Posts Tagged ‘Mathematics’
“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 [10 raised to the 10th power, raised again to the 120th power] 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 [10 raised to the 10th power, again to the 10th power, then to the 120th power] 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).
“Pure mathematics is, in its way, the poetry of logical ideas”*…
Alexander Grothendieck is revered in the world of math; outside of it, he’s known for his unusual life, if he’s known at all. Konstantin Kakaes outlines his actual mathematical contributions…
What Albert Einstein was to 20th-century physics, Alexander Grothendieck was to 20th-century mathematics. He is much less well known because math gets technical even more quickly than physics does. But as with Einstein, Grothendieck’s impact came not just from his own results, revolutionary though they were. His work also reoriented his entire discipline in radical new directions.
Grothendieck was intense and ascetic from his early days. Starting in the early 1950s, when he was in his 20s, he produced thousands of pages of formal and informal notes that changed the course of mathematics. Then in 1970, he quit. He left his post at a prestigious research institute just outside of Paris to teach at the provincial university in Montpellier where he studied as an undergraduate. He mostly stopped talking to other mathematicians. In the early 1990s, he moved to a small village in the Pyrenees, where he lived as a hermit.
Mathematicians are still grappling with the innovations he made half a century ago. His work pushed mathematics to a new level of abstraction by focusing on the relationships between objects rather than the objects themselves. “If there is one thing in mathematics which fascinates me more than any other (and undoubtedly always has), it is neither ‘number’ nor ‘size,’ but invariably shape,” he wrote in his memoirs. “And among the thousand and one faces under which shape chooses to reveal itself to us, that which has fascinated me more than any other and continues to do so is the structure hidden in mathematical things.”
His revolutionary mathematics centered around that search for hidden structure…
Read on: “How Alexander Grothendieck Revolutionized 20th-Century Mathematics,” from @kkakaes.bsky.social in @quantamagazine.bsky.social.
For more, see the section on Grothendieck in Benjamin Labatut‘s remarkable When We Cease To Understand the World.
* Albert Einstein
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As we study shape, we might send speculative birthday greetings to a man who, while not technically a mathematician, nonetheless created a famous equation: Frank Drake; he was born this date in 1930. An astronomer and astrophysicist, he formulated the Drake Equation in 1961 to estimate the number of technological civilizations that might exist in the Milky Way galaxy, N = R* × fp × ne× fl × fi × fc × L. Using plausible guesses for the parameters, Drake concluded perhaps 10 planets in our galaxy may have life originating detectable signals. In 1960, Drake led the first search, the two-month Project Ozma to listen for patterns in radio waves with a complex, ordered pattern that might be assumed to represent messages from some extraterrestrial intelligence.
Carl Sagan and Drake designed the plaques on Pioneer 10 and Pioneer 11 for the purpose of greeting and informing any extraterrestrial life that might find the vessels after they left the solar system.
“Something that doesn’t actually exist can still be useful”*…
Gregory Barber on ultrafinitism, a philosophy that rejects the infinite. Ultrafinitism has long been dismissed as mathematical heresy, but it is also producing new insights in math and beyond…
Doron Zeilberger is a mathematician who believes that all things come to an end. That just as we are limited beings, so too does nature have boundaries — and therefore so do numbers. Look out the window, and where others see reality as a continuous expanse, flowing inexorably forward from moment to moment, Zeilberger sees a universe that ticks. It is a discrete machine. In the smooth motion of the world around him, he catches the subtle blur of a flip-book.
To Zeilberger, believing in infinity is like believing in God. It’s an alluring idea that flatters our intuitions and helps us make sense of all sorts of phenomena. But the problem is that we cannot truly observe infinity, and so we cannot truly say what it is. Equations define lines that carry on off the chalkboard, but to where? Proofs are littered with suggestive ellipses. These equations and proofs are, according to Zeilberger — a longtime professor at Rutgers University and a famed figure in combinatorics — both “very ugly” and false. It is “completely nonsense,” he said, huffing out each syllable in a husky voice that seemed worn out from making his point.
As a matter of practicality, infinity can be scrubbed out, he contends. “You don’t really need it.” Mathematicians can construct a form of calculus without infinity, for instance, cutting infinitesimal limits out of the picture entirely. Curves might look smooth, but they hide a fine-grit roughness; computers handle math just fine with a finite allowance of digits. (Zeilberger lists his own computer, which he named “Shalosh B. Ekhad,” as a collaborator on his papers.) With infinity eliminated, the only thing lost is mathematics that was “not worth doing at all,” Zeilberger said.
Most mathematicians would say just the opposite — that it’s Zeilberger who spews complete nonsense. Not just because infinity is so useful and so natural to our descriptions of the universe, but because treating sets of numbers (like the integers) as actual, infinite objects is at the very core of mathematics, embedded in its most fundamental rules and assumptions.
At the very least, even if mathematicians don’t want to think about infinity as an actual entity, they acknowledge that sequences, shapes, and other mathematical objects have the potential to grow indefinitely. Two parallel lines can in theory go on forever; another number can always be added to the end of the number line.
Zeilberger disagrees. To him, what matters is not whether something is possible in principle, but whether it is actually feasible. What this means, in practice, is that not only is infinity suspect, but extremely large numbers are as well. Consider “Skewes’ number,” eee79. This is an exceptionally large number, and no one has ever been able to write it out in decimal form. So what can we really say about it? Is it an integer? Is it prime? Can we find such a number anywhere in nature? Could we ever write it down? Perhaps, then, it is not a number at all.
This raises obvious questions, such as where, exactly, we will find the end point. Zeilberger can’t say. Nobody can. Which is the first reason that many dismiss his philosophy, known as ultrafinitism. “When you first pitch the idea of ultrafinitism to somebody, it sounds like quackery — like ‘I think there’s a largest number’ or something,” said Justin Clarke-Doane, a philosopher at Columbia University.
“A lot of mathematicians just find the whole proposal preposterous,” said Joel David Hamkins, a set theorist at the University of Notre Dame. Ultrafinitism is not polite talk at a mathematical society dinner. Few (one might say an ultrafinite number) work on it. Fewer still are card-carrying members, like Zeilberger, willing to shout their views out into the void. That’s not just because ultrafinitism is contrarian, but because it advocates for a mathematics that is fundamentally smaller, one where certain important questions can no longer be asked.
And yet it gives Hamkins and others a good deal to think about. From one angle, ultrafinitism can be seen as a more realistic mathematics. It is math that better reflects the limits of what people can create and verify; it may even better reflect the physical universe. While we might be inclined to think of space and time as eternally expansive and divisible, the ultrafinitist would argue that these are assumptions that science has increasingly brought into question — much as, Zeilberger might say, science brought doubt to God’s doorstep.
“The world that we’re describing needs to be honest through and through,” said Clarke-Doane, who in April 2025 convened a rare gathering of experts to explore ultrafinitist ideas. “If there might only be finitely many things, then we’d better also be using a math that doesn’t just assume that there are infinitely many things at the get-go.” To him, “it sure seems like that should be part of the menu in the philosophy of math.”
For mathematicians to take it seriously, though, ultrafinitists first need to agree on what they’re talking about — to turn arguments that sound like “bluster,” as Hamkins puts it, into an official theory. Mathematics is steeped in formal systems and common frameworks. Ultrafinitism, meanwhile, lacks such structure.
It is one thing to tackle problems piecemeal. It is quite another to rewrite the logical foundations of mathematics itself. “I don’t think the reason ultrafinitism has been dismissed is that people have good arguments against it,” Clarke-Doane said. “The feeling is that, oh, well, it’s hopeless.”
That’s a problem that some ultrafinitists are still trying to address.
Zeilberger, meanwhile, is prepared to abandon mathematical ideals in favor of a mathematics that’s inherently messy — just like the world is. He is less a man of foundational theories than a man of opinions, of which he lists 195 on his website. “I cannot be a tenured professor without doing this crackpot stuff,” he said. But one day, he added, mathematicians will look back and see that this crackpot, like those of yore who questioned gods and superstitions, was right. “Luckily, heretics are no longer burned at the stake.”…
Read on for the history of ultrafinitism, the critical dialogue surrounding it, and its implications: “What Can We Gain by Losing Infinity?” from @gregbarber.bsky.social in @quantamagazine.bsky.social.
* Ian Stewart (whose point was somewhat different from Zeilberger’s :-), Infinity: A Very Short Introduction
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As we engage the endless, we might spare a thought for a man whose work touched on the infinitesimal, Isaac Barrow; he died on this date in 1677. A theologian and mathematician, he played a key role in the development of infinitesimal calculus (in particular, for a proof of the fundamental theorem of calculus). Barrow was the inaugural holder of the prestigious Lucasian Professorship of Mathematics at the University of Cambridge, a post later held by his student, Isaac Newton (who, of course, shares primary credit for the development of calculus with Gottfried Wilhelm Leibniz).
“The present is pregnant with the future”*…
The estimable Tim O’Reilly uses scenario planning to create an insightful look at AI, our futures, and the choices that will define them…
We all read it in the daily news. The New York Times reports that economists who once dismissed the AI job threat are now taking it seriously. In February, Jack Dorsey cut 40% of Block’s workforce, telling shareholders that “intelligence tools have changed what it means to build and run a company.” Block’s stock rose 20%. Salesforce has shed thousands of customer support workers, saying AI was already doing half the work. And a Stanford study found that software developers aged 22 to 25 saw employment drop nearly 20% from its peak, while developers over 26 were doing fine.
But how are we to square this news with a Vanguard study that found that the 100 occupations most exposed to AI were actually outperforming the rest of the labor market in both job growth and wages, and a rigorous NBER study of 25,000 Danish workers that found zero measurable effect of AI on earnings or hours?
Other studies could contribute to either side of the argument. For example, PwC’s 2025 Global AI Jobs Barometer, analyzing close to a billion job ads across six continents, found that workers with AI skills earn a 56% wage premium, and that productivity growth has nearly quadrupled in the industries most exposed to AI.
This is exactly the kind of contradictory, uncertain landscape that scenario planning was designed for. Scenario planning doesn’t ask you to predict what the future will be. It asks you to imagine divergent possible futures and to develop a strategy that improves your odds of success across all of them. I’ve used it many times at O’Reilly and have written about it before with COVID and climate change as illustrative examples. The argument between those who say AI will cause mass unemployment and those who insist technology always creates more jobs than it destroys is a debate that will only be resolved by time. Both sides have evidence. Both are probably right at some level. And both framings are not terribly helpful for anyone trying to figure out what to do next…
[O’Reilly explains the scenario approach, then applies it to our future with AI (see the image above), astutely assessing the conflicting signals that we’ve experiencing; he explores the “robust strategy” for our uncertian future (strategic choices that make sense regardless of which future unfolds); then he concludes…
… I’ll return to the theme that I sounded in my book WTF? What’s the Future and Why It’s Up To Us.
Every time a company uses AI to do what it was already doing with fewer people, it is making a choice for the lower half of the scenario grid. Every time a company uses AI to do something that wasn’t previously possible, to serve a customer who wasn’t previously served, to solve a problem that wasn’t previously solvable, it is making a choice for the upper half. These choices compound, for good or ill. An economy that uses AI primarily for efficiency will slowly hollow itself out.
Looking at the news from the future, both sets of signals are present. The question is which will dominate. AI will give us both the Augmentation Economy and the Displacement Crisis, in different measures in different places, depending on the choices we make.
Scenario planning teaches us that we don’t have to predict which future we’ll get. We do have to prepare for a very uncertain future. But the robust strategy, the one that works across every quadrant, is to focus on doing more, not just doing the same with less, and to find ways that human taste still matters in what is created. As long as there is unmet demand, as long as there are problems we haven’t solved and people we haven’t served, AI will augment human work rather than replacing it. It’s only when we stop looking for new things to do that the machines come for the jobs…
Eminently worth reading in full. Indeed, speaking as a long-time scenario planner, your correspondent can only wish that everyone who wields “scenarios” applies the approach as appropriately, adriotly, and acutely as Tim has: “Scenario Planning for AI and the ‘Jobless Future‘,” from @timoreilly.bsky.social.
* Voltaire
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As we take the long view, we might send formative birthday greetings to Mark Pinsker; he was born on this date in 1923. A mathematician, he made impoprtant contributions to the fields of information theory, probability theory, coding theory, ergodic theory, mathematical statistics, and communication networks. This work, which helped lay the foundation for AI-as-we-know-it, earned him the IEEE Claude E. Shannon Award in 1978, and the IEEE Richard W. Hamming Medal in 1996, among other honors.
“It’s the bell curve again”*…
Joseph Howlett on how the central limit theorem, which started as a bar trick for 18th-century gamblers, became something on which scientists rely every day…
No matter where you look, a bell curve is close by.
Place a measuring cup in your backyard every time it rains and note the height of the water when it stops: Your data will conform to a bell curve. Record 100 people’s guesses at the number of jelly beans in a jar, and they’ll follow a bell curve. Measure enough women’s heights, men’s weights, SAT scores, marathon times — you’ll always get the same smooth, rounded hump that tapers at the edges.
Why does the bell curve pop up in so many datasets?
The answer boils down to the central limit theorem, a mathematical truth so powerful that it often strikes newcomers as impossible, like a magic trick of nature. “The central limit theorem is pretty amazing because it is so unintuitive and surprising,” said Daniela Witten, a biostatistician at the University of Washington. Through it, the most random, unimaginable chaos can lead to striking predictability.
It’s now a pillar on which much of modern empirical science rests. Almost every time a scientist uses measurements to infer something about the world, the central limit theorem is buried somewhere in the methods. Without it, it would be hard for science to say anything, with any confidence, about anything.
“I don’t think the field of statistics would exist without the central limit theorem,” said Larry Wasserman, a statistician at Carnegie Mellon University. “It’s everything.”
Perhaps it shouldn’t come as a surprise that the push to find regularity in randomness came from the study of gambling…
Read on for the fascinating story of: “The Math That Explains Why Bell Curves Are Everywhere,” from @quantamagazine.bsky.social.
Howlett concludes by observing that “The central limit theorem is a pillar of modern science, ultimately, because it’s a pillar of the world around us. When we combine lots of independent measurements, we get clusters. And if we’re clever enough, we can use those clusters to find out something interesting about the processes that made them”– which follows from the story he shares.
Still, we’d do well to remember that there are limits to its applicability, both descriptively (as Nassim Nicholas Taleb points out, “because the bell curve ignores large deviations, cannot handle them, yet makes us confident that we have tamed uncertainty”) and prescriptively (as Benjamim Bloom argues, “The bell-shaped curve is not sacred. It describes the outcome of a random process. Since education is a purposeful activity….the achievement distribution should be very different from the normal curve if our instruction is effective).
For (much) more, see Peter Bernstein‘s wonderful Against the Gods: The Remarkable Story of Risk
* Robert A. Heinlein, Time Enough for Love
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As we noodle on the normal distribution, we might send curve-shattering birthday greetings to Norman Borlaug; he was born on ths date in 1914. An agronomist, he developed and led initiatives worldwide that contributed to the voluminous increases in agricultural production we call “the Green Revolution.” Borlaug was awarded multiple honors for his work, including the Nobel Peace Prize, the Presidential Medal of Freedom, and the Congressional Gold Medal; he’s one of only seven people to have received all three of those awards.
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