“Things that are so far removed from our daily experience… are inherently hard to understand”*…
That’s certainly true of numbers. And as the numbers grow, the cognitive challenges grow with them. (Indeed, by way of example: 1 million seconds, is roughly 11.5 days; 1 billion seconds is almost 32 years.)
We’ve looked before at the mysterious extremes of math: zero and infinity [and here]. But as Dan Falk reminds us, the numbers in between can seem pretty strange as well– especially the extremely large ones. In a review of Richard Elwes‘ Huge Numbers: A Story of Counting Ambitiously, From 4½ to Fish 7, Falk spotlights some of the largest numbers humans have ever contemplated…
… Aficionados of huge numbers are called “googologists,” a reference to the number 10100, known as a googol. Such numbers have a peculiar sort of existence. For the vast majority of us, they’re of limited everyday value. Calculations at the supermarket checkout, or at tax time in April, typically involve far more modest figures. Perhaps we’ve read that the U.S. national debt is in excess of $38 trillion — a mind-numbing figure, to be sure, but it’s not as though any one individual needs to count it up in stacks of $20 bills.
And yet, much larger numbers await those who seek them out. Consider the kinds of numbers that crop up in problems involving combinations and permutations. For example, in how many distinct ways can one shuffle a deck of cards? Elwes takes us through the calculation, and we end up with a figure of about 8×1067. Compared to that number, the odds of getting a royal flush when dealt a five-card poker hand seem pretty decent, sitting at a mere 1 in 649,740 (still rare enough that many poker players have never held such a hand). Or consider that famous 1980s cultural touchstone, the Rubik’s cube. In how many ways can one scramble the cube? It turns out that the figure is about 43 quintillion, or 4.3×1019 — but in spite of that ridiculously large figure, people do routinely solve the puzzle, and champions can do it in mere seconds. In fact, as Elwes explains, no Rubik’s cube arrangement is more than 20 moves away from any other arrangement.
Or consider the age of the universe, estimated to be about 13.8 billion years. This may seem like a lengthy span of time, but our cosmic future is where the really big numbers come up. Elwes examines the so-called heat death of the universe, in which all matter has broken down into subatomic particles. We may reach this point in years — this dizzying figure is 10 raised to the power of 10120 — at which point, Elwes says, the universe will have ballooned up to a diameter of 10 to the power of 10 to the power of 10120 light years. (Yes, that’s light years.) Elwes adds a footnote: “At this point, the choice of units hardly matters; the distance is so immense that whether we choose to measure it in Planck lengths or giga-light years makes little difference.” Let that sink in!
As mind numbing as such figures are, the highest numbers contemplated by humans come not from physics but from pure mathematics and computer science. Like “Graham’s number” — an immense figure put forward as the upper-bound for solutions to a problem in a branch of mathematics known as Ramsey theory. Some readers may find the ensuing discussion of multi-dimensional hypercubes a bit challenging, but one can enjoy the payoff regardless: We end up with a number that can’t even be expressed in conventional notation, and which earned a mention in the 1980 edition of the “Guinness Book of World Records” as “the highest number ever used in a mathematical proof.”
Reading this book is a little bit like sitting in the back row of an auction house where a rare Picasso (let’s say) is up for grabs: How high is this thing going to go? And indeed, Elwes keeps going. We eventually meet the so-called busy beaver numbers, a set of numbers that crop up in theoretical computer science, when one tries to deduce whether a particular computer program will eventually stop, or keep going forever — a conundrum known as the “halting problem.” As Elwes explains, it’s not at all straightforward to distinguish the two types of programs (and if it was, it would help mathematicians tackle some of the most vexing problems in their field).
The fifth busy beaver number, known as BB(5) — associated with a computer program that can access five internal states — works out to 47,176,870. And that’s as far as we’ve gotten, Elwes explains. No one has worked out the value of BB(6), but he assures us that it’s beyond the range of any physical computer; and BB(16) leaves even Graham’s number in the dust.
But wait, there’s more! “Rayo’s number,” concocted by Agustín Rayo — a dean and professor at MIT — using set theory, is bigger still (here’s a fun video about it); and “Fish 7,” mentioned in the book’s subtitle, named for a Japanese googologist who goes by the pseudonym “Fish,” builds on Rayo’s number, and … well, the details are not easily digested, but the mind-melting nature of these numbers comes across as a feature, not a bug, of Elwes’s story… the narrative is enlivened by explorations of the peculiarities of math history…
… Archimedes tried to estimate how many grains of sand would be needed to fill up the known universe, back in the third century B.C. Did he simply have too much time on his hands? Not at all, insists Elwes: The Greek thinker was articulating an important idea — that no matter how unfathomably large a quantity may be, we can describe it with precision, thanks to mathematics. “Archimedes,” he writes, “was penning a manifesto for the expressive power of large numbers.”…
… [Elwes focuses] on numbers that are ridiculously large and yet finite. In the end, perhaps this is the most mind-boggling fact of all: that these enormous numbers, from Graham’s number to Fish 7 and beyond, fall as far short of infinity as does the humble number 1…
The mysteries of the massive: “The Mind-Boggling Science of Enormous Numbers,” @danfalk.bsky.social on @richardelwes.bsky.social in @undark.org.
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As we enumerate enormity, we might spare a thought for a seminal mathematician, Alan Turing; he died on this date in 1954. He was a foundational computer science pioneer (inventor of the Turing Machine (an influential model for the general-purpose computer), creator of the “Turing Test” (only too relevant in these AI-infected times), inspiration for “The Turing Award” (the “Nobel Prize of computing“), and cryptographer (leading member of the team that cracked the Enigma code during WWII).
“Here comes the sun”*…
Further, in a fashion, to last Wednesday’s post… We’ve looked before (e.g., here) at the potential havoc that solar storms could wreak on our electified lives. Now, as Paul Voosen reports, scientists are speculating on a defense, suggesting that gases released from satellites could slash the threat of severe “space weather”…
When violent eruptions from the Sun slam into Earth’s magnetic field, they do more than paint aurorae across the night sky. They can scramble the electronics of satellites and induce powerful ground currents that knock out electrical grids. It’s been estimated that a one-in-a-100-year solar storm like the 1859 Carrington Event could cause more than $3 trillion of damage to the power grid alone. [See here.]
Yet for decades, society’s only defenses have been better space weather forecasts and more durable technology on the ground and in space. Now, a small group of space physicists says humanity should intervene and weaken solar storms in real time. In a study published [recently] in Space Weather, the researchers describe a provocative proposal called “StormWall”: a fleet of satellites that would release hundreds of tons of gases into space just before a solar storm strikes Earth. Computer simulations suggest the artificial cloud could cut the intensity of a major solar storm by half or more. “It’s as if you could install an airbag in the magnetosphere,” says Daniel Welling, a co-author and space physicist at the University of Michigan.
Call it “helioengineering”—a deliberate intervention in the near-Earth space environment. But unlike controversial geoengineering proposals to mitigate global warming, which would inject long-lived Sun-blocking particles into the atmosphere, StormWall’s protective gases would dissipate within hours, says Brian Walsh, the study’s lead author and a space physicist at Boston University. “It’s waiting for us to do some temporary modification.”
The proposal would require more extensive simulations and testing. But it is “highly innovative and appears to be quite feasible in the near term,” says Allison Jaynes, a space physicist at the University of Iowa. It’s a “laudable idea,” adds Gurudas Ganguli, a space physicist at the U.S. Naval Research Laboratory (NRL)…
[Voosen explains the technology proposed and considers the challenges in its implementation…]
… Of course, like an airbag, StormWall would have to be replaced if deployed. But just as NASA and other space agencies are studying how to protect the planet from asteroids [and here], Walsh says there’s a good argument for fortifying an electronics-dependent society against massive solar eruptions. “If we lose all our power grids and can’t use the internet for 6 years, it would be traumatic.”
“Radical proposal would block solar storms with orbital ‘airbag’” from @science.org.
* George Harrison
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As we apply sunscreen, we might send bright birthday greetings to Godfried Wendelen; he was born on this date in 1580. And astronomer (and Catholic priest) known as “the Ptolemy of his time.” Despite the tenets of his church, Wendelen was an audacious proponent of the Copernican theory that the planets orbit around the sun. He made more accurate measurements of the distance to the sun than those previously made by Aristachus (2,000 years earlier) from the geometrical relationships at the exact time of a half-moon.
Wendelen is considered by many as a precursor of Kepler and Newton, and was in fact cited by Newton in his Principia. The crater Vendelinus on the Moon is named after him
“Too much of a good thing can be wonderful”*…
On the rise and fall of the Las Vegas casino buffet…
… With the May 31 closure of the MGM Grand Buffet, the Strip is down to about half a dozen all-you-can-eat buffets. It was once home to more than 10 times that many.
Excluding the sushi bar, the MGM Grand’s $44 Sunday mimosa brunch might have looked about like it did when the resort opened in 1993. It offered crispy brisket at the carving station, biscuits, scrambled eggs and sauteed vegetables. Most of the meats had a tub of gravy next to them, either dark brown or as beige as the decor. The anachronistic vibe at the 535-seat establishmentstood in contrast with more expensive buffets at nearby Caesars Palace and Wynn, overflowing with luxury offerings like turmeric grilled baby octopus, Peking duck and lobster toasts garnished with caviar.
“Young people complain that it looks old,” says Shaunell Samano, the MGM Grand Buffet’s assistant general manager. She has a job lined up at the nearby Luxor. All five of the servers hustling the floor had worked there since the resort’s opening. Most of the staff had been prepping the buffet for at least 26 years. Samano recalled guests even visiting twice a day, including retired boxer Evander Holyfield and his wife a few years ago.
The vanishing old-school Vegas buffet comes as Americans rethink their relationship to food and travel. A 2025 Cornell University study found that the proliferation of GLP-1 drugs is driving down demand for the kinds of indulgent foods available at all-you-can-eat buffets, and several studies show that gastronomic experiences are fundamental to choosing a vacation destination. Still, a 2025 Pew Research Center study indicated that even if consumers are more health-conscious than ever, taste and affordability remain the most important factors in deciding what to eat.
All-you-can-eat buffets may be receding from their spiritual home of Las Vegas, but the country isn’t abandoning them yet.
Golden Corral Chief Executive Officer Lance Trenary told Bloomberg Intelligence in November that his company’s restaurants were averaging the same number of meals served as they were pre-pandemic. The all-you-can-eat Korean BBQ and hot-pot chain KPOT had three locations in 2020; it plans to have more than 150 open by the end of the calendar year. Yelp’s 2026 Trends forecast cited a 252% increase in searches for “all you can eat buffet.”
“Customers like buffets,” says Eric Chiang, a University of Nevada at Las Vegas economics professor who loves using buffets as a way to explain economics. “It’s a flat price with no risk involved and no surprise at the end,” he says.
The novelty of all-you-can-eat dining is rooted in contradictory American lifestyles: One diner sees freedom and abundance, while another sees waste and gluttony. They’re rare restaurants where, at least for an hour or two, anyone can eat like royalty…
… the all-you-can-eat buffet is inextricably linked to the glamorous excesses of Las Vegas, where famed promoter Herb McDonald hired Norwegian chef Arne Hansen Rom in 1946 to tailor the European smorgasbord to the tastes of the Western Yankee. The Midnight Chuckwagon, later known as the Buckaroo Buffet, lured gamblers at the El Rancho hotel and its previous incarnation, the Thunderbird. Along with a lounge act came unlimited food ranging from deviled eggs to shrimp cocktail to Rom’s specialty: barbecue spareribs. The all-you-can-eat buffet evolved into a signature loss leader for resorts competing to attract a new wave of Las Vegas tourists: families and international travelers.
When John Curtas recalls his first visit to a Las Vegas buffet as a 10-year-old in the early 1960s, the veteran Las Vegas food critic remembers a haunch of beef that looked 12 feet tall manned by a chef wielding a carving knife like a scimitar. Beside the beef sat piles of shrimp, whole-cooked turkeys, potato salad and cowboy beans. It cost just $1, and he could return for more without embarrassment.
“Buffets gave you such a dazzle factor and eye candy,” Curtas says. “But they also gave a lot of perceived value for people and for families.”…
More on the social psychology and economics of buffets: “The Quintessential Old-School Las Vegas Buffet Bids Farewell” gift link from @bloomberg.com.
* Mae West
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As we go back for seconds, we might spare a thought for Edwin Traisman; he died on this date in 2007. A food scientist, he is best remembered for helping to create Cheez Whiz for Kraft, then for perfecting the method used by McDonalds standardize their french fries (by freezing partially-cooked fries for transport and storage). But relevantly to the piece above, he also helped initiate research on E. coli 0157:H7, which was at the time (1987) a little known pathogen.
“A free press can, of course, be both good and bad; but, most certainly, without freedom, the press will never be anything but bad”*…
The Fourth Estate is, of course, hugely influential in civic and political life; a free press is essential to the healthy functioning of a democracy, in the U.S. and around the world— more generally to effective self-determination in any society. So the latest World Press Freedom Index from Reporters Without Borders (RSF) is concerning. Indeed, the global state of press freedom has reached a 25-year low.
RSF has been compiling the Index since 2002; as of this year:
• Less than 1% of the global population lives in a country rated as having “good” press freedom.
• More than half of countries and territories now fall into the “difficult” or “very serious” categories, up from 13.7% in 2002.
• The U.S. ranks 64th globally in 2026, down from 17th when the index began.
The index ranks 180 countries and territories based on five indicators: political context, legal framework, economic context, sociocultural context, and journalist safety.
This world map shows press freedom scores around the world in 2026, revealing a widening divide between Europe, the only region with countries rated “good,” and much of the rest of the world.
More on what’s happening and why: “Mapped: Press Freedom Around the World in 2026,” from @voronoiapp.bsky.social (and, of course, much more in the RSF Index)
(Image above: source)
* Albert Camus
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As we challenge censorship (and oligopolistic control), we might recall that it was on this date in 1917 that the first Pulitzer Prizes were awarded. In his will, Joseph Pultizer specified solely four awards for excellence in journalism, four in books and drama, one for education, and five traveling scholarships.
In journalism, prizes were to recognize “the most disinterested and meritorious public service rendered by any American newspaper during the preceding year” (a gold medal worth $500 with no monetary component); “the best editorial article written during the year, the test of excellence being clearness of style, moral purpose, sound reasoning, and power to influence public opinion in the right direction” ($500); and “the best example of a reporter’s work during the year, the test being strict accuracy, terseness, the accomplishment of some public good commanding public attention and respect” ($1,000). (A $1,000 prize for the best history of services rendered to the public by the American press in the preceding year was only awarded once; similarly, a $1,000 prize for a paper on the development of the School of Journalism was never awarded due to a dearth of competitors.)…
… the Pulitzer Prize Board, has increased the number of awards to 23 and introduced poetry, music, photography, memoir and audio journalism as subjects, while adhering to the spirit of the founder’s will and its intent…
– source
The awards were administered/bestowed by Columbia University (the journalism school at which Pulitzer had endowed). Herbert B. Swope received the first Pulitzer for journalism (the only one awarded in that first year of the program) for his series “Inside the German Empire” for the New York World… as it happens, a Pulitzer paper.
The Internet Archive has the book that Swope’s series became)
“By failing to prepare, you are preparing to fail”*…
Further, in a fashion, to yesterday’s post, a piece by Nate Hagens on preparing for what promises to be (to put it politely) a challenging future. It’s via (R)D‘s old friend Patrick Tanguay (and his wonderful newsletter Sentiers), who does the introductory honors…
Nate Hagens has spent two decades mapping what he calls the “more-than-human predicament,” the interlocking crises of fossil fuel depletion, ecological overshoot, and economic fragility. This piece marks a shift in his focus: the diagnostic work is largely done, and the current moment demands a framework for action rather than further description of the problem. It’s quite long and perhaps a bit dry, but considering the complexity of everything he’s talking about, I think it reaches a nice balance between all the things and “ok, I can read this, sit with it, and have a well ordered map of what needs doing and a framework from which to work.” The framework has four levels—personal psychological grounding, trusted network-building, six broad intervention fronts, and a timeline—premised on the idea that none of the material work is possible without first stabilising the individual (“being human”) and building shared understanding among people who see the situation clearly.
The six fronts—physical infrastructure, ecological intervention, dignity systems for the dispossessed, governance, culture and meaning, and economic transition—are not a menu but an interdependent set of domains where work is needed simultaneously. Some are familiar territory for anyone thinking about collapse and resilience; others are less obvious. Hagens insists, correctly, on including dignity infrastructure for people who will lose livelihoods as supply chains contract and jobs are automated, and treats culture and collective meaning-making as essential rather than supplementary. He also argues directly that ideological critique, however accurate, is not a plan. That the moment calls for moving from naming what’s wrong to building what comes next.
What gives the framework its structure is the timeline underneath it all: three overlapping phases. Phase A is the stability window that still exists in much of the world, and the one in which trust, infrastructure, and institutions must be built while surplus and coordination capacity remain. Phase B is the period of shocks and triage, already beginning in places, where the goal is to hold systems together and prevent cascades. Phase C is the stable destination—regenerative, locally embedded, equitable—that gives the earlier phases their direction. Hagens’ central argument is that what gets built now sets the initial conditions for everything that follows, and that path dependence operates at a civilisational scale.
Most of the positive climate outcomes we are likely to see in the next twenty years will not come from technology, they will come from curtailed economic expansion driven by the very forces I described at the beginning of this essay: war, debt, and energy depletion. We already got a preview of this during the pandemic as economic activity halted. Industrial activity contracting is not a climate policy, but it is a climate outcome. […]
Subsidiarity and local governance capacity: decisions made at the lowest appropriate level, which in many cases is probably much more local than we currently assume. Communities need the ability to govern their own resource allocation when higher-level institutions can’t or won’t. […]
Collective imagination and sensemaking: the role of arts and creative work in helping communities grieve, adapt, and imagine. This is not a luxury, it is how human groups have always metabolized disruption to continue working together. […]
Shared reality and sovereign visioning: the capacity of communities to tell their own story and find their own vision for the future rather than have it told for them by algorithms, demagogues, or strangers with large followings. In a period of disruption, the communities that hold together will have a strong enough shared cultural narrative to metabolize hardship without breaking apart. This is not a soft category, it is essential and has the ability to bear weight. […]
This window is finite, and many of us – especially in the last few weeks – are increasingly aware that it is closing. We just don’t know exactly how fast. But everything that can only be built in stability – institutional trust, physical infrastructure, knowledge transfer, and relationships – has to be built now, in this window, before conditions change. […]
What does it look like? Regenerative, resilient, human-scale, embedded in local ecology, equitable in a way that does not depend on infinite growth to fund redistribution, and rich in meaning, social connection, and all the things that actually make human life good.
Eminently worth reading in full. The logic of Pascal’s wager suggests that we take Hagens’ advice seriously: “What to Do as the World Falls Apart: A Framework for Action,” from @natehagens.bsky.social (via @inevernu.bsky.social).
Evidence that taking action can matter: “Scientists have scrapped the worst‑case climate scenario – because action is making a difference,” from Andrew King.
And a reminder that there’s more action to take: “The world is heading toward a financial crisis – the state of US politics has left us ill-prepared,” from Eduardo Porter.
* Widely (but incorrectly) attributed to Benjamin Franklin
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As we prepare, we might recall that the Academy of the Distrustful was founded in the library room of the Palau Dalmases in Barcelona. A Baroque literary and musical academy with the aim of promoting the study of classical and Catalan history and poetry, mostly in Spanish, by fourteen scholars headed by the noble Pau Ignasi de Dalmases i Ros. In the event, it lasted only several years: during the War of the Spanish Succession several of its members supported Charles III of Austria over (the man who became) Philip V of Spain— and disbanded the group.
On this date in 2005, 2025, the 325th anniversary of its foundation, the Academy was reconstituted as a humanities academy (in the broadest sense of the term) in the Sala Dalmases of the Historical Archive of the City of Barcelona.
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