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"What is mathematics? It's neither physical nor mental, it's social. It's part of culture, it's part of history. It's like law, like religion, like money, like all those other things which are very real, but only as part of collective human consciousness. That's what math is. For mathematician Reuben Hersh, mathematics has existence or reality only as part of human culture. Despite its seeming timelessness and infallibility, it is a social-cultural- historic phenomenon. He takes the long view. He thinks a lot about the ancient problems. What are numbers? What are triangles, squares and circles? What are infinite sets? What is the fourth dimension? What is the meaning and nature of mathematics? In so doing he explains and criticizes current and past theories of the nature of mathematics. His main purpose is to confront philosophical problems: In what sense do mathematical objects exist? How can we have knowledge of them? Why do mathematicians think mathematical entities exist forever, independent of human action and knowledge? "

December 2005 Over the millennia, many mathematicians have hoped that mathematics would one day produce a Theory of Everything (TOE); a finite set of axioms and rules from which every mathematical truth could be derived. But in 1931 this hope received a serious blow: Kurt Gödel published his famous Incompleteness Theorem, which states that in every mathematical theory, no matter how extensive, there will always be statements which can't be proven to be true or false. Gregory Chaitin has been fascinated by this theorem ever since he was a child, and now, in time for the centenary of Gödel's birth in 2006, he has published his own book, called Meta Math! on the subject (you can read a review in this issue of Plus). It describes his journey, which, from the work of Gödel via that of Leibniz and Turing, led him to the number Omega, which is so complex that no mathematical theory can ever describe it. In this article he explains what Omega is all about, why maths can have no Theory of Everything, and what this means for mathematicians."

"Melvyn Bragg and guests John Barrow, Colva Roney-Dougal and Marcus du Sautoy explore the unintended consequences of mathematical discoveries, from the computer to online encryption, to alternating current and predicting the path of asteroids. In his book The Mathematician's Apology (1941), the Cambridge mathematician GH Hardy expressed his reverence for pure maths, and celebrated its uselessness in the real world. Yet one of the branches of pure mathematics in which Hardy excelled was number theory, and it was this field which played a major role in the work of his younger colleague, Alan Turing, as he worked first to crack Nazi codes at Bletchley Park and then on one of the first computers. Melvyn Bragg and guests explore the many surprising and completely unintended uses to which mathematical discoveries have been put."

"Grisha Perelman, where are you? Three years ago, a Russian mathematician by the name of Grigory Perelman, a k a Grisha, in St. Petersburg, announced that he had solved a famous and intractable mathematical problem, known as the Poincaré conjecture, about the nature of space. After posting a few short papers on the Internet and making a whirlwind lecture tour of the United States, Dr. Perelman disappeared back into the Russian woods in the spring of 2003, leaving the world's mathematicians to pick up the pieces and decide if he was right. Now they say they have finished his work, and the evidence is circulating among scholars in the form of three book-length papers with about 1,000 pages of dense mathematics and prose between them. As a result there is a growing feeling, a cautious optimism that they have finally achieved a landmark not just of mathematics, but of human thought."

"Dr. Mandelbrot coined the term "fractal" to refer to a new class of mathematical shapes whose uneven contours could mimic the irregularities found in nature. "Applied mathematics had been concentrating for a century on phenomena which were smooth, but many things were not like that: the more you blew them up with a microscope the more complexity you found," said David Mumford, a professor of mathematics at Brown University. "He was one of the primary people who realized these were legitimate objects of study." "

"Female physicists, astronomers and mathematicians are up against more than 2,000 years of convention that has long portrayed these fields as inherently male. Though women are no longer barred from university laboratories and scientific societies, the idea that they are innately less suited to mathematical science is deeply ingrained in our cultural genes. The problem goes back to the ancient Greeks, particularly to Pythagoras, the philosophical giant who dreamed the dream that became modern physics. Pythagoras almost certainly learned his famous theorem about right-angled triangles from the Babylonians, but we owe to him a far greater idea: "All is number," he declared, becoming the first person to say that the physical world could be described by the language of mathematics. Pythagoras also gave us the idea of the "music of the spheres," a set of mathematical relationships that would describe the structure of the universe itself. His vision would eventually give rise to the scientific revolution led by Copernicus, Kepler, Galileo and Newton. The search for a theory of everything today is the latest version of the ancient Pythagorean quest for divine "cosmic harmonies.""

"What can mathematics say about history? According to TED Fellow Jean-Baptiste Michel, quite a lot. From changes to language to the deadliness of wars, he shows how digitized history is just starting to reveal deep underlying patterns. Jean-Baptiste Michel looks at how we can use large volumes of data to better understand our world."

"HERE IS A mathematician's nightmare I heard in the 1980s when that irritating, unconforming, self-regarding provocateur Benoît Mandelbrot was suddenly famous - fractals, fractals everywhere. The mathematician dreamed that Mandelbrot died, and God spoke: "You know, there really was something to that Mandelbrot." Sure enough. Mandelbrot created nothing less than a new geometry, to stand side by side with Euclid's - a geometry to mirror not the ideal forms of thought but the real complexity of nature. He was a mathematician who was never welcomed into the fraternity ("Fortress Mathematics," he said, where "the highest ambition is to wall off the windows and preserve only one door"), and he pretended that was fine with him. When Yale first hired him to teach, it was in engineering and applied science; for most of his career he was supported at I.B.M.'s Westchester research lab. He called himself a "nomad by choice." He considered himself an experienced refugee: born to a Jewish family in Warsaw in 1924, he immigrated to Paris ahead of the Nazis, then fled farther and farther into the French countryside."

From sociology to mathematics, the academic world as viewed through the lens of "purity."

Thanks to Catherine T. (Oberlin Class of 2014) for this one.

"Economics aspires to be a science. But in this it does not succeed. Neither does finance. This despite the fact that there is an annual, optimistically named Nobel Prize in "Economic Sciences." Financial crises keep happening-the list is long. Could they be avoided if economics and finance were science? To paraphrase financial observer James Grant: science is progressive, but finance is cyclical. But why should this be? Do we not learn from experience? Does economic knowledge not increase? And how about having computers, vast amounts of data and information, and new mathematical models to guide lending and investing decisions?"

" By applying electrical current to the brain, researchers can enhance a person's mathematical ability for up to six months. "

"It is difficult to get through a day in an American school without hearing maxims such as these: "To succeed, you must believe in yourself," and "To teach, you must relate the subject to the lives of students." But the Brookings Institution is reporting today that countries such as the United States that embrace self-esteem, joy and real-world relevance in learning mathematics are lagging behind others that don't promote all that self-regard. Consider Korea and Japan. According to the Washington think tank's annual Brown Center report on education, 6 percent of Korean eighth-graders surveyed expressed confidence in their math skills, compared with 39 percent of U.S. eighth-graders. But a respected international math assessment showed Koreans scoring far ahead of their peers in the United States, raising questions about the importance of self-esteem. In Japan, the report found, 14 percent of math teachers surveyed said they aim to connect lessons to students' lives, compared with 66 percent of U.S. math teachers. Yet the U.S. scores in eighth-grade math trail those of the Japanese, raising similar questions about the importance of practical relevance. "

"True art would never be mistaken for a crude, paint-by-the-numbers copy. But a researcher has developed a statistical tool for determining whether a purported masterpiece is only a skilled imitation, suggesting that art may be a numbers game after all. Using high-resolution digital images and complex mathematical formulas, associate professor Hany Farid of Dartmouth College analyzed works by Renaissance artists to determine their authenticity. His computer program was able to accurately separate eight drawings by 16th century Flemish artist Pieter Bruegel the Elder from five drawings by imitators. It also found that portions of a painting by Italian artist Pietro di Cristoforo Vannucci, known as Perugino, were probably done by Perugino's apprentices. Farid described his work, presented Monday in the Proceedings of the National Academy of Sciences, as "simply another tool that is contributing to the dialogue of art authenticating" and said more work is needed before digital analysis of art could be done on a wider scale. Art experts reacted warily to the prospect that a masterpiece could be reduced to the sum of its digital parts. "

"In the beauty and geometric complexity of tile mosaics on walls of medieval Islamic buildings, scientists have recognized patterns suggesting that the designers had made a conceptual breakthrough in mathematics beginning as early as the 13th century. A new study shows that the Islamic pattern-making process, far more intricate than the laying of one's bathroom floor, appears to have involved an advanced math of quasi crystals, which was not understood by modern scientists until three decades ago. The findings, reported in the current issue of the journal Science, are a reminder of the sophistication of art, architecture and science long ago in the Islamic culture. They also challenge the assumption that the designers somehow created these elaborate patterns with only a ruler and a compass. Instead, experts say, they may have had other tools and concepts."

"BUSINESSES in nearly every industry were caught off guard by the Great Recession. Few leaders in business - or government, for that matter - seem to have even considered the possibility that an economic downturn of this magnitude could happen. "

To better understand the world in which they will live, students need foundations in economics, statistics, finance and psychology.