Tag Archives: mathematics

Idea Soup

The Birthday Problem

birthday_paradox

I first came across the Birthday Problem in my first few days of my first year of secondary school in London [that would be 6th grade for my US readers]. My mathematics teacher at the time realized the need to discuss abstract problems in concrete terms, especially statistics and probability. So, he wowed many of us — in a class of close to 30 kids — by firmly stating that there was a better than even chance that two of us shared the same birthday. In a class of 30, the actual probability is 60 percent, and rises to close to 100 percent is a group of only 60.

Startlingly, two in our class did indeed share the same birthday. How could that be possible, I wondered?

Well, the answer is grounded in the simple probability of large populations. But, it is also colored by our selective biases to remember “remarkable” coincidences and to ignore the much, much larger number of instances where there is no coincidence at all.

From the Washington Post.

Mathematician Joseph Mazur was in the back of a van snaking through the mountains of Sardinia when he heard one of his favorite coincidence stories. The driver, an Italian language teacher named Francesco, told of meeting a woman named Manuela who had come to study at his school. Francesco and Manuela met for the first time in a hotel lobby, and then went to have coffee.

They spoke for an hour, getting acquainted, before the uncomfortable truth came out. Noting Manuela’s nearly perfect Italian, Francesco finally asked why she decided to come to his school.

“She said, ‘Italian? What are you talk about? I’m not here to learn Italian,’” Mazur relates. “And then it dawned on both of them that she was the wrong Manuela and he was the wrong Francesco.” They returned to the hotel lobby where they had met to find a different Francesco offering a different Manuela a job she didn’t want or expect.

The tale is one of the many stories that populate Mazur’s new book, “Fluke,” in which he explores the probability of coincidences.

Read the entire article here.

Image: The computed probability of at least two people sharing a birthday versus the number of people. Courtesy: Rajkiran g / Wikipedia. CC BY-SA 3.0.

Idea Soup

When 8 Equals 16

commercial-standard-cs215-58

I’m sure that most, if not all, mathematicians would tell you that their calling is at the heart of our understanding of the universe. Mathematics describes our world precisely and logically. But, mix it with the world of women’s fashion and this rigorous discipline becomes rather squishy, and far from absolute. A case in point: a women’s size 16 today is equivalent to a women’s size 8 from 1958.

This makes me wonder what the fundamental measurements and equations describing our universe would look like if controlled by advertisers and marketers. Though, Einstein’s work on Special and General Relativity may seem to fit the fashion industry quite well: one of the central tenets of relativity holds that measurements of various quantities (read: dress size) are relative to the velocities (market size) of observers (retailers). In particular, space (dress size) contracts and time (waist size) dilates.

From the Washington Post:

Here are some numbers that illustrate the insanity of women’s clothing sizes: A size 8 dress today is nearly the equivalent of a size 16 dress in 1958. And a size 8 dress of 1958 doesn’t even have a modern-day equivalent — the waist and bust measurements of a Mad Men-era 8 come in smaller than today’s size 00.

These measurements come from official sizing standards once maintained by the National Bureau of Standards (now the National Institute of Standards and Technology) and taken over in recent years by the American Society of Testing and Materials. Data visualizer Max Galka recently unearthed them for a blog post on America’s obesity epidemic.

Centers for Disease Control and Prevention data show that the average American woman today weighs about as much as the average 1960s man. And while the weight story is pretty straightforward — Americans got heavier — the story behind the dress sizes is a little more complicated, as any woman who’s ever shopped for clothes could probably tell you.

As Julia Felsenthal detailed over at Slate, today’s women’s clothing sizes have their roots in a depression-era government project to define the “Average American Woman” by sending a pair of statisticians to survey and measure nearly 15,000 women. They “hoped to determine whether any proportional relationships existed among measurements that could be broadly applied to create a simple, standardized system of sizing,” Felsenthal writes.

Sadly, they failed. Not surprisingly, women’s bodies defied standardization. The project did yield one lasting contribution to women’s clothing: The statisticians were the first to propose the notion of arbitrary numerical sizes that weren’t based on any specific measurement — similar to shoe sizes.

The government didn’t return to the project until the late 1950s, when the National Bureau of Standards published “Body Measurements for the Sizing of Women’s Patterns and Apparel” in 1958. The standard was based on the 15,000 women interviewed previously, with the addition of a group of women who had been in the Army during World War II. The document’s purpose? “To provide the consumer with a means of identifying her body type and size from the wide range of body types covered, and enable her to be fitted properly by the same size regardless of price, type of apparel, or manufacturer of the garment.”

Read the entire article here.

Image: Diagram from “Body Measurements for the Sizing of Women’s Patterns and Apparel”, 1958. Courtesy of National Bureau of Standards /  National Institute of Standards and Technology (NIST).

BigBang

Emmy Noether, Mathematician

Emmy-NoetherMost non-mathematicians have probably heard of Euclid, Pythagoras, Poincaré, Gauss, Lagrange, de Fermat, and Hilbert,  to name but a few. All giants in their various mathematical specialties. But, I would hazard a wager that even most mathematicians have never heard of Noether. Probably because Emmy Noether is a woman.

Yet learning of her exploits in the early 20th century, I can see how far we still have to travel to truly recognize the contributions of women in academia and science — and everywhere else for that matter — as on a par with those of men. Women like Noether succeeded despite tremendous (male) pressure against them, which makes their achievements even more astonishing.

From ars technica:

By 1915, any list of the world’s greatest living mathematicians included the name David Hilbert. And though Hilbert previously devoted his career to logic and pure mathematics, he, like many other critical thinkers at the time, eventually became obsessed with a bit of theoretical physics.

With World War I raging on throughout Europe, Hilbert could be found sitting in his office at the great university at Göttingen trying and trying again to understand one idea—Einstein’s new theory of gravity.

Göttingen served as the center of mathematics for the Western world by this point, and Hilbert stood as one of its most notorious thinkers. He was a prominent leader for the minority of mathematicians who preferred a symbolic, axiomatic development in contrast to a more concrete style that emphasized the construction of particular solutions. Many of his peers recoiled from these modern methods, one even calling them “theology.” But Hilbert eventually won over most critics through the power and fruitfulness of his research.

For Hilbert, his rigorous approach to mathematics stood out quite a bit from the common practice of scientists, causing him some consternation. “Physics is much too hard for physicists,” he famously quipped. So wanting to know more, he invited Einstein to Göttingen to lecture about gravity for a week.

Before the year ended, both men would submit papers deriving the complete equations of general relativity. But naturally, the papers differed entirely when it came to their methods. When it came to Einstein’s theory, Hilbert and his Göttingen colleagues simply couldn’t wrap their minds around a peculiarity having to do with energy. All other physical theories—including electromagnetism, hydrodynamics, and the classical theory of gravity—obeyed local energy conservation. With Einstein’s theory, one of the many paradoxical consequences of this failure of energy conservation was that an object could speed up as it lost energy by emitting gravity waves, whereas clearly it should slow down.

Unable to make progress, Hilbert turned to the only person he believed might have the specialized knowledge and insight to help. This would-be-savior wasn’t even allowed to be a student at Göttingen once upon a time, but Hilbert had long become a fan of this mathematician’s highly “abstract” approach (which Hilbert considered similar to his own style). He managed to recruit this soon-to-be partner to Göttingen about the same time Einstein showed up.

And that’s when a woman—one Emmy Noether—created what may be the most important single theoretical result in modern physics.

 …

During Noether’s stay at Göttingen, Hilbert contrived a way to allow her to lecture unofficially. He repeatedly attempted to get her hired as a Privatdozent, or an officially recognized lecturer. The science and mathematics faculty was generally in favor of this, but Hilbert could not overcome the resistance of the humanities professors, who simply could not stomach the idea of a female teacher. At one meeting of the faculty senate, frustrated again in his attempts to get Noether a job, he famously remarked, “I do not see that the sex of a candidate is an argument against her admission as Privatdozent. After all, we are a university, not a bathing establishment.”

Social barriers aside, Noether immediately grasped the problem with Einstein’s theory. Over the course of three years, she not only solved it, but in doing so she proved a theorem that simultaneously reached back to the dawn of physics and pushed forward to the physics of today. Noether’s Theorem, as it is now called, lies at the heart of modern physics, unifying everything from the orbits of planets to the theories of elementary particles.

Read the entire story here.

Image: Emmy Noether (1882-1935). Public domain.

 

BigBang

A Godless Universe: Mind or Mathematics

In his science column for the NYT George Johnson reviews several recent books by noted thinkers who for different reasons believe science needs to expand its borders. Philosopher Thomas Nagel and physicist Max Tegmark both agree that our current understanding of the universe is rather limited and that science needs to turn to new or alternate explanations. Nagel, still an atheist, suggests in his book Mind and Cosmos that the mind somehow needs to be considered a fundamental structure of the universe. While Tegmark in his book Our Mathematical Universe: My Quest for the Ultimate Nature of Reality suggests that mathematics is the core, irreducible framework of the cosmos. Two radically different ideas — yet both are correct in one respect: we still know so very little about ourselves and our surroundings.

From the NYT:

Though he probably didn’t intend anything so jarring, Nicolaus Copernicus, in a 16th-century treatise, gave rise to the idea that human beings do not occupy a special place in the heavens. Nearly 500 years after replacing the Earth with the sun as the center of the cosmic swirl, we’ve come to see ourselves as just another species on a planet orbiting a star in the boondocks of a galaxy in the universe we call home. And this may be just one of many universes — what cosmologists, some more skeptically than others, have named the multiverse.

Despite the long string of demotions, we remain confident, out here on the edge of nowhere, that our band of primates has what it takes to figure out the cosmos — what the writer Timothy Ferris called “the whole shebang.” New particles may yet be discovered, and even new laws. But it is almost taken for granted that everything from physics to biology, including the mind, ultimately comes down to four fundamental concepts: matter and energy interacting in an arena of space and time.

There are skeptics who suspect we may be missing a crucial piece of the puzzle. Recently, I’ve been struck by two books exploring that possibility in very different ways. There is no reason why, in this particular century, Homo sapiens should have gathered all the pieces needed for a theory of everything. In displacing humanity from a privileged position, the Copernican principle applies not just to where we are in space but to when we are in time.

Since it was published in 2012, “Mind and Cosmos,” by the philosopher Thomas Nagel, is the book that has caused the most consternation. With his taunting subtitle — “Why the Materialist Neo-Darwinian Conception of Nature Is Almost Certainly False” — Dr. Nagel was rejecting the idea that there was nothing more to the universe than matter and physical forces. He also doubted that the laws of evolution, as currently conceived, could have produced something as remarkable as sentient life. That idea borders on anathema, and the book quickly met with a blistering counterattack. Steven Pinker, a Harvard psychologist, denounced it as “the shoddy reasoning of a once-great thinker.”

What makes “Mind and Cosmos” worth reading is that Dr. Nagel is an atheist, who rejects the creationist idea of an intelligent designer. The answers, he believes, may still be found through science, but only by expanding it further than it may be willing to go.

“Humans are addicted to the hope for a final reckoning,” he wrote, “but intellectual humility requires that we resist the temptation to assume that the tools of the kind we now have are in principle sufficient to understand the universe as a whole.”

Dr. Nagel finds it astonishing that the human brain — this biological organ that evolved on the third rock from the sun — has developed a science and a mathematics so in tune with the cosmos that it can predict and explain so many things.

Neuroscientists assume that these mental powers somehow emerge from the electrical signaling of neurons — the circuitry of the brain. But no one has come close to explaining how that occurs.

Continue reading the main story Continue reading the main story
Continue reading the main story

That, Dr. Nagel proposes, might require another revolution: showing that mind, along with matter and energy, is “a fundamental principle of nature” — and that we live in a universe primed “to generate beings capable of comprehending it.” Rather than being a blind series of random mutations and adaptations, evolution would have a direction, maybe even a purpose.

“Above all,” he wrote, “I would like to extend the boundaries of what is not regarded as unthinkable, in light of how little we really understand about the world.”

Dr. Nagel is not alone in entertaining such ideas. While rejecting anything mystical, the biologist Stuart Kauffman has suggested that Darwinian theory must somehow be expanded to explain the emergence of complex, intelligent creatures. And David J. Chalmers, a philosopher, has called on scientists to seriously consider “panpsychism” — the idea that some kind of consciousness, however rudimentary, pervades the stuff of the universe.

Some of this is a matter of scientific taste. It can be just as exhilarating, as Stephen Jay Gould proposed in “Wonderful Life,” to consider the conscious mind as simply a fluke, no more inevitable than the human appendix or a starfish’s five legs. But it doesn’t seem so crazy to consider alternate explanations.

Heading off in another direction, a new book by the physicist Max Tegmark suggests that a different ingredient — mathematics — needs to be admitted into science as one of nature’s irreducible parts. In fact, he believes, it may be the most fundamental of all.

In a well-known 1960 essay, the physicist Eugene Wigner marveled at “the unreasonable effectiveness of mathematics” in explaining the world. It is “something bordering on the mysterious,” he wrote, for which “there is no rational explanation.”

The best he could offer was that mathematics is “a wonderful gift which we neither understand nor deserve.”

Dr. Tegmark, in his new book, “Our Mathematical Universe: My Quest for the Ultimate Nature of Reality,” turns the idea on its head: The reason mathematics serves as such a forceful tool is that the universe is a mathematical structure. Going beyond Pythagoras and Plato, he sets out to show how matter, energy, space and time might emerge from numbers.

Read the entire article here.

Technica

I Think, Therefore I am, Not Robot

Robbie_the_Robot_2006

A sentient robot is the long-held dream of both artificial intelligence researcher and science fiction author. Yet, some leading mathematicians theorize it may never happen, despite our accelerating technological prowess.

From New Scientist:

So long, robot pals – and robot overlords. Sentient machines may never exist, according to a variation on a leading mathematical model of how our brains create consciousness.

Over the past decade, Giulio Tononi at the University of Wisconsin-Madison and his colleagues have developed a mathematical framework for consciousness that has become one of the most influential theories in the field. According to their model, the ability to integrate information is a key property of consciousness. They argue that in conscious minds, integrated information cannot be reduced into smaller components. For instance, when a human perceives a red triangle, the brain cannot register the object as a colourless triangle plus a shapeless patch of red.

But there is a catch, argues Phil Maguire at the National University of Ireland in Maynooth. He points to a computational device called the XOR logic gate, which involves two inputs, A and B. The output of the gate is “0” if A and B are the same and “1” if A and B are different. In this scenario, it is impossible to predict the output based on A or B alone – you need both.

Memory edit

Crucially, this type of integration requires loss of information, says Maguire: “You have put in two bits, and you get one out. If the brain integrated information in this fashion, it would have to be continuously haemorrhaging information.”

Maguire and his colleagues say the brain is unlikely to do this, because repeated retrieval of memories would eventually destroy them. Instead, they define integration in terms of how difficult information is to edit.

Consider an album of digital photographs. The pictures are compiled but not integrated, so deleting or modifying individual images is easy. But when we create memories, we integrate those snapshots of information into our bank of earlier memories. This makes it extremely difficult to selectively edit out one scene from the “album” in our brain.

Based on this definition, Maguire and his team have shown mathematically that computers can’t handle any process that integrates information completely. If you accept that consciousness is based on total integration, then computers can’t be conscious.

Open minds

“It means that you would not be able to achieve the same results in finite time, using finite memory, using a physical machine,” says Maguire. “It doesn’t necessarily mean that there is some magic going on in the brain that involves some forces that can’t be explained physically. It is just so complex that it’s beyond our abilities to reverse it and decompose it.”

Disappointed? Take comfort – we may not get Rosie the robot maid, but equally we won’t have to worry about the world-conquering Agents of The Matrix.

Neuroscientist Anil Seth at the University of Sussex, UK, applauds the team for exploring consciousness mathematically. But he is not convinced that brains do not lose information. “Brains are open systems with a continual turnover of physical and informational components,” he says. “Not many neuroscientists would claim that conscious contents require lossless memory.”

Read the entire story here.

Image: Robbie the Robot, Forbidden Planet. Courtesy of San Diego Comic Con, 2006 / Wikipedia.

Arts and Letters, Idea Soup

Beauty Of and In Numbers

golden-ratio

Mathematics seems to explain and underlie much of our modern world: manufacturing, exploration, transportation, logistics, healthcare, technology — all depend on numbers in one form or another. So it should come as no surprise that there are mathematical formulae that describe our notions of beauty. This would seem to be counter-intuitive since beauty is a very subjective experience for all of us — one person’s colorful, blotted mess is another’s Jackson Pollock masterpiece. Yet, mathematicians have known for some time that there is a certain composition of sizes that are more frequently characterized as beautiful than others. Known as the golden ratio, architects, designers and artists have long exploited this mathematical formula to render their works more appealing.

From Wired:

Mathematical concepts can be difficult to grasp, but given the right context they can help explain some of the world’s biggest mysteries. For instance, what is it about a sunflower that makes it so pleasing to look at? Or why do I find the cereal box-shaped United Nations building in New York City to be so captivating?

Beauty may very well be subjective, but there’s thought to be mathematical reasoning behind why we’re attracted to certain shapes and objects. Called the golden ratio, this theory states there’s a recurring proportion of arrangement that lends certain things their beauty. Represented as an equation: a/b = (a+b)/a, the golden ratio is all around us—conical sea shells, human faces, flower petals, buildings—we just don’t always know we’re looking at it. In Golden Meaning, a new book from London publisher GraphicDesign&, 55 designers aim to demystify the golden ratio using clever illustrations and smart graphic design.

GraphicDesign& founders Lucienne Roberts and Rebecca Wright partnered up with math evangelist Alex Bellos to develop the book, with the main goal of making math accessible through design. “We want this to be a useful tool to demonstrate something that often makes people anxious,” explains Roberts. “We hope it’s as interesting to people who are interested in math as it is to the people who are interested in the visual.”

Each designer came at the problem from a different angle, but in order to appreciate the cleverness found in the book, it’s important to have a little background on the golden mean. Bellos uses this line to illustrate the concept at its most basic.

n Golden Meaning he writes: “The line is separated into two sections in such a way that the ratio of the whole line to the larger section is equal to the ratio of the larger section to the smaller section.” This ratio ends up being 1.618.

Salvador Dali and Le Corbusier have used the golden mean as a guiding principle in their work, the Taj Mahal was designed with it in mind, and it’s thought that many of the faces of attractive people follow these proportions. The golden ratio then is essentially a formula for beauty.

With this in mind, Robert and Wright gave designers a simple brief: To explore, explain and communicate the golden ratio however they see fit. There’s a recipe for golden bars that requires bakers to parcel out ingredients based the ratio instead of exact measurements, an illustration that shows a bottle of wine being poured into glasses using the ratio. The book itself is actually a golden rectangle. “You get it much more than looking at an equation,” says Roberts.

A particular favorite shows two side-by-side images of British designer Oli Kellett. On the left is his normal face, on the right is the same face after he rearranged his features in accordance to the golden ratio. So is he really more beautiful after his mathematical surgery? “We liked him as he is,” says Roberts. “In a way it disproves the theory.”

Read the entire article here.

Image: Several examples of the golden ratio at work, from the book Golden Meaning by GraphicDesign&. Courtesy of GraphicDesign&.

BigBang

The Universe of Numbers

There is no doubt that mathematics — some very complex — has been able to explain much of what we consider the universe. In reality, and perhaps surprisingly, only a small subset of equations is required to explain everything around us from the atoms and their constituents to the vast cosmos. Why is that? And, what is the fundamental relationship between mathematics and our current physical understanding of all things great and small?

From the New Scientist:

When Albert Einstein finally completed his general theory of relativity in 1916, he looked down at the equations and discovered an unexpected message: the universe is expanding.

Einstein didn’t believe the physical universe could shrink or grow, so he ignored what the equations were telling him. Thirteen years later, Edwin Hubble found clear evidence of the universe’s expansion. Einstein had missed the opportunity to make the most dramatic scientific prediction in history.

How did Einstein’s equations “know” that the universe was expanding when he did not? If mathematics is nothing more than a language we use to describe the world, an invention of the human brain, how can it possibly churn out anything beyond what we put in? “It is difficult to avoid the impression that a miracle confronts us here,” wrote physicist Eugene Wigner in his classic 1960 paper “The unreasonable effectiveness of mathematics in the natural sciences” (Communications on Pure and Applied Mathematics, vol 13, p 1).

The prescience of mathematics seems no less miraculous today. At the Large Hadron Collider at CERN, near Geneva, Switzerland, physicists recently observed the fingerprints of a particle that was arguably discovered 48 years ago lurking in the equations of particle physics.

How is it possible that mathematics “knows” about Higgs particles or any other feature of physical reality? “Maybe it’s because math is reality,” says physicist Brian Greene of Columbia University, New York. Perhaps if we dig deep enough, we would find that physical objects like tables and chairs are ultimately not made of particles or strings, but of numbers.

“These are very difficult issues,” says philosopher of science James Ladyman of the University of Bristol, UK, “but it might be less misleading to say that the universe is made of maths than to say it is made of matter.”

Difficult indeed. What does it mean to say that the universe is “made of mathematics”? An obvious starting point is to ask what mathematics is made of. The late physicist John Wheeler said that the “basis of all mathematics is 0 = 0”. All mathematical structures can be derived from something called “the empty set”, the set that contains no elements. Say this set corresponds to zero; you can then define the number 1 as the set that contains only the empty set, 2 as the set containing the sets corresponding to 0 and 1, and so on. Keep nesting the nothingness like invisible Russian dolls and eventually all of mathematics appears. Mathematician Ian Stewart of the University of Warwick, UK, calls this “the dreadful secret of mathematics: it’s all based on nothing” (New Scientist, 19 November 2011, p 44). Reality may come down to mathematics, but mathematics comes down to nothing at all.

That may be the ultimate clue to existence – after all, a universe made of nothing doesn’t require an explanation. Indeed, mathematical structures don’t seem to require a physical origin at all. “A dodecahedron was never created,” says Max Tegmark of the Massachusetts Institute of Technology. “To be created, something first has to not exist in space or time and then exist.” A dodecahedron doesn’t exist in space or time at all, he says – it exists independently of them. “Space and time themselves are contained within larger mathematical structures,” he adds. These structures just exist; they can’t be created or destroyed.

That raises a big question: why is the universe only made of some of the available mathematics? “There’s a lot of math out there,” Greene says. “Today only a tiny sliver of it has a realisation in the physical world. Pull any math book off the shelf and most of the equations in it don’t correspond to any physical object or physical process.”

It is true that seemingly arcane and unphysical mathematics does, sometimes, turn out to correspond to the real world. Imaginary numbers, for instance, were once considered totally deserving of their name, but are now used to describe the behaviour of elementary particles; non-Euclidean geometry eventually showed up as gravity. Even so, these phenomena represent a tiny slice of all the mathematics out there.

Not so fast, says Tegmark. “I believe that physical existence and mathematical existence are the same, so any structure that exists mathematically is also real,” he says.

So what about the mathematics our universe doesn’t use? “Other mathematical structures correspond to other universes,” Tegmark says. He calls this the “level 4 multiverse”, and it is far stranger than the multiverses that cosmologists often discuss. Their common-or-garden multiverses are governed by the same basic mathematical rules as our universe, but Tegmark’s level 4 multiverse operates with completely different mathematics.

All of this sounds bizarre, but the hypothesis that physical reality is fundamentally mathematical has passed every test. “If physics hits a roadblock at which point it turns out that it’s impossible to proceed, we might find that nature can’t be captured mathematically,” Tegmark says. “But it’s really remarkable that that hasn’t happened. Galileo said that the book of nature was written in the language of mathematics – and that was 400 years ago.”

Read the entire article here.

Idea Soup, tD

Elite Mediocrity

Yet another survey of global education attainment puts the United States firmly in yet another unenviable position. US students ranked a mere 28th in science and came further down the scale on math, at 36th, out of 65 nations. So, it’s time for another well-earned attack on the system that is increasingly nurturing mainstream mediocrity and dumbing-down education to mush. In fact, some nameless states seem to celebrate the fact by re-working textbooks and curricula to ensure historic fact and scientific principles are distorted to promote a religious agenda. And, for those who point to the US as a guiding light in all things innovative, please don’t forget that a significant proportion of the innovators gained their educational credentials elsewhere, outside the US.

As the news Comedy Central faux-news anchor and satirist Stephen Colbert recently put it:

“Like all great theologies, Bill [O’Reilly]’s can be boiled down to one sentence: there must be a God, because I don’t know how things work.”

From the Huffington Post:

The 2012 Programme for International Student Assessment, or PISA, results are in, and there’s some really good news for those that worry about the U.S. becoming a nation of brainy elitists. Of the 65 countries that participated in the PISA assessment, U.S. students ranked 36th in math, and 28th in science. When it comes to elitism, the U.S. truly has nothing to worry about.

For those relative few Americans who were already elite back when the 2009 PISA assessment was conducted, there’s good news for them too: they’re even more elite than they were in 2009, when the US ranked 30th in math and 23rd in science. Educated Americans are so elite, they’re practically an endangered species.

The only nagging possible shred of bad news from these test scores comes in the form of a question: where will the next Internet come from? Which country will deliver the next great big, landscape-changing, technological innovation that will propel its economy upward? The country of bold, transformative firsts, the one that created the world’s first nuclear reactor and landed humans on the moon seems very different than the one we live in today.

Mediocrity in science education has metastasized throughout the American mindset, dumbing down everything in its path, including the choices made by our elected officials. A stinging byproduct of America’s war on excellence in science education was the loss of its leadership position in particle physics research. On March 14 of this year, CERN, the European Organization for Nuclear Research, announced that the Higgs Boson, aka the “God particle,” had been discovered at the EU’s Large Hadron Collider. CERN describes itself as “the world’s leading laboratory for particle physics” — a title previously held by America’s Fermilab. Fermilab’s Tevatron particle accelerator was the world’s largest and most powerful until eclipsed by CERN’s Large Hadron Collider. The Tevatron was shut down on September 30th, 2011.

The Tevatron’s planned replacement, Texas’ Superconducting Super Collider (SSC), would have been three times the size of the EU’s Large Hadron Collider. Over one third of the SSC’s underground tunnel had been bored at the time of its cancellation by congress in 1993. As Texas Monthly reported in “How Texas Lost the World’s Largest Super Collider,” “Nobody doubts that the 40 TeV Superconducting Super Collider (SSC) in Texas would have discovered the Higgs boson a decade before CERN.” Fighting to save the SSC in 1993, its director, Dr. Roy Schwitters, said in a New York Times interview, “The SSC is becoming a victim of the revenge of the C students.”

Ever wonder about the practical benefits of theoretical physics? Consider this: without Einstein’s theory of general relativity, GPS doesn’t work. That’s because time in those GPS satellites whizzing above us in space is slightly different than time for us terrestrials. Without compensating for the difference, our cars would end up in a ditch instead of Starbucks. GPS would also not have happened without advances in US space technology. Consider that, in 2013, there are two manned spacefaring nations on Earth – the US isn’t one of them. GPS alone is estimated to generate $122.4 billion annually in direct and related benefits according to an NDP Consulting Group report. The Superconducting Super Collider would have cost $8.4 billion.

‘C’ students’ revenge doesn’t stop with crushing super colliders or grounding our space program. Fox News’ Bill O’Reilly famously translated his inability to explain 9th grade astronomy into justification for teaching creationism in public schools, stating that we don’t know how tides work, or where the sun or moon comes from, or why the Earth has a moon and Mars doesn’t (Mars actually has two moons).

Read the entire article here.

Idea Soup

Linguistic Vectors

Our friends at Google have transformed the challenge of language translation from one of linguistics to mathematics.

By mapping parts of the linguistic structure of one language in the form of vectors in a mathematical space and comparing those to the structure of a few similar words in another they have condensed the effort to equations. Their early results of an English to Spanish translation seem very promising. (Now, if they could only address human conflict, aging and death.)

Visit arXiv for a pre-print of their research.

From Technology Review:

Computer science is changing the nature of the translation of words and sentences from one language to another. Anybody who has tried BabelFish or Google Translate will know that they provide useful translation services but ones that are far from perfect.

The basic idea is to compare a corpus of words in one language with the same corpus of words translated into another. Words and phrases that share similar statistical properties are considered equivalent.

The problem, of course, is that the initial translations rely on dictionaries that have to be compiled by human experts and this takes significant time and effort.

Now Tomas Mikolov and a couple of pals at Google in Mountain View have developed a technique that automatically generates dictionaries and phrase tables that convert one language into another.

The new technique does not rely on versions of the same document in different languages. Instead, it uses data mining techniques to model the structure of a single language and then compares this to the structure of another language.

“This method makes little assumption about the languages, so it can be used to extend and re?ne dictionaries and translation tables for any language pairs,” they say.

The new approach is relatively straightforward. It relies on the notion that every language must describe a similar set of ideas, so the words that do this must also be similar. For example, most languages will have words for common animals such as cat, dog, cow and so on. And these words are probably used in the same way in sentences such as “a cat is an animal that is smaller than a dog.”

The same is true of numbers. The image above shows the vector representations of the numbers one to five in English and Spanish and demonstrates how similar they are.

This is an important clue. The new trick is to represent an entire language using the relationship between its words. The set of all the relationships, the so-called “language space”, can be thought of as a set of vectors that each point from one word to another. And in recent years, linguists have discovered that it is possible to handle these vectors mathematically. For example, the operation ‘king’ – ‘man’ + ‘woman’ results in a vector that is similar to ‘queen’.

It turns out that different languages share many similarities in this vector space. That means the process of converting one language into another is equivalent to finding the transformation that converts one vector space into the other.

This turns the problem of translation from one of linguistics into one of mathematics. So the problem for the Google team is to find a way of accurately mapping one vector space onto the other. For this they use a small bilingual dictionary compiled by human experts–comparing same corpus of words in two different languages gives them a ready-made linear transformation that does the trick.

Having identified this mapping, it is then a simple matter to apply it to the bigger language spaces. Mikolov and co say it works remarkably well. “Despite its simplicity, our method is surprisingly effective: we can achieve almost 90% precision@5 for translation of words between English and Spanish,” they say.

The method can be used to extend and refine existing dictionaries, and even to spot mistakes in them. Indeed, the Google team do exactly that with an English-Czech dictionary, finding numerous mistakes.

Finally, the team point out that since the technique makes few assumptions about the languages themselves, it can be used on argots that are entirely unrelated. So while Spanish and English have a common Indo-European history, Mikolov and co show that the new technique also works just as well for pairs of languages that are less closely related, such as English and Vietnamese.

Read the entire article here.

BigBang, tD

Big Data Versus Talking Heads

With the election in the United States now decided, the dissection of the result is well underway. And, perhaps the biggest winner of all is the science of big data. Yes, mathematical analysis of vast quantities of demographic and polling data won over the voodoo proclamations and gut felt predictions of the punditocracy. Now, that’s a result truly worth celebrating.

[div class=attrib]From ReadWriteWeb:[end-div]

Political pundits, mostly Republican, went into a frenzy when Nate Silver, a New York Times pollster and stats blogger, predicted that Barack Obama would win reelection.

But Silver was right and the pundits were wrong – and the impact of this goes way beyond politics.

Silver won because, um, science. As ReadWrite’s own Dan Rowinski noted,  Silver’s methodology is all based on data. He “takes deep data sets and applies logical analytical methods” to them. It’s all just numbers.

Silver runs a blog called FiveThirtyEight, which is licensed by the Times. In 2008 he called the presidential election with incredible accuracy, getting 49 out of 50 states right. But this year he rolled a perfect score, 50 out of 50, even nailing the margins in many cases. His uncanny accuracy on this year’s election represents what Rowinski calls a victory of “logic over punditry.”

In fact it’s bigger than that. Bear in mind that before turning his attention to politics in 2007 and 2008, Silver was using computer models to make predictions about baseball. What does it mean when some punk kid baseball nerd can just wade into politics and start kicking butt on all these long-time “experts” who have spent their entire lives covering politics?

It means something big is happening.

Man Versus Machine

This is about the triumph of machines and software over gut instinct.

The age of voodoo is over. The era of talking about something as a “dark art” is done. In a world with big computers and big data, there are no dark arts.

And thank God for that. One by one, computers and the people who know how to use them are knocking off these crazy notions about gut instinct and intuition that humans like to cling to. For far too long we’ve applied this kind of fuzzy thinking to everything, from silly stuff like sports to important stuff like medicine.

Someday, and I hope it’s soon, we will enter the age of intelligent machines, when true artificial intellgence becomes a reality, and when we look back on the late 20th and early 21st century it will seem medieval in its simplicity and reliance on superstition.

What most amazes me is the backlash and freak-out that occurs every time some “dark art” gets knocked over in a particular domain. Watch Moneyball (or read the book) and you’ll see the old guard (in that case, baseball scouts) grow furious as they realize that computers can do their job better than they can. (Of course it’s not computers; it’s people who know how to use computers.)

We saw the same thing when IBM’s Deep Blue defeated Garry Kasparov in 1997. We saw it when Watson beat humans at Jeopardy.

It’s happening in advertising, which used to be a dark art but is increasingly a computer-driven numbers game. It’s also happening in my business, the news media, prompting the same kind of furor as happened with the baseball scouts in Moneyball.

[div class=attrib]Read the entire article following the jump.[end-div]

[div class=attrib]Political pundits, Left to right: Mark Halperin, David Brooks, Jon Stewart, Tim Russert, Matt Drudge, John Harris & Jim VandeHei, Rush Limbaugh, Sean Hannity, Chris Matthews, Karl Rove. Courtesy of Telegraph.[end-div]

BigBang

As Simple as abc; As Difficult as ABC

As children we all learn our abc’s; as adults very few ponder the ABC Conjecture in mathematics. The first is often a simple task of rote memorization; the second is a troublesome mathematical problem with a fiendishly complex solution (maybe).

[div class=attrib]From the New Scientist:[end-div]

?Whole numbers, addition and multiplication are among the first things schoolchildren learn, but a new mathematical proof shows that even the world’s best minds have plenty more to learn about these seemingly simple concepts.

Shinichi Mochizuki of Kyoto University in Japan has torn up these most basic of mathematical concepts and reconstructed them as never before. The result is a fiendishly complicated proof for the decades-old “ABC conjecture” – and an alternative mathematical universe that should prise open many other outstanding enigmas.

To boot, Mochizuki’s proof also offers an alternative explanation for Fermat’s last theorem, one of the most famous results in the history of mathematics but not proven until 1993 (see “Fermat’s last theorem made easy”, below).

The ABC conjecture starts with the most basic equation in algebra, adding two whole numbers, or integers, to get another: a + b = c. First posed in 1985 by Joseph Oesterlé and David Masser, it places constraints on the interactions of the prime factors of these numbers, primes being the indivisible building blocks that can be multiplied together to produce all integers.

Dense logic

Take 81 + 64 = 145, which breaks down into the prime building blocks 3 × 3 × 3 × 3 + 2 × 2 × 2 × 2 × 2 × 2 = 5 × 29. Simplified, the conjecture says that the large amount of smaller primes on the equation’s left-hand side is always balanced by a small amount of larger primes on the right – the addition restricts the multiplication, and vice versa.

“The ABC conjecture in some sense exposes the relationship between addition and multiplication,” says Jordan Ellenberg of the University of Wisconsin-Madison. “To learn something really new about them at this late date is quite startling.”

Though rumours of Mochizuki’s proof started spreading on mathematics blogs earlier this year, it was only last week that he posted a series of papers on his website detailing what he calls “inter-universal geometry”, one of which claims to prove the ABC conjecture. Only now are mathematicians attempting to decipher its dense logic, which spreads over 500 pages.

So far the responses are cautious, but positive. “It will be fabulously exciting if it pans out, experience suggests that that’s quite a big ‘if’,” wrote University of Cambridge mathematician Timothy Gowers on Google+.

Alien reasoning

“It is going to be a while before people have a clear idea of what Mochizuki has done,” Ellenberg told New Scientist. “Looking at it, you feel a bit like you might be reading a paper from the future, or from outer space,” he added on his blog.

Mochizuki’s reasoning is alien even to other mathematicians because it probes deep philosophical questions about the foundations of mathematics, such as what we really mean by a number, says Minhyong Kim at the University of Oxford. The early 20th century saw a crisis emerge as mathematicians realised they actually had no formal way to define a number – we can talk about “three apples” or “three squares”, but what exactly is the mathematical object we call “three”? No one could say.

Eventually numbers were redefined in terms of sets, rigorously specified collections of objects, and mathematicians now know that the true essence of the number zero is a set which contains no objects – the empty set – while the number one is a set which contains one empty set. From there, it is possible to derive the rest of the integers.

But this was not the end of the story, says Kim. “People are aware that many natural mathematical constructions might not really fall into the universe of sets.”

Terrible deformation

Rather than using sets, Mochizuki has figured out how to translate fundamental mathematical ideas into objects that only exist in new, conceptual universes. This allowed him to “deform” basic whole numbers and push their innate relationships – such as multiplication and addition – to the limit. “He is literally taking apart conventional objects in terrible ways and reconstructing them in new universes,” says Kim.

These new insights led him to a proof of the ABC conjecture. “How he manages to come back to the usual universe in a way that yields concrete consequences for number theory, I really have no idea as yet,” says Kim.

Because of its fundamental nature, a verified proof of ABC would set off a chain reaction, in one swoop proving many other open problems and deepening our understanding of the relationships between integers, fractions, decimals, primes and more.

Ellenberg compares proving the conjecture to the discovery of the Higgs boson, which particle physicists hope will reveal a path to new physics. But while the Higgs emerged from the particle detritus of a machine specifically designed to find it, Mochizuki’s methods are completely unexpected, providing new tools for mathematical exploration.

[div class=attrib]Read the entire article after the jump.[end-div]

[div class=attrib]Image courtesy of Clare College Cambridge.[end-div]

BigBang

A Most Beautiful Equation

Many mathematicians and those not mathematically oriented would consider Albert Einstein’s equation stating energy=mass equivalence to be singularly simple and beautiful. Indeed, e=mc2 is perhaps one of the few equations to have entered the general public consciousness. However, there are a number of other less well known mathematical constructs that convey this level of significance and fundamental beauty as well. Wired lists several to consider.

[div class=attrib]From Wired:[end-div]

Even for those of us who finished high school algebra on a wing and a prayer, there’s something compelling about equations. The world’s complexities and uncertainties are distilled and set in orderly figures, with a handful of characters sufficing to capture the universe itself.

For your enjoyment, the Wired Science team has gathered nine of our favorite equations. Some represent the universe; others, the nature of life. One represents the limit of equations.

We do advise, however, against getting any of these equations tattooed on your body, much less branded. An equation t-shirt would do just fine.

The Beautiful Equation: Euler’s Identity

ei? + 1 = 0

Also called Euler’s relation, or the Euler equation of complex analysis, this bit of mathematics enjoys accolades across geeky disciplines.

Swiss mathematician Leonhard Euler first wrote the equality, which links together geometry, algebra, and five of the most essential symbols in math — 0, 1, i, pi and e — that are essential tools in scientific work.

Theoretical physicist Richard Feynman was a huge fan and called it a “jewel” and a “remarkable” formula. Fans today refer to it as “the most beautiful equation.”

[div class=attrib]Read more here.[end-div]

[div class=attrib]Image: Euler’s Relation. Courtesy of Wired.[end-div]

Idea Soup

Ultimate logic: To infinity and beyond

[div class=attrib]From the New Scientist:[end-div]

WHEN David Hilbert left the podium at the Sorbonne in Paris, France, on 8 August 1900, few of the assembled delegates seemed overly impressed. According to one contemporary report, the discussion following his address to the second International Congress of Mathematicians was “rather desultory”. Passions seem to have been more inflamed by a subsequent debate on whether Esperanto should be adopted as mathematics’ working language.

Yet Hilbert’s address set the mathematical agenda for the 20th century. It crystallised into a list of 23 crucial unanswered questions, including how to pack spheres to make best use of the available space, and whether the Riemann hypothesis, which concerns how the prime numbers are distributed, is true.

Today many of these problems have been resolved, sphere-packing among them. Others, such as the Riemann hypothesis, have seen little or no progress. But the first item on Hilbert’s list stands out for the sheer oddness of the answer supplied by generations of mathematicians since: that mathematics is simply not equipped to provide an answer.

This curiously intractable riddle is known as the continuum hypothesis, and it concerns that most enigmatic quantity, infinity. Now, 140 years after the problem was formulated, a respected US mathematician believes he has cracked it. What’s more, he claims to have arrived at the solution not by using mathematics as we know it, but by building a new, radically stronger logical structure: a structure he dubs “ultimate L”.

The journey to this point began in the early 1870s, when the German Georg Cantor was laying the foundations of set theory. Set theory deals with the counting and manipulation of collections of objects, and provides the crucial logical underpinnings of mathematics: because numbers can be associated with the size of sets, the rules for manipulating sets also determine the logic of arithmetic and everything that builds on it.

[div class=attrib]More from theSource here.[end-div]

[div class=attrib]Image courtesy of Wikipedia / Creative Commons.[end-div]

Book Reviews

Book Review: The Drunkard’s Walk: How Randomness Rules Our Lives. Leonard Mlodinow

Leonard Mlodinow weaves a compelling path through the world of statistical probability showing us how the laws of chance affect our lives on personal and grande scales. Mlodinow skillfully illustrates randomness and its profound implications by presenting complex mathematical constructs in language for the rest of us (non-mathematicians), without dumbing-down this important subject.

The book defines many of the important mathematical concepts behind randomness and exposes the key fallacies that often blind us as we wander through life on our “drunkard’s walk”. The law of large numbers, the prosecutor’s fallacy, conditional probability, the availability bias and bell curves were never so approachable.

Whether it’s a deluded gambler, baseball star on a “winning streak” or a fortunate CEO wallowing in the good times, Mlodinow debunks the common conceptions that skill, planning and foresight result in any significant results beyond pure chance. With the skill of a storyteller Mlodinow shows us how polls, grades, ratings and even measures of corporate success are far less objective and reliable than we ought to believe. Lords of Wall Street take notice, the secrets of your successes are not all that they seem.

BigBang

The Evolution of the Physicist’s Picture of Nature

[div class=attrib]From Scientific American:[end-div]

Editor’s Note: We are republishing this article by Paul Dirac from the May 1963 issue of Scientific American, as it might be of interest to listeners to the June 24, 2010, and June 25, 2010 Science Talk podcasts, featuring award-winning writer and physicist Graham Farmelo discussing The Strangest Man, his biography of the Nobel Prize-winning British theoretical physicist.

In this article I should like to discuss the development of general physical theory: how it developed in the past and how one may expect it to develop in the future. One can look on this continual development as a process of evolution, a process that has been going on for several centuries.

The first main step in this process of evolution was brought about by Newton. Before Newton, people looked on the world as being essentially two-dimensional-the two dimensions in which one can walk about-and the up-and-down dimension seemed to be something essentially different. Newton showed how one can look on the up-and-down direction as being symmetrical with the other two directions, by bringing in gravitational forces and showing how they take their place in physical theory. One can say that Newton enabled us to pass from a picture with two-dimensional symmetry to a picture with three-dimensional symmetry.

Einstein made another step in the same direction, showing how one can pass from a picture with three-dimensional symmetry to a picture with four­dimensional symmetry. Einstein brought in time and showed how it plays a role that is in many ways symmetrical with the three space dimensions. However, this symmetry is not quite perfect. With Einstein’s picture one is led to think of the world from a four-dimensional point of view, but the four dimensions are not completely symmetrical. There are some directions in the four-dimensional picture that are different from others: directions that are called null directions, along which a ray of light can move; hence the four-dimensional picture is not completely symmetrical. Still, there is a great deal of symmetry among the four dimensions. The only lack of symmetry, so far as concerns the equations of physics, is in the appearance of a minus sign in the equations with respect to the time dimension as compared with the three space dimensions [see top equation in diagram].

four-dimensional symmetry equation and Schrodinger's equationsWe have, then, the development from the three-dimensional picture of the world to the four-dimensional picture. The reader will probably not be happy with this situation, because the world still appears three-dimensional to his consciousness. How can one bring this appearance into the four-dimensional picture that Einstein requires the physicist to have?

What appears to our consciousness is really a three-dimensional section of the four-dimensional picture. We must take a three-dimensional section to give us what appears to our consciousness at one time; at a later time we shall have a different three-dimensional section. The task of the physicist consists largely of relating events in one of these sections to events in another section referring to a later time. Thus the picture with four­dimensional symmetry does not give us the whole situation. This becomes particularly important when one takes into account the developments that have been brought about by quantum theory. Quantum theory has taught us that we have to take the process of observation into account, and observations usually require us to bring in the three-dimensional sections of the four-dimensional picture of the universe.

The special theory of relativity, which Einstein introduced, requires us to put all the laws of physics into a form that displays four-dimensional symmetry. But when we use these laws to get results about observations, we have to bring in something additional to the four-dimensional symmetry, namely the three-dimensional sections that describe our consciousness of the universe at a certain time.

Einstein made another most important contribution to the development of our physical picture: he put forward the general theory of relativity, which requires us to suppose that the space of physics is curved. Before this physicists had always worked with a flat space, the three-dimensional flat space of Newton which was then extended to the four­dimensional flat space of special relativity. General relativity made a really important contribution to the evolution of our physical picture by requiring us to go over to curved space. The general requirements of this theory mean that all the laws of physics can be formulated in curved four-dimensional space, and that they show symmetry among the four dimensions. But again, when we want to bring in observations, as we must if we look at things from the point of view of quantum theory, we have to refer to a section of this four-dimensional space. With the four-dimensional space curved, any section that we make in it also has to be curved, because in general we cannot give a meaning to a flat section in a curved space. This leads us to a picture in which we have to take curved three­dimensional sections in the curved four­dimensional space and discuss observations in these sections.

During the past few years people have been trying to apply quantum ideas to gravitation as well as to the other phenomena of physics, and this has led to a rather unexpected development, namely that when one looks at gravitational theory from the point of view of the sections, one finds that there are some degrees of freedom that drop out of the theory. The gravitational field is a tensor field with 10 components. One finds that six of the components are adequate for describing everything of physical importance and the other four can be dropped out of the equations. One cannot, however, pick out the six important components from the complete set of 10 in any way that does not destroy the four-dimensional symmetry. Thus if one insists on preserving four-dimensional symmetry in the equations, one cannot adapt the theory of gravitation to a discussion of measurements in the way quantum theory requires without being forced to a more complicated description than is needed bv the physical situation. This result has led me to doubt how fundamental the four-dimensional requirement in physics is. A few decades ago it seemed quite certain that one had to express the whole of physics in four­dimensional form. But now it seems that four-dimensional symmetry is not of such overriding importance, since the description of nature sometimes gets simplified when one departs from it.

Now I should like to proceed to the developments that have been brought about by quantum theory. Quantum theory is the discussion of very small things, and it has formed the main subject of physics for the past 60 years. During this period physicists have been amassing quite a lot of experimental information and developing a theory to correspond to it, and this combination of theory and experiment has led to important developments in the physicist’s picture of the world.

[div class=attrib]More from theSource here.[end-div]

BigBang

The Limits of Reason

[div class=attrib]From Scientific American:[end-div]

Ideas on complexity and randomness originally suggested by Gottfried W. Leibniz in 1686, combined with modern information theory, imply that there can never be a “theory of everything” for all of mathematics.

In 1956 Scientific American published an article by Ernest Nagel and James R. Newman entitled “Gödel’s Proof.” Two years later the writers published a book with the same title–a wonderful work that is still in print. I was a child, not even a teenager, and I was obsessed by this little book. I remember the thrill of discovering it in the New York Public Library. I used to carry it around with me and try to explain it to other children.

It fascinated me because Kurt Gödel used mathematics to show that mathematics itself has limitations. Gödel refuted the position of David Hilbert , who about a century ago declared that there was a theory of everything for math, a finite set of principles from which one could mindlessly deduce all mathematical truths by tediously following the rules of symbolic logic. But Gödel demonstrated that mathematics contains true statements that cannot be proved that way. His result is based on two self-referential paradoxes: “This statement is false” and “This statement is unprovable.”.

[div class=attrib]More from theSource here.[end-div]