(Roughly) Daily

Posts Tagged ‘Mathematics

“The seen is the changing, the unseen is the unchanging”*…

Pharmacist’s Spatula, by William Toogood Ltd, English

We start 2021 with three big milestones for the Science Museum Group Collection.

100,000 incredible objects now have a photograph online, the online collection regularly receives 100,000 views each month and we’ve just recorded 3,000,000 visitors since launching the website in late 2016.

Each time you visit our online collection you can see more than ever before. Almost a quarter of the remarkable objects we care for (24.9% or 105,715 objects to be exact) have a photograph online, with hundreds of new photographs added each month as we digitise our vast collection.

You can explore photographs of artworks, tools and video games, or items from astronomy, firefighting and printing to give a few examples from the collection…

n the past we’ve released digital tools to help you explore the collection, including our Random Object Generator, Museum in a Tab (a Google Chrome extension) and What the machine saw (a machine learning experiment). You can even add our objects to the popular game Animal Crossing.

However, it can be difficult to spot recently photographed objects in the collection. So today we have published a new tool to help you explore these new items.

Never Been Seen shows objects from the Science Museum Group Collection that have never been seen online before. Each time you refresh this webpage an object with zero views is shown, making you the very first person to see it…

The spatula at the top of this post is no longer in that category, as your correspondent has seen (and now shared) it. But there’s so much more! Explore as yet unnoticed items in the collection of the Science Museum (London): “Never Been Seen.”

* Plato

###

As we uncover the unobserved, we might spare a thought for a man who saw much that had hitherto been unseen: Frank Plumpton Ramsey, a philosopher, mathematician, and economist who made major contributions to all three fields before his death (at the age of 26) on this date in 1930.

For more on Ramsey and his thought, see “One of the Great Intellects of His Time,” “The Man Who Thought Too Fast,” and Ramsey’s entry in the Stanford Encyclopedia of Philosophy.

source

Written by LW

January 19, 2021 at 1:01 am

“Mathematics is the art of giving the same name to different things”*…

Freeman Dyson was a translator: he turned physics into math, and those subjects into English for the general public.

Mathematics, like life, is complicated. But, for those who do mathematics, it is a source of joy. “The main thing is just astonishment that there’s such a rich world out there—a wonderful, abstract, very beautiful, simple world,” [John] Conway said. “It’s like Pizarro standing on the shores of the Pacific or whatever. . . . I can sit here in this chair and go on a voyage of exploration. A very different voyage of exploration, but, still, there are things to be discovered, things to be seen, that you can quite easily be the first person ever to see.”

So many of us now sit in our rooms, bound in space while time drips away. It can be a bit of a comfort to know that, as long as you are able to sit still and think, your creative spirit can be an engine of exploration. On their journeys, these playful, curious mathematicians discovered Monsters and numbers so large that they can hardly be written down. We’re grateful for the lively stories of their expeditions, and for the thinkers who led them. They’ll be missed…

John Conway, Ronald Graham, and Freeman Dyson all explored the world with their minds. Dan Rockmore (@dan_rockmore) celebrates “Three Mathematicians We Lost in 2020.”

Special bonus: an interview with an heir to Dyson– that’s to say, an important mathematician who’s also a gifted “translator”– Steven Strogatz.

* Henri Poincaré

###

As we share their amazement, we might we might spare a thought for Max Born; he died on this date in 1970.  A German physicist and Nobel Laureate, he coined the phrase “quantum mechanics” to describe the field in which he made his greatest contributions.  But beyond his accomplishments as a practitioner, he was a master teacher whose students included Enrico Fermi and Werner Heisenberg– both of whom became Nobel Laureates before their mentor– and  J. Robert Oppenheimer.

Less well-known is that Born, who died in 1970, was the grandfather of Australian phenom and definitive Sandy-portrayer Olivia Newton-John.

 source

“In the space between chaos and shape there was another chance”*…

Prince Hamlet spent a lot of time pondering the nature of chance and probability in William Shakespeare’s tragedy. In the famous “To be or not to be” speech, he notes that we helplessly face “the slings and arrows of outrageous fortune” — though a little earlier in the play he declares that “there’s a special providence in the fall of a sparrow,” suggesting that everything happens because God wills it to be so.

We can hardly fault the prince for holding two seemingly contradictory views about the nature of chance; after all, it is a puzzle that has vexed humankind through the ages. Why are we here? Or to give the question a slightly more modern spin, what sequence of events brought us here, and can we imagine a world in which we didn’t arrive on the scene at all?

It is to biologist Sean B. Carroll’s credit that he’s found a way of taking a puzzle that could easily fill volumes (and probably has filled volumes), and presenting it to us in a slim, non-technical, and fun little book, “A Series of Fortunate Events: Chance and the Making of the Planet, Life, and You.”

Carroll (not to be confused with physicist and writer Sean M. Carroll) gets the ball rolling with an introduction to the key concepts in probability and game theory, but quickly moves on to the issue at the heart of the book: the role of chance in evolution. Here we meet a key historical figure, the 20th-century French biochemist Jacques Monod, who won a Nobel Prize for his work on genetics. Monod understood that genetic mutations play a critical role in evolution, and he was struck by the random nature of those mutations…

Carroll quotes Monod: “Pure chance, absolutely free and blind, at the very root of the stupendous edifice of evolution: This central concept of modern biology is no longer one among other possible or even conceivable hypotheses. It is today the sole conceivable hypothesis, the only one that squares with observed and tested fact.”

“There is no scientific concept, in any of the sciences,” Monod concludes, “more destructive of anthropocentrism than this one.”

From there, it’s a short step to the realization that we humans might never have evolved in the first place…

Preview(opens in a new tab)

The profound impact of randomness in determining destiny: “The Power of Chance in Shaping Life and Evolution.”

See also: “Survival of the Luckiest.”

* Jeanette Winterson, The World and Other Places

###

As we blow on the dice, 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

source

“Those who wish to know the art of calculating, its subtleties and ingenuities, must know computing with hand figures”*…

Scholars at an Abbasid library. Maqamat of al-Hariri Illustration by Yahyá al-Wasiti, 1237 [source]

The House of Wisdom sounds a bit like make believe: no trace remains of this ancient library, destroyed in the 13th Century, so we cannot be sure exactly where it was located or what it looked like.

But this prestigious academy was in fact a major intellectual powerhouse in Baghdad during the Islamic Golden Age, and the birthplace of mathematical concepts as transformative as the common zero and our modern-day “Arabic” numerals.

Founded as a private collection for caliph Harun Al-Rashid in the late 8th Century then converted to a public academy some 30 years later, the House of Wisdom appears to have pulled scientists from all over the world towards Baghdad, drawn as they were by the city’s vibrant intellectual curiosity and freedom of expression (Muslim, Jewish and Christian scholars were all allowed to study there).

An archive as formidable in size as the present-day British Library in London or the Bibliothèque Nationale of Paris, the House of Wisdom eventually became an unrivalled centre for the study of humanities and sciences, including mathematics, astronomy, medicine, chemistry, geography, philosophy, literature and the arts – as well as some more dubious subjects such as alchemy and astrology.

To conjure this great monument thus requires a leap of imagination (think the Citadel in Westeros, or the library at Hogwarts), but one thing is certain: the academy ushered in a cultural Renaissance that would entirely alter the course of mathematics.

The House of Wisdom was destroyed in the Mongol Siege of Baghdad in 1258 (according to legend, so many manuscripts were tossed into the River Tigris that its waters turned black from ink), but the discoveries made there introduced a powerful, abstract mathematical language that would later be adopted by the Islamic empire, Europe, and ultimately, the entire world.

Tracing the House of Wisdom’s mathematical legacy involves a bit of time travel back to the future, as it were. For hundreds of years until the ebb of the Italian Renaissance, one name was synonymous with mathematics in Europe: Leonardo da Pisa, known posthumously as Fibonacci. Born in Pisa in 1170, the Italian mathematician received his primary instruction in Bugia, a trading enclave located on the Barbary coast of Africa (coastal North Africa). In his early 20s, Fibonacci traveled to the Middle East, captivated by ideas that had come west from India through Persia. When he returned to Italy, Fibonacci published Liber Abbaci, one of the first Western works to describe the Hindu-Arabic numeric system.

When Liber Abbaci first appeared in 1202, Hindu-Arabic numerals were known to only a few intellectuals; European tradesmen and scholars were still clinging to Roman numerals, which made multiplication and division extremely cumbersome (try multiplying MXCI by LVII!). Fibonacci’s book demonstrated numerals’ use in arithmetic operations – techniques which could be applied to practical problems like profit margin, money changing, weight conversion, barter and interest…

Fibonacci’s great genius was not just his creativity as a mathematician, however, but his keen understanding of the advantages known to Muslim scientists for centuries: their calculating formulas, their decimal place system, their algebra. In fact, Liber Abbaci relied almost exclusively on the algorithms of 9th-Century mathematician Al-Khwarizmi. His revolutionary treatise presented, for the first time, a systematic way of solving quadratic equations. Because of his discoveries in the field, Al-Khwarizmi is often referred to as the father of algebra – a word we owe to him, from the Arabic al-jabr, “the restoring of broken parts”—and in 821 he was appointed astronomer and head librarian of the House of Wisdom…

Centuries ago, a prestigious Islamic library (tragically burned in the the Siege of Baghdad) brought Arabic numerals to the world; its mathematical revolution changed our world: “How modern mathematics emerged from a lost Islamic library.”

For more on The House of Wisdom– and the sad stories of other libraries and archives that have been destroyed through the ages– see Richard Ovenden‘s remarkable new Burning the Books- a History of the Deliberate Destruction of Knowledge.

* Leonardo da Pisa, known posthumously as Fibonacci [see here]

###

As we count our blessings, we might spare a thought for John Pell; he died on this date in 1685.  An English mathematician, he is perhaps best remembered for having introduced the “division sign”– the “obelus,” a short line with dots above and below– into use in English.  It was first used in German by Johann Rahn in 1659 in Teutsche Algebra; Pell’s translation brought the symbol to English-speaking mathematicians.  But Pell was an important influence on Rahn, and edited his book– so may well have been, many scholars believe, the originator of the symbol for this use.  (In any case the symbol wasn’t new to them:  the obelus [derived from the word for “roasting spit” in Greek] had already been used to mark passages in writings that were considered dubious, corrupt or spurious…. a use that surely seems only too appropriate to legions of second and third grade math students.)

source

“If and when all the laws governing physical phenomena are finally discovered, and all the empirical constants occurring in these laws are finally expressed through the four independent basic constants, we will be able to say that physical science has reached its end”*…

The fine-structure constant was introduced in 1916 to quantify the tiny gap between two lines in the spectrum of colors emitted by certain atoms. The closely spaced frequencies are seen here through a Fabry-Pérot interferometer.

As fundamental constants go, the speed of light, c, enjoys all the fame, yet c’s numerical value says nothing about nature; it differs depending on whether it’s measured in meters per second or miles per hour. The fine-structure constant, by contrast, has no dimensions or units. It’s a pure number that shapes the universe to an astonishing degree — “a magic number that comes to us with no understanding,” as Richard Feynman described it. Paul Dirac considered the origin of the number “the most fundamental unsolved problem of physics.”

Numerically, the fine-structure constant, denoted by the Greek letter α (alpha), comes very close to the ratio 1/137. It commonly appears in formulas governing light and matter. “It’s like in architecture, there’s the golden ratio,” said Eric Cornell, a Nobel Prize-winning physicist at the University of Colorado, Boulder and the National Institute of Standards and Technology. “In the physics of low-energy matter — atoms, molecules, chemistry, biology — there’s always a ratio” of bigger things to smaller things, he said. “Those ratios tend to be powers of the fine-structure constant.”

The constant is everywhere because it characterizes the strength of the electromagnetic force affecting charged particles such as electrons and protons. “In our everyday world, everything is either gravity or electromagnetism. And that’s why alpha is so important,” said Holger Müller, a physicist at the University of California, Berkeley. Because 1/137 is small, electromagnetism is weak; as a consequence, charged particles form airy atoms whose electrons orbit at a distance and easily hop away, enabling chemical bonds. On the other hand, the constant is also just big enough: Physicists have argued that if it were something like 1/138, stars would not be able to create carbon, and life as we know it wouldn’t exist.

Physicists have more or less given up on a century-old obsession over where alpha’s particular value comes from; they now acknowledge that the fundamental constants could be random, decided in cosmic dice rolls during the universe’s birth. But a new goal has taken over.

Physicists want to measure the fine-structure constant as precisely as possible. Because it’s so ubiquitous, measuring it precisely allows them to test their theory of the interrelationships between elementary particles — the majestic set of equations known as the Standard Model of particle physics. Any discrepancy between ultra-precise measurements of related quantities could point to novel particles or effects not accounted for by the standard equations. Cornell calls these kinds of precision measurements a third way of experimentally discovering the fundamental workings of the universe, along with particle colliders and telescopes…

In a new paper in the journal Nature, a team of four physicists led by Saïda Guellati-Khélifa at the Kastler Brossel Laboratory in Paris reported the most precise measurement yet of the fine-structure constant. The team measured the constant’s value to the 11th decimal place, reporting that α = 1/137.03599920611. (The last two digits are uncertain.)

With a margin of error of just 81 parts per trillion, the new measurement is nearly three times more precise than the previous best measurement in 2018 by Müller’s group at Berkeley, the main competition. (Guellati-Khélifa made the most precise measurement before Müller’s in 2011.) Müller said of his rival’s new measurement of alpha, “A factor of three is a big deal. Let’s not be shy about calling this a big accomplishment”… largely ruling out some proposals for new particles

A team in Paris has made the most precise measurement yet of the fine-structure constant, killing hopes for a new force of nature: “Physicists Nail Down the ‘Magic Number’ That Shapes the Universe.”

[TotH to MK]

* George Gamow

###

As we ponder precision, we might spare a thought for Persian polymath Omar Khayyam; the mathematician, philosopher, astronomer, epigrammatist, and poet died on this date in 1131.  While he’s probably best known to English-speakers as a poet, via Edward FitzGerald’s famous translation of the quatrains that comprise the Rubaiyat of Omar Khayyam, Omar was one of the major mathematicians and astronomers of the medieval period.  He is the author of one of the most important works on algebra written before modern times, the Treatise on Demonstration of Problems of Algebra (which includes a geometric method for solving cubic equations by intersecting a hyperbola with a circle).  His astronomical observations contributed to the reform of the Persian calendar.  And he made important contributions to mechanics, geography, mineralogy, music, climatology, and Islamic theology.

 source

%d bloggers like this: