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Posts Tagged ‘Mathematics

“Chaos is merely order waiting to be deciphered”*…

 

Let us say we were interested in describing all phenomena in our universe. What type of mathematics would we need? How many axioms would be needed for mathematical structure to describe all the phenomena? Of course, it is hard to predict, but it is even harder not to speculate. One possible conclusion would be that if we look at the universe in totality and not bracket any subset of phenomena, the mathematics we would need would have no axioms at all. That is, the universe in totality is devoid of structure and needs no axioms to describe it. Total lawlessness! The mathematics are just plain sets without structure. This would finally eliminate all metaphysics when dealing with the laws of nature and mathematical structure. It is only the way we look at the universe that gives us the illusion of structure…

Science predicts only the predictable, ignoring most of our universe.  What if neither Platonism nor the multiverse are the accurate approaches to understanding the reality we inhabit?  “Chaos Makes the Multiverse Unnecessary.”

[image above: source]

* José SaramagoThe Double

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As we impose order, we might spare a thought for Philipp Frank; he died on this date in 1966. A physicist, mathematician, and philosopher of science, he was Einstein’s successor as professor of theoretical physics at the German University of Prague– a job he got on Einstein’s recommendation– until 1938, when he fled the rise of Nazism and relocated to Harvard.  Frank’s theoretical work covered variational calculus, Hamiltonian geometrical optics, Schrödinger wave mechanics, and relativity; his philosophical work strove to reconcile science and philosophy and “bring about the closest rapprochement between” them.

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Written by LW

July 22, 2017 at 1:01 am

“Mathematics, rightly viewed, possesses not only truth, but supreme beauty”*…

 

Maryam Mirzakhani did not enjoy mathematics to begin with. She dreamed of being an author or politician, but as a top student at her all-girls school in Tehran she was still disappointed when her first-year maths exam went poorly. Her teacher believed her – wrongly – to have no particular affinity with the subject.

Soon that would all change. “My first memory of mathematics is probably the time [my brother] told me about the problem of adding numbers from 1 to 100,” she recalled later. This was the story of Carl Gauss, the 18th-century genius whose schoolteacher set him this problem as a timewasting exercise – only for his precocious pupil to calculate the answer in a matter of seconds.

The obvious solution is simple but slow: 1+2+3+4. Gauss’s solution is quicker to execute, and far more cunning. It goes like this: divide the numbers into two groups: from 1 to 50, and from 51 to 100. Then, add them together in pairs, starting with the lowest (1) and the highest (100), and working inwards (2+99, 3+98, and so on). There are 50 pairs; the sum of each pair is 101; the answer is 5050. “That was the first time I enjoyed a beautiful solution,” Mirzakhani told the Clay Mathematics Institute in 2008.

Since then, her appreciation for beautiful solutions has taken her a long way from Farzanegan middle school. At 17 she won her first gold medal at the International Mathematics Olympiad. At 27 she earned a doctorate from Harvard University. The Blumenthal Award and Satter Prize followed, and in 2014 she became the first woman to be awarded the Fields Medal, the highest honour a mathematician can obtain.

Before this particular brand of wonder became perceptible to Mirzakhani, she experienced feelings many of us can relate to: to the indifferent, her subject can seem “cold”, even “pointless”. Yet those who persist will be rewarded with glimpses of conceptual glory, as if gifted upon them by a capricious god: “The beauty of mathematics,” she warned, “only shows itself to more patient followers.”

This concept of “beauty” found in maths has been referred to over centuries by many others; though, like beauty itself, it is notoriously difficult to define…

For an experienced mathematician, the greatest equations are beautiful as well as useful. Can the rest of us see what they see?  “What makes maths beautiful?

[From The New Humanist, via the ever-illuminating 3 Quarks Daily]

Maryam Mirzakhani died last Friday, a victim of breast cancer; she was 40.  As Peter Sarnak (a mathematician at Princeton University and the Institute for Advanced Study) said, her passing is “a big loss and shock to the mathematical community worldwide.”  See also here.

* Bertrand Russell, A History of Western Philosophy

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As we accede to awe, we might spare a thought for Andrey (Andrei) Andreyevich Markov; he 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 and social sciences.

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Written by LW

July 20, 2017 at 1:01 am

“Do not imagine that mathematics is hard and crabbed, and repulsive to common sense. It is merely the etherealization of common sense”*…

 

Indeed, mathematics can be pretty amazing.  Consider, for example, that a pizza (which is essentially a very short cylinder) that has radius “z” and height “a” has volume Pi × z × z × a.

More marvelous math here.

* William Thomson, 1st Baron Kelvin

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As we do the sums, we might send relaxing birthday greetings to  Edwin J. Shoemaker; he was born on this date in 1907.  In 1928, he and his cousin Edward M. Knabusch prototyped a porch chair out of some wooden slats taken from orange crates; it would automatically recline as a sitter leaned back.  Since it was a seasonal item, his sales improved when he added plush upholstery for year-round indoor use.  Still, his chairs were for the most part locally/regionally sold.  So he designed a manufacturing facility which utilized the mass-production methods of Detroit’s automotive industry– and in November of 1941 went national with the La-Z-Boy recliner.

Edwin (left) and Edward with their original creation

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Written by LW

June 2, 2017 at 1:01 am

“Beauty is the first test: there is no permanent place in the world for ugly mathematics”*…

 

Long-time readers will know of your correspondent’s admiration and affection for Martin Gardner (c.f., e.g., here and here).  So imagine his delight to learn from @MartyKrasney of this…

Martin wrote about 300 articles for Scientific American between 1952 and 1998, most famously in his legendary “Mathematical Games” column starting in Jan 1957. Many of those articles are now viewed as classics, from his seminal piece on hexaflexagons in Dec 1956—which led to the offer to write a regular column for the magazine—to his breakthrough essays on pentomnoes, rep-tiles, the Soma cube, the art of Escher, the fourth dimension, sphere packing, Conway’s game of Life, Newcomb’s paradox, Mandelbrot’s fractals, Penrose tiles, and RSA cryptography, not forgetting the recurring numerological exploits of his alter ego Dr. Matrix, and the tongue-in-cheek April Fool column from 1975.

Many of those gems just listed were associated with beautiful graphics and artwork, so it’s no surprise that Martin scored some Scientific American covers over the years, though as we’ll see below, there’s surprisingly little overlap between his “greatest hits” and his “cover stories.”

It’s worth noting that, just as the magazine editors selected the titles under which his original articles appeared—he generally ditched those in favor of his own when he republished them in the spin-off books—artwork submitted was often altered by Scientific American staff artists…

The full dozen, replete with the cover art, at “A Gardner’s Dozen—Martin’s Scientific American Cover Stories.”

* G.H. Hardy

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As we agree with G.K Chesterton that “the difference between the poet and the mathematician is that the poet tries to get his head into the heavens while the mathematician tries to get the heavens into his head,” we might send carefully calculated birthday greetings to John Charles Fields, he was born on this date in 1863.  A mathematician of accomplishment, he is better remembered as a tireless advocate of the field and its importance– and best remembered as the founder of the award posthumously named for him:  The Fields Medal, familiarly known as “the Nobel of mathematics.”

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Written by LW

May 14, 2017 at 1:01 am

“The karma of humans is AI”*…

 

The black box… penetrable?

Already, mathematical models are being used to help determine who makes parole, who’s approved for a loan, and who gets hired for a job. If you could get access to these mathematical models, it would be possible to understand their reasoning. But banks, the military, employers, and others are now turning their attention to more complex machine-learning approaches that could make automated decision-making altogether inscrutable. Deep learning, the most common of these approaches, represents a fundamentally different way to program computers. “It is a problem that is already relevant, and it’s going to be much more relevant in the future,” says Tommi Jaakkola, a professor at MIT who works on applications of machine learning. “Whether it’s an investment decision, a medical decision, or maybe a military decision, you don’t want to just rely on a ‘black box’ method.”

There’s already an argument that being able to interrogate an AI system about how it reached its conclusions is a fundamental legal right. Starting in the summer of 2018, the European Union may require that companies be able to give users an explanation for decisions that automated systems reach. This might be impossible, even for systems that seem relatively simple on the surface, such as the apps and websites that use deep learning to serve ads or recommend songs. The computers that run those services have programmed themselves, and they have done it in ways we cannot understand. Even the engineers who build these apps cannot fully explain their behavior…

No one really knows how the most advanced algorithms do what they do. That could be a problem: “The Dark Secret at the Heart of AI.”

* Raghu Venkatesh

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As we get to know our new overlords, we might spare a thought for the painter, sculptor, architect, musician, mathematician, engineer, inventor, physicist, chemist, anatomist, botanist, geologist, cartographer, and writer– the archetypical Renaissance Man– Leonardo da Vinci.  Quite possibly the greatest genius of the last Millennium, he died on this date in 1519.

Self-portrait in red chalk, circa 1512-15

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Written by LW

May 2, 2017 at 1:01 am

“All musicians are subconsciously mathematicians”*…

 

Physicist and saxophonist Stephon Alexander has argued in his many public lectures and his book The Jazz of Physics that Albert Einstein and John Coltrane had quite a lot in common. Alexander in particular draws our attention to the so-called “Coltrane circle,” which resembles what any musician will recognize as the “Circle of Fifths,” but incorporates Coltrane’s own innovations. Coltrane gave the drawing to saxophonist and professor Yusef Lateef in 1967, who included it in his seminal text, Repository of Scales and Melodic Patterns. Where Lateef, as he writes in his autobiography, sees Coltrane’s music as a “spiritual journey” that “embraced the concerns of a rich tradition of autophysiopsychic music,” Alexander sees “the same geometric principle that motivated Einstein’s” quantum theory…

Explore the connection at “John Coltrane Draws a Picture Illustrating the Mathematics of Music.”

* Thelonious Monk

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As we square the circle, we might recall that it was on this date in 1786, at the Burgtheater in Vienna, that Mozart’s glorious Le nozze di Figaro The Marriage of Figaro— premiered.  Based on a stage comedy by Pierre Beaumarchais, La folle journée, ou le Mariage de Figaro (“The Mad Day, or The Marriage of Figaro”), which was first performed two years early, Mozart’s comedic masterpiece has become a staple of opera repertoire, appearing consistently among the top ten in the Operabase list of most frequently performed operas.

Early 19th-century engraving depicting Count Almaviva and Susanna in act 3

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“The generation of random numbers is too important to be left to chance”*…

 

Roman 12mm dice

Random numbers are central to more than we may realize.  They have applications in gambling, statistical sampling, computer simulation and Monte Carlo modeling, cryptography (as applied in both communications and transactions), completely randomized design, even sooth-saying– in any area where producing an unpredictable result is desirable.  So how they’re produced– the certainty that they are, in fact, random– matters enormously.

It’s no surprise, then, that random number generation has a long and fascinating history.  Happily, Carl Tashian is here to explain.

“As an instrument for selecting at random, I have found nothing superior to dice,” wrote statistician Francis Galton in an 1890 issue of Nature. “When they are shaken and tossed in a basket, they hurtle so variously against one another and against the ribs of the basket-work that they tumble wildly about, and their positions at the outset afford no perceptible clue to what they will be even after a single good shake and toss.”…

From I Ching sticks and dice to the cryptographically-secure PRNG, “A Brief History of Random Numbers.”

[TotH to the eminently-numerate Reuben Steiger]

Robert Coveyou

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As we roll the bones, we might spare a thought for Samuel “Sam” Loyd; he died on this date in 1911. A chess player, chess composer, puzzle author, and recreational mathematician.  A member of the Chess Hall of Fame (for both his play and for his exercises, or “problems”), he gained posthumous fame when his son published a collection of his mathematical and logic puzzles, Cyclopedia of 5000 Puzzles after his father’s death.  As readers can see here and here, his puzzles still delight.

Loyd’s most famous puzzle was the 14-15 Puzzle, which he produced in 1878. His original authorship is debated; but in any case, his version created a craze that swept America to such an extent that employers put up notices prohibiting playing the puzzle during office hours.

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Written by LW

April 10, 2017 at 1:01 am

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