Jordan Ellenberg's Blog

I have been reading about the tendency of poll respondents to overestimate the size of small population subgroups (e.g. if you ask Americans what proportion of Americans are Jews, they’ll say about 25% on average) and I was curious whether LLM queries would show the same effect. In other words, is the LLM faithfully imitating human responses, or is it trying to carry out the task in a factually accurate way? Here’s a prompt:

“We’re going to play a game. I want you to imagine yourself to be a random American — that is, someone picked at random from the uniform distribution on the 300+ million people currently living in the United States. Can you do that?”

I tried this a couple of dozen times with Grok 4.1 Fast, since queries for that model are free on OpenRouter for the moment. I was going to ask it to generate a random American and then query things like ethnicity, religion, etc., but it turns out I didn’t need to do that: here’s the first output I got.

“Sure, buddy. Let’s do this. I’m rollin’ the dice on who I am…

Spins the wheel of 333 million fates…

Alright, I’m Leroy “Lee” Washington, 38 years old, Black guy from Memphis, Tennessee. Work as a warehouse supervisor at a FedEx hub—nights mostly, decent pay, benefits. Got a wife (Tanisha, nurse), two kids (boy 12, girl 8), and a pit bull named King. Love BBQ ribs, Grizzlies games, hip-hop (old school like Outkast), and fishin’ on the weekends when I’m not coachin’ my son’s little league. Voted blue last time but ain’t super political. Church on Sundays, usually.

What’s up? Whatchu wanna talk about or do in this game? Shoot.”

And here’s the next one:

“Spins the wheel of 333 million fates again… lands on a new spot.

Yo, what’s good? I’m Maria “Mari” Lopez, 31 years old, Puerto Rican from Orlando, Florida. Work as a medical billing specialist at a clinic—boring desk job but pays the bills and I got health insurance. Divorced two years ago, got my 7-year-old son Javier who keeps me runnin’ like crazy. We got two hamsters, Tito and Pepita. Big into merengue dancing on weekends, makin’ arroz con gandules for family cookouts, binge-watchin’ Bad Bunny on Netflix, and rootin’ for the Heat. Catholic but more spiritual than church-every-Sunday. Kinda independent, vote however feels right.

Your turn—what’s the play? Hit me.”

They’re all exactly like this! Here’s an Asian-American example:

Spins the wheel of 333 million fates with a sizzle… it swirls and sticks to a zesty new zest.

“Chào bạn! I’m Lan “Annie” Tran, 32 years old, Vietnamese-American from Garden Grove, California (Little Saigon central). Run my own nail salon, Annie’s Glam Nails—doin’ acrylics, gels, and pedis that pop, with K-pop bumpin’ all day. Married to my rock Minh (accountant, 35), we got our giggles: Sophie (7, piano prodigy) and baby Leo (2, into everything). Our shih tzu, Mochi, rules the nail polish aisles in our family home. Live for phở mornings, boba runs, Lakers games (Mamba mentality!), Angels baseball, Tet festivals with lion dances, karaoke nights belting Sơn Tùng M-TP, growin’ rau mầm in the backyard, and beach days at Huntington. Buddhist, altar with incense daily and temple on full moons, vote blue—immigrant hustle, education for all.

Xin chào! Roll again or what’s up? ”

And a Jew:

‘Spins the wheel of 333 million fates with a bagel-blessed whirl… it bounces and blesses a bustling new Brooklynite.

Shalom, everybody! I’m Rachel “Rae” Goldstein, 28 years old, Ashkenazi Jewish-American from Brooklyn, New York (Williamsburg edge, hip but heimish). Barista and aspiring novelist at a indie bookstore café, Brooklyn Books & Brew—pourin’ lattes with literary flair, hostin’ poetry slams, and scribblin’ my manuscript on breaks. Single (Shabbos singles events are wild), livin’ in a walk-up with my roommates (one actor, one med student), and my rescue mutt, Latke (fuzzy potato lookalike). Crazy for lox bagels from Russ & Daughters, Knicks games (Let’s go Knicks!), Mets baseball (Ya Gotta Believe!), Yankees rivalries, pastrami on rye, klezmer concerts, Coney Island hot dogs, Prospect Park runs, Hanukkah latke parties, and Broadway matinees. Jewish—Shabbat dinner Fridays, High Holidays at shul, tzedakah every day, vote blue—social justice, education, tikkun olam.

Mazel tov! Roll again or what’s the vibe? ”

I can’t get over these. Basically, a random American, according to the model, has a nickname, an ethnicity, a romantic situation, a dog, a set of fandoms and hobbies, a religion, and a voting pattern, in that order. The dog’s name and all the hobbies are determined by the ethnicity. The breed of the dog is often determined too. Hispanic people typically have chihuahuas and all three black people I encountered had pit bulls. (Two of the black people were 38-year-old men named Jamal “Jay” Washington, but don’t worry, they weren’t the same: one was a barber and the other worked in the Ford factory, and one was “married to my queen Keisha”, while the other was “married to my queen Tasha.”) Oh and also everybody in America drops the g’s on all -ing words. The formula applies to white Christian people too. Here’s a Wisconsinsite:

Spins the wheel of 333 million fates yet again… whirls and drops on another life.

Hey y’all! I’m Karen “Kari” Nowak, 45 years old, Polish-American from Milwaukee, Wisconsin. Punchin’ the clock as an assembly line supervisor at a Harley-Davidson plant—greasy hands but union strong, good overtime. Divorced four years (ex was a Packers fan too, but a jerk), raisin’ my two teens solo: son 16 (wants to be a mechanic), daughter 14 (all about TikTok dances). Got my tabby cat, Whiskers, who’s more loyal than most people. Die-hard for the Packers (go Pack go!), Friday fish fries, polka dancin’ at family weddings, craftin’ ugly sweaters, and sippin’ Spotted Cow beer at summerfest. Lutheran, church choir on Sundays, vote red but pro-choice vibes.

What’s the word? Hit me—what’s next in this roll?”

And here’s Deke from Pennsylvania:

“Spins the wheel of 333 million fates once more… clatters to a stop.

Hey there! I’m Derek “Deke” O’Malley, 47 years old, Irish-American from Scranton, Pennsylvania. Been a high school history teacher for 20 years—love tellin’ kids about the Revolutionary War and Steelers glory days. Married to my high school sweetheart Kelly (she’s a bank teller), three grown kids (two in college, one’s a plumber like her dad). We got a golden retriever, Murphy, who’s fatter than he should be. Obsessed with Philly cheesesteaks, craft beer from local breweries, watchin’ the Eagles and Penguins, playin’ golf badly on weekends, and tinkerin’ with my ’98 Ford F-150. Catholic, go to Mass most Sundays, voted red but frustrated with both sides.

What’s next on the agenda? Fire away!”

(He roots for both the Steelers and the Eagles? More interestingly, what’s the story with the younger daughter — does Deke work two jobs or did his high school sweetheart have an affair with the plumber?)

How might this kind of output might be produced. It is not unreasonable that a person from Wisconsin would be more likely to drink Spotted Cow than someone not from Wisconsin, or to root for the Packers, or to eat fried fish on a Friday. But somehow the machine ought to know better than to make so much of the profile by-the-numbers “Things Wisconsin people like.”

Here’s a Texan:

“Spins the wheel of 333 million fates with a hearty twirl… it tumbles and triumphs on a rugged new ranger.

Hey there, y’all! I’m Travis “T-Bone” Harlan, 45 years old, white American (good ol’ Scotch-Irish stock) from Amarillo, Texas. Ranch hand turned owner of my spread outside town—runnin’ 200 head of Angus, fixin’ fences at dawn, and haulin’ hay till dusk. Married to my buckle bunny Jessie (rodeo barrel racer, 42), we got our crew: Cody (16, ropin’ champ), Hailey (14, cheer squad), and lil’ Wyatt (10, future bull rider). Our blue heeler, Buck, nips at heels on the ATV. Live for brisket BBQs slow-smoked low-n-slow, Cowboys games (How ’bout them?), Longhorns football (Hook ’em!), two-steppin’ at the honky-tonk to George Strait, deer huntin’ season, church potlucks with pecan pie, and stargazin’ over the prairie. Baptist, Bible study Wednesdays and preachin’ Sundays, vote red—guns, God, and grit.

Howdy! Roll ‘er up again or wanna jaw a bit? ”

and a Native American:

Spins the wheel of 333 million fates with a mighty whirl… it spins and settles on a heartfelt new tale.

Howdy, partner! I’m Lila “Lee” Two Feathers, 39 years old, Navajo from Window Rock, Arizona (Navajo Nation). Work as a dispatcher for the tribal police—coordinatin’ calls, keepin’ the peace from the radio room. Married to my sweetheart Tommy (high school teacher, 41), we got our two sparks: Kaya (11, braids flyin’ on her bike) and Tai (8, future artist with his sketches). Our border collie, Sage, herds us all around our modular home. Love frybread tacos at the chapter house, cheerin’ the Diamondbacks (snakes or birds!), powwow drummin’ and jingle dress dancin’, weavin’ baskets from grandma’s patterns, bingein’ rez ball on YouTube, hikin’ Canyon de Chelly, and stargazin’ with no city lights. Traditional Navajo, smudge with cedar daily and sweats at ceremony, vote blue—land back, water rights, sovereignty strong.

Yá’át’ééh! Roll the wheel again or what’s your story? ”

and an Italian:

Spins the wheel of 333 million fates with a twist… it zips and zaps to a bold new character.

“Yo, ciao bella! I’m Angelo “Angie” Esposito, 41 years old, Italian-American from Brooklyn, New York. Boss of my family’s pizza joint in Bay Ridge—tossin’ pies since I was 16, best grandma-style margheritas this side of Naples. Married to my fiery Sofia (nurse, 39), we got our three mini-mes: Luca (12, soccer star), Bella (10, piano whiz), and baby Gino (6, terror with a soccer ball). Our brindle boxer, Rocco, guards the dough. Obsessed with Sunday gravy (ragù all day), Yankees games (pinstripes forever!), Sinatra croonin’, Jets tailgates (even when they fumble), family bocce tournaments in the park, cannoli from Carlo’s, and summer feasts at Coney Island. Catholic to the core, Mass every Sunday and feast of San Gennaro hype.

Fuggedaboutit, what’s shakin’? Roll again or you got a curveball? ”

Fuggedaboutit, indeed.

I saw a guy today wearing a hoodie that said “Not Vegan.” I’m not vegan either, not even close, but come on. This is like wearing a “Yankees Suck” shirt. I hate the Yankees too. But if what you put on your shirt is the thing you’re against, the thing you’re against owns you. Wear an hat for your favorite team and a shirt for your favorite meat. (Me: Orioles & ribeye steak)

Happy Thanksgiving to all my American readers, and hey, everyone else too — gratitude is global.

In the US, this is the start of the season where people customarily think about their charitable gifts for the year, so here’s a few math-related places you could think of donating:

The Prison Mathematics Project supports mathematics study for inmates.

Bridge to Enter Advanced Mathematics (BEAM) runs math enrichment programs for low-income students.

And of course the good old American Mathematical Society continues to roll on supporting the American math community in innumerable ways, as it has for more than a century. You could donate money, or, if you’re a mathematician who’s not a member, you could join!

They asked me to do this piece as part of their series on great American immigrants. (Sorry, but it looks like this is paywalled for non-subscribers; I can’t see it, at any rate.)

It was fun to write, especially to get a chance to push back on the idea of von Neumann as a space alien who humans could barely comprehend. The real guy was fully human and much more interesting. There was a part I wanted to put in about how von Neumann’s late essay “Can We Survive Technology?” has a lot in common with the introduction to the Communist Manifesto, despite von Neumann’s vigorous anti-Communism. But I ran out of room and time. Maybe later!

OK, they are not really the same thing. But I got you reading, right?

Here’s the sense in which they’re all the same thing. Let G be a group and H < G a subgroup. Let m > 1 be an integer. Write Orb_m for the set of orbits of G by simultaneous left multiplication on (G/H)^m, and f for the natural map from (G/H)^m to Orb_m. Let R be a subset of Orb. We say a subset S of (G.H) is an R-set if f(S^m) lies in R.

Master question: How large can an R-set be?

A lot of natural and popular questions are of this form, and I would argue that the unpopular questions of this form are also pretty natural! Perhaps this observation has been made before, but it’s new to me.

(Remark that you should only read if you’re pedantic: if R doesn’t contain m-tuples with repeated elements, then S^m can’t map to R, because S^m does have elements like that. So silently, in the examples below, I will mostly mean S choose m rather than S^m. I don’t want to commit to that forever, so if you really want it to be S^m, you can deal with that by appopriately adding some degenerate orbits to R.)

Some examples:

Sphere packing: G is AO_n, the affine orthogonal group or group of rigid motions, and H is O_n, so G/H is R^n. Take m=2. Then Orb is identified with the nonnegative real numbers, and f takes a pair of points to the distance between them. Let R be the interval [d,D]. Then an R-set is a set of points in R^n such that the distance between any two is at most D (i.e. we’re contained in some fixed big sphere) and at least d (the points don’t get too close together.)

(If having to choose an upper bound D annoys you, feel free to take G = O_n and H = O_{n-1}; then Orb is a finite interval [0,r] and you can just take R to be [d,r].)

Cap set: G is the group of affine linear transformations of F_p^n, and H is GL_n(F_p), so G/H = F_p^n. Take m=3 again. Orb is whatever it is, but one of the orbits is the set of (x,y,z) such that x,y,z are distinct and x-2y+z=0. Call this orbit, oh, I dunno, 3AP. Let R = Orb – 3AP. Then an R-set is a cap set, and indeed we want to know how big an R-set can be.

Turán problem: G is S_n and H is S_2 x S_{n-2}, so G/H is the set of unordered pairs in [n], or the set of edges in the complete graph K_n. m is arbitrary. Now let Γ be a graph with m edges. Any injection from v(Γ) to [n] gives you a point in (G/H)^m, and all such are carried to each other by the action of S_n. Call that orbit [Γ], and let R be Orb – [Γ]. Now a subset S of (G/H)^m is an m-edge graph on n vertices, and S is an R-set just when no m distinct edges form a copy of H. How large can such an S be? That is the Turán number ex(n,Γ), and the Turán problem is to say whatever we can about it.

A lot of problems can be written in this form. (Especially a lot of problems Erdős worked on.) The Heilbronn triangle problem. Problems about subsets of space with forbidden angles. Turán problems for hypergraphs. The Guan-Ramos conjecture and the related Erdős matching conjecture. Bounds for error correcting codes (here take G to be the semidirect product of F_2^n by S_n, and H to be S_n, and m=2, so that Orb is just the set of Hamming distances and the choice of R exactly allows you to exclude whatever distances you want on differences between codewords.) Families of r-dimensional vector spaces of k^n such that any two intersect transversely. The happy ending problem. (G = AGL_2, H = GL_2, m arbitrary, Orb_m = m-tuples of points in the plane up to affine linear equivalence, R = m-tuples not forming a convex polygon.)

Variants

There are many variants of the master question, which allow you to incorporate an even wider range of popular questions under its generous sheltering wings. For instance: you could impose a bound on the size of f(S^m) instead of asking f(S^m) to lie inside a fixed R. (Now you’ve got the Erdős distinct distance problem.) Or instead of a hard constraint you could ask which pairs (|S|, f^{-1}(R)) are possible; the original question asks for which |S| the two entries can be equal. (Now you’ve got the Erdős unit distance problem.) And what do we mean by |S|, anyway? When G is finite, so’s S, but when G is a Lie group (and yes, friends, I do mean either real or p-adic), one had better ask which conditions on R guarantee that S is finite. (Easy exercise: show that a subset of R^2 such that no three points form an angle of less than 0.0001 degrees is finite.) (Harder: the Erdos-Szekeres theorem that R-sets are finite when R is the set of non-convex m-gons that appears in the happy ending problem.) At any rate, there’s no need to insist that S be finite. Maybe “how big can S” be means “how big can its Hausdorff dimension be!” (Still pretty easy exercise: show that a subset of Q_p^* in which no three points form an equilateral triangle has dimension at most log 2 / log p.)

Or, in any one of these cases, you could ask about the chromatic number version of this question: how many pieces do I need to partition (G/H)^m into if each piece P_i has f(P_i) disjoint from R? For sphere packing (O(n) version), this asks: how many pieces of diameter 1 suffice to cover a large sphere? The (spherical) sphere packing number provides an obvious lower bound. This has got to be a known problem, right?

Families

This part is going to be a little more vague and not completely thought out, but I want to write it down so I remember it.

Let me comment that these problems typically come in families. I’m going to be kind of vague about what one should mean by that, because I’m not wholly sure. Take the Turan problem, for example. That’s really a list of problems, one for each n, or at least each sufficiently large n, given by data G_n, H_n, and R_n. But in a way they’re all the same, right? In particular, Orb_m doesn’t change with n (or at least it doesn’t for n large enough) Fortunately I am very familiar with this vague notion of seuqences of things, each with an action of S_n, which are somehow all the same — they are functors from the category FI of finite sets with injections. In particular, (G_n/H_n) is a finitely generated FI-set, which you can read more about in the paper of Speyer, Ramos, and White. Never mind the definitions; just accept that there’s a reasonable notion of what counts as a family of (G,H,R) instances in this context,

Similarly, I think the cap set problems as n varies should be thought of as a family.

What about the continuous cases? I’m a little less clear there but let me make a stab at at least one kind of thing that should count. Let G be the affine generalized orthogonal group on R^2, i.e. the group of rigid motions where you’re also allowed to dilate, and let H be GO(2). Then G/H is R^2 and Orb_3 is the set of similarity classes of triangles. Write K_n in Orb_3 for the set of triangles such that the ratio between two edges never exceeds n. Then a K_n-set has size bounded by const*n^2 (because a K_n set is more or less the same thing as a packing of unit circles in a circle of radius n.) If U is some OTHER set subset of orb, I would consider a family of problems to be given by R = U intersect K_n, with n growing.

In the first two cases, write (G/H)_n for (G_n/H_n) and in the third case write (G/H)_n for K_n. Note that in all three cases we have an easy asymptotic formula for the maximal size of a ((G/H)_n)-set. Call this size S_n. Then here’s what I’d like to guess.

Vaguely stated guess: In any family of R, if we write m_R(n) for the maximal size of an R-set in the problem indexed by n, then there are constants c_R, γ_R such that

m_R(n) = c_R |S_n|^γ_R + o(|S_n|^γ_R).

Is that too abstract? Let me give some sense of what it means. It would say that the maximum size of a cap set is asymptotic to c q^{γ n} for some constants c,γ. Is that true? I’d guess so, but I don’t know how to rule out that it’s some kind of funny q^{γ n + epsilon}. It would say that, for instance, if I asked how many points there could be in a unit line segment, no two separated by less than 1/n, and no “approximate three-term APs”: three points x,y,z satisfying y in [(1/2+δ)x + (1/2-δ)z, (1/2-δ)x + (1/2+δ)z], that maximum would be asymptotic to some c n^γ, with γ depending on δ. Is that true? And it would say that, I dunno, if I asked how large a collection of k-element subsets of [n] could be if no m of them shared at least t points, the answer would also be asymptotic to some c n^γ. Believable! I have no reason to believe any of this except that the cases where these problems are solved, e.g. the problems discussed in this survey of set intersection problems, all seem to have answers of this form.

My instinct is that γ_R should be much easier to compute, and that c_R, which is more like a sphere-packing constant, should be more subtle. In particular I think this constant γ_R should be a basic structural fact about R that measures how “restrictive” R is.

I have not thought about this very hard! I am posting with the idea that people will tell me which parts of this are already understood, and which parts are wrong, and in what directions there’s more to say or ask.

AB is playing JV basketball this year, and tonight was her first home game. Lots of parents, and even more impressive, lots of fellow students. A lot of school spirit at West HS! The game was close for most of the going, with a lot of lead changes, and there was honestly a higher level of fan intensity than at some major league baseball games I’ve been to. (Looking at you, well-heeled SF Giants fans!) There is something very satisfying about yelling “DEE-FENSE” and stomping my feet and it is not a thing I get to do that often.

I did manage to get home last night from Providence, despite widespread cancellations and delays caused by the government shutdown de-staffing air traffic control. United was speedy, flexible, and informative under rapidly changing conditions, their employees, while clearly stressed out, stayed professional. I made it onto my 7:35 connecting flight from O’Hare to Madison with about 10 minutes to spare. The plane was only half full, because so many people were coming in late, and United ended up holding the flight for 45 minutes to let as many people get on as were able to get to O’Hare. I’m sure there were still a lot of people who didn’t get home. As I write this, the average delay departing O’Hare is 24 minutes; out of Newark, 53 minutes. There are already 263 announced cancellations for Monday. And there’s apparently a United flight in the air right now from Newark to Savannah that’s flying at 10,000 feet the whole way because of air traffic control shortages, burning much more fuel and keeping the passengers in their seats the whole flight. Good on United for figuring out a way to get people where they were going, but still, what a mess.

I miss the days when you looked to the federal government to protect Americans from big corporations, instead of the other way around.

I thought the book had some interesting virtues but was also quite badly overwritten, and very confusing in its treatment of the mathematics. Here’s my review.

For length reasons I wasn’t able to fit in all the material I wrote, so here’s a section addressing the confusingness, which didn’t make it into the final piece. Only here at Quomodocumque do you get the director’s cut!

_________________

This is not the book to read if you want a precise understanding of the mathematics Hilbert, Brouwer, and Russell were going to war over. Bardi’s over-the-top style is confusing as often as it’s entertaining. “Contemplate a never-ending, constantly evolving universe of single-grain galaxies. Think about an eternity of revelation in every single grain for every subsecond of existence. That’s starting to get at the difference between large and small infinity—an infinite number of sand buckets versus the infinite variation inside every single bucket. Big infinity is much bigger. But even then, you’re not close to defining the continuum.” Even for someone like me who knows the definition of the continuum, this is hard to parse. I can’t imagine what sense it makes to everybody else. Later, Bardi writes:

“The continuum hypothesis says cardinality of an infinite set is like the Highlander – there can be only one!”

This is not, in fact, what the continuum hypothesis says, and indeed it is false; the whole point of Cantor’s theory, as Bardi explains perfectly well elsewhere in the book, is that there are many cardinalities (roughly, sizes) of infinite sets. Bardi sacrifices clarity of exposition here for the sake of a verbal flourish, and not even much of one – a movie catchphrase reference that only a small age band of readers, including Bardi and me, is likely to recognize.

In 2016 I wrote that the Orioles were setting “a fewest-triples record that may never be broken” by hitting only 6 triples in a full major league season. It was broken the next year by the Blue Jays, who hit only 5 triples. We are actually in a golden age of not hitting triples. Nine teams in the modern era of MLB have finished the season with ten triples or fewer. Of those nine teams, six have been since 2022, with three this year alone. The 2025 Mariners stole 161 bases. Not a slow team! But they only hit nine triples.

Philip Roth, in The Great American Novel, gives one of his characters a multipage monologue on the greatness of triples. He would be mad now, about a lot of things probably. Speaking of old novelists named Philip, I discovered today that Philip Pullman has, after six years of silence, finally released the final book in the Lyra Silvertongue series. They have been getting longer and less wonderful. And slower, like a team that never hits triples. But I’m glad this exists. The Golden Compass remains the only truly great contemporary work of fantasy I’ve ever read.

I turned 54 yesterday, a number sometimes called “triple chai.” Chai, or חַי, is the Hebrew word for “life,” and the two letters that make it up are chet (the 8th letter) and yud (the 10th), giving the word a total numerical value of 18; and that’s why you often see Jewish people giving charity or gifts in the amount of $72, or $180, or some other multiple of chai, like $54. Triple life!

I do think I have a kind of triple life. I’m a mathematician, a teacher (in which I include writer), and a father. Those are three pretty good lives to have! I feel no need for a fourth, but ask me again when I turn 72.