## Posts Tagged ‘**Mathematics**’

## “Nothing is more wonderful than the art of being free, but nothing is harder to learn how to use than freedom”*…

Lynn Hunt on Alexis de Tocqueville, who left France to study the American prison system and returned with the material that would become *Democracy in America*…

Alexis de Tocqueville was a study in contradictions: a French aristocrat of proud heritage who trumpeted the inevitable, salutary rise of democracy, using the United States as his exemplar; a cosmopolitan with an English wife and many friends in the Anglo-American world who brandished a fervent French nationalism; an antislavery advocate who felt no discomfort in supporting the French colonization of Algeria and hired as his main assistant Arthur de Gobineau, who later published one of the founding texts of white supremacy; and finally a man of delicate constitution who undertook an arduous trip on horseback into the wilderness of northern Michigan in order to see Native Americans and new settler communities for himself. Such inconsistencies make for a fascinating story, and in

The Man Who Understood Democracy, Olivier Zunz, a French-educated historian who has taught US history for decades at the University of Virginia, shows that he is ideally suited to tell it.Tocqueville’s

Democracy in America, published in two volumes in 1835 and 1840, became an instant classic and has remained one to this day. On its hundredth anniversary in 1935, the French government presented a bust of the author to Franklin D. Roosevelt, and an article at the time referred to the book as “perhaps the greatest, most lucid, and most impartial commentary that free institutions in general, and American self-government in particular, had ever received.”Democracy in Americaserved as a kind of textbook for US students for many generations, but it is now more often cited than read. That dutiful disregard may be the fate of all such masterworks, especially one that runs about eight hundred pages, but Zunz has succeeded in restoring its appeal, first by vividly retracing its origins and then by skillfully evoking the enduring excitement and relevance of its analysis…

Alexis de Tocqueville, the Frenchman who unpacked the tension between freedom and equality in the United States: “‘A Great Democratic Revolution’.”

* Alexis de Tocqueville– who went on to observe that “Americans are so enamored of equality, they would rather be equal in slavery than unequal in freedom.”

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**As we dedicate ourselves to democracy,** we might note that today is Fibonacci Day, as today’s date is often rendered 11/23, and the Fibonacci sequence (also here and here) begins 1, 1, 2, 3…

Five Ways to Celebrate Fibonacci Day.

## “Whoever wishes to keep a secret must hide the fact that he possesses one”*…

… or, as Sheon Han explains, maybe not…

Imagine you had some useful knowledge — maybe a secret recipe, or the key to a cipher. Could you prove to a friend that you had that knowledge, without revealing anything about it? Computer scientists proved over 30 years ago that you could, if you used what’s called a zero-knowledge proof.

For a simple way to understand this idea, let’s suppose you want to show your friend that you know how to get through a maze, without divulging any details about the path. You could simply traverse the maze within a time limit, while your friend was forbidden from watching. (The time limit is necessary because given enough time, anyone can eventually find their way out through trial and error.) Your friend would know you could do it, but they wouldn’t know how.

Zero-knowledge proofs are helpful to cryptographers, who work with secret information, but also to researchers of computational complexity, which deals with classifying the difficulty of different problems. “A lot of modern cryptography relies on complexity assumptions — on the assumption that certain problems are hard to solve, so there has always been some connections between the two worlds,” said Claude Crépeau, a computer scientist at McGill University. “But [these] proofs have created a whole world of connection.”…

More about how zero-knowledge proofs allow researchers conclusively to demonstrate their knowledge without divulging the knowledge itself: “How Do You Prove a Secret?,” from @sheonhan in @QuantaMagazine.

* Johann Wolfgang von Goethe

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**As we stay sub rosa,** we might recall that today (All Saints Day) is the (fictional) birthday of Hello Kitty (full name: Kitty White); she was born in a suburb of London. A cartoon character designed by Yuko Shimizu (currently designed by Yuko Yamaguchi), she is the property of the Japanese company Sanrio. An avatar of *kawaii* (cute) culture, Hello Kitty is one of the highest-grossing media franchises of all time; Hello Kitty product sales and media licensing fees have run as high as $8 billion a year.

## “Advantage! What is advantage?”*…

Pradeep Mutalik unpacks the magic and math of how to win games when your opponent goes first…

Most games that pit two players or teams against each other require one of them to make the first play. This results in a built-in asymmetry, and the question arises: Should you go first or second?

Most people instinctively want to go first, and this intuition is usually borne out. In common two-player games, such as chess or tennis, it is a real, if modest, advantage to “win the toss” and go first. But sometimes it’s to your advantage to let your opponent make the first play.

In our February Insights puzzle, we presented four disparate situations in which, counterintuitively, the obligation to move is a serious and often decisive disadvantage. In chess, this is known as zugzwang — a German word meaning “move compulsion.”…

Four fascinating examples: “The Secrets of Zugzwang in Chess, Math and Pizzas,” from @PradeepMutalik.

* Fyodor Dostoyevsky, *Notes from Underground*

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**As we play to win,** we might recall that it was on this date in 2011 that scientists involved in the OPERA experiment (a collaboration between CERN and the Laboratori Nazionali del Gran Sasso) mistakenly observed neutrinos appearing to travel faster than light. OPERA scientists announced the results with the stated intent of promoting further inquiry and debate. Later the team reported two flaws in their equipment set-up that had caused errors far outside their original confidence interval: a fiber optic cable attached improperly, which caused the apparently faster-than-light measurements, and a clock oscillator ticking too fast; accounting for these two sources of error eliminated the faster-than-light results. But even before the sources of the error were discovered, the result was considered anomalous because speeds higher than that of light in a vacuum are generally thought to violate special relativity, a cornerstone of the modern understanding of physics for over a century.

## “Why, sometimes I’ve believed as many as six impossible things before breakfast”*…

Imaginary numbers were long dismissed as mathematical “bookkeeping.” But now, as Karmela Padavic-Callaghan explains, physicists are proving that they describe the hidden shape of nature…

Many science students may imagine a ball rolling down a hill or a car skidding because of friction as prototypical examples of the systems physicists care about. But much of modern physics consists of searching for objects and phenomena that are virtually invisible: the tiny electrons of quantum physics and the particles hidden within strange metals of materials science along with their highly energetic counterparts that only exist briefly within giant particle colliders.

In their quest to grasp these hidden building blocks of reality scientists have looked to mathematical theories and formalism. Ideally, an unexpected experimental observation leads a physicist to a new mathematical theory, and then mathematical work on said theory leads them to new experiments and new observations. Some part of this process inevitably happens in the physicist’s mind, where symbols and numbers help make invisible theoretical ideas visible in the tangible, measurable physical world.

Sometimes, however, as in the case of imaginary numbers – that is, numbers with negative square values – mathematics manages to stay ahead of experiments for a long time. Though imaginary numbers have been integral to quantum theory since its very beginnings in the 1920s, scientists have only recently been able to find their physical signatures in experiments and empirically prove their necessity…

Learn more at “Imaginary numbers are real,” from @Ironmely in @aeonmag.

* The Red Queen, in Lewis Carroll’s *Through the Looking Glass*

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**As we get real,** we might spare a thought for two great mathematicians…

Georg Friedrich Bernhard Riemann died on this date in 1866. A mathematician who made contributions to analysis, number theory, and differential geometry, he is remembered (among other things) for his 1859 paper on the prime-counting function, containing the original statement of the Riemann hypothesis, regarded as one of the most influential papers in analytic number theory.

Andrey (Andrei) Andreyevich Markov died on this date in 1922. A Russian mathematician, he helped to develop the theory of stochastic processes, especially those now called Markov chains: sequences of random variables in which the future variable is determined by the present variable but is independent of the way in which the present state arose from its predecessors. (For example, the probability of winning at the game of *Monopoly* can be determined using Markov chains.) His work on the study of the probability of mutually-dependent events has been developed and widely applied to the biological, physical, and social sciences, and is widely used in Monte Carlo simulations and Bayesian analyses.

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