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Posts Tagged ‘Mathematics

“With the sextant he made obeisance to the sun-god”*…

 

exam-final-Fig5_Boombaar_HSM

A practice exam in the navigation workbook of C. J. Boombaar (1727–32)

 

In 1673, in a North Sea skirmish that killed nearly 150 men, the French privateer Jean-François Doublet took a bullet that tossed him from the forecastle and broke his arm in two places. How did the precocious young second lieutenant choose to spend his convalescence? Doublet repaired to the French port city of Dieppe, where he signed up for three months of navigation lessons…

During the 16th to 18th centuries, Europeans embarked on thousands of long-distance sea voyages around the world. These expeditions in the name of trade and colonisation had irreversible, often deadly, impacts on peoples around the globe. Heedless of those consequences, Europeans focused primarily on devising new techniques to make their voyages safer and faster. They could no longer sail along the coasts, taking their directional cues from prominent landmarks (as had been common in the preceding centuries). Nor did they have sophisticated knowledge of waves and currents, as did their counterparts in the Pacific. They had no choice but to figure out new methods of navigating across the open water. Instead of memorising the shoreline, they looked to the heavens, calculating time and position from the sun and the stars.

Celestial navigation was certainly feasible, but it required real technical skills as well as fairly advanced mathematics. Sailors needed to calculate the angle of a star’s elevation using a cross-staff or quadrant. They needed to track the direction of their ship’s course relative to magnetic north. Trigonometry and logarithms offered the best way to make these essential measurements: for these, a sailor needed to be adept at using dense numerical tables. All of a sudden, a navigator’s main skill wasn’t his memory – it was his mathematical ability.

To help the average sailor with these technical computations, maritime administrators and entrepreneurs opened schools in capital cities and port towns across Europe. Some were less formal arrangements, where small groups of men gathered in the teacher’s home, paying for a series of classes over the course of a winter when they were on shore…

How did the sailors of early modern Europe learn to traverse the world’s seas? By going to school and doing maths problems: “When pirates studied Euclid.”

* “With the sextant he made obeisance to the sun-god, he consulted ancient tomes and tables of magic characters, muttered prayers in a strange tongue that sounded like Indexerrorparallaxrefraction, made cabalistic signs on paper, added and carried one, and then, on a piece of holy script called the Grail – I mean, the Chart – he placed his finger on a certain space conspicuous for its blankness and said, ‘Here we are.’ When we looked at the blank space and asked, “And where is that?” he answered in the cipher-code of the higher priesthood, “31 -15 – 47 north, 133 – 5 – 30 west.” And we said, ‘Oh,’ and felt mighty small.”                           – Jack London, The Cruise of the Snark

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As we find our way, we might send carefully-calculated birthday greetings to John Locke; he died on this date in 1856.  A namesake of the famous philosopher, Locke trained as a doctor, but turned to geology– and to the invention of scientific, surveying, and navigational instruments, including a surveyor’s compass, a collimating level (Locke’s Hand Level), and a gravity escapement for regulator clocks.  The electro-chronograph he constructed (1844-48) for the United States Coast Survey was installed in the Naval Observatory, Washington, in 1848.  It improved determination of longitudes, as it was able to make a printed record on a time scale of an event to within one one-hundredth of a second.  When connected via the nation’s telegraph system, astronomers could record the time of events they observed from elsewhere in the country, by pressing a telegraph key.

Locke,_John source

 

Written by LW

July 10, 2019 at 1:01 am

“Information is a difference that makes a difference”*…

 

Shannon information

 

Information was something guessed at rather than spoken of, something implied in a dozen ways before it was finally tied down. Information was a presence offstage. It was there in the studies of the physiologist Hermann von Helmholtz, who, electrifying frog muscles, first timed the speed of messages in animal nerves just as Thomson was timing the speed of messages in wires. It was there in the work of physicists like Rudolf Clausius and Ludwig Boltzmann, who were pioneering ways to quantify disorder—entropy—little suspecting that information might one day be quantified in the same way. Above all, information was in the networks that descended in part from the first attempt to bridge the Atlantic with underwater cables. In the attack on the practical engineering problems of connecting Points A and B—what is the smallest number of wires we need to string up to handle a day’s load of messages? how do we encrypt a top-secret telephone call?—the properties of information itself, in general, were gradually uncovered.

By the time of Claude Shannon’s childhood, the world’s communications networks were no longer passive wires acting as conduits for electricity, a kind of electron plumbing. They were continent-spanning machines, arguably the most complex machines in existence. Vacuum-tube amplifiers strung along the telephone lines added power to voice signals that would have otherwise attenuated and died out on their thousand-mile journeys. A year before Shannon was born, in fact, Bell and Watson inaugurated the transcontinental phone line by reenacting their first call, this time with Bell in New York and Watson in San Francisco. By the time Shannon was in elementary school, feedback systems managed the phone network’s amplifiers automatically, holding the voice signals stable and silencing the “howling” or “singing” noises that plagued early phone calls, even as the seasons turned and the weather changed around the sensitive wires that carried them. Each year that Shannon placed a call, he was less likely to speak to a human operator and more likely to have his call placed by machine, by one of the automated switchboards that Bell Labs grandly called a “mechanical brain.” In the process of assembling and refining these sprawling machines, Shannon’s generation of scientists came to understand information in much the same way that an earlier generation of scientists came to understand heat in the process of building steam engines.

It was Shannon who made the final synthesis, who defined the concept of information and effectively solved the problem of noise. It was Shannon who was credited with gathering the threads into a new science…

The story of Claude Shannon, his colorful life–  and the birth of the Information Age: “How Information Got Re-Invented.”

* Gregory Bateson

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As we separate the signal from the noise, we might send communicative birthday greetings to the subject of today’s main post, Claude Elwood Shannon; he was born on this date in 1916.  A mathematician, electrical engineer, and cryptographer, he is, for reasons explained in the article featured above, known as “the father of information theory.”  But he is also remembered for his contributions to digital circuit design theory and for his cryptanalysis work during World War II, both as a codebreaker and as a designer of secure communications systems.

220px-ClaudeShannon_MFO3807 source

 

“The laws of nature are but the mathematical thoughts of God”*…

 

euclid

2,300 years ago, Euclid of Alexandria sat with a reed pen–a humble, sliced stalk of grass–and wrote down the foundational laws that we’ve come to call geometry. Now his beautiful work is available for the first time as an interactive website.

Euclid’s Elements was first published in 300 B.C. as a compilation of the foundational geometrical proofs established by the ancient Greek. It became the world’s oldest, continuously used mathematical textbook. Then in 1847, mathematician Oliver Byrne rereleased the text with a new, watershed use of graphics. While Euclid’s version had basic sketches, Byrne reimagined the proofs in a modernist, graphic language based upon the three primary colors to keep it all straight. Byrne’s use of color made his book expensive to reproduce and therefore scarce, but Byrne’s edition has been recognized as an important piece of data visualization history all the same…

Explore elemental beauty at “A masterpiece of ancient data viz, reinvented as a gorgeous website.”

* Euclid, Elements

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As we appreciate the angles, we might spare a thought for Kurt Friedrich Gödel; he died on this date in 1978.  A  logician, mathematician, and philosopher, he is considered (along with Aristotle, Alfred Tarski— whose birthday this also is– and Gottlob Frege) to be one of the most important logicians in history.  Gödel had an immense impact upon scientific and philosophical thinking in the 20th century.  He is, perhaps, best remembered for his Incompleteness Theorems, which led to (among other important results) Alan Turing’s insights into computational theory.

Kurt Gödel’s achievement in modern logic is singular and monumental – indeed it is more than a monument, it is a landmark which will remain visible far in space and time. … The subject of logic has certainly completely changed its nature and possibilities with Gödel’s achievement.                  — John von Neumann

kurt_gödel source

 

Written by LW

January 14, 2019 at 1:01 am

“I see the beard and cloak, but I don’t yet see a philosopher”*…

 

beards

Victorian taste-maker Thomas Gowing:

The Beard, combining beauty with utility, was intended to impart manly grace and free finish to the male face. To its picturesqueness, Poets and Painters, the most competent judges, have borne universal testimony. It is indeed impossible to view a series of bearded portraits, however indifferently executed, without feeling that they possess dignity, gravity, freedom, vigor, and completeness; while in looking on a row of razored faces, however illustrious the originals, or skillful the artists, a sense of artificial conventional bareness is experienced…

More from Gowing’s masterwork, The Philosophy of Beards, at “The argument we need for the universal wearing of beards.”

* Aulus Gellius

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As we let ’em grow, we might send carefully-calculated birthday greetings to Vladimir Andreevich Steklov; he was born on this date in 1864.  An important Russian mathematician and physicist, he made important contributions to set theory, hydrodynamics, and the theory of elasticity, and wrote widely on the history of science.  But he is probably best remembered as the honored namesake of the Russian Institute of Physics and Mathematics (for which he was the original petitioner); its math department is now known as the Steklov Institute of Mathematics.

220px-steklov source

 

Written by LW

January 9, 2019 at 1:01 am

“Control of consciousness determines the quality of life”*…

 

2D02EB62-9945-4BB8-A5E9507B396FEF67_source

Peter Carruthers, Distinguished University Professor of Philosophy at the University of Maryland, College Park, is an expert on the philosophy of mind who draws heavily on empirical psychology and cognitive neuroscience. He outlined many of his ideas on conscious thinking in his 2015 book The Centered Mind: What the Science of Working Memory Shows Us about the Nature of Human Thought. More recently, in 2017, he published a paper with the astonishing title of “The Illusion of Conscious Thought.”…

Philosopher Peter Carruthers insists that conscious thought, judgment and volition are illusions. They arise from processes of which we are forever unaware.  He explains to Steve Ayan the reasons for his provocative proposal: “There Is No Such Thing as Conscious Thought.”

See also: “An Anthropologist Investigates How We Think About How We Think.”

* Mihaly Csikszentmihalyi, Flow: The Psychology of Optimal Experience

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As we think about thought, we might spare one for Georg Ferdinand Ludwig Philipp Cantor; he died on this date in 1918.  Cantor was the mathematician who created set theory, now fundamental to math,  His proof that the real numbers are more numerous than the natural numbers implies the existence of an “infinity of infinities”… a result that generated a great deal of resistance, both mathematical (from the likes of Henri Poincaré) and philosophical (most notably from Wittgenstein).  Some Christian theologians (particularly neo-Scholastics) saw Cantor’s work as a challenge to the uniqueness of the absolute infinity in the nature of God – on one occasion equating the theory of transfinite numbers with pantheism – a proposition that Cantor, a devout Lutheran, vigorously rejected.

These harsh criticisms fueled Cantor’s bouts of depression (retrospectively judged by some to have been bipolar disorder); he died in a mental institution.

220px-Georg_Cantor2 source

 

Written by LW

January 6, 2019 at 1:01 am

“I have had my results for a long time, but I do not yet know how to arrive at them”*…

 

Fermat

Andrew Wiles gave a series of lectures cryptically titled “Modular Forms, Elliptic Curves, and Galois Representations” at a mathematics conference in Cambridge, England, in June 0f 1993. His argument was long and technical. Finally, 20 minutes into the third talk, he came to the end. Then, to punctuate the result, he added:

=> FLT

“Implies Fermat’s Last Theorem.” The most famous unverified conjecture in the history of mathematics. First proposed by the 17th-century French jurist and spare-time mathematician Pierre de Fermat, it had remained unproven for more than 350 years. Wiles, a professor at Princeton University, had worked on the problem, alone and in secret in the attic of his home, for seven years. Now he was unveiling his proof.

His announcement electrified his audience—and the world. The story appeared the next day on the front page of The New York Times. Gap, the clothing retailer, asked him to model a new line of jeans, though he demurred. People Weekly named him one of “The 25 Most Intriguing People of the Year,” along with Princess Diana, Michael Jackson, and Bill Clinton. Barbara Walters’ producers reached out to him for an interview, to which Wiles responded, “Who’s Barbara Walters?”

But the celebration didn’t last. Once a proof is proposed, it must be checked and verified before it is accepted as valid. When Wiles submitted his 200-page proof to the prestigious journal Inventiones Mathematicae, its editor divvied up the manuscript among six reviewers. One of them was Nick Katz, a fellow Princeton mathematician.

For two months, Katz and a French colleague, Luc Illusie, scrutinized every logical step in Katz’s section of the proof. From time to time, they would come across a line of reasoning they couldn’t follow. Katz would email Wiles, who would provide a fix. But in late August, Wiles offered an explanation that didn’t satisfy the two reviewers. And when Wiles took a closer look, he saw that Katz had found a crack in the mathematical scaffolding. At first, a repair seemed straightforward. But as Wiles picked at the crack, pieces of the structure began falling away…

How mistakes– first Fermat’s, then Wiles’– reinvigorated a field, then led to fundamental insight: “How Math’s Most Famous Proof Nearly Broke.”

* Karl Friedrich Gauss

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As we ponder proof, we might we might spare a thought for Josiah Wedgwood; he died on this date in 1795. An English potter and businessman (he founded the Wedgwood company), he is credited, via his technique of “division of labor,” with the industrialization of the manufacture of pottery– and via his example, much of British (and thus American) manufacturing.

Wedgwood was a member of the Lunar Society, the Royal Society, and was an ardent abolitionist.  His daughter, Susannah, was the mother of Charles Darwin.

 source

 

“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

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