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Posts Tagged ‘Math

“Do not worry about your difficulties in Mathematics. I can assure you mine are still greater.”*…

No scripture is as old as mathematics is. All the other sciences are younger, most by thousands of years. More than history, mathematics is the record that humanity is keeping of itself. History can be revised or manipulated or erased or lost. Mathematics is permanent. A² + B² = C² was true before Pythagoras had his name attached to it, and will be true when the sun goes out and no one is left to think of it. It is true for any alien life that might think of it, and true whether they think of it or not. It cannot be changed. So long as there is a world with a horizontal and a vertical axis, a sky and a horizon, it is inviolable and as true as anything that can be thought.

As precise as mathematics is, it is also the most explicit language we have for the description of mysteries. Being the language of physics, it describes actual mysteries—things we can’t see clearly in the natural world but suspect are true and later confirm—and imaginary mysteries, things that exist only in the minds of mathematicians. A question is where these abstract mysteries exist, what their home range is. Some people would say that they reside in the human mind, that only the human mind has the capacity to conceive of what are called mathematical objects, meaning numbers and equations and formulas and so on—the whole glossary and apparatus of mathematics—and to bring these into being, and that such things arrive as they do because of the way our minds are structured. We are led to examine the world in a way that agrees with the tools that we have for examining it. (We see colors as we do, for example, because of how our brains are structured to receive the reflection of light from surfaces.) This is a minority view, held mainly by neuroscientists and a certain number of mathematicians disinclined toward speculation. The more widely held view is that no one knows where math resides. There is no mathematician/naturalist who can point somewhere and say, “That is where math comes from” or “Mathematics lives over there,” say, while maybe gesturing toward magnetic north and the Arctic, which I think would suit such a contrary and coldly specifying discipline.

The belief that mathematics exists somewhere else than within us, that it is discovered more than created, is called Platonism, after Plato’s belief in a non-spatiotemporal realm that is the region of the perfect forms of which the objects on earth are imperfect reproductions. By definition, the non-spatiotemporal realm is outside time and space. It is not the creation of any deity; it simply is. To say that it is eternal or that it has always existed is to make a temporal remark, which does not apply. It is the timeless nowhere that never has and never will exist anywhere but that nevertheless is. The physical world is temporal and declines; the non-spatiotemporal one is ideal and doesn’t.

A third point of view, historically and presently, for a small but not inconsequential number of mathematicians, is that the home of mathematics is in the mind of a higher being and that mathematicians are somehow engaged with Their thoughts. Georg Cantor, the creator of set theory—which in my childhood was taught as a part of the “new math”—said, “The highest perfection of God lies in the ability to create an infinite set, and its immense goodness leads Him to create it.” And the wildly inventive and self-taught mathematician Srinivasa Ramanujan, about whom the movie “The Man Who Knew Infinity” was made, in 2015, said, “An equation for me has no meaning unless it expresses a thought of God.”

In Book 7 of the Republic, Plato has Socrates say that mathematicians are people who dream that they are awake. I partly understand this, and I partly don’t.

Mathematics has been variously described as an ideal reality, a formal game, and the poetry of logical ideas… an excerpt from “What is Mathematics?” from Alec Wilkinson— eminently worthy of reading in full.

* Albert Einstein

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As we sum it up, we might send carefull-calcuated birthday greetings to Georgiy Antonovich Gamov; he was born on this date in 1904. Better known by the name he adopted on immigrating to the U.S., George Gamow, he was a physicist and cosmologist whose early work was instrumental in developing the Big Bang theory of the universe; he also developed the first mathematical model of the atomic nucleus.

But mid-career Gamow began to shift his energy to teaching and to writing popular books on science… one of which, One Two Three… Infinity, inspired legions of young scientists-to-be and kindled a life-long interest in science in an even larger number of other youngsters (like your correspondent).

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“Time is the longest distance between two places”*…

 

In quantum mechanics, time is universal and absolute; its steady ticks dictate the evolving entanglements between particles. But in general relativity (Albert Einstein’s theory of gravity), time is relative and dynamical, a dimension that’s inextricably interwoven with directions x, y and z into a four-dimensional “space-time” fabric. The fabric warps under the weight of matter, causing nearby stuff to fall toward it (this is gravity), and slowing the passage of time relative to clocks far away. Or hop in a rocket and use fuel rather than gravity to accelerate through space, and time dilates; you age less than someone who stayed at home.

Unifying quantum mechanics and general relativity requires reconciling their absolute and relative notions of time. Recently, a promising burst of research on quantum gravity has provided an outline of what the reconciliation might look like — as well as insights on the true nature of time…

The effort to unify quantum mechanics and general relativity means reconciling totally different notions of time; catch up on the state of play at “Quantum Gravity’s Time Problem.”

* Tennessee Williams, The Glass Menagerie

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As we set our watches, we might send carefully-calculated birthday greetings to Gabrielle-Émilie Le Tonnelier de Breteuil, Marquise du Châtelet, the French mathematician and physicist who is probably (if unfairly) better known as Voltaire’s mistress; she was born on this date in 1706.  Fascinated by the work of Newton and Leibniz, she dressed as a man to frequent the cafes where the scientific discussions of the time were held.  Her major work was a translation of Newton’s Principia, for which Voltaire wrote the preface; it was published a decade after her death, and was for many years the only translation of the Principia into French.

Judge me for my own merits, or lack of them, but do not look upon me as a mere appendage to this great general or that great scholar, this star that shines at the court of France or that famed author. I am in my own right a whole person, responsible to myself alone for all that I am, all that I say, all that I do. it may be that there are metaphysicians and philosophers whose learning is greater than mine, although I have not met them. Yet, they are but frail humans, too, and have their faults; so, when I add the sum total of my graces, I confess I am inferior to no one.
– Mme du Châtelet to Frederick the Great of Prussia

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Written by LW

December 17, 2016 at 1:01 am

“Math is sometimes called the science of patterns”*…

 

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From Katie Steckles, help for the Holidays…

Special Holiday bonus:  the story behind those massive bows that bedeck cars given as Holiday presents.

* Ronald Graham

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As we fold with care, we might recall that it was on this date in 1937 that Walt Disney released the first full-length animated feature film produced in the U.S. (and the first produced anywhere in full color), Snow White and the Seven Dwarfs.

The original theatrical one-sheet

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Written by LW

December 21, 2015 at 1:01 am

“You can’t criticize geometry. It’s never wrong.”*…

 

In the world of mathematical tiling, news doesn’t come bigger than this.  In the world of bathroom tiling – I bet they’re interested too.

If you can cover a flat surface using only identical copies of the same shape leaving neither gaps nor overlaps, then that shape is said to “tile the plane.” Every triangle can tile the plane. Every four-sided shape can also tile the plane.

Things get interesting with pentagons. The regular pentagon cannot tile the plane. (A regular pentagon has equal side lengths and equal angles between sides, like, say, a cross section of okra, or, erm, the Pentagon). But some non-regular pentagons can.

The hunt to find and classify the pentagons that can tile the plane has been a century-long mathematical quest, begun by the German mathematician Karl Reinhardt, who in 1918 discovered five types of pentagon that do tile the plane…

Pentagons remain the area of most mathematical interest when it comes to tilings since it is the only of the ‘-gons’ that is not yet totally understood…

Read the whole story– and see all 15 types of pentagonal tilings discovered so far– at “Attack on the pentagon results in discovery of new mathematical tile.”

* Paul Rand

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As we grab the grout, we might recall that it was on this date in 1953, after a year of experimentation, that marine engineer and retired semi-pro baseball player David Mullany, Sr. invented the Wiffleball.  (He patented it early the following year.)  Watching his 13-year-old son play with a broomstick and a plastic golf ball ball in the confines of their backyard, Mullany worried that the effort to throw a curve would damage his young arm.  So he fabricated a full- (baseball-)sized ball from the plastic used in perfume packaging, with oblong holes on one side… a ball that would naturally curve.  The balls had the added advantage, given their light weight, that they’d not break windows.

David Jr. came up with the name: he was fond of saying that he had “whiffed” the batters that he struck out with his curves.  The “h” was dropped, the name trademarked, and (after Woolworth’s adopted the item) a generation of young ballplayers– and their parents– converted.

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Written by LW

August 14, 2015 at 1:01 am

“Everyone’s quick to blame the alien”*…

 

In August of 1977, volunteer astronomer Jerry Ehman reviewed readings from Ohio State’s Big Ear Radio Observatory (that’s a scan, above, of Ehman’s notations on the print-out he was assessing)…

He was sitting in his kitchen when he spotted a pattern that a couple of physicists had theorized 18 years earlier would signify alien chatter, according to Michael Brooks, the author of 13 Things That Don’t Make Sense . The printout read 6EQUJ5, a human way of cataloging the 72-second burst of sound registering at a frequency of 1420 MHz. The significance? E.T. may have phoned our home long before Spielberg set otherworldly hearts aglow with his 1982 film…

Scientists have rules out terrestrial sources– the signal came from “out there.”  So there are two possibilities: It was an actual alien communication, or Ehman stumbled across a previously undiscovered natural astrophysical phenomenon.  And as H. Paul Shuch (an engineer, radio astronomer, and executive director emeritus of the SETI League) observes, “either one would be worthy of a Nobel Prize, if only we knew which.”

Read the whole story at “The ‘Wow’ Signal. or That Time Jerry Ehman May Have Heard From Aliens.”

* Aeschylus, The Suppliant Maidens

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As we phone home, we might spare a thought for Martin Gardner; he died on this date in 2010.  Though not an academic, nor ever a formal student of math or science, he wrote widely and prolifically on both subjects in such popular books as The Ambidextrous Universe and The Relativity Explosion and as the “Mathematical Games” columnist for Scientific American.  Indeed, his elegant– and understandable– puzzles delighted professional and amateur readers alike, and helped inspire a generation of young mathematicians.

Gardner’s interests were wide; in addition to the math and science that were his power alley, he studied and wrote on topics that included magic, philosophy, religion, and literature (c.f., especially his work on Lewis Carroll– including the delightful Annotated Alice— and on G.K. Chesterton).  And he was a fierce debunker of pseudoscience: a founding member of CSICOP, and contributor of a monthly column (“Notes of a Fringe Watcher,” from 1983 to 2002) in Skeptical Inquirer, that organization’s monthly magazine.

 

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Written by LW

May 22, 2015 at 1:01 am

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