Culture
Could Monkeys Really Type All of Shakespeare?

Science doesn’t usually tolerate frivolity, but the infinite monkey theorem enjoys an exception. The question it poses is thoroughly outlandish: Could an infinite number of monkeys, each given an infinite amount of time to peck away at a typewriter (stocked with an infinite supply of paper, presumably) eventually produce, by pure chance, the complete works of William Shakespeare?
The problem was first described in a 1913 paper by the French mathematician Émile Borel, a pioneer of probability theory. As modernity opened new scientific fronts, approaches to the theorem also evolved. Today, the problem pulls in computer science and astrophysics, among other disciplines.
In 1979, The New York Times reported on a Yale professor who, using a computer program to try to prove this “venerable hypothesis,” managed to produce “startlingly intelligible, if not quite Shakespearean” strings of text. In 2003, British scientists put a computer into a monkey cage at the Paignton Zoo. The outcome was “five pages of text, primarily filled with the letter S,” according to news reports. In 2011, Jesse Anderson, an American programmer, ran a computer simulation with much better results, albeit under conditions that — like the Yale professor’s — mitigated chance.
A new paper by Stephen Woodcock, a mathematician at the University of Technology Sydney, suggests that those efforts may have been for naught: It concludes that there is simply not enough time until the universe expires for a defined number of hypothetical primates to produce a faithful reproduction of “Curious George,” let alone “King Lear.” Don’t worry, scientists believe that we still have googol years — 10¹⁰⁰, or 1 followed by 100 zeros — until the lights go out. But when the end does come, the typing monkeys will have made no more progress than their counterparts at the Paignton Zoo, according to Dr. Woodcock.
“It’s not happening,” Dr. Woodcock said in an interview. The odds of a monkey typing out the first word of Hamlet’s famous “To be or not to be” soliloquy on a 30-key keyboard was 1 in 900, he said. Not bad, one could argue — but every new letter offers 29 fresh opportunities for error. The chances of a monkey spelling out “banana” are “approximately 1 in 22 billion,” Dr. Woodcock said.
The idea for the paper came to Dr. Woodcock during a lunchtime discussion with Jay Falletta, a water-usage researcher at the University of Technology Sydney. The two were working on a project about washing machines, which strain Australia’s extremely limited water resources. They were “a little bit bored” by the task, Dr. Woodcock acknowledged. (Mr. Falletta is a co-author on the new paper.)
If resources for washing clothes are limited, why shouldn’t typing monkeys be similarly constrained? By neglecting to impose a time or monkey limit on the experiment, the infinite monkey theorem essentially contains its own cheat code. Dr. Woodcock, on the other hand, opted for a semblance of reality — or as much reality as a scenario featuring monkeys trying to write in iambic pentameter would allow — in order to say something about the interplay of order and chaos in the real world.
Even if the life span of the universe were extended billions of times, the monkeys would still not accomplish the task, the researchers concluded. Their paper calls the infinite monkey theorem “misleading” in its fundamental assumptions. It is a fitting conclusion, perhaps, for a moment when human ingenuity seems to be crashing hard against natural constraints.
Low as the chances are of a monkey spelling out “banana,” they are still “an order of magnitude which is in the realm of our universe,” Dr. Woodcock said. Not so with longer material such as the children’s classic “Curious George” by Margret Rey and H.A. Rey, which contains about 1,800 words. The chances of a monkey replicating that book are 1 in 10¹⁵⁰⁰⁰ (a 1 followed by 15,000 zeros). And, at nearly 836,000 words, the collected plays of Shakespeare are about 464 times longer than “Curious George.”
“If we replaced every atom in the universe with a universe the size of ours, it would still be orders of magnitude away from making the monkey typing likely to succeed,” Dr. Woodcock said.
Like other monkey theorem enthusiasts, Dr. Woodcock mentioned a famous episode of “The Simpsons,” in which the crusty plutocrat C. Montgomery Burns tries the experiment, only to fly into a fury when a monkey mistypes the opening sentence of Charles Dickens’s “A Tale of Two Cities.” In reality, the monkey’s achievement (“It was the best of times, it was the blurst of times”) would have been a stunning triumph over randomness.
Outside cartoons, such successes are unlikely. First, there is cosmic death to consider. Many physicists believe that in 10¹⁰⁰ years — a much larger number than it might seem in type — entropy will have caused all the heat in the universe to dissipate. Far away as that moment may be, experts do think it is coming.
Then there’s the availability of monkeys. Of the more than 250 possible species, Dr. Woodcock selected chimpanzees, our closest genomic kin, to mimic the Bard. He enlisted 200,000 — the entire population of chimps currently on Earth — until the end of time. (Optimistically, he did not to plan for the species’ dwindling or extinction. Nor did he consider constraints like the availability of paper or electricity; the study does not specify which platform the monkeys might use.)
Monkeys intent on recreating Shakespeare would also need editors, with a strict reinforcement training regimen that allowed for learning — and a lot of it, since Dr. Woodcock set each monkey’s life span at 30 years. “If it’s cumulative, obviously, you can get somewhere,” said Richard Dawkins, the evolutionary biologist, who discusses the typing monkeys in “The Blind Watchmaker,” his 1986 book about evolution. Unless the typing were “iterative,” though, Dr. Dawkins said in an interview, progress would be impossible.
The new paper has been mocked online because the authors purportedly fail to grapple with infinity. Even the paper’s title, “A numerical evaluation of the Finite Monkeys Theorem,” seems to be a mathematical bait-and-switch. Isn’t infinity a basic condition of the infinite monkey theorem?
It shouldn’t be, Dr. Woodcock seems to be saying. “The study we did was wholly a finite calculation on a finite problem,” he wrote in an email. “The main point made was just how constrained our universe’s resources are. Mathematicians can enjoy the luxury of infinity as a concept, but if we are to draw meaning from infinite-limit results, we need to know if they have any relevance in our finite universe.”
This conclusion circles back to the French mathematician Borel, who took an unlikely turn into politics, eventually fighting against the Nazis as part of the French Resistance. It was during the war that he introduced an elegant and intuitive law that now bears his name, and which states: “Events with a sufficiently small probability never occur.” That is where Dr. Woodcock lands, too. (Mathematicians who believe the infinite monkey theorem holds true cite two related, minor theorems known as the Borel-Cantelli lemmas, developed in the prewar years.)
The new paper offers a subtle comment on the seemingly unbridled optimism of some proponents of artificial intelligence. Dr. Woodcock and Mr. Falletta note, without truly elaborating, that the monkey problem could be “very pertinent” to today’s debates about artificial intelligence.
For starters, just as the typing monkeys will never write “Twelfth Night” without superhuman editors looking over their shoulders, so increasingly powerful artificial intelligences will require increasingly intensive human input and oversight. “If you live in the real world, you have to do real-world limitation,” said Mr. Anderson, who conducted the 2011 monkey experiment.
There is no free lunch, so to speak, said Eric Werner, a research scientist who runs the Oxford Advanced Research Foundation and has studied various forms of complexity. In a 1994 paper about ants, Dr. Werner laid out a guiding principle that, in his view, applies equally well to typing monkeys and today’s language-learning models: “Complex structures can only be generated by more complex structures.” Lacking constant curation, the result will be a procession of incoherent letters or what has come to be known as “A.I. slop.”
A monkey will never understand Hamlet’s angst or Falstaff’s bawdy humor. But the limits of A.I. cognition are less clear. “The big question in the industry is when or if A.I. will understand what it is writing,” Mr. Anderson said. “Once that happens, will A.I. be able to surpass Shakespeare in artistic merit and create something as unique as Shakespeare created?”
And when that day comes, “Do we become the monkeys to the A.I.?”

Culture
Book Review: ‘How to Be Well,’ by Amy Larocca

HOW TO BE WELL: Navigating Our Self-Care Epidemic, One Dubious Cure at a Time, by Amy Larocca
Oh, the irony of cracking open “How to Be Well” while on vacation in Italy. There, on an island off the coast of Naples, a breakfast buffet included three varieties of tiramisu. Wine was poured not to a stingy fingertip’s depth but from a bottomless carafe — at lunchtime, no less. And when stores closed in observance of an afternoon siesta, the only people on the streets were American tourists, jogging. (I was on the prowl for a postage stamp because, yes, I still send postcards.)
It was from this place of abundance and balance that I followed Amy Larocca, a veteran journalist, into the hellscape of stringent food plans, cultish exercise routines and medical quackery that have, over the past decade or so, constituted healthy living in some of the wealthiest enclaves of the United States. Blame social media, political turmoil or the pandemic — no matter how you slice it, the view is dispiriting. But Larocca’s tour is a lively one, full of information and humor.
The book begins with a colonic, “the flossing of the wellness world,” Larocca writes. We find the author herself on an exam table, “white-knuckled and curled up like a baby shrimp, naked from the waist down.” She recalls her doctor’s disapproval of the procedure — a sort of power washing of the colon — and its risks, including rectal perforation, juxtaposed with one woman’s claim that a colonic made her feel like she could fly, like it was “rinsing out the corners of her psyche.”
Where did we get the idea that the body — specifically a woman’s body — is unclean inside? A problem to be solved? And how did the concept of wellness bloom “like a rash,” Larocca writes, into a $5.6 trillion global industry?
These are the questions she seeks to answer, using data, history, medicine, pop culture and her own experience. She parses fads and trends, clean beauty and athleisure wear, the gospel of SoulCycle and the world according to Goop. She weighs the advantages and disadvantages of micro-dosing and biohacking. She too goes to Italy, where she attends a Global Wellness Summit featuring a spandex and sneaker fashion show and a presentation on ending preventable chronic disease the world over.
At times, Larocca seems to approach her own subject with the same sweep. The second half of “How to Be Well” reads like a survey course, cramming the industry’s relationship to politics, men and the environment into single chapters when each could fill a whole semester. As for why meditation merits more real estate than vaccines, I can only assume that the book was already at the printer when Robert F. Kennedy Jr. was confirmed as health secretary.
But when Larocca goes deep, as she does on self-care, body confidence and sex positivity, she’s at her best — authoritative and witty, personal without being chummy.
She debunks the cockamamie but persistent notion that “feeling old is not an inevitable byproduct of aging but something easily avoided by paying attention.” (And by forking over gobs of cash; more on this shortly.) After attending an Oprah-sponsored conference on menopause, a subject Larocca has covered for The New York Times, she realizes that “aging is different from disease” and “isn’t necessarily something to be cured,” let alone through “neat, tidy, attainable solutions.”
Then there’s the sneaky rebranding of old-school dieting for “detoxification,” another wolf in sheep’s clothing. Think fasting, juicing, abstaining from all manner of verboten foods. Even if the professed endgame is “glow,” Larocca makes clear, “part of the promise is still, always, to rid us of a bit of ourselves.”
And finally, refreshingly, she’s honest about the money at stake for the wellness-industrial complex — not just for stylists turned wellness coaches or models turned nutritionists, but for massive corporations cashing in on an age of worry.
“None of these institutions is nonprofit; none of these institutions is altruistic at its core,” Larocca writes, in a passage reminiscent of Carol Channing’s monologue from “Free to Be You and Me,” in which she reminded us that happy people doing housework on TV tend to be paid actors.
“It is their job to persuade me to come back,” Larocca continues, “to spend more money on what they’ve got to give, to serve their investors, to serve themselves.”
And that, as “How to Be Well” wisely shows us, is the bottom line.
HOW TO BE WELL: Navigating Our Self-Care Epidemic, One Dubious Cure at a Time | By Amy Larocca | Knopf | 291 pp. | $28
Culture
New Crime Novels With Unexpected Twists

By Jeffery Deaver
Colter Shaw is a professional “rewards seeker,” a skilled tracker who specializes in finding missing people — usually for the reward money, though sometimes out of the goodness of his heart. It’s a simple enough vocation and yet, as the suspense veteran Deaver has demonstrated in four prior Shaw novels (and the TV adaptation “Tracker”), the ways in which Shaw finds peril — or peril finds him — keep multiplying. In SOUTH OF NOWHERE (Putnam, 403 pp., $30), his sister Dorion implores him to help to locate potential survivors after a levee collapses in a small Northern California town.
From here, Deaver is off to the proverbial races. Does every chapter have a twist? Pretty much. Is Colter just likable enough to brush off needless conflict and still find time for romance? Definitely. Is the writing a little too reminiscent of detailed outlines like the ones Deaver is known to fashion before writing a first draft? You bet. Could I put the book down? Not a chance.
The Colter Shaw series prioritizes action and the constant possibility of calamity, leaving only the barest amount of room for character development, like Colter’s continued grappling with the effects of his survivalist upbringing. The books don’t measure up to the best of Deaver’s Lincoln Rhyme novels, but they all accomplish their mission: thrilling engagement.
by Uzma Jalaluddin
Kausar Khan, introduced in DETECTIVE AUNTY (Harper Perennial, 326 pp., paperback, $17.99), has spent the past 20 years relishing the stability of placid North Bay, where she and her husband moved after fleeing busy, bustling Toronto in the wake of a family tragedy.
But then her husband dies shortly after being diagnosed with pancreatic cancer, and her daughter, Sana, calls with upsetting news: “I’m in trouble. There’s been a murder, and I’m the prime suspect.” It seems Sana’s landlord has been found inside her clothing store with a dagger in his chest. Kausar returns to Toronto’s Golden Crescent neighborhood as both a concerned mother and a tenacious amateur sleuth.
The case against Sana is strong, but as Kausar discovers, the murder victim had many enemies. If only the ghosts of Kausar’s past would stop haunting her present-day investigation!
Jalaluddin, who has crossed into crime fiction from the romantic comedy genre, doesn’t skimp on plotting — the whodunit twist caught me pleasingly flat-footed — but shines most with character and community, showing the complexities of mother-daughter relationships and the variability of longtime friendships. “Detective Aunty” is the first in a new series and I certainly welcome more installments.
by Michael McGarrity
Reading Michael McGarrity’s noir novel NIGHT IN THE CITY (Norton, 263 pp., $28.99), about the midcentury death of a Manhattan socialite named Laura Neilson, I found it difficult to avoid thinking about Vera Caspary’s 1943 classic suspense novel “Laura” (and the equally classic film adaptation featuring Gene Tierney and Dana Andrews). While I wished for more structural innovation along the lines of what Caspary accomplished, I did enjoy McGarrity’s more conventional narrative here: A man finds his ex-lover murdered and must clear his name, rooting out widespread corruption as the atmosphere thickens.
The man is the assistant district attorney Sam Monroe, who dated Laura for a time and never really got over the way she broke up with him by bringing a new flame to the local bar that was “their private haunt and rendezvous.” So when she summons him to her Manhattan penthouse, off Sam goes, waved in by an expectant doorman — only to find her body, his Army dog tags wrapped around her neck. One thing is clear: He’s been set up. With the help of an intrepid private eye and his former lover’s diary, Sam sets out to find her killer.
McGarrity paints a seedy portrait of a bygone New York that pulses with life, lust and larceny.
by Leonie Swann; translated by Amy Bojang
Finally, it gives me great pleasure that Swann’s exceedingly delightful Sheep Detective books are once again available for American audiences. “Three Bags Full,” first published in 2005 and reissued in February, introduced an intrepid flock on the case of who had killed their beloved shepherd. In BIG BAD WOOL (Soho Crime, 384 pp., $28.95), the sheep — including Zora, “a Blackface sheep with a weakness for the abyss,” Ramesses, a “nervous young ram full of good ideas,” and Miss Maple, “the cleverest sheep in the flock and maybe even the world” — return with a new minder, Rebecca.
They’re wintering next to a French château, which sounds idyllic, but the disappearance of other sheep, the mounting deaths of deer and, eventually, a human, strike fear in the hearts of the flock, who are worried they or their shepherd may be next. Is it a werewolf, the shape-shifting creature called Garou, as the local goats seem to believe? Or a more prosaic yet sinister culprit? How the sheep discover the truth will enchant readers who pay close attention.
Culture
Book Review: ‘Warhol’s Muses,’ by Laurence Leamer

Leamer is undeniably excellent at setting a scene, especially a louche one. He knows just when to have someone wonder if he’s caught crabs from a couch or a crotch. And Leamer is very good on rich people playing at being disheveled, tuned to the comic possibilities of that particular brand of tourism. (Holzer, of Florida real estate wealth, announces after seeing the Stones for the first time that “they’re all from the lower classes. … There is no class anymore. Everyone is equal.” Leamer adds that Holzer’s “maid and butler might have disagreed.”) Nearly every page has at least one great sleazy anecdote or pinch of gossip.
The problem is that so many of these scenes, however expertly set, are variations on the same stale theme of boomers getting up to wild stuff because the times they were a-changin’. Does anyone still need reminding that “the ’60s was a decade of radical political and cultural dissent”? Or that it was once considered shocking that a high-culture figure such as Rudolf Nureyev could go straight from a performance of “Swan Lake” to dancing “to rock ’n’ roll in a nightclub wearing dungarees. Dungarees! Not a suit and tie like some uptight New York businessman”? Reading this book felt akin to being trapped in an endless Time-Life loop of jingle jangle mornings, lazy Sunday afternoons and warm San Franciscan nights, the author providing the stentorian voice-over as the usual footage rolls by: Bob Dylan “would soon emerge as the poetic troubadour of the ’60s”; Brian Jones, “addicted to drugs and sex … was on a short road to an early death”; Jim Morrison, “a troubadour of the counterculture … wrote poetic lyrics that chronicled the lives of his generation.”
Such minor sins might have been forgiven had I ultimately gleaned some deep or unforeseen insight into the lives of the book’s subjects — a group that includes Ultra Violet, Ingrid Superstar, Brigid Berlin and other Factory figures — or, failing that, into Andy Warhol’s work. But I got neither. Nor was I convinced by the whopping claim that “without his Superstars, Warhol might never have become a world-celebrated artist.”
Meeting these 10 historical actors in roughly chronological order as they enter Warhol’s life, one has a view of the artist and his milieu that actually narrows rather than widens. Warhol, a shape-shifter so manic and intense that he could slide into several personas in the span of a single season, is here reduced to a necessarily static figure so that the women can bounce off him. Which is fine as a narrative strategy, but then not much happens to the women, either. As each one flickers into view, her upbringing (often troubled) is dutifully covered before she provides some service to Warhol — as entertainment, as emotional consort, as visual material, as key holder to Park Avenue penthouses — and then fades out to make room for the next one. (Sedgwick is the exception, a frequent and always beguiling presence; Solanas, the would-be assassin, and not one of the 10 Superstars, stands out as foil rather than helpmeet, but appears only briefly.)
Rarely is there any sense of genuine collaboration or exchange. The book’s subtitle gives away the game: In the end, these women of varied backgrounds, with their respective dreams and desires, are all here to play the same passive role — to be inevitably and unsurprisingly “destroyed by the Factory fame machine.”
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