(Roughly) Daily

Posts Tagged ‘astonomy

“Truth is ever to be found in the simplicity, and not in the multiplicity and confusion of things”*…

Knots with 8 crossings

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.

source

Written by (Roughly) Daily

March 20, 2025 at 1:00 am

“The real war will never get in the books”*…

Still, historians try. And as Anton Jäger argues in his consideration of Charles S. Maier‘s The Project-State and Its Rivals- A New History of the Twentieth and Twenty-First Centuries, that’s a challenging, frustrating, but ultimately very useful thing…

“We thought we knew the story of the twentieth century,” Charles Maier notes in an announcement for his new book The Project State and Its Rivals. Both haunting and tantalizing, the sentence’s past tense speaks to a profoundly contemporary mood. As the twenty-first century progresses, confident visions about the previous century conceived from the vantage point of the 1990s—the “age of extremes” resolved by a set of liberal settlements—no longer seem safe and secure. In 2023, the European extreme Right is establishing itself as a force of government, populism is going global, and inter-imperial tensions have ushered in a new arms race. In Germany, the Alternative für Deutschland is currently polling above 20 percent, while Modi is set to win another term in India with an approval rating near 80 percent. To the desperation of liberals nostalgic for the 1990s, the “end of the end of history” has arrived.

As Maier surmises, there might be a connection between this sense of surprise and the comfortable judgments we tend to make about humani­ty’s last hundred years. “If the twentieth century meant the triumph of liberalism,” he asks, “why have the era’s darker impulses—ethnic nationalism, racist violence, and populist authoritarianism—revived?” The question provides the working hypothesis for Maier’s new mono­graph, a self-described “rethinking of the long twentieth century,” which aims to “explain the fraying of our own civic culture” while also “allowing hope for its recovery.” Provocatively, Maier’s focus is on “both democracies and dictatorships that sought not just to retain power but to transform their societies,” next to “new forms of imperial domination,” “global networks of finance,” and “international associations” that both challenged and shaped the state. The ambition is nothing less than a new general theory of the twentieth century, one that would allow us to deal with an unmastered past, but also to gain proper self-understanding in a new and confusing century.

Readers would be hard-pressed to find a more suitable candidate for the task than Charles S. Maier. At eighty-four, Maier—still teaching European and international history at Harvard—remains a scholar with panoramic disciplinary reach. His 1975 debut, Recasting Bourgeois Europe: Stabilization in France, Germany, and Italy in the Decade after World War I, swiftly established itself a masterpiece of comparative political history. Based on a prior Harvard dissertation complemented with a decade of additional archival research, it examined the fraught resolution of the crises of liberalism after 1918, and what factors deter­mined the potential emergence and stymieing of authoritarian regimes. After works on Germany’s collective memory of the Holocaust and an elite-driven account of the fall of East Germany, he waded into histori­cal political science with Leviathan 2.0: Inventing Modern Statehood in 2014, followed by Once within Borders: Territories of Power, Wealth, and Belonging since 1500 in 2016. Clearly a product of a buoyant Cold War academe, Maier has always been locked in an uneasy pas-à-deux with Marxism: attentive to the class content of political life, but never taken to monolithic views of business interests and overly abstract notions of capital. His work on political economy looked closely at the class coalitions that gave way to divergent corporatist settlements in the 1920s and ’30s, and how these national blocs interlocked with differing international arrangements—a Marxist historiography despite itself. He also took the force of ideas seriously, weaving a tapestry of conceptual, political, and economic history, which explains the unique force of his writing. Yet unlike cultural historians, Maier has retained an interest in causality through the construction of comparative counterfactuals—what Britain and Germany shared in 1918, for instance, or why English Tories did not need a Duce and why the American South was different from the Mezzogiorno—a sensibility that also informed his consistently transnational approach to the twentieth century.

The Project State and Its Rivals exudes a similarly boundless ambition. As the book’s announcements make clear, Maier is on the lookout for a unifying category to cohere our historical experience of the twentieth century—or, more specifically, the forms of statehood that emerged in the interwar period, and that still present such vexing challenges to our intellectual imagination…

A critical account of Maier’s hypothesis, eminently worth reading in full: “The Rise and Fall of the Project State: Rethinking the Twentieth Century,” from @AntonJaegermm in @AmericanAffrs. Via Adam Tooze/@adam_tooze.

* Walt Whitman

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As we keep searching, we might pause to contrast the rigorously serious with the frivolously venal: it was on this date in 1835 that the New York Sun began a series of six articles detailing the discovery of civilized life on the moon; circulation soared.  Now known as “The Great Moon Hoax,” the articles attributed the “discovery” to Sir John Herschel, the greatest living astronomer of the day.  Herschel was initially amused, wryly noting that his own real observations could never be as exciting.  But ultimately he tired of having to answer questioners who believed the story.  The series was not discovered to be a hoax for several weeks after its publication and, even then, while the paper did admit (on September 16, 1835) that the whole thing was a “satire,” it never issued a retraction (and didn’t suffer a drop in sales).

The “ruby amphitheater” on the Moon, per the New York Sun (source)