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Posts Tagged ‘Mathematics

“Men of broader intellect know that there is no sharp distinction betwixt the real and the unreal”*…



Colored cubes — known as “Tesseracts” — as depicted in the frontispiece to Hinton’s The Fourth Dimension (1904)


During the period we now call the fin de siècle, worlds collided. Ideas were being killed off as much as being born. And in a sort of Hegelian logic of thesis/antithesis/synthesis, the most interesting ones arose as the offspring of wildly different parents. In particular, the last gasp of Victorian spirituality infused cutting-edge science with a certain sense of old-school mysticism. Theosophy was all the rage; Huysmans dragged Satan into modern Paris; and eccentric poets and scholars met in the British Museum Reading Room under the aegis of the Golden Dawn for a cup of tea and a spot of demonology. As a result of all this, certain commonly-accepted scientific terms we use today came out of quite weird and wonderful ideas being developed at the turn of the century. Such is the case with space, which fascinated mathematicians, philosophers, and artists with its unfathomable possibilities…

In April 1904, C. H. Hinton published The Fourth Dimension, a popular maths book based on concepts he had been developing since 1880 that sought to establish an additional spatial dimension to the three we know and love. This was not understood to be time as we’re so used to thinking of the fourth dimension nowadays; that idea came a bit later. Hinton was talking about an actual spatial dimension, a new geometry, physically existing, and even possible to see and experience; something that linked us all together and would result in a “New Era of Thought.”…

Hinton’s ideas gradually pervaded the cultural milieu over the next thirty years or so — prominently filtering down to the Cubists and Duchamp. The arts were affected by two distinct interpretations of higher dimensionality: on the one hand, the idea as a spatial, geometric concept is readily apparent in early Cubism’s attempts to visualise all sides of an object at once, while on the other hand, it becomes a kind of all-encompassing mystical codeword used to justify avant-garde experimentation. “This painting doesn’t make sense? Ah, well, it does in the fourth dimension…” It becomes part of a language for artists exploring new ideas and new spaces…

By the late 1920s, Einsteinian Space-Time had more or less replaced the spatial fourth dimension in the minds of the public. It was a cold yet elegant concept that ruthlessly killed off the more romantic idea of strange dimensions and impossible directions. What had once been the playground of spiritualists and artists was all too convincingly explained. As hard science continued to rise in the early decades of the twentieth century, the fin-de-siècle’s more outré ideas continued to decline. Only the Surrealists continued to make reference to it, as an act of rebellion and vindication of the absurd. The idea of a real higher dimension linking us together as One sounded all a bit too dreamy, a bit too old-fashioned for a new century that was picking up speed, especially when such vague and multifarious explanations were trumped by the special theory of relativity. Hinton was as much hyperspace philosopher as scientist and hoped humanity would create a more peaceful and selfless society if only we recognised the unifying implications of the fourth dimension. Instead, the idea was banished to the realms of New Age con-artists, reappearing these days updated and repackaged as the fifth dimension. Its shadow side, however, proved hopelessly alluring to fantasy writers who have seen beyond the veil, and bring back visions of horror from an eldritch land outside of time and space that will haunt our nightmares with its terrible geometry, where tentacles and abominations truly horrible sleep beneath the Pacific Ocean waiting to bring darkness to our world… But still we muddle on through.

Hyperspace, tesseracts, ghosts, and colorful cubesJon Crabb, Editor, British Library Publishing, on the work of Charles Howard Hinton and the cultural history of higher dimensions: “Notes on the Fourth Dimension.”

[TotH to MK]

* H.P. Lovecraft, The Tomb


As we get high(er), we might recall that it was on this date in 1946 that Al Gross went public with his invention of the walkie talkie.  Gross had developed it as a top secret project during World War II; he went on to develop the circuitry that opened the way to personal pocket paging systems, CB radio, and patented precursors of the cell phone and the cordless phone.  Sadly for him, his patents expired before they became commercially viable.  ”Otherwise,” Gross said, after winning the M.I.T. lifetime achievement award, ”I’d be as rich as Bill Gates.”

While Gross himself is almost unknown to the general public, he did achieve one-step-removed notoriety in 1948 when he “gifted” his friend Chester Gould the concept of miniaturized radio transceivers, which Gross had just patented.  Gould put it to use as the two-way wrist radio in his comic strip Dick Tracy.

200px-ALGROS2 source


Written by LW

November 20, 2019 at 1:01 am

“It is still an unending source of surprise for me how a few scribbles on a blackboard or on a piece of paper can change the course of human affairs”*…




For the last year, Jessica Wynne, a photographer and professor at the Fashion Institute of Technology in New York, has been photographing mathematicians’ blackboards, finding art in the swirling gangs of symbols sketched in the heat of imagination, argument and speculation. “Do Not Erase,” a collection of these images, will be published by Princeton University Press in the fall of 2020…

This is what thought looks like.

Ideas, and ideas about ideas. Suppositions and suspicions about relationships among abstract notions — shape, number, geometry, space — emerging through a fog of chalk dust, preferably of the silky Hagoromo chalk, originally from Japan, now made in South Korea.

In these diagrams, mysteries are being born and solved…


More (and larger) examples from this photo survey of the blackboards of mathematicians at “Where Theory Meets Chalk, Dust Flies.”

* Stanislaw Ulam


As we scribble “do not erase,” we might spare a thought for Herbert Aaron Hauptman; he died on this date in 2011.  A mathematician, he pioneered and developed a mathematical method that has changed the whole field of chemistry and opened a new era in research in determination of molecular structures of crystallized materials.  Today, Hauptman’s “direct methods,” which he continued to improve and refine, are routinely used to solve complicated structures… work for which he shared the the 1985 Nobel Prize in Chemistry.

R source


Written by LW

October 24, 2019 at 1:01 am

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



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


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


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”*…



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


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”*…



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


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”*…



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


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

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