(Roughly) Daily

Posts Tagged ‘Mathematics

“There are 10 kinds of people in the world: those who understand binary numerals, and those who don’t”*…

 

Guide to Computing

From a collection of vintage photos of computing equipment by “design and tech obsessive” James Ball…

Guide to Computing

More at Docubyte

[TotH to Kottke]

* vernacular joke, as invoked by Ian Stewart in Professor Stewart’s Cabinet of Mathematical Curiosities

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As we rewind, we might spare a thought for Christian Goldbach; he died on this date in 1764.  A mathematician, lawyer, and historian who studied infinite sums, the theory of curves and the theory of equations, he is best remembered for his correspondence with Leibniz, Euler, and Bernoulli, especially his 1742 letter to Euler containing what is now known as “Goldbach’s conjecture.”

In that letter he outlined his famous proposition:

Every even natural number greater than 2 is equal to the sum of two prime numbers.

It has been checked by computer for vast numbers– up to at least 4 x 1014– but remains unproved.

(Goldbach made another conjecture that every odd number is the sum of three primes; it has been checked by computer for vast numbers, but also remains unproved.)

Goldbach’s letter to Euler (source, and larger view)

(Roughly) Daily is headed into a Thanksgiving hiatus; regular service will resume when the tryptophan haze clears…  probably around Monday, November 26.  Thanks for reading– and have Happy Holidays!

Written by LW

November 20, 2018 at 1:01 am

“The cyclical rebirth of caste in America is a recurring racial nightmare”*…

 

griffith

Dorothy and Lillian Gish and D.W. Griffith at the White House, 1922. Library of Congress

 

The year 1915 marked the fiftieth anniversary of the end of the Civil War. Monuments to Confederate and Union heroes were being dedicated all over the country. Woodrow Wilson, a fan of Jim Crow laws, was president. He had allowed federal workplaces to segregate again.

Enter Thomas Dixon Jr., Wilson’s classmate from Johns Hopkins. A film had just been made of Dixon’s second novel, “the true story” of the South under Reconstruction. Would the president, he wondered, be interested in viewing it? (He would.)

“History written with lightning,” Wilson declared of The Clansman, the second film ever to be screened in the White House. It was an endorsement guaranteed to head off resistance from town censor boards charged with shutting down entertainment deemed unsuitable or incendiary to the public…

The Clansman was a silent movie with title cards. It depicted whites as victims and blacks as villains. Benevolent former masters were denied votes and subjugated by newly freed blacks taking over the country. In an early scene, black legislators sit at desks, shoeless and drunk, too busy stuffing their faces with fried chicken to work. The title card read: “An historical facsimile of the State House of Representatives of South Carolina in 1870.” South Carolina had been the first state to elect a majority-black legislature and that the card implied that the apish behavior depicted was historically accurate, too.

In a later scene, the white heroine (played by Lillian Gish) is threatened by a black man unable to contain his urge to “mongrelize” the white race. Before she is ravaged, a savior army rides in: The Ku Klux Klan. The title-card copy comes straight from the president’s five-volume History of the American People, published in 1902:

440px-wilson-quote-in-birth-of-a-nation

More of this sad story, and its aftermath, at “Hatred Endorsed by a President.”

* Michelle Alexander, author of The New Jim Crow: Mass Incarceration in the Age of Colorblindness

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As we ruminate on recurrence, we might send never-ending birthday greetings to August Ferdinand Möbius; he was born on this date in 1790.  A German mathematician and theoretical astronomer, he is best remembered as a topologist, more specifically for his discovery of the Möbius strip (a two-dimensional surface with only one side… or more precisely, a non-orientable two-dimensional surface with only one side when embedded in three-dimensional Euclidean space).  See ““It might help to think of the universe as a rubber sheet, or perhaps not.”

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Written by LW

November 17, 2018 at 1:01 am

“The information revolution came without an instruction manual”*…

 

Machines

In my graduate seminar we’ve recently been thinking a bit about machines. Given that our focus has been on the 19th Century, attention has been directed toward ergodic machines (from the root ergon meaning work). Ergodic machines are machines that run on heat and energy. Such machines are essentially mechanical in nature. They deal with basic physical mechanics like levers and pulleys, and questions of mass, weight, and counter-balance. Ergodic machines adhere to the laws of motion and inertia, the conservation of energy, and the laws of thermodynamics governing heat, pressure, and energy…

Still, ergodic machines do not account for all machines. Informatic machines, those devices dominating contemporary life, have in many ways taken over from their 19th-century counterparts. Informatic machines have physical bodies, of course, and they frequently require electricity or other forms of power to operate. However the essence of the informatic machine is not found in motion, unrest, heat, or energy. The essence of the informatic machine is found in form, not energy or presence. From the perspective of philosophy, computers are therefore quite classical, even conservative. They follow that most basic law of Western idealism, that the formal determines the physical

The anti-computer has yet to be invented. But traces of it are found everywhere. Even Bitcoin, that most miserable invention, relies on an anti-computational infrastructure. In order to mine coins, one must expend energy. Hence these twenty-first-century machines are yoked to a nineteenth-century mandate: burn fuel to release value. Bitcoin may run on a computer but it is anti-computational at heart. Bitcoin only works because it is grounded in an anti-computer (energy). It is thus a digital machine made subsidiary to an analog foundation, a twenty-first-century future tied to a nineteenth-century past.

The encryption algorithms at the heart of Bitcoin are anti-computational as well. Cryptography deploys form as a weapon against form. Such is the magic of encryption. Encryption is a kind of structure that makes life difficult for other competing structures. Encryption does not promote frictionlessness, on the contrary it produces full and complete friction at all levels. Not the quotidian friction of everyday life, but a radical friction frustrating all expression. What used to be a marginal activity practiced by hackers — cracking password hashes — is now the basis of an entire infrastructure. Earn a buck by cracking hashes using “brute force.” Turn your computer into an anti-computer.

A friend of Marshall McLuhan’s, Father John Culkin, SJ, a Professor of Communication at Fordham University, observed that “we shape our tools and then our tools shape us” (though the quote is often attributed to McLuhan, who may in fact have inspired it).   Alexander R. Galloway ponders the tools that dominate our lives these days: “Anti-Computer.”

* “The central paradox of the machines that have made our lives so much brighter, quicker, longer and healthier is that they cannot teach us how to make the best use of them; the information revolution came without an instruction manual”  — Pico Iyer

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As we muse on machines, we might spare a thought for James Clerk Maxwell; he died on this date in 1879.  a mathematician and and physicist, his work in uniting electricity, magnetism, and light– that’s to say, formulating the classical theory of electromagnetic radiation— is considered the “second great unification in physics” (after the first, realized by Isaac Newton), and laid the foundation for modern physics, starting the search for radio waves and paving the way for such fields as special relativity and quantum mechanics.  In the millennium poll – a survey of the 100 most prominent physicists at the turn of the 21st century – Maxwell was voted the third greatest physicist of all time, behind only Newton and Einstein.

225px-James_Clerk_Maxwell source

 

Written by LW

November 5, 2018 at 2:01 am

“The Bible shows the way to go to heaven, not the way the heavens go”*…

 

cellarius-seasons-banner-1024x689

 

Harmonia Macrocosmica (1660), an atlas of the stars from the Dutch Golden Age of cartography, maps the structure of the heavens in twenty-nine extraordinary double-folio spreads. We are presented with the motions of the celestial bodies, the stellar constellations of the northern hemisphere, the old geocentric universe of Ptolemy, the newish heliocentric one of Copernicus [as above], and Tycho Brahe’s eccentric combination of the two — in which the Moon orbits the Earth, and the planets orbit the Sun, but the Sun still orbits the Earth. The marginal area of each brightly coloured map is a hive of activity: astronomers bent over charts debate their findings, eager youngsters direct their quadrants skywards, and cherubs fly about with pet birds in tow…

northern stars

The Northern Stellar Hemisphere of Antiquity

More marvelous maps of the heavens at “The Celestial Atlas of Andreas Cellarius (1660)

* Galileo (quoting a librarian at the Vatican)

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As we look to the stars, we might recall that it was on this date in 1843 that Sir William Rowan Hamilton conceived the theory of quaternions.  A physicist, astronomer, and mathematician who made important contributions to classical mechanics, optics, and algebra, he had been working since the late 1830s on the basic principles of algebra, resulting in a theory of conjugate functions, or algebraic couples, in which complex numbers are expressed as ordered pairs of real numbers.  But he hadn’t succeeded in developing a theory of triplets that could be applied to three-dimensional geometric problems.  Walking with his wife along the Royal Canal in Dublin, Hamilton realized that the theory should involve quadruplets, not triplets– at which point he stopped to carve carve the underlying equations in a nearby bridge lest he forget them.

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Written by LW

October 16, 2018 at 1:01 am

“It might help to think of the universe as a rubber sheet, or perhaps not”*…

 

mobius strip

You have most likely encountered one-sided objects hundreds of times in your daily life – like the universal symbol for recycling, found printed on the backs of aluminum cans and plastic bottles.

This mathematical object is called a Mobius strip. It has fascinated environmentalists, artists, engineers, mathematicians and many others ever since its discovery in 1858 by August Möbius, a German mathematician who died 150 years ago, on Sept. 26, 1868.

Möbius discovered the one-sided strip in 1858 while serving as the chair of astronomy and higher mechanics at the University of Leipzig. (Another mathematician named Listing actually described it a few months earlier, but did not publish his work until 1861.)…

The discovery of the Möbius strip in the mid-19th century launched a brand new field of mathematics: topology: “The Mathematical Madness of Möbius Strips and Other One-Sided Objects.”

* Terry Pratchett, Hogfather

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As we return from whence we came, we might wish a Joyeux Anniversaire to Denis Diderot, contributor to and the chief editor of the Encyclopédie (“All things must be examined, debated, investigated without exception and without regard for anyone’s feelings.”)– and thus towering figure in the Enlightenment; he was born on this date in 1713.  Diderot was also a novelist (e.g., Jacques le fataliste et son maître [Jacques the Fatalist and his Master])…  and no mean epigramist:

From fanaticism to barbarism is only one step.

We swallow greedily any lie that flatters us, but we sip only little by little at a truth we find bitter.

Man will never be free until the last king is strangled with the entrails of the last priest.

A thing is not proved just because no one has ever questioned it.

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Written by LW

October 5, 2018 at 1:01 am

“To change something, build a new model that makes the existing model obsolete”*…

 

science

There is remarkably little reflection taking place about the state of science today, despite significant challenges, rooted in globalization, the digitization of knowledge, and the growing number of scientists.

At first glance, all of these seem to be positive trends. Globalization connects scientists worldwide, enabling them to avoid duplication and facilitating the development of universal standards and best practices. The creation of digital databases allows for systematic mining of scientific output and offers a broader foundation for new investigations. And the rising number of scientists means that more science is being conducted, accelerating progress.

But these trends are Janus-faced…

Jeremy Baumberg argues that we live in an age of hyper-competitive, trend-driven, and herd-like approach to scientific research: “What Is Threatening Science?

* Buckminster Fuller

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As we rethink research, we might spare a thought for Paul Erdős; he died on this date in 1996.  One of the most prolific mathematicians of the 20th century (he published around 1,500 mathematical papers during his lifetime, a figure that remains unsurpassed), he is remembered both for his “social practice” of mathematics (he engaged more than 500 collaborators) and for his eccentric lifestyle (he spent his waking hours virtually entirely on math; he would typically show up at a colleague’s doorstep and announce “my brain is open”, staying long enough to collaborate on a few papers before moving on a few days later).

Erdős’s prolific output with co-authors prompted the creation of the Erdős number, the number of steps in the shortest path between a mathematician and Erdős in terms of co-authorships.  Low numbers are a badge of pride– and a usual marker of accomplishment: as of 2016, all Fields Medalists have a finite Erdős number, with values that range between 2 and 6, and a median of 3.  Physics Nobelists Einstein and Sheldon Glashow have an Erdős number of 2.   Baseball Hall of Famer Hank Aaron can be considered to have an Erdős number of 1 because they both autographed the same baseball (for number theorist Carl Pomerance).  Natalie Portman’s undergraduate collaboration with a Harvard professor earned her an Erdős number of 5; Danica McKellar (“Winnie Cooper” in The Wonder Years) has an Erdős number of 4, for a mathematics paper coauthored while an undergraduate at UCLA.

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Written by LW

September 20, 2018 at 1:01 am

“I am incapable of conceiving infinity, and yet I do not accept finity”*…

 

hilberts_hotel

Suppose you’re working at a hotel with infinitely many rooms in it, numbered 1, 2, 3, 4, 5, … all the way up forever and ever. (This is known as a Hilbert Hotel.) One evening when every single room is occupied, a traveler arrives and requests to be accommodated too. You’re the manager. What do you do to help the traveler?

Simple. You just ask each occupant to one room forward. 1 goes to 2, and 2 goes to 3, and so on. Every previous occupant gets a new room. And the first room is now open for the traveler.

The procedure above is characterized by an infinite number of actions or tasks to be carried out in a finite amount of time. Procedures with this character are known as supertasks…

More on the ins and outs of infinities at “Introducing Supertasks.” (More fun musings on infinity here and here; and more on Hilbert’s Hotel here.)

* Simone de Beauvoir, La Vieillesse

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As we muse on many, we might spare a thought for Hermann Hankel; he died on this date in 1873.  A mathematician who worked with Möbius, Riemann, Weierstrass,  and Kronecker (among others), he made important contributions to the understanding of complex numbers and quaternions… and to work begun by Bernard Bolzano on infinite series.

220px-Hankel source

 

Written by LW

August 29, 2018 at 1:01 am

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