Tagged: Mathematics Toggle Comment Threads | Keyboard Shortcuts

  • feedwordpress 09:01:26 on 2019/01/14 Permalink
    Tags: , Elements, Euclid, , , , incompleteness theorems, , Mathematics, ,   

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


    Warning: preg_match_all(): Compilation failed: invalid range in character class at offset 7 in /homepages/23/d339537987/htdocs/pb/wp-content/themes/p2/inc/mentions.php on line 77

     

    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

    ###

    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

     

     
  • feedwordpress 09:01:35 on 2019/01/09 Permalink
    Tags: , grooming, , , Mathematics, , , Steklov, The Philosophy of Beards, ,   

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


    Warning: preg_match_all(): Compilation failed: invalid range in character class at offset 7 in /homepages/23/d339537987/htdocs/pb/wp-content/themes/p2/inc/mentions.php on line 77

     

    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

    ###

    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

     

     
  • feedwordpress 09:01:22 on 2019/01/06 Permalink
    Tags: , Georg Cantor, , , , Mathematics, Peter Carruthers, , set theory,   

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


    Warning: preg_match_all(): Compilation failed: invalid range in character class at offset 7 in /homepages/23/d339537987/htdocs/pb/wp-content/themes/p2/inc/mentions.php on line 77

     

    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

    ###

    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

     

     
  • feedwordpress 09:01:53 on 2019/01/03 Permalink
    Tags: Andrew Wiles, , Fermat, Fermat's Last Theorem, , Josiah Wedgwood, manuafacturing, Mathematics, number theiry, ,   

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


    Warning: preg_match_all(): Compilation failed: invalid range in character class at offset 7 in /homepages/23/d339537987/htdocs/pb/wp-content/themes/p2/inc/mentions.php on line 77

     

    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

    ###

    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

     

     
  • feedwordpress 09:01:59 on 2018/11/20 Permalink
    Tags: , , , , , , Mathematics, ,   

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


    Warning: preg_match_all(): Compilation failed: invalid range in character class at offset 7 in /homepages/23/d339537987/htdocs/pb/wp-content/themes/p2/inc/mentions.php on line 77

     

    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

    ###

    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!

     
  • feedwordpress 09:01:54 on 2018/11/17 Permalink
    Tags: D.W. Griffith, Jim Crow, Mathematics, , , , The Clansman, Thomas Dixon Jr., , , Woodrow Wilson   

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


    Warning: preg_match_all(): Compilation failed: invalid range in character class at offset 7 in /homepages/23/d339537987/htdocs/pb/wp-content/themes/p2/inc/mentions.php on line 77

     

    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

    ###

    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.”

     source

     source

     

     
  • feedwordpress 10:01:14 on 2018/11/05 Permalink
    Tags: electromagnetic radiation, , , , Mathematics, , , ,   

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


    Warning: preg_match_all(): Compilation failed: invalid range in character class at offset 7 in /homepages/23/d339537987/htdocs/pb/wp-content/themes/p2/inc/mentions.php on line 77

     

    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

    ###

    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

     

     
  • feedwordpress 08:01:49 on 2018/10/16 Permalink
    Tags: Andreas Cellarius, , , Harmonia Macrocosmica, , , Mathematics, , Sir William Rowan Hamilton,   

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


    Warning: preg_match_all(): Compilation failed: invalid range in character class at offset 7 in /homepages/23/d339537987/htdocs/pb/wp-content/themes/p2/inc/mentions.php on line 77

     

    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)

    ###

    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.

     source

     

     
  • feedwordpress 08:01:41 on 2018/10/05 Permalink
    Tags: , , , , , Mathematics, , , ,   

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


    Warning: preg_match_all(): Compilation failed: invalid range in character class at offset 7 in /homepages/23/d339537987/htdocs/pb/wp-content/themes/p2/inc/mentions.php on line 77

     

    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

    ###

    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.

     source

     

     
  • feedwordpress 08:01:55 on 2018/09/20 Permalink
    Tags: , , Jeremy Baumberg, Mathematics, Paul Erdos, , ,   

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


    Warning: preg_match_all(): Compilation failed: invalid range in character class at offset 7 in /homepages/23/d339537987/htdocs/pb/wp-content/themes/p2/inc/mentions.php on line 77

     

    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

    ###

    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.

     source

     

     
c
compose new post
j
next post/next comment
k
previous post/previous comment
r
reply
e
edit
o
show/hide comments
t
go to top
l
go to login
h
show/hide help
esc
cancel