Posts Tagged ‘order’
“As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality”*…
As Gregory Barber explains, two new notions of infinity challenge a long-standing plan to define the mathematical universe…
It was minus 20 degrees Celsius, and while some went cross-country skiing, Juan Aguilera, a set theorist at the Vienna University of Technology, preferred to linger in the cafeteria, tearing pieces of pulla pastry and debating the nature of two new notions of infinity. The consequences, Aguilera believed, were grand. “We just don’t know what they are yet,” he said.
Infinity, counterintuitively, comes in many shapes and sizes. This has been known since the 1870s, when the German mathematician Georg Cantor proved that the set of real numbers (all the numbers on the number line) is larger than the set of whole numbers, even though both sets are infinite. (The short version: No matter how you try to match real numbers to whole numbers, you’ll always end up with more real numbers.) The two sets, Cantor argued, represented entirely different flavors of infinity and therefore had profoundly different properties.
From there, Cantor constructed larger infinities, too. He took the set of real numbers, built a new set out of all of its subsets, then proved that this new set was bigger than the original set of real numbers. And when he took all the subsets of this new set, he got an even bigger set. In this way, he built infinitely many sets, each larger than the last. He referred to the different sizes of these infinite sets as cardinal numbers (not to be confused with the ordinary cardinals 1, 2, 3…).
Set theorists have continued to define cardinals that are far more exotic and difficult to describe than Cantor’s. In doing so, they’ve discovered something surprising: These “large cardinals” fall into a surprisingly neat hierarchy. They can be clearly defined in terms of size and complexity. Together, they form a massive tower of infinities that set theorists then use to probe the boundaries of what’s mathematically possible.
But the two new cardinals that Aguilera was pondering in the Arctic cold behaved oddly. He had recently constructed them, along with Joan Bagaria of the University of Barcelona and Philipp Lücke of the University of Hamburg, only to find that they didn’t quite fit into the usual hierarchy. Instead, they “exploded,” Aguilera said, creating a new class of infinities that their colleagues hadn’t bargained on — and implying that far more chaos abounds in mathematics than expected.
It’s a provocative claim. The prospect is, to some, exciting. “I love this paper,” said Toby Meadows, a logician and philosopher at the University of California, Irvine. “It seems like real progress — a really interesting insight that we didn’t have before.”
But it’s also difficult to really know whether the claim is true. That’s the nature of studying infinity. If mathematics is a tapestry sewn together by traditional assumptions that everyone agrees on, the higher reaches of the infinite are its tattered fringes. Set theorists working in these extreme areas operate in a space where the traditional axioms used to write mathematical proofs do not always apply, and where new axioms must be written — and often break down.
Up here, most questions are fundamentally unprovable, and uncertainty reigns. And so to some, the new cardinals don’t change anything. “I don’t buy it at all,” said Hugh Woodin, a set theorist at Harvard University who is currently leading the quest to fully define the mathematical universe. Woodin was Bagaria’s doctoral adviser 35 years ago and Aguilera’s in the 2010s. But his students are cutting their own path through infinity’s thickets. “Your children grow up and defy you,” Woodin said…
More on the fascinating state of play at: “Is Mathematics Mostly Chaos or Mostly Order?” from @GregoryJBarber in @quantamagazine.bsky.social.
* Albert Einstein
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As we get down with Gödel, we might send insightful birthday greetings to John Allen Paulos; he was born on this date in 1945. A mathematician, he is best known as an advocate for– and a skilled teacher of– mathematical literacy. His book Innumeracy: Mathematical Illiteracy and its Consequences (1988) was a bestseller, and A Mathematician Reads the Newspaper (1995) extended the critique. Paulos was a regular columinst for both The Guardian and ABC News. And in 2001 he created and taught a course on quantitative literacy for journalists at the Columbia University School of Journalism– an exercise that stimulated further programs at Columbia and elsewhere in precision and data-driven journalism.
Happy 4th of July to readers in the U.S… but are we commemorating the right day?
“Tennyson said that if we could understand a single flower we would know who we are and what the world is”*…
Reality feels “stable” enough to talk about it– though all logic seems to point away from that possibility. Marco Giancotti unpacks what he suggests is the only line of reasoning that resolves that paradox…
What is the source of what we call order? Why do many things look too complex, too perfectly organized to arise unintentionally from chaos? How can something as special as a star or a flower even happen? And, for that matter, why do some natural phenomena seem designed for a purpose?
We live in a universe of forces eternally straining to crush things together or tear them apart. There is no physical law for “forming shapes”, no law for being separated from other things, no law for staying still.
Boundaries are in the eye of the beholder, not in the world out there. Out there is only tumult, clashing, and shuffling of everything with everything else.
And yet, our familiar world is filled with things stable and consistent enough for us to give them names—and to live our whole lives with.
In this essay we’ll tackle these questions at the very root. We need good questions to get good answers, so we’ll begin by clarifying the problem. It has to do with probabilities—we’ll see why those natural objects seem so utterly unlikely to happen by chance, and we’ll find the fundamental process that solves the dilemma.
This will take us most of the way, but we’ll have one final obstacle to overcome, a cognitive Last Boss: living things still feel a little magical in some way, imbued with a mysterious substance called “purpose” that feels qualitatively different from how inanimate things work. This kind of confusion runs very deep in our culture. To remove it, I’ll give a name to something that, as far as I know, hasn’t been named before: phenomena that I’ll be calling—enigmatically, for now—“Water Lilies.”…
Applying systems dynamics, complexity, and emergence to understanding reality itself: “Recursion, Tidy Stars, and Water Lilies,” from @marco_giancotti (the second in a trilogy of essays: part one here; subscribe to his newsletter for Part Three when it drops).
* Jorge Luis Borges, “The Zahir“
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As we explore existence, we might spare a thought for Francis Simpson; he died on this date in 2003. An English naturalist, conservationist, and chronicler of the countryside and wild flowers of his native Suffolk, he became a botanist at Ipswich Museum, where he worked until his retirement in 1977.
He published one of the most highly regarded county floras, simply entitled Simpson’s Flora of Suffolk, and in 1938 saved a small meadow, famous for its snakeshead fritillaries, from being drained and ploughed into farmland. Using donations amounting to £75, he was able to purchase the field, Mickfield Meadow, for the Society for the Promotion of Nature Reserves. Today, it is one of the oldest nature reserves in the country, protecting the meadow flowers now surrounded by farmland.
“Two dangers constantly threaten the world: order and disorder”*…
After two days of posts on the state of our civil society, a palette-cleanser: Jordana Cepelewicz with a possibly-consoling reminder…
When he died in 1930 at just 26 years old, Frank Ramsey [see here] had already made transformative contributions to philosophy, economics and mathematics. John Maynard Keynes sought his insights; Ludwig Wittgenstein admired him and considered him a close friend. In his lifetime, Ramsey published only eight pages on pure math: the beginning of a paper about a problem in logic. But in that work, he proved a theorem that ultimately led to a whole new branch of mathematics — what would later be called Ramsey theory.
His theorem stated that if a system is large enough, then no matter how disordered it might be, it’s always bound to exhibit some sort of regular structure. Order inevitably emerges from chaos; patterns are unavoidable. Ramsey theory is the study of when this happens — in sets of numbers, in collections of vertices and edges called graphs, and in other systems. The mathematicians Ronald Graham and Joel Spencer likened it to how you can always pick out patterns among the stars in the night sky…
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… In fact, Ramsey theory isn’t just about inevitable patterns found in graphs. Hidden structure emerges in lists of numbers, strings of beads and even card games. In 2019, for example, mathematicians studied collections of sets that can always be arranged to resemble the petals of a sunflower. That same year, Quanta reported on research into sets of numbers that are guaranteed to contain numerical patterns called polynomial progressions. And last year, mathematicians proved a similar result, about sets of integers that must always include three evenly spaced numbers, called arithmetic progressions.
In its hunt for patterns, Ramsey theory gets to the core of what mathematics is all about: finding beauty and order in the most unexpected places…
Finding order in chaos: “Why Complete Disorder Is Mathematically Impossible,” from @jordanacep in @QuantaMagazine.
* Paul Valery
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As we ponder patterns, we might send paradigm-shaping birthday greetings to a woman who found order and pattern of a different– and world-changing– sort: Rosalind Franklin; she was born on this date in 1920. A biophysicist and X-ray crystallographer, Franklin captured the X-ray diffraction images of DNA that were, in the words of Francis Crick, “the data we actually used” when he and James Watson developed their “double helix” hypothesis for the structure of DNA. Indeed, it was Franklin who argued to Crick and Watson that the backbones of the molecule had to be on the outside (something that neither they nor their competitor in the race to understand DNA, Linus Pauling, had understood). Franklin never received the recognition she deserved for her independent work– her paper was published in Nature after Crick and Watson’s, which barely mentioned her– and she died of cancer four years before Crick, Watson, and their lab director Maurice Wilkins won the Nobel Prize for the discovery.
“Words are sacred. They deserve respect. If you get the right ones, in the right order, you can nudge the world a little.”*…
And as Gail Sherman observes, that principle operates at a pretty basic level…
There is a Royal Order of Adjectives, and you follow it without knowing what it is—a particular sequence to use when more than one adjective precedes a noun. There are exceptions, of course, because English is three languages in a trenchcoat. According to the Cambridge Dictionary, in general, the proper order is:
Opinion
Size
Physical quality
Shape
Age
Color
Origin
Material
Type
PurposeMost people couldn’t tell you this rule, but everyone follows it. If you use the wrong order, it just sounds weird. If you have a fancy new blue metal lunchbox but call it a metal new fancy blue lunchbox, people might be worried you are having a stroke…
“There is a Royal Order of Adjectives, and you follow it without knowing what it is,” from @CambridgeWords via @BoingBoing.
* Tom Stoppard
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As we parse, we might send powerfully-phrased birthday greetings to a spare but graceful user of adjectives, Seymour Wilson “Budd” Schulberg; he was born on this date in 1914. A screenwriter, television producer, novelist, and sportswriter, Schulberg is best remembered for his novels What Makes Sammy Run? (1941) and The Harder They Fall (1947), as well as his screenplays for On the Waterfront (1954, for which he received an Academy Award) and A Face in the Crowd (1957).
As a sportswriter, Schulberg was most famously chief boxing correspondent for Sports Illustrated. He wrote some well-received books on boxing, including Sparring with Hemingway and was inducted into the International Boxing Hall of Fame (in 2002).
The son of B. P. Schulberg, head of Paramount Studios in its golden age, Budd wrote Moving Pictures: Memoirs of a Hollywood Prince, an autobiography covering his youth in Hollywood, growing up in the 1920s and 1930s among the famous.
“For what are myths if not the imposing of order on phenomena that do not possess order in themselves? And all myths, however they differ from philosophical systems and scientific theories, share this with them, that they negate the principle of randomness in the world.”*…
And we humans are, as Kit Yates explains, myth-making animals…
Unfortunately, when it comes to understanding random phenomena, our intuition often lets us down. Take a look at the image below. Before you read the caption, see if you can pick out the data set generated using truly uniform random numbers for the coordinates of the dots (i.e., for each point, independent of the others, the horizontal coordinate is equally likely to fall anywhere along the horizontal axis and the vertical coordinate is equally likely to fall anywhere along the vertical).
The truly randomly distributed points in the figure are those in the left-most image. The middle image represents the position of ants’ nests that, although distributed with some randomness, demonstrate a tendency to avoid being too close together in order not to overexploit the same resources. The territorial Patagonian seabirds’ nesting sites, in the right-most image, exhibit an even more regular and well-spaced distribution, preferring not to be too near to their neighbors when rearing their young. The computer-generated points, distributed uniformly at random in the left-hand image, have no such qualms about their close proximity.
If you chose the wrong option, you are by no means alone. Most of us tend to think of randomness as being “well spaced.” The tight clustering of dots and the frequent wide gaps of the genuinely random distribution seem to contradict our inherent ideas of what randomness should look like…
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… As a case in point, after noticing a disproportionate number of Steely Dan songs playing on his iPod shuffle, journalist Steven Levy questioned Steve Jobs directly about whether “shuffle” was truly random. Jobs assured him that it was and even got an engineer on the phone to confirm it. A follow-up article Levy wrote in Newsweek garnered a huge response from readers having similar experiences, questioning, for example, how two Bob Dylan songs shuffled to play one after the other (from among the thousands of songs in their collections) could possibly be random.
We ascribe meaning too readily to the clustering that randomness produces, and, consequently, we deduce that there is some generative force behind the pattern. We are hardwired to do this. The “evolutionary” argument holds that tens of thousands of years ago, if you were out hunting or gathering in the forest and you heard a rustle in the bushes, you’d be wise to play it safe and to run away as fast as you could. Maybe it was a predator out looking for their lunch and by running away you saved your skin. Probably, it was just the wind randomly whispering in the leaves and you ended up looking a little foolish—foolish, but alive and able to pass on your paranoid pattern-spotting genes to the next generation…
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This… is just one example of the phenomenon known in the psychology literature as pareidolia, in which an observer interprets an ambiguous auditory or visual stimulus as something they are familiar with. This phenomenon, otherwise known as “patternicity,” allows people to spot shapes in the clouds and is the reason why people think they see a man in the moon. Pareidolia is itself an example of the more general phenomenon of apophenia, in which people mistakenly perceive connections between and ascribe meaning to unrelated events or objects. Apophenia’s misconstrued connections lead us to validate incorrect hypotheses and draw illogical conclusions. Consequently, the phenomenon lies at the root of many conspiracy theories—think, for example, of extraterrestrial seekers believing that any bright light in the sky is a UFO.
Apophenia sends us looking for the cause behind the effect when, in reality, there is none at all. When we hear two songs by the same artist back-to-back, we are too quick to cry foul in the belief that we have spotted a pattern, when in fact these sorts of clusters are an inherent feature of randomness. Eventually, the dissatisfaction caused by the clustering inherent to the iPod’s genuinely random shuffle algorithm led Steve Jobs to implement the new “Smart Shuffle” feature on the iPod, which meant that the next song played couldn’t be too similar to the previous song, better conforming to our misconceived ideas of what randomness looks like. As Jobs himself quipped, “We’re making it less random to make it feel more random.”…
“Why Randomness Doesn’t Feel Random,” an excerpt from How to Expect the Unexpected: The Science of Making Predictions—and the Art of Knowing When Not To, by @Kit_Yates_Maths in @behscientist.
* Stanislaw Lem
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As we ponder purported patterns, we might send carefully-discerned birthday greetings to a man who did in fact find a pattern (or at least a meaning) in what might have seemed random and meaningless: Robert Woodrow Wilson; he was born on this date in 1936. An astronomer, he detected– with Bell Labs colleague Arno Penzias– cosmic microwave background radiation: “relic radiation”– that’s to say, the “sound “– of the Big Bang… familiar to those of us old enough to remember watching an old-fashioned television after the test pattern was gone (when there was no broadcast signal received): the “fuzz” we saw and the static-y sounds we heard, were that “relic radiation” being picked up.
Their 1964 discovery earned them the 1978 Nobel Prize in Physics.
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