Today, unfortunately, the traditional place of mathematics in education is in grave danger. The teaching and learning of mathematics has degenerated into the realm of rote memorization, the outcome of which leads to satisfactory formal ability but does not lead to real understanding or to greater intellectual independence. This new edition of Richard Courant's and Herbert Robbins's classic work seeks to address this problem. Its goal is to put the meaning back into mathematics.Written for beginners and scholars, for students and teachers, for philosophers and engineers, What is Mathematics? Second Edition is a sparkling collection of mathematical gems that offers an entertaining and accessible portrait of the mathematical world. Covering everything from natural numbers and the number system to geometrical constructions and projective geometry, from topology and calculus to matters of principle and the Continuum Hypothesis, this fascinating survey allows readers to delve into mathematics as an organic whole rather than an empty drill in problem solving. With chapters largely independent of one another and sections that lead upward from basic to more advanced discussions, readers can easily pick and choose areas of particular interest without impairing their understanding of subsequent parts.Brought up to date with a new chapter by Ian Stewart, What is Mathematics? Second Edition offers new insights into recent mathematical developments and describes proofs of the Four-Color Theorem and Fermat's Last Theorem, problems that were still open when Courant and Robbins wrote this masterpiece, but ones that have since been solved.Formal mathematics is like spelling and grammar - a matter of the correct application of local rules. Meaningful mathematics is like journalism - it tells an interesting story. But unlike some journalism, the story has to be true. The best mathematics is like literature - it brings a story to life before your eyes and involves you in it, intellectually and emotionally. What is Mathematics is like a fine piece of literature - it opens a window onto the world of mathematics for anyone interested to view.

    From Quantum Mechanics to Natural Selection, what we have instead are theories - ideas explain why things happen the way they do. We don't know for certain these are correct - no one ever saw the Big Bang - but with them we can paint beautiful, breathtaking pictures of everything from human behaviour to what the future may hold. Profiling the key scientists behind each theory, "30-Second Theories" presents each entry in a unique, eye-catching full-colour design, with thought-provoking extras and stylish illustrations. It is essential for anyone keen on expanding their mind with science's most thrilling ideas.

    Plait created his popular web site: http://www.badastronomy.com/index.html, to debunk bad astronomy in popular culture. This website proved popular, which led to this first book by Plait, that carries on from the website and in a detailed and clear fashion criticises and disproves popular myths and misconceptions relating to astronomy, and promotes science as a means of explaining the skies. The work describes 24 common astronomical fallacies, including the beliefs that the Coriolis effect determines the direction that water drains in a bathtub, and that planetary alignments can cause disaster on Earth. The author sharply and convincingly dismisses astrology, creationism, and UFO sightings, and explains the principles behind basic general concepts (the Big Bang, why the sky is blue, etc.).

    This book reveals the most exhilarating oddities from Professor Stewart's legendary cabinet.Inside, you will find hidden gems of logic, geometry, and probability-like how to extract a cherry from a cocktail glass (harder than you think), a pop-up dodecahedron, and the real reason why you can't divide anything by zero. Scattered among these are keys to Fermat's last theorem, the Poincaréonjecture, chaos theory, and the P=NP problem (you'll win a million dollars if you solve it). You never know what enigmas you'll find in the Stewart cabinet, but they're sure to be clever, mind-expanding, and delightfully fun.

    Randall Fuller takes us back to one of those turning points, in 1860, with the story of the influence of Charles Darwin's just-published On the Origin of Species on five American intellectuals, including Bronson Alcott, Henry David Thoreau, the child welfare reformer Charles Loring Brace, and the abolitionist Franklin Sanborn.Each of these figures seized on the book's assertion of a common ancestry for all creatures as a powerful argument against slavery, one that helped provide scientific credibility to the cause of abolition. Darwin's depiction of constant struggle and endless competition described America on the brink of civil war. But some had difficulty aligning the new theory to their religious convictions and their faith in a higher power. Thoreau, perhaps the most profoundly affected all, absorbed Darwin's views into his mysterious final work on species migration and the interconnectedness of all living things.Creating a rich tableau of nineteenth-century American intellectual culture, as well as providing a fascinating biography of perhaps the single most important idea of that time, The Book That Changed America is also an account of issues and concerns still with us today, including racism and the enduring conflict between science and religion.

    If you want to understand the greatest ideas that shaped our current cosmic cartography, read this book.”—Adam G. Riess, Nobel Laureate in Physics, 2011   This book provides a tour of the “greatest hits” of cosmological discoveries—the ideas that reshaped our universe over the past century. The cosmos, once understood as a stagnant place, filled with the ordinary, is now a universe that is expanding at an accelerating pace, propelled by dark energy and structured by dark matter. Priyamvada Natarajan, our guide to these ideas, is someone at the forefront of the research—an astrophysicist who literally creates maps of invisible matter in the universe. She not only explains for a wide audience the science behind these essential ideas but also provides an understanding of how radical scientific theories gain acceptance.   The formation and growth of black holes, dark matter halos, the accelerating expansion of the universe, the echo of the big bang, the discovery of exoplanets, and the possibility of other universes—these are some of the puzzling cosmological topics of the early twenty-first century. Natarajan discusses why the acceptance of new ideas about the universe and our place in it has never been linear and always contested even within the scientific community. And she affirms that, shifting and incomplete as science always must be, it offers the best path we have toward making sense of our wondrous, mysterious universe.

    Inspired by Einstein, Coltrane put physics and geometry at the core of his music. Physicist and jazz musician Stephon Alexander follows suit, using jazz to answer physics' most vexing questions about the past and future of the universe. Following the great minds that first drew the links between music and physics-a list including Pythagoras, Kepler, Newton, Einstein, and Rakim-The Jazz of Physics reveals that the ancient poetic idea of the Music of the Spheres," taken seriously, clarifies confounding issues in physics. The Jazz of Physics will fascinate and inspire anyone interested in the mysteries of our universe, music, and life itself.

    Of these, gravity may the most obvious, but it is also the most mysterious. Newton managed to predict the force of gravity but couldn't explain how it worked at a distance. Einstein picked up on the simple premise that gravity and acceleration are interchangeable to devise his mind-bending general relativity, showing how matter warps space and time. Not only did this explain how gravity worked – and how apparently simple gravitation has four separate components – but it predicted everything from black holes to gravity's effect on time. Whether it's the reality of anti-gravity or the unexpected discovery that a ball and a laser beam drop at the same rate, gravity is the force that fascinates.

    When temperatures drop below 56°F, tin crumbles into powder. Were the soldiers of the Grande Armée acutee fatally weakened by cold because the buttons of their uniforms fell apart? How different our world might be if tin did not disintegrate at low temperatures and the French had continued their eastward expansion! This fascinating book tells the stories of seventeen molecules that, like the tin of those buttons, greatly influenced the course of history. These molecules provided the impetus for early exploration and made possible the ensuing voyages of discovery. They resulted in grand feats of engineering and spurred advances in medicine; lie behind changes in gender roles, in law, and in the environment; and have determined what we today eat, drink, and wear. Showing how a change as small as the position of an atom can lead to enormous differences in the properties of a substance, the authors reveal the astonishing chemical connections among seemingly unrelated events. Napoleon's Buttons offers a novel way to understand how our contemporary world works and how our civilization has been shaped over time.

    Now he brings his considerable talents to the history of one of math's most enduring puzzles: the seemingly paradoxical nature of infinity.Is infinity a valid mathematical property or a meaningless abstraction? The nineteenth-century mathematical genius Georg Cantor's answer to this question not only surprised him but also shook the very foundations upon which math had been built. Cantor's counterintuitive discovery of a progression of larger and larger infinities created controversy in his time and may have hastened his mental breakdown, but it also helped lead to the development of set theory, analytic philosophy, and even computer technology.Smart, challenging, and thoroughly rewarding, Wallace's tour de force brings immediate and high-profile recognition to the bizarre and fascinating world of higher mathematics.

    From the classroom to the courtroom and from financial markets to supermarkets, Mlodinow's intriguing and illuminating look at how randomness, chance, and probability affect our daily lives will intrigue, awe, and inspire.

    It's ever-present. Without it, you would die in minutes. And it has an epic story to tell.In Caesar's Last Breath, New York Times bestselling author Sam Kean takes us on a journey through the periodic table, around the globe, and across time to tell the story of the air we breathe, which, it turns out, is also the story of earth and our existence on it.With every breath, you literally inhale the history of the world. On the ides of March, 44 BC, Julius Caesar died of stab wounds on the Senate floor, but the story of his last breath is still unfolding; in fact, you're probably inhaling some of it now. Of the sextillions of molecules entering or leaving your lungs at this moment, some might well bear traces of Cleopatra's perfumes, German mustard gas, particles exhaled by dinosaurs or emitted by atomic bombs, even remnants of stardust from the universe's creation.Tracing the origins and ingredients of our atmosphere, Kean reveals how the alchemy of air reshaped our continents, steered human progress, powered revolutions, and continues to influence everything we do. Along the way, we'll swim with radioactive pigs, witness the most important chemical reactions humans have discovered, and join the crowd at the Moulin Rouge for some of the crudest performance art of all time. Lively, witty, and filled with the astounding science of ordinary life, Caesar's Last Breath illuminates the science stories swirling around us every second.

    Yet in The Science of Interstellar, Kip Thorne, the physicist who assisted Nolan on the scientific aspects of Interstellar, shows us that the movie’s jaw-dropping events and stunning, never-before-attempted visuals are grounded in real science. Thorne shares his experiences working as the science adviser on the film and then moves on to the science itself. In chapters on wormholes, black holes, interstellar travel, and much more, Thorne’s scientific insights—many of them triggered during the actual scripting and shooting of Interstellar—describe the physical laws that govern our universe and the truly astounding phenomena that those laws make possible.Interstellar and all related characters and elements are trademarks of and © Warner Bros. Entertainment Inc. (s14).

    In this book, he expands the points he has made, and claims that this debate is not so much about religion versus science, as about the nature of science itself. With infectious enthusiasm, he reveals the mechanics of evolutionary theory, explains how it is rooted in the testable and verifiable scientific method, and why it is therefore a sound explanation of our beginning. He argues passionately that to continue to assert otherwise, to continue to insist that creationism has a place in the science classroom is harmful not only to our children, but to the future of the greater world as well.

    It defines our experience of the world; it echoes through our every waking hour. Time is the very foundation of conscious experience. Yet as familiar as it is, time is also deeply mysterious. We cannot see, hear, smell, taste, or touch it. Yet we do "feel" it--or at least we "think" we feel it. No wonder poets, writers, philosophers, and scientists have grappled with time for centuries.In his latest book, award-winning science writer Dan Falk chronicles the story of how humans have come to understand time over the millennia, and by drawing from the latest research in physics, psychology, and other fields, Falk shows how that understanding continues to evolve. "In Search of Time" begins with our earliest ancestors' perception of time and the discoveries that led--with much effort--to the Gregorian calendar, atomic clocks, and "leap seconds." Falk examines the workings of memory, the brain's remarkable "bridge across time," and asks whether humans are unique in their ability to recall the past and imagine the future. He explores the possibility of time travel, and the paradoxes it seems to entail. Falk looks at the quest to comprehend the beginning of time and how time--and the universe--may end. Finally, he examines the puzzle of time's "flow," and the remarkable possibility that the passage of time may be an illusion.Entertaining, illuminating, and ultimately thought provoking, "In Search of Time "reveals what some of our most insightful thinkers have had to say about time, from Aristotle to Kant, from Newton to Einstein, and continuing with the brightest minds of today.