Posts Tagged ‘Royal Society’
“If geometry is dressed in a suit coat, topology dons jeans and a T-shirt”*…
Paulina Rowińska on how, in the mid-19th century, Bernhard Riemann conceived of a new way to think about mathematical spaces, providing the foundation for modern geometry and physics…
Standing in the middle of a field, we can easily forget that we live on a round planet. We’re so small in comparison to the Earth that from our point of view, it looks flat.
The world is full of such shapes — ones that look flat to an ant living on them, even though they might have a more complicated global structure. Mathematicians call these shapes manifolds. Introduced by Bernhard Riemann in the mid-19th century, manifolds transformed how mathematicians think about space. It was no longer just a physical setting for other mathematical objects, but rather an abstract, well-defined object worth studying in its own right.
This new perspective allowed mathematicians to rigorously explore higher-dimensional spaces — leading to the birth of modern topology, a field dedicated to the study of mathematical spaces like manifolds. Manifolds have also come to occupy a central role in fields such as geometry, dynamical systems, data analysis and physics.
Today, they give mathematicians a common vocabulary for solving all sorts of problems. They’re as fundamental to mathematics as the alphabet is to language. “If I know Cyrillic, do I know Russian?” said Fabrizio Bianchi, a mathematician at the University of Pisa in Italy. “No. But try to learn Russian without learning Cyrillic.”
So what are manifolds, and what kind of vocabulary do they provide?…
[Rowińska explains manifolds and the history of the development of our understanding of them, concentrating on the pivotal role of Riemann…]
… Manifolds are crucial to our understanding of the universe… In his general theory of relativity, Einstein described space-time as a four-dimensional manifold, and gravity as that manifold’s curvature. And the three-dimensional space we see around us is also a manifold — one that, as manifolds do, appears Euclidean to those of us living within it, even though we’re still trying to figure out its global shape.
Even in cases where manifolds don’t seem to be present, mathematicians and physicists try to rewrite their problems in the language of manifolds to make use of their helpful properties. “So much of physics comes down to understanding geometry,” said Jonathan Sorce, a theoretical physicist at Princeton University. “And often in surprising ways.”
Consider a double pendulum, which consists of one pendulum hanging from the end of another. Small changes in the double pendulum’s initial conditions lead it to carve out very different trajectories through space, making its behavior hard to predict and understand. But if you represent the configuration of the pendulum with just two angles (one describing the position of each of its arms), then the space of all possible configurations looks like a doughnut, or torus — a manifold. Each point on this torus represents one possible state of the pendulum; paths on the torus represent the trajectories the pendulum might follow through space. This allows researchers to translate their physical questions about the pendulum into geometric ones, making them more intuitive and easier to solve. This is also how they study the movements of fluids, robots, quantum particles and more.
Similarly, mathematicians often view the solutions to complicated algebraic equations as a manifold to better understand their properties. And they analyze high-dimensional datasets — such as those recording the activity of thousands of neurons in the brain — by looking at how those data points might sit on a lower-dimensional manifold.
Asking how scientists use manifolds is akin to asking how they use numbers, Sorce said. “They are at the foundation of everything.”…
“What Is a Manifold?” from @quantamagazine.bsky.social.
Apposite: Rowińska in conversation with Ira Flatow on Science Friday: “How Math Helps Us Map The World.”
* David S. Richeson, Euler’s Gem: The Polyhedron Formula and the Birth of Topology (Riemann’s work was an advance on the foundation that Euler laid in his 1736 paper on the Seven Bridges of Königsberg, which led to his polyhedron formula)
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As we get down with geometry, we might spare a thought for John Wallis; he died on this date in 1703. A clergyman and mathematician, he served as chief cryptographer for Parliament (decoding Royalist messages during the Civil War) and, later (as Savilian Chair of geometry at Oxford after the hostilities), for the the royal court. Wallis is credited with introducing the symbol ∞ to represent the concept of infinity, and used 1/∞ for an infinitesimal… which earned him (along with his contemporaries Isaac Newton and Gottfried Wilhelm Leibniz) a share of the credit for the development of infinitesimal calculus. He was a founding member of the Royal Society and one of its first Fellows.
“Where all think alike there is little danger of innovation”*…
Last week, Northwestern Professor Joel Mokyr was awarded a half-share in The Nobel Prize in Economic Sciences (AKA The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel) “for having identified the prerequisites for sustained growth through technological progress.” Anton Howes explains why this is noteworthy…
Among today’s winners of the Nobel prize in Economics is Joel Mokyr, the professor at Northwestern whose name is indelibly associated with the primacy of innovation to modern economic growth – the gradual, sustained, and unprecedented improvement in living standards that first Britain, and then country after country, have enjoyed over the past few hundred years. It was reading Mokyr’s The Enlightened Economy that first opened my eyes to the importance of studying the history of invention to explaining the causes of the Industrial Revolution, which I have since made my career.
What makes this Nobel win so remarkable, and so pleasantly surprising, is that Mokyr’s work is not the kind that is often published by economics journals, or even many economic history journals anymore. Over the past few decades, journal editors and peer-reviewers have increasingly insisted that papers must present large datasets that have been treated using complex statistical methods in order to make even the mildest claims about what caused what. Although Mokyr is a master of such methods – he was one of the early pioneers of economic history’s quantitative turn – the work for which he has won the prize is firmly and necessarily qualitative.
Mokyr’s is the economic history that gets written up in books – his classics are The Lever of Riches, The Gifts of Athena, The Enlightened Economy, and A Culture of Growth – and in readable papers shorn of unnecessary formulae. His is history accessible to the layman, though rigorously applying the insights of economics. The prize is a clear signal from the economics profession that it doesn’t just value the application of fancy statistical methods; its highest prize can go to works of history.
Whereas most of the public, and even many historians, think of the causes of modern economic growth – the beginnings of the Industrial Revolution – as being rooted in material factors, like conquest, colonialism, or coal, Mokyr tirelessly argued that it was rooted in ideas, in the intellectual entrepreneurship of figures like Francis Bacon and Isaac Newton, and in the uniquely precocious accumulation in eighteenth-century Britain of useful, often mechanically actionable knowledge. Britain, he argued, through its scientific and literary societies, and its penchant for publications and sharing ideas, was the site of a world-changing Industrial Enlightenment – the place where progress was thoughtpossible, and then became real.
One of Mokyr’s big early insights, first appearing in Lever of Riches, was that many inventions could not be predicted by economic factors. Society could enjoy remarkable productivity improvements from simply increasing the size of the market, leading to division of labour and specialization – what he labelled ‘micro-inventions’ – in the vein popularised by Adam Smith. But this could not explain an invention that appeared out of the blue, like Montgolfier’s hot air balloon in the 1780s – what he called a ‘macro-invention’, not for the magnitude of its impact, but for its novelty. Macro-inventions often required further development to make them important, but the original breakthrough could not be predicted by looking at changes in prices or the availability of resources. It ultimately came down to advances in our understanding of the world. Mokyr put the Scientific Revolution – and the factors that contributed to it – on the economist’s map.
Mokyr also looked at the relationship between different kinds of knowledge. A scientist might know, through observation, that the air has a weight. A craftsman might know, through long training and experience with glass, how to make a long glass tube. Each could not get far alone. But combining them, by creating means to ensure that scientists and craftsmen talked with one another and collaborated – through connecting their propositional and prescriptive knowledge, their heads and hands – very quickly led to the invention of thermometers, barometers, and much more besides, in an ever expanding field of knowledge. What Mokyr taught economists is that it’s not knowledge per se that makes the difference, but the way it is organized. Much of his later work has shown just how deep a pool Britain’s scientists could draw on, of skilled artisans.
In a way, Mokyr himself has practised what he preached. As editor of Princeton University Press’s book series on the Economic History of the Western World, Mokyr has for decades provided an all-important space for economists and historians to write the kinds of research that would never have been publishable in economics journals – including of explanations of the Industrial Revolution that are the polar opposite to his own. He helped keep the connection between history and economics alive.
Mokyr’s case for the primacy of knowledge and ideas was not an easy one to make to economists. They are naturally drawn to data that can be counted, and not to narrative, often no matter how well evidenced. But it appears that Mokyr’s persistence, elevated by his infectious, irrepressible sprightliness, has paid off. His prize is a long overdue recognition of the historyin economic history, and a remarkable testament to the power of ideas to persuade…
A triumph for history and the importance of ideas: “Joel Mokyr’s Nobel,” from @antonhowes.bsky.social.
See also: “Why Joel Mokyr deserves his Nobel prize,” gift article from The Economist.
* Edward Abbey, Desert Solitaire
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As we ponder the process of progress, we might send creative birthday greetings to one of the subjects Mokyr’s study, Sir Christopher Wren; he born on this date in 1632. A mathematician and astronomer (who co-founded and later served as president of the Royal Society), he is better remembered as one of the most highly acclaimed English architects in history; he was given responsibility for rebuilding 52 churches in the City of London after the Great Fire in 1666, including what is regarded as his masterpiece, St. Paul’s Cathedral, on Ludgate Hill.
Wren, whose scientific work ranged broadly– e.g., he invented a “weather clock” similar to a modern barometer, new engraving methods, and helped develop a blood transfusion technique– was admired by Isaac Newton, as Newton noted in the Principia.
“Every picture tells a story”*…
The world’s populations is unevenly spread across the globe. But, plotted by latitude (as per this visualization from Engaging Data), it’s a little more concentrated…
… which is interesting (perhaps better said, “bracing”) to consider aside this illustration from NOAA…
Global warming is coming for most of us: “World Population Distribution by Latitude and Longitude,” from @engagingdata.bsky.social and @climate.noaa.gov.
See also: “The world is heating up. How much can our bodies handle?” from @gristnews.bsky.social and “Understanding Climate Migration,” from RAND.
* traditional saying
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As we feel the heat, we might spare a thought for John Graunt; he died on this date in 1674. A haberdasher turned statistician, he is considered by many to be the father of demography (the statistical study of human populations).
A charter member of The Royal Society, Graunt distributed a 90-page book, Natural and Political Observations Mentioned in a Following Index, and Made upon the Bills of Mortality at the February, 1662 Society meeting. He described his work as having “reduced several great confused volumes” of parish records into a few easily to understood tables, and “abridged such Observations… into a few succinct Paragraphs.” He initiated “life tables” of life expectancy. His use of demographics was further pioneered by his friend Sir William Petty and Edmond Halley, the Astronomer Royal.
Graunt’s work also gives him some claim to having been the first epidemiologist.
“Nature uses only the longest threads to weave her patterns, so that each small piece of her fabric reveals the organization of the entire tapestry”*…
More than 70 years ago, mathematician Alan Turing proposed a mechanism that explained how patterns could emerge from bland uniformity. As Amber Dance explains, scientists are still using his model — and adding new twists — to gain a deeper understanding of animal markings…
There’s a reason fashion designers look to animal prints for inspiration. Creatures have evolved a dizzying array of patterns: stripes, spots, diamonds, chevrons, hexagons and even mazelike designs. Some, like peacocks, want to be seen, to attract a mate or scare off a rival or predator. Others, like tigers or female ducks, need to blend in, either to sneak up on prey or to avoid becoming lunch themselves.
Some patterns arise simply or randomly, but others develop via complex, precise interactions of pattern-generating systems. Their beauty aside, the intricacies of these systems are inspiring the scientists who aim to elucidate how the tiger got its stripes, the cheetah its spots and more besides.
Mammals like cats and dogs can have white tummies. They get them in a straightforward way: As the embryo develops, pigment-making cells originate along the site of the future spine and migrate down and around toward the belly. But sometimes they don’t make it all the way. Where the pigment cells run out of steam, the white begins.
The black dots on Dalmatians are generated randomly. So are the black-and-orange splotches on calico cats.
But the stripes of chipmunks and tigers, the speckles on fishes and chickens, and many other glorious animal features are laid down with exquisite precision. In a remarkable feat of self-organization, a uniform surface becomes patterned.
The person who figured out how this happens was Alan Turing [here]. You may know him as the 20th century mathematician who broke Nazi codes during World War II and developed early concepts in artificial intelligence.Turing also turned his math skills to understanding how regular features could emerge on the developing embryo. Scientists since then have applied his equations to the development of such patterns as fingerprint ridges, the places where hairs will sprout, and color patterns like stripes and spots.And it turns out he was really onto something: Today, scientists studying animal patterns still find Turing’s ideas to be remarkably effective — especially when combined with other factors that elaborate the patterns further….
A colorful tour of what scientists are learning today, starting with Turing’s theory: “Spots, stripes and more: Working out the logic of animal patterns,” from @amberldance in @KnowableMag.
* Richard Feynman
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As we contemplate coloring, we might spare a thought for Emanuel Mendes da Costa; he died on this date in 1791. A naturalist, he published A Natural History of Fossils in 1757 and served as clerk (from 1763) to the Royal Society– from which he embezzled membership funds to indulge his reckless penchant for collecting. When caught in 1767, the treasury was short by £1500—a substantial amount in those years. He confessed; his collections were auctioned to make restitution; but he was still sentenced for five years to debtor’s prison. After release he scraped by, with lecturing about fossils, translating, and trading in mineral, fossil and shell specimens. He wrote two books on shells and was perhaps the first to coin the word conchology. Still impoverished, he died in penury.
“I was reading the dictionary. I thought it was a poem about everything.”*…
That seminal semanticist Samuel Johnson suggested, “dictionaries are like watches, the worst is better than none and the best cannot be expected to go quite true.” From “unabridged” to “slanguage,” Madeline Kripke’s library of lexicons is a logophile’s heaven (or hell)…
Madeline Kripke’s first dictionary was a copy of Webster’s Collegiate that her parents gave her when she was a fifth grader in Omaha in the early 1950s. By the time of her death in 2020, at age 76, she had amassed a collection of dictionaries that occupied every flat surface of her two-bedroom Manhattan apartment—and overflowed into several warehouse spaces. Many believe that this chaotic, personal library is the world’s largest compendium of words and their usage.
“We don’t really know how many books it is,” says Michael Adams, a lexicographer and chair of the English department at Indiana University Bloomington. More than 1,500 boxes, with vague labels such as “Kripke documents” or “Kripke: 17 books,” arrived at the school’s Lilly Library on two tractor-trailers in late 2021. The delivery was accompanied by a nearly 2,000-page catalog detailing some 6,000 volumes. But that’s only a fraction of the total. In summer 2023, the library hired a group of students to simply open each box and list its contents. By the fall, their count stood at about 9,700. “And they’ve got a long way to go,” says Adams. “20,000 sounds like a pretty good estimate.”
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“This is my favorite wall,” Madeline Kripke told Narratively reporter Daniel Kreiger when he visited her West Village apartment in 2013. She shined a flashlight on glass-fronted shelves jammed with dictionaries full of the slanguage and cryptolect of small and likely overlooked communities. Kreiger listed some of the groups represented at that time, among them cowboys and flappers, mariners and gamblers, soldiers, circus workers, and thieves.
Among the first tomes Adams pulled from the boxes was a well-known example of the slang genre: The Canting Academy. This 17th-century dictionary by Richard Head is a guide to “cant,” the jargon of London’s criminal class or, as the subtitle to the second edition puts it, “The Mysterious and Villainous Practices Of that wicked Crew, commonly known by the Names of Hectors, Treppaners, Gults, &c.” (Adams wonders if a first edition is also hidden in the banker’s boxes.) With The Canting Academy, one can learn to translate the cant of the “priggs” (“all sorts of thieves”) to English: “lour” to “money,” “pannam” to “bread,” “lage” to “water.” Most of the language is indecipherable without this key, but Adams notes some usages that are common today. “To plant” something is, in centuries-old cant or modern-day English, “to lay, place, or hide.”
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Much of what Adams has unpacked has a far less storied (and pricey) past, but, he says, the quirky and unexpected volumes in Kripke’s collection might be the most valuable to future lexicographers and historians. A bright red pamphlet with a doodle of heart on the cover might seem disposable, but it is an artifact of a particular place and time, Adams says. “Dictionaries are made by people, so they’re not just language books,” he says, “they’re culture books.”
Printed in 1962 as a marketing tool for a CBS sitcom, that slim pamphlet featuring a big heart around the faces of two 20-something actors is Dobie Gillis: Teenage Slanguage Dictionary, filled with “teen-age antics and terms.” It’s the type of thing that might have been stuffed into a cereal box or inserted in a teen magazine, says Adams. “I’m pretty sure that most people threw the copy they had away, and so this one is a fairly rare item that says something important about the representation of teen language and culture in the 1950s and 1960s.” Thanks to Kripke’s copy we know that this, at least according to the marketers behind The Many Loves of Dobie Gillis, was the era of the “keen teen” (“well-liked person”), the “cream puff” (“conceited person”), the “meatball” (“a dull guy”), and the “mathematician” (“teen who can put two and two together and get SEX”).
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Kripke—“the mistress of slang,” in the words of one colleague—dedicated decades of her life to curating this collection of words, including countless ones we might like to forget. When she passed away without a will, the fate of her overwhelming library, plus a trove of documents on the history of dictionary making, was uncertain. Auctioning it off in lots could have brought the highest bids, but Kripke’s family worked in conjunction with the lexicographic community to preserve what Adams calls “her legacy.” That it was ultimately purchased in total by Indiana University Bloomington, a state university that committed to making the works accessible to the public, seems in keeping with the way Kripke herself viewed the collection, as a resource for the curious.
“You would go to see her in her Village apartment, and it was filled from top to bottom and side to side with books,” Adams says. It would have taken some digging but, “she would have the book that you need to see out for you and always some other specimens, too.”…
“The Low Down on the Greatest Dictionary Collection in the World,” in @atlasobscura.
* Steven Wright
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As we look it up, we might recall that it was on this date in 1660, at Gresham College in London, that twelve men, including Christopher Wren, Robert Boyle, John Wilkins, and Sir Robert Moray decided to found a “Colledge for the Promoting of Physico-Mathematicall Experimentall Learning” to promote “experimental philosophy” (which became science-as-we-know-it). Six months later, Robert Hooke‘s first publication, a pamphlet on capillary action, was read to the group.
The Society subsequently petitioned King Charles II to recognize it and to make a royal grant of incorporation. The Royal Charter, which was passed in July, 1662 created the Royal Society of London.
In 1665, the society introduced the world’s first journal exclusively devoted to science in 1665, Philosophical Transactions (and in so doing originated the peer review process now widespread in scientific journals). Its founding editor was Henry Oldenburg, the society’s first secretary. It remains the oldest and longest-running scientific journal in the world.
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