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  • feedwordpress 08:01:27 on 2018/08/06 Permalink
    Tags: , , , , Johann Bernoulli, Mathematics, , ,   

    “Once is happenstance. Twice is coincidence. Three times, it’s enemy action.”*… 

     

    complexity

    A couple of weeks ago, we considered the human urge to find significance, meaning in everyday occurrences: “All mystical experience is coincidence; and vice versa, of course.” Today, we consider the same phenomena from a more mathematical point-of-view…

    Was it a chance encounter when you met that special someone or was there some deeper reason for it? What about that strange dream last night—was that just the random ramblings of the synapses of your brain or did it reveal something deep about your unconscious? Perhaps the dream was trying to tell you something about your future. Perhaps not. Did the fact that a close relative developed a virulent form of cancer have profound meaning or was it simply a consequence of a random mutation of his DNA?

    We live our lives thinking about the patterns of events that happen around us. We ask ourselves whether they are simply random, or if there is some reason for them that is uniquely true and deep. As a mathematician, I often turn to numbers and theorems to gain insight into questions like these. As it happens, I learned something about the search for meaning among patterns in life from one of the deepest theorems in mathematical logic. That theorem, simply put, shows that there is no way to know, even in principle, if an explanation for a pattern is the deepest or most interesting explanation there is. Just as in life, the search for meaning in mathematics knows no bounds…

    Noson Yanofsky on what math can teach us about finding order in our chaotic lives.

    * Ian Fleming

    ###

    As we consider the odds, we might send carefully-calculated birthday greetings to Johann Bernoulli; he was born on this date in 1667.  A member of the mathematically-momentous Bernoulli family, Johann (also known as Jean or John) discovered the exponential calculus and (with Leibniz and Huygens) the equation of the catenary.  Still, he be best remembered as teacher and mentor of Leonhard Euler.

    220px-Johann_Bernoulli2 source

     

     
  • feedwordpress 08:01:18 on 2018/07/31 Permalink
    Tags: , , , , , , Mathematics, sentimental cartography, Seymour Papert,   

    “I have an existential map. It has ‘you are here’ written all over it.”*… 

     

    map

    A detail from illustrator James Turner‘s Map of Humanity.

     

    A long time ago, I made a map of the rationalist community.  This is in the same geographic-map-of-something-non-geographic tradition as the Greater Ribbonfarm Cultural Region or xkcd’s map of the Internet. There’s even some sort of therapy program that seems to involve making a map like this of your life, though I don’t know how seriously they take it.

    There’s no good name for this art and it’s really hard to Google. If you try “map of abstract concept” you just get a bunch of concept maps. It seems the old name, from back when this was a popular Renaissance amusement, is “sentimental cartography”, since it was usually applied to sentiments like love or sorrow. This isn’t great – the Internet’s not a sentiment – but it’s what we’ve got and I’ll do what I can to try to make it catch on…

    See the marvelous examples (like the one above) collected by Scott Alexander at “Sentimental Cartography.”

    * Steven Wright

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    As we find our place, we might spare a thought for Seymour Papert; he died on this date in 2016.  Trained as a mathematician, Papert was a pioneer of computer science, and in particular, artificial intelligence. He created the Epistemology and Learning Research Group at the MIT Architecture Machine Group (which later became the MIT Media Lab); he directed MIT’s Artificial Intelligence Laboratory; he authored the hugely-influential LOGO computer language; and he was a principal of the One Laptop Per Child Program.  Called by Marvin Minsky “the greatest living mathematics educator,” Papert won a Guggenheim fellowship (1980), a Marconi International fellowship (1981), the Software Publishers Association Lifetime Achievement Award (1994), and the Smithsonian Award (1997).

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  • feedwordpress 08:01:10 on 2018/07/12 Permalink
    Tags: , , , Fullerenes, , hole, Mathematics, nothing, , , zero   

    “I love to talk about nothing. It’s the only thing I know anything about.”*… 

     

    zero

    The computer you’re reading this article on right now runs on a binary — strings of zeros and ones. Without zero, modern electronics wouldn’t exist. Without zero, there’s no calculus, which means no modern engineering or automation. Without zero, much of our modern world literally falls apart.

    Humanity’s discovery of zero was “a total game changer … equivalent to us learning language,” says Andreas Nieder, a cognitive scientist at the University of Tübingen in Germany.

    But for the vast majority of our history, humans didn’t understand the number zero. It’s not innate in us. We had to invent it. And we have to keep teaching it to the next generation.

    Other animals, like monkeys, have evolved to understand the rudimentary concept of nothing. And scientists just reported that even tiny bee brains can compute zero. But it’s only humans that have seized zero and forged it into a tool.

    So let’s not take zero for granted. Nothing is fascinating. Here’s why…

    It is indeed fascinating, as you’ll see at “The mind-bendy weirdness of the number zero, explained.”

    Pair with: “Is a hole a real thing, or just a place where something isn’t?” and with The Ministry of Ideas’ podcast “Nothing Matters.”

    * Oscar Wilde

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    As we obsess about absence, we might box a dome-shaped birthday cake for inventor, educator, author, philosopher, engineer, and architect R(ichard) Buckminster Fuller; he was born on this date in 1895.  “Bucky” most famously developed the geodesic dome, the only large dome that can be set directly on the ground as a complete structure, and the only practical kind of building that has no limiting dimensions (i.e., beyond which the structural strength must be insufficient).  But while he never got around to frankfurters, he was sufficiently prolific to have scored over 2,000 patents.

    “Fullerenes” (molecules composed entirely of carbon, in the form of a hollow spheres, ellipsoids, or tubes), key components in many nanotechnology applications, were named for Fuller, as their structure mimes that of the geodesic dome.  Spherical fullerenes (resembling soccer balls) are also called “buckyballs”; cylindrical ones, carbon nanotubes or “buckytubes.”

    I have to say, I think that we are in some kind of final examination as to whether human beings now, with this capability to acquire information and to communicate, whether we’re really qualified to take on the responsibility we’re designed to be entrusted with. And this is not a matter of an examination of the types of governments, nothing to do with politics, nothing to do with economic systems. It has to do with the individual. Does the individual have the courageto really go along with the truth?

    God, to me, it seems
    is a verb,
    not a noun,
    proper or improper.

    For more, see “And that’s a lot.”

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  • feedwordpress 08:01:40 on 2018/07/01 Permalink
    Tags: , , , Mathematics, , practical joke, , Pythagoras, Pythagorean Cup,   

    “All practical jokes, friendly, harmless or malevolent, involve deception, but not all deceptions are practical jokes”*… 

     

    28134677457_5f485d8423_z

    When you think of the ancient Greeks, practical jokes might not be the first thing that comes to mind. But along with art, architecture, and philosophy, you can add trick cups to their list of accomplishments.

    The Pythagorean cup is so-named because it was allegedly invented by Pythagoras of Samos (yes, the same guy who gave us theories about right triangles). It’s a small cup with a column in its center. It doesn’t look like much, but when an unsuspecting drinker fills it past a designated level, the liquid mysteriously drains out. Legend has it that Pythagoras used it as a way to punish greedy drinkers who poured themselves too much wine…

    A timeless practical joke, brought to you by the ancient Greeks: more merriment at “Pythagorean Cup.”

    * W. H. Auden, The Dyer’s Hand

    ###

    As we ponders pranks, we might send a “Alles Gute zum Geburtstag” to the polymathic Gottfried Wilhelm Leibniz, the philosopher, mathematician, and political adviser, who was important both as a metaphysician and as a logician, but who is probably best remembered for his independent invention of the calculus; he was born on this date in 1646.  Leibniz independently discovered and developed differential and integral calculus, which he published in 1684;  but he became involved in a bitter priority dispute with Isaac Newton, whose ideas on the calculus were developed earlier (1665), but published later (1687).

    As it happens, Leibnitz was no mean humorist.  Consider, e.g…

    If geometry conflicted with our passions and our present concerns as much as morality does, we would dispute it and transgress it almost as much–in spite of all Euclid’s and Archimedes’ demonstrations, which would be treated as fantasies and deemed to be full of fallacies. [Leibniz, New Essays, p. 95]

    28134677537_d79a889e6a_o source

     

     
  • feedwordpress 08:01:48 on 2018/06/03 Permalink
    Tags: , , , mathematical economics, Mathematics, merit, RGD Allen, , , wealth inequality   

    “Oh, I am fortune’s fool!”*… 

     

    The distribution of wealth follows a well-known pattern sometimes called an 80:20 rule: 80 percent of the wealth is owned by 20 percent of the people. Indeed, a report last year concluded that just eight men had a total wealth equivalent to that of the world’s poorest 3.8 billion people.

    This seems to occur in all societies at all scales. It is a well-studied pattern called a power law that crops up in a wide range of social phenomena. But the distribution of wealth is among the most controversial because of the issues it raises about fairness and merit. Why should so few people have so much wealth?

    The conventional answer is that we live in a meritocracy in which people are rewarded for their talent, intelligence, effort, and so on. Over time, many people think, this translates into the wealth distribution that we observe, although a healthy dose of luck can play a role.

    But there is a problem with this idea: while wealth distribution follows a power law, the distribution of human skills generally follows a normal distribution that is symmetric about an average value. For example, intelligence, as measured by IQ tests, follows this pattern. Average IQ is 100, but nobody has an IQ of 1,000 or 10,000.

    The same is true of effort, as measured by hours worked. Some people work more hours than average and some work less, but nobody works a billion times more hours than anybody else.

    And yet when it comes to the rewards for this work, some people do have billions of times more wealth than other people. What’s more, numerous studies have shown that the wealthiest people are generally not the most talented by other measures.

    What factors, then, determine how individuals become wealthy? Could it be that chance plays a bigger role than anybody expected? And how can these factors, whatever they are, be exploited to make the world a better and fairer place?…

    A new computer model of wealth creation confirms that the most successful people are not the most talented, just the luckiest. Learn more at: “If you’re so smart, why aren’t you rich? Turns out it’s just chance.

    * Shakespeare, Romeo and Juliet

    ###

    As we muse on merit, we might send carefully-calculated birthday greetings to a forbearer of the researchers who did the work recounted above, Sir Roy George Douglas Allen; he was born on this date in 1906.  A mathematician and statistician turned economist, he was a leader in the field of mathematical economics, writing a number of influential texts including  Mathematical Analysis for EconomistsStatistics for Economists, and Mathematical Economics.

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  • feedwordpress 08:01:22 on 2018/03/14 Permalink
    Tags: , , Mathematics, , , ,   

    “All numbers are by their nature correct. Well, except for Pi, of course. I can’t be doing with Pi. Gives me a headache just thinking about it, going on and on and on and on and on…”*… 

     

    It’s Pi Day!

    In celebration, a few amusing– and illuminating– links:

    The history of pi

    Pi day magic revealed

    10 stunning images show the beauty hidden in pi

    The history of Pi Day

    How to Memorize Pi if You’re a Word Person (from whence, the image above)

    * Neil Gaiman, Anansi Boys

    ###

    As we enumerate endlessly, we might pause for a piece of pi(e)…

     source

    … in celebration of Albert Einstein’s birthday; he was born on this date in 1879.

    source

    “Everything should be made as simple as possible, but not simpler.”

     

     
  • feedwordpress 09:01:09 on 2018/02/05 Permalink
    Tags: Carlyle Circle, Great Man theory, , , , Mathematics, , simile, the simile museum, Thomas Carlyle,   

    “A metaphor is like a simile”*… 

     

    “Fiction is like a spider’s web, attached ever so lightly perhaps, but still attached to life at all four corners.”

    -Virginia Woolf

    Just one of the “exhibits” in “an ongoing collection of the world’s most likable literary device”:  The Simile Museum.

    [source of the image above]

    * Steven Wright

    ###

    As we remember that “liking” has a very long history, we might spare a thought for Thomas Carlyle; he died on this date in 1881.  A Victorian polymath, he was an accomplished philosopher, satirical writer, essayist, translator, historian, mathematician, and teacher.  While he was an enormously popular lecturer in his time, and his contributions to mathematics earned him eponymous fame (the Carlyle circle), he may be best remembered as a historian (and champion of the “Great Man” theory of history)… and as the coiner of phrases like “the dismal science” (to describe economics)

    “A well-written Life is almost as rare as a well-spent one.”   – Thomas Carlyle

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  • feedwordpress 09:01:30 on 2018/01/29 Permalink
    Tags: Bernard Brunhes, , brachistochrone problem, , , , , , Mathematics, ,   

    “We never cease to stand like curious children before the great mystery into which we were born”*… 

     

    This animation shows the movement of the north magnetic pole at 10-year intervals from 1970 to 2020. The red and blue lines indicate “declination,” the difference between magnetic north and true north depending on where one is standing; on the green line, a compass would point to true north. Visual by NOAA National Centers for Environmental Information

    In scenario planning, one tries to identify the “driving forces”– the social, political, ecological, technical, and economic dynamics afoot– in the environment that are both likely to impact our future materially and outside our control; one then to knits the possible outcomes of those forces into alternative futures, plausible sketches of the opportunities and challenges that one might face.

    There is a special class of driving force, what scenario planners call a wild card: a possibility that has relative low probability in the (usually 10 year) time horizon, but that, should it occur, would have massive consequence.  Wild cards are often things like major earthquakes or geo-political conflicts… or environmental catastrophes.  While one plans for the implications of the scenarios and their defining driving forces, one plans against wild cards; one creates action plans for the scenarios, contingency plans for the wild cards.

    As climate change is slowly but surely converting yesterday’s wildcards (sustained droughts, regular, catastrophic wildfires and storms, etc.) into “regular” driving forces, it is perhaps prudent to look at some of the wildest cards that remain…

    One day in 1905, the French geophysicist Bernard Brunhes brought back to his lab some rocks he’d unearthed from a freshly cut road near the village of Pont Farin. When he analyzed their magnetic properties, he was astonished at what they showed: Millions of years ago, the Earth’s magnetic poles had been on the opposite sides of the planet. North was south and south was north. The discovery spoke of planetary anarchy. Scientists had no way to explain it.

    Today, we know that the poles have changed places hundreds of times, most recently 780,000 years ago. (Sometimes, the poles try to reverse positions but then snap back into place, in what is called an excursion. The last time was about 40,000 years ago.) We also know that when they flip next time, the consequences for the electrical and electronic infrastructure that runs modern civilization will be dire. The question is when that will happen…

    The shield that protects the Earth from solar radiation is under attack from within. We can’t prevent it, but we ought to prepare. Learn more at “The Magnetic Field Is Shifting. The Poles May Flip. This Could Get Bad.”

    * Albert Einstein

    ###

    As we ponder powerlessness, we might recall that it was on this date in 1697 that Isaac Newton received a copy of Johann Bernoulli’s long-standing mathematical challenge, the brachistochrone problem: “To determine the curved line joining two given points, situated at different distances from the horizontal and not in the same vertical line, along which the mobile body, running down by its own weight and starting to move from the upper point, will descend most quickly to the lower point.” (Bernoulli coined the name from Gr. brachistos, shortest; and chronos, time.)

    Newton solved it the same day, and forwarded his solution to the Royal Society—anonymously.  When Bernoulli read the solution, he shrewdly guessed it was Newton’s work.  By legend, he said, “I recognize the lion by his paw.”

    Bernoulli and Newton

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  • feedwordpress 09:01:50 on 2017/11/17 Permalink
    Tags: , Mathematics, , Pablo Iglesias Maurer, , post cards, ,   

    “Woe, destruction, ruin, and decay; the worst is death and death will have his day”*… 

     

    Grossinger’s outdoor pool, olympic sized, built in 1949 at a cost of $400,000 (about $5 million in today’s market.) Long gone are the private cabanas, changing room and lounges that used to surround it.

    Not long ago an old matchbook laying on photographer Pablo Iglesias Maurer‘s desk caught his eye. Or rather, it was the postcard-like picture on it, of a resort complex built in the 1960s. It got Pablo wondering how the place looked now, and the answer has led him to make an amazing photo series called Abandoned States.

    The picture came with the title How to Run A Successful Golf Course, but when Maurer got to the place, it was clear the owner of Penn Hills Resort didn’t follow that advice. He pointed the camera at the decaying building at roughly the same spot and did a ‘5-decades-after’ shot of the place.

    Ever since then, Pablo was hooked. He ordered more 60s postcards from eBay and started going around the country capturing these once beautiful buildings that now stand abandoned only as faint memories of what once was…

    * Shakespeare, Richard II

    ###

    As we contemplate continuity, 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).

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  • feedwordpress 08:01:01 on 2017/10/16 Permalink
    Tags: , , baryons, , , Mathematics, missing matter, quaternions, , , William Rowan Hamilton   

    “Oh, there you are Peter”*… 

     

    The missing links between galaxies have finally been found. This is the first detection of the roughly half of the normal matter in our universe – protons, neutrons and electrons – unaccounted for by previous observations of stars, galaxies and other bright objects in space.

    You have probably heard about the hunt for dark matter, a mysterious substance thought to permeate the universe, the effects of which we can see through its gravitational pull. But our models of the universe also say there should be about twice as much ordinary matter out there, compared with what we have observed so far.

    Two separate teams found the missing matter – made of particles called baryons rather than dark matter – linking galaxies together through filaments of hot, diffuse gas

    Get galactic at: “Half the universe’s missing matter has just been finally found.”

    * meme

    ###

    As we heed E.M. Forster, 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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