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Posts Tagged ‘Mathematics

“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

“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”*…

 

74. People once believed that the number of grains of sand is limitless. However, Archimedes argued in The Sand Reckoner that the number of grains of sand is not infinite. He gave a method for calculating the highest number of grains of sand that can fit into the universe– approximately 1063

100 other titillating tidbits at “101 Mathematical Trivia.”

* G.K. Chesterton

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As we count our blessings, we might spare a thought for Sir Christopher Wren; he died on this date in 1723.  A mathematician and astronomer, he became one of the most highly acclaimed English architects in history when he was was accorded responsibility for rebuilding 52 churches in the City of London after the Great Fire in 1666, including what is regarded as his masterpiece, St. Paul’s Cathedral, on Ludgate Hill.

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Written by LW

February 25, 2017 at 1:01 am

“For, after all, how do we know that two and two make four?”*…

 

“The teaching of Logic or Dialetics” from a collection of scientific, philosophical and poetic writings, French, 13th century; Bibliotheque Sainte-Genevieve, Paris, France. Photo by Bridgeman

Is logical thinking a way to discover or to debate? The answers from philosophy and mathematics define human knowledge..

The history of logic should be of interest to anyone with aspirations to thinking that is correct, or at least reasonable. This story illustrates different approaches to intellectual enquiry and human cognition more generally. Reflecting on the history of logic forces us to reflect on what it means to be a reasonable cognitive agent, to think properly. Is it to engage in discussions with others? Is it to think for ourselves? Is it to perform calculations?…

The rise and fall and rise of logic: “What is logic?

* George Orwell, 1984

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As we ruminate on reason, we might send enlightened birthday greetings to Benjamin Franklin; he was born on this date in 1706.  One of the Founding Fathers of the United States, Franklin was a renowned polymath: a leading author, printer, political theorist, politician, freemason, postmaster, scientist, inventor, civic activist, statesman, and diplomat. As a scientist, he was a major figure in the American Enlightenment and the history of physics for his discoveries and theories regarding electricity. As an inventor, he is known for the lightning rod, bifocals, and the Franklin stove, among other innovations.  And as a social entrepreneur (who grasped the fact that by united effort a community could have amenities which only the wealthy few can afford for themselves), he helped establish several institutions people now take for granted: a fire company (1736), a library (1731), an insurance company (1752), an academy (the University of Pennsylvania, 1751), a hospital (1751), and the U.S. Postal Service (starting as postmaster of the Colonies in 1753, then becoming U.S. Postmaster during the Revolution).  In most cases these foundations were the first of their kind in North America.

In a Franklin could be merged the virtues of Puritanism without its defects, the illumination of the Enlightenment without its heat.

– Henry Steele Commager

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Written by LW

January 17, 2017 at 1:01 am

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