Posts Tagged ‘calculus’
“Truth is ever to be found in the simplicity, and not in the multiplicity and confusion of things”*…
From Kim (Scott) Morrison‘s and Dror Bar-Natan‘s, The Knot Atlas, “a complete user-editable knot atlas, in the wiki spirit of Wikipedia“– a marvelous example of a wide-spread urge in mathematics to find order through classification. As Joseph Howlett explains, that quest continues, even as it proves vexatious…
Biology in the 18th century was all about taxonomy. The staggering diversity of life made it hard to draw conclusions about how it came to be. Scientists first had to put things in their proper order, grouping species according to shared characteristics — no easy task. Since then, they’ve used these grand catalogs to understand the differences among organisms and to infer their evolutionary histories. Chemists built the periodic table for the same purpose — to classify the elements and understand their behaviors. And physicists made the Standard Model to explain how the fundamental particles of the universe interact.
In his book The Order of Things, the philosopher Michel Foucault describes this preoccupation with sorting as a formative step for the sciences. “A knowledge of empirical individuals,” he wrote, “can be acquired only from the continuous, ordered and universal tabulation of all possible differences.”
Mathematicians never got past this obsession. That’s because the menagerie of mathematics makes the biological catalog look like a petting zoo. Its inhabitants aren’t limited by physical reality. Any conceivable possibility, whether it lives in our universe or in some hypothetical 200-dimensional one, needs to be accounted for. There are tons of different classifications to try — groups, knots, manifolds and so on — and infinitely many objects to sort in each of those classifications. Classification is how mathematicians come to know the strange, abstract world they’re studying, and how they prove major theorems about it.Take groups, a central object of study in math. The classification of “finite simple groups” — the building blocks of all groups — was one of the grandest mathematical accomplishments of the 20th century. It took dozens of mathematicians nearly 100 years to finish. In the end, they figured out that all finite simple groups fall into three buckets, except for 26 itemized outliers. A dedicated crew of mathematicians has been working on a “condensed” proof of the classification since 1994 — it currently comprises 10 volumes and several thousand pages, and still isn’t finished. But the gargantuan undertaking continues to bear fruit, recently helping to prove a decades-old conjecture that you can infer a lot about a group by examining one small part of it.
Mathematics, unfettered by the typical constraints of reality, is all about possibility. Classification gives mathematicians a way to start exploring that limitless potential…[Howlett reviews attempts to classify numbers by “type” (postive/negative, rational/irrational), and mathematical objects by “equivalency” (shapes that can be stretched or squeezed into the other without breaking or tearing, like a doughnut and and coffee cup (see here)…]
… Similarly, classification has played an important role in knot theory. Tie a knot in a piece of string, then glue the string’s ends together — that’s a mathematical knot. Knots are equivalent if one can be tangled or untangled, without cutting the string, to match the other. This mundane-sounding task has lots of mathematical uses. In 2023, five mathematicians made progress on a key conjecture in knot theory that stated that all knots with a certain property (being “slice”) must also have another (being “ribbon”), with the proof ruling out a suspected counterexample. (As an aside, I’ve often wondered why knot theorists insist on using nouns as adjectives.)
Classifications can also get more meta. Both theoretical computer scientists and mathematicians classify problems about classification based on how “hard” they are.
All these classifications turn math’s disarrayed infinitude into accessible order. It’s a first step toward reining in the deluge that pours forth from mathematical imaginings…
“The Never-Ending Struggle to Classify All Math,” from @quantamagazine.bsky.social.
* Isaac Newton
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As we sort, we might spare a thought for the author of our title quote, Sir Isaac Newton; he died in this date in 1727. A polymath, Newton excelled in– and advanced– mathematics, physics, and astronomy; he was a theologian and a government offical (Master of the Mint)… and a dedicated alchemist. He was key to the Scientific Revolution and the Enlightenment that followed.
Newton’s book Philosophiæ Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy), first published in 1687, achieved the first great unification in physics and established classical mechanics (e.g., the Laws of Motion and the principle of universal gravitation). He also made seminal contributions to optics, and shares credit with German mathematician Gottfried Wilhelm Leibniz for formulating infinitesimal calculus. Indeed, Newton contributed to and refined the scientific method to such an extent that his work is considered the most influential in the development of modern science.
“Speed and acceleration are merely the dream of making time reversible”*…
In the early 20th century, there was Futurism…
The Italian Futurists, from the first half of the twentieth century… wanted to drive modernisation in turn-of-the-century Italy at a much faster pace. They saw the potential in machines, and technology, to transform the country, to demand progress. It was not however merely an incrementalist approach they were after: words like annihilation, destruction and apocalypse appear in the writings of the futurists, including the author of The Futurist Manifesto, Filippo Tomasso Marinetti. ‘We want to glorify war – the only cure for the world…’ Marinetti proclaimed – this was not for the faint hearted! That same Marinetti was the founder of the Partito Politico Futuristo in 1918, which became part of Mussolini’s Fascist party in 1919. Things did not go well after that.
“Beautiful Ideas Which Kill: Accelerationism, Futurism and Bewilderment“
And now, in the early 21st century, there is Accelerationism…
These [politically-motivated mass] killings were often linked to the alt-right, described as an outgrowth of the movement’s rise in the Trump era. But many of these suspected killers, from Atomwaffen thugs to the New Zealand mosque shooter to the Poway synagogue attacker, are more tightly connected to a newer and more radical white supremacist ideology, one that dismisses the alt-right as cowards unwilling to take matters into their own hands.
It’s called “accelerationism,” and it rests on the idea that Western governments are irreparably corrupt. As a result, the best thing white supremacists can do is accelerate their demise by sowing chaos and creating political tension. Accelerationist ideas have been cited in mass shooters’ manifestos — explicitly, in the case of the New Zealand killer — and are frequently referenced in white supremacist web forums and chat rooms.
Accelerationists reject any effort to seize political power through the ballot box, dismissing the alt-right’s attempts to engage in mass politics as pointless. If one votes, one should vote for the most extreme candidate, left or right, to intensify points of political and social conflict within Western societies. Their preferred tactic for heightening these contradictions, however, is not voting, but violence — attacking racial minorities and Jews as a way of bringing us closer to a race war, and using firearms to spark divisive fights over gun control. The ultimate goal is to collapse the government itself; they hope for a white-dominated future after that…
“Accelerationism: the obscure idea inspiring white supremacist killers around the world” (and source of the image above)
See also: “A Year After January 6, Is Accelerationism the New Terrorist Threat?“
For a look at the “intellectual” roots of accelerationism, see “Accelerationism: how a fringe philosophy predicted the future we live in.”
For a powerful articulation of the dangers of Futurism (and even more, Acclerationism), see “The Perils of Smashing the Past.”
And for a reminder of the not-so-obvious ways that movements like these live on, see “The Intentionally Scandalous 1932 Cookbook That Stands the Test of Time,” on The Futurist Cookbook, by Futurist Manifesto author Filippo Tommaso Marinetti… which foreshadowed the “food as fuel” culinary movements that we see today.
* Jean Baudrillard
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As we slow down, we might send a “Alles Gute zum Geburtstag” to the polymathic Gottfried Wilhelm Leibniz, the philosopher, mathematician, and political adviser, who was important both as a metaphysician and as a logician, but who is probably best remembered for his independent invention of the calculus; he was born on this date in 1646. Leibniz discovered and developed differential and integral calculus on his own, which he published in 1684; but he became involved in a bitter priority dispute with Isaac Newton, whose ideas on the calculus were developed earlier (1665), but published later (1687).
As it happens, Leibnitz was a wry and incisive political and cultural observer. Consider, e.g…
If geometry conflicted with our passions and our present concerns as much as morality does, we would dispute it and transgress it almost as much–in spite of all Euclid’s and Archimedes’ demonstrations, which would be treated as fantasies and deemed to be full of fallacies. [Leibniz, New Essays, p. 95]
“All practical jokes, friendly, harmless or malevolent, involve deception, but not all deceptions are practical jokes”*…
When you think of the ancient Greeks, practical jokes might not be the first thing that comes to mind. But along with art, architecture, and philosophy, you can add trick cups to their list of accomplishments.
The Pythagorean cup is so-named because it was allegedly invented by Pythagoras of Samos (yes, the same guy who gave us theories about right triangles). It’s a small cup with a column in its center. It doesn’t look like much, but when an unsuspecting drinker fills it past a designated level, the liquid mysteriously drains out. Legend has it that Pythagoras used it as a way to punish greedy drinkers who poured themselves too much wine…
A timeless practical joke, brought to you by the ancient Greeks: more merriment at “Pythagorean Cup.”
* W. H. Auden, The Dyer’s Hand
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As we ponders pranks, we might send a “Alles Gute zum Geburtstag” to the polymathic Gottfried Wilhelm Leibniz, the philosopher, mathematician, and political adviser, who was important both as a metaphysician and as a logician, but who is probably best remembered for his independent invention of the calculus; he was born on this date in 1646. Leibniz independently discovered and developed differential and integral calculus, which he published in 1684; but he became involved in a bitter priority dispute with Isaac Newton, whose ideas on the calculus were developed earlier (1665), but published later (1687).
As it happens, Leibnitz was no mean humorist. Consider, e.g…
If geometry conflicted with our passions and our present concerns as much as morality does, we would dispute it and transgress it almost as much–in spite of all Euclid’s and Archimedes’ demonstrations, which would be treated as fantasies and deemed to be full of fallacies. [Leibniz, New Essays, p. 95]
“I go to seek a Great Perhaps”*…
As we’ve noted before, 2016 seemed a bumper year for the Grim Reaper. Jason Crease tested that perception against the data…
It’s become cliché that unusually many prominent people died in 2016. Is this true?…
Find out at “Was 2016 especially dangerous for celebrities? An empirical analysis.“
[Image above: source]
* François Rabelais
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As we usher in the new, we might spare a thought for the first woman in the Western world considered to be a mathematician: Maria Gaetana Agnesi, she died this date in 1799. While she thought and wrote broadly about natural science and philosophy, she is best remembered for her work in differential calculus– perhaps most particularly for her work on the cubic curve now know as the “witch of Agnesi.”
“All opinions are not equal. Some are a very great deal more robust, sophisticated and well supported in logic and argument than others”*…
Now more than ever: Get a free logical fallacy poster.
* Douglas Adams, The Salmon of Doubt
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As we dedicate ourselves to discipline, we might send carefully-calculated birthday greetings to John Wallis; he was born on this date in 1616. An English mathematician who served as chief cryptographer for Parliament and, later, the royal court, he helped develop infinitesimal calculus and is credited with introducing the symbol ∞ for infinity.
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