(Roughly) Daily

Posts Tagged ‘Math

“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 America served 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.”


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


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.


Written by (Roughly) Daily

November 1, 2022 at 1:00 am

“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


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.

The Large Hadron Collider at CERN


Written by (Roughly) Daily

September 22, 2022 at 1:00 am

“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


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.


“Nature is pleased with simplicity”*…

As Clare Booth Luce once said, sometimes “simplicity is the ultimate sophistication”…

… The uniformity of the cosmic microwave background (CMB) tells us that, at its birth, ‘the Universe has turned out to be stunningly simple,’ as Neil Turok, director emeritus of the Perimeter Institute for Theoretical Physics in Ontario, Canada, put it at a public lecture in 2015. ‘[W]e don’t understand how nature got away with it,’ he added. A few decades after Penzias and Wilson’s discovery, NASA’s Cosmic Background Explorer satellite measured faint ripples in the CMB, with variations in radiation intensity of less than one part in 100,000. That’s a lot less than the variation in whiteness you’d see in the cleanest, whitest sheet of paper you’ve ever seen.

Wind forward 13.8 billion years, and, with its trillions of galaxies and zillions of stars and planets, the Universe is far from simple. On at least one planet, it has even managed to generate a multitude of life forms capable of comprehending both the complexity of our Universe and the puzzle of its simple origins. Yet, despite being so rich in complexity, some of these life forms, particularly those we now call scientists, retain a fondness for that defining characteristic of our primitive Universe: simplicity.

The Franciscan friar William of Occam (1285-1347) wasn’t the first to express a preference for simplicity, though he’s most associated with its implications for reason. The principle known as Occam’s Razor insists that, given several accounts of a problem, we should choose the simplest. The razor ‘shaves off’ unnecessary explanations, and is often expressed in the form ‘entities should not be multiplied beyond necessity’. So, if you pass a house and hear barking and purring, then you should think a dog and a cat are the family pets, rather than a dog, a cat and a rabbit. Of course, a bunny might also be enjoying the family’s hospitality, but the existing data provides no support for the more complex model. Occam’s Razor says that we should keep models, theories or explanations simple until proven otherwise – in this case, perhaps until sighting a fluffy tail through the window.

Seven hundred years ago, William of Occam used his razor to dismantle medieval science or metaphysics. In subsequent centuries, the great scientists of the early modern era used it to forge modern science. The mathematician Claudius Ptolemy’s (c100-170 CE) system for calculating the motions of the planets, based on the idea that the Earth was at the centre, was a theory of byzantine complexity. So, when Copernicus (1473-1543) was confronted by it, he searched for a solution that ‘could be solved with fewer and much simpler constructions’. The solution he discovered – or rediscovered, as it had been proposed in ancient Greece by Aristarchus of Samos, but then dismissed by Aristotle – was of course the solar system, in which the planets orbit around the Sun. Yet, in Copernicus’s hands, it was no more accurate than Ptolemy’s geocentric system. Copernicus’s only argument in favour of heliocentricity was that it was simpler.

Nearly all the great scientists who followed Copernicus retained Occam’s preference for simple solutions. In the 1500s, Leonardo da Vinci insisted that human ingenuity ‘will never devise any [solutions] more beautiful, nor more simple, nor more to the purpose than Nature does’. A century or so later, his countryman Galileo claimed that ‘facts which at first seem improbable will, even on scant explanation, drop the cloak which has hidden them and stand forth in naked and simple beauty.’ Isaac Newton noted in his Principia (1687) that ‘we are to admit no more causes of natural things than such as are both true and sufficient to explain their appearances’; while in the 20th century Einstein is said to have advised that ‘Everything should be made as simple as possible, but not simpler.’ In a Universe seemingly so saturated with complexity, what work does simplicity do for us?

Part of the answer is that simplicity is the defining feature of science. Alchemists were great experimenters, astrologers can do maths, and philosophers are great at logic. But only science insists on simplicity…

Just why do simpler laws work so well? The statistical approach known as Bayesian inference, after the English statistician Thomas Bayes (1702-61), can help explain simplicity’s power. Bayesian inference allows us to update our degree of belief in an explanation, theory or model based on its ability to predict data. To grasp this, imagine you have a friend who has two dice. The first is a simple six-sided cube, and the second is more complex, with 60 sides that can throw 60 different numbers. Suppose your friend throws one of the dice in secret and calls out a number, say 5. She asks you to guess which dice was thrown. Like astronomical data that either the geocentric or heliocentric system could account for, the number 5 could have been thrown by either dice. Are they equally likely? Bayesian inference says no, because it weights alternative models – the six- vs the 60-sided dice – according to the likelihood that they would have generated the data. There is a one-in-six chance of a six-sided dice throwing a 5, whereas only a one-in-60 chance of the 60-sided dice throwing a 5. Comparing likelihoods, then, the six-sided dice is 10 times more likely to be the source of the data than the 60-sided dice.

Simple scientific laws are preferred, then, because, if they fit or fully explain the data, they’re more likely to be the source of it.

In my latest book, I propose a radical, if speculative, solution for why the Universe might in fact be as simple as it’s possible to be. Its starting point is the remarkable theory of cosmological natural selection (CNS) proposed by the physicist Lee Smolin. CNS proposes that, just like living creatures, universes have evolved through a cosmological process, analogous to natural selection.

Smolin came up with CNS as a potential solution to what’s called the fine-tuning problem: how the fundamental constants and parameters, such as the masses of the fundamental particles or the charge of an electron, got to be the precise values needed for the creation of matter, stars, planets and life. CNS first notes the apparent symmetry between the Big Bang, in which stars and particles were spewed out of a dimensionless point at the birth of our Universe, and the Big Crunch, the scenario for the end of our Universe when a supermassive black hole swallows up stars and particles before vanishing back into a dimensionless point. This symmetry has led many cosmologists to propose that black holes in our Universe might be the ‘other side’ of Big Bangs of other universes, expanding elsewhere. In this scenario, time did not begin at the Big Bang, but continues backwards through to the death of its parent universe in a Big Crunch, through to its birth from a black hole, and so on, stretching backward in time, potentially into infinity. Not only that but, since our region of the Universe is filled with an estimated 100 billion supermassive black holes, Smolin proposes that each is the progenitor of one of 100 billion universes that have descended from our own.

The model Smolin proposed includes a kind of universal self-replication process, with black holes acting as reproductive cells. The next ingredient is heredity. Smolin proposes that each offspring universe inherits almost the same fundamental constants of its parent. The ‘almost’ is there because Smolin suggests that, in a process analogous to mutation, their values are tweaked as they pass through a black hole, so baby universes become slightly different from their parent. Lastly, he imagines a kind of cosmological ecosystem in which universes compete for matter and energy. Gradually, over a great many cosmological generations, the multiverse of universes would become dominated by the fittest and most fecund universes, through their possession of those rare values of the fundamental constants that maximise black holes, and thereby generate the maximum number of descendant universes.

Smolin’s CNS theory explains why our Universe is finely tuned to make many black holes, but it does not account for why it is simple. I have my own explanation of this, though Smolin himself is not convinced. First, I point out that natural selection carries its own Occam’s Razor that removes redundant biological features through the inevitability of mutations. While most mutations are harmless, those that impair vital functions are normally removed from the gene pool because the individuals carrying them leave fewer descendants. This process of ‘purifying selection’, as it’s known, maintains our genes, and the functions they encode, in good shape.

However, if an essential function becomes redundant, perhaps by a change of environment, then purifying selection no longer works. For example, by standing upright, our ancestors lifted their noses off the ground, so their sense of smell became less important. This means that mutations could afford to accumulate in the newly dispensable genes, until the functions they encoded were lost. For us, hundreds of smell genes accumulated mutations, so that we lost the ability to detect hundreds of odours that we no longer need to smell. This inevitable process of mutational pruning of inessential functions provides a kind of evolutionary Occam’s Razor that removes superfluous biological complexity.

Perhaps a similar process of purifying selection operates in cosmological natural selection to keep things simple…

It’s unclear whether the kind of multiverse envisaged by Smolin’s theory is finite or infinite. If infinite, then the simplest universe capable of forming black holes will be infinitely more abundant than the next simplest universe. If instead the supply of universes is finite, then we have a similar situation to biological evolution on Earth. Universes will compete for available resources – matter and energy – and the simplest that convert more of their mass into black holes will leave the most descendants. For both scenarios, if we ask which universe we are most likely to inhabit, it will be the simplest, as they are the most abundant. When inhabitants of these universes peer into the heavens to discover their cosmic microwave background and perceive its incredible smoothness, they, like Turok, will remain baffled at how their universe has managed to do so much from such a ‘stunningly simple’ beginning.

The cosmological razor idea has one further startling implication. It suggests that the fundamental law of the Universe is not quantum mechanics, or general relativity or even the laws of mathematics. It is the law of natural selection discovered by Darwin and Wallace. As the philosopher Daniel Dennett insisted, it is ‘The single best idea anyone has ever had.’ It might also be the simplest idea that any universe has ever had.

Does the existence of a multiverse hold the key for why nature’s laws seem so simple? “Why simplicity works,” from JohnJoe McFadden (@johnjoemcfadden)

* “Nature does nothing in vain when less will serve; for Nature is pleased with simplicity and affects not the pomp of superfluous causes.” – Isaac Newton, The Mathematical Principles of Natural Philosophy


As we emphasize the essential, we might spare a thought for Martin Gardner; he died on this date in 2010. Though not an academic, nor ever a formal student of math or science, he wrote widely and prolifically on both subjects in such popular books as The Ambidextrous Universe and The Relativity Explosion and as the “Mathematical Games” columnist for Scientific American. Indeed, his elegant– and understandable– puzzles delighted professional and amateur readers alike, and helped inspire a generation of young mathematicians.

Gardner’s interests were wide; in addition to the math and science that were his power alley, he studied and wrote on topics that included magic, philosophy, religion, and literature (c.f., especially his work on Lewis Carroll– including the delightful Annotated Alice— and on G.K. Chesterton).  And he was a fierce debunker of pseudoscience: a founding member of CSICOP, and contributor of a monthly column (“Notes of a Fringe Watcher,” from 1983 to 2002) in Skeptical Inquirer, that organization’s monthly magazine.


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