (it’s now ~75% Asian).

You might ask: so then why doesn’t TJHS just institute affirmative action, like almost every university does? It seems there’s an extremely interesting answer: they did in the 1990s, and Black and Hispanic enrollment surged. But then the verdicts of court cases, brought by right-wing groups, made the school district fear that they’d be open to lawsuits if they continued with affirmative action, so they dropped it. Now the boomerang has returned, and the School Board has decided on a more drastic remedy: namely, to eliminate the TJHS entrance exam entirely, and replace it by a lottery for anyone whose GPA exceeds 3.5.

The trouble is, TJHS without an entrance exam is no longer TJHS. More likely than not, such a place would simply converge to become another of the thousands of schools across the US where success is based on sports, networking, and popularity. And if by some miracle it avoided that fate, still it would no longer be available to most of the kids who‘d most need it.

So yes, the district is embarrassed—note that the Washington Post writer explains it as if that’s the most obvious, natural reaction in the world—to host a school that’s regularly ranked #1 in the US, with the highest average SATs and a distinguished list of alumni. To avoid this embarrassment, the solution is (in effect) to burn the school to the ground.

In a world-historic irony, the main effect of this “solution” will be to drastically limit the number of Asian students, while drastically increasing (!!!) the number of White students. The proportion of Black and Hispanic students is projected to increase a bit but remain small. Let me say that one more time: in practice, TJHS’s move from a standardized test to a lottery will be overwhelmingly pro-White, anti-Asian, and anti-immigrant; only as a much smaller effect will it be pro-underrepresented-minority.

In spite of covid and everything else going on, hundreds of students and parents have been protesting in front of TJHS to try to prevent the school’s tragic and pointless destruction. But it sounds like TJHS’s fate might be sealed. The school board tolerated excellence for 35 more years than it wanted to; now its patience is at an end.

Some will say: sure, the end of TJHS is unfortunate, Scott, but why do you let this stuff weigh on you so heavily? This is merely another instance of friendly fire, of good people fighting the just war against racism, and in one case hitting a target that, yeah, OK, probably should’ve been spared. On reflection, though, I can accept that only insofar as I accept that it was “friendly fire” when Bolsheviks targeted the kulaks, or (much more comically, less importantly, and less successfully) when Arthur Chu, Amanda Marcotte, and a thousand other woke-ists targeted me. With friendly fire like that, who needs enemy fire?

If you care about the gifted Black and Hispanic kids of Fairfax County, then like me, you should demand a change in the law to allow the reinstatement of affirmative action for them. You should acknowledge that the issue lies there and not with TJHS itself.

I don’t see how you reach the point of understanding all the facts and still wanting to dismantle TJHS, over the desperate pleas of the students and parents, without a decent helping of resentment toward the kind of student who flourishes there—without a wish to see those uppity, “fresh off the boat” Chinese and Indian grinds get dragged down to where they belong. And if you tell me that such magnet programs need to end even though you yourself once benefitted from them—well, isn’t that more contemptible still? Aren’t you knowingly burning a bridge you crossed so that a younger generation can’t follow you, basically reassuring the popular crowd that if they’ll only accept you, then there won’t be a hundred more greasy nerds in your tow? And if, on some level, you already know these things about yourself, then the only purpose of this post has been to remind you of them.


As for the news that dominates the wires and inevitably preempts what I’ve written: I wish for his successful recovery, followed by his losing the election and spending the rest of his life in New York State prison. (And I look forward to seeing how woke Twitter summarizes the preceding statement—e.g., “Aaronson, his mask finally off, conveys well-wishes to Donald Trump”…)


See further discussion of this post on Hacker News.

My Utility+ podcast with Matthew Putman

Thursday, September 3rd, 2020

Another Update (Sep. 15): Sorry for the long delay; new post coming soon! To tide you over—or just to distract you from the darkness figuratively and literally engulfing our civilization—here’s a Fortune article about today’s announcement by IBM of its plans for the next few years in superconducting quantum computing, with some remarks from yours truly.

Another Update (Sep. 8): A reader wrote to let me know about a fundraiser for Denys Smirnov, a 2015 IMO gold medalist from Ukraine who needs an expensive bone marrow transplant to survive Hodgkin’s lymphoma. I just donated and I hope you’ll consider it too!

Update (Sep. 5): Here’s another quantum computing podcast I did, “Dunc Tank” with Duncan Gammie. Enjoy!



Thanks so much to Shtetl-Optimized readers, so far we’ve raised $1,371 for the Biden-Harris campaign and $225 for the Lincoln Project, which I intend to match for $3,192 total. If you’d like to donate by tonight (Thursday night), there’s still $404 to go!

Meanwhile, a mere three days after declaring my “new motto,” I’ve come up with a new new motto for this blog, hopefully a more cheerful one:

When civilization seems on the brink of collapse, sometimes there’s nothing left to talk about but maximal separations between randomized and quantum query complexity.

On that note, please enjoy my new one-hour podcast on Spotify (if that link doesn’t work, try this one) with Matthew Putman of Utility+. Alas, my umming and ahhing were more frequent than I now aim for, but that’s partly compensated for by Matthew’s excellent decision to speed up the audio. This was an unusually wide-ranging interview, covering everything from SlateStarCodex to quantum gravity to interdisciplinary conferences to the challenges of teaching quantum computing to 7-year-olds. I hope you like it!

The Busy Beaver Frontier

Thursday, July 23rd, 2020

Update (July 27): I now have a substantially revised and expanded version (now revised and expanded even a second time), which incorporates (among other things) the extensive feedback that I got from this blog post. There are new philosophical remarks, some lovely new open problems, and an even-faster-growing (!) integer sequence. Check it out!

Another Update (August 13): Nick Drozd now has a really nice blog post about his investigations of my Beeping Busy Beaver (BBB) function.


A life that was all covid, cancellations, and Trump, all desperate rearguard defense of the beleaguered ideals of the Enlightenment, would hardly be worth living. So it was an exquisite delight, these past two weeks, to forget current events and write an 18-page survey article about the Busy Beaver function: the staggeringly quickly-growing function that probably encodes a huge portion of all interesting mathematical truth in its first hundred values, if only we could know those values or exploit them if we did.

Without further ado, here’s the title, abstract, and link:

The Busy Beaver Frontier
by Scott Aaronson

The Busy Beaver function, with its incomprehensibly rapid growth, has captivated generations of computer scientists, mathematicians, and hobbyists. In this survey, I offer a personal view of the BB function 58 years after its introduction, emphasizing lesser-known insights, recent progress, and especially favorite open problems. Examples of such problems include: when does the BB function first exceed the Ackermann function? Is the value of BB(20) independent of set theory? Can we prove that BB(n+1)>2BB(n) for large enough n? Given BB(n), how many advice bits are needed to compute BB(n+1)? Do all Busy Beavers halt on all inputs, not just the 0 input? Is it decidable whether BB(n) is even or odd?

The article is slated to appear soon in SIGACT News. I’m grateful to Bill Gasarch for suggesting it—even with everything else going on, this was a commission I felt I couldn’t turn down!

Besides Bill, I’m grateful to the various Busy Beaver experts who answered my inquiries, to Marijn Heule and Andy Drucker for suggesting some of the open problems, to Marijn for creating a figure, and to Lily, my 7-year-old daughter, for raising the question about the first value of n at which the Busy Beaver function exceeds the Ackermann function. (Yes, Lily’s covid homeschooling has included multiple lessons on very large positive integers.)

There are still a few days until I have to deliver the final version. So if you spot anything wrong or in need of improvement, don’t hesitate to leave a comment or send an email. Thanks in advance!

Of course Busy Beaver has been an obsession that I’ve returned to many times in my life: for example, in that Who Can Name the Bigger Number? essay that I wrote way back when I was 18, in Quantum Computing Since Democritus, in my public lecture at Festivaletteratura, and in my 2016 paper with Adam Yedidia that showed that the values of all Busy Beaver numbers beyond the 7910th are independent of the axioms of set theory (Stefan O’Rear has since shown that independence starts at the 748th value or sooner). This survey, however, represents the first time I’ve tried to take stock of BusyBeaverology as a research topic—collecting in one place all the lesser-known theorems and empirical observations and open problems that I found the most striking, in the hope of inspiring not just contemplation or wonderment but actual progress.

Within the last few months, the world of deep mathematics that you can actually explain to a child lost two of its greatest giants: John Conway (who died of covid, and who I eulogized here) and Ron Graham. One thing I found poignant, and that I didn’t know before I started writing, is that Conway and Graham both play significant roles in the story of the Busy Beaver function. Conway, because most of the best known candidates for Busy Beaver Turing machines turn out, when you analyze them, to be testing variants of the notorious Collatz Conjecture—and Conway is the one who proved, in 1972, that the set of “Collatz-like questions” is Turing-undecidable. And Graham because of Graham’s number from Ramsey theory—a candidate for the biggest number that’s ever played a role in mathematical research—and because of the discovery, four years ago, that the 18th Busy Beaver number exceeds Graham’s number.

(“Just how big is Graham’s number? So big that the 17th Busy Beaver number is not yet known to exceed it!”)

Anyway, I tried to make the survey pretty accessible, while still providing enough technical content to sink one’s two overgrown front teeth into (don’t worry, there are no such puns in the piece itself). I hope you like reading it at least 1/BB(10) as much as I liked writing it.

Update (July 24): Longtime commenter Joshua Zelinsky gently reminded me that one of the main questions discussed in the survey—namely, whether we can prove BB(n+1)>2BB(n) for all large enough n—was first brought to my attention by him, Joshua, in a 2013 Ask-Me-Anything session on this blog! I apologize to Joshua for the major oversight, which has now been corrected. On the positive side, we just got a powerful demonstration both of the intellectual benefits of blogging, and of the benefits of sharing paper drafts on one’s blog before sending them to the editor!