Really big. You just won't believe how vastly, hugely, mind-bogglingly big it is. I mean, you may think it's a long way down the street to the chemist, but that's just peanuts to space.' Douglas Adams, Hitch-hiker's Guide to the GalaxyWe human beings have trouble with infinity - yet infinity is a surprisingly human subject. Philosophers and mathematicians have gone mad contemplating its nature and complexity - yet it is a concept routinely used by schoolchildren. Exploring the infinite is a journey into paradox. Here is a quantity that turns arithmetic on its head, making it feasible that 1 = 0. Here is a concept that enables us to cram as many extra guests as we like into an already full hotel. Most bizarrely of all, it is quite easy to show that there must be something bigger than infinity - when it surely should be the biggest thing that could possibly be. Brian Clegg takes us on a fascinating tour of that borderland between the extremely large and the ultimate that takes us from Archimedes, counting the grains of sand that would fill the universe, to the latest theories on the physical reality of the infinite. Full of unexpected delights, whether St Augustine contemplating the nature of creation, Newton and Leibniz battling over ownership of calculus, or Cantor struggling to publicise his vision of the transfinite, infinity's fascination is in the way it brings together the everyday and the extraordinary, prosaic daily life and the esoteric.Whether your interest in infinity is mathematical, philosophical, spiritual or just plain curious, this accessible book offers a stimulating and entertaining read.

    16 years later, the French Ambassador Hugues de Terlon secretly unearthed Descartes' bones & transported them to France. Why would this devoutly Catholic official care so much about the remains of a philosopher who was hounded from country to country on charges of atheism? Why would Descartes' bones take such a strange, serpentine path over the next 350 years—a path intersecting some of the grandest events imaginable: the birth of science, the rise of democracy, the mind-body problem, the conflict between faith & reason? Their story involves people from all walks of life—Louis XIV, a Swedish casino operator, poets & playwrights, philosophers & physicists, as these people used the bones in scientific studies, stole them, sold them, revered them as relics, fought over them, passed them surreptitiously from hand to hand. The answer lies in Descartes’ famous phrase: Cogito ergo sum—"I think, therefore I am." In his deceptively simple 78-page essay, Discourse on the Method, this small, vain, vindictive, peripatetic, ambitious Frenchman destroyed 2000 years of received wisdom & laid the foundations of the modern world. At the root of Descartes’ method was skepticism: "What can I know for certain?" Like-minded thinkers around Europe passionately embraced the book--the method was applied to medicine, nature, politics & society. The notion that one could find truth in facts that could be proved, & not in reliance on tradition & the Church's teachings, would become a turning point in human history. In an age of faith, what Descartes was proposing seemed like heresy. Yet Descartes himself was a good Catholic, who was spurred to write his incendiary book for the most personal of reasons: He'd devoted himself to medicine & the study of nature, but when his beloved daughter died aged 5, he took his ideas deeper. To understand the natural world one needed to question everything. Thus the scientific method was created & religion overthrown. If the natural world could be understood, knowledge could be advanced, & others might not suffer as his child did. The great controversy Descartes ignited continues to our era: where Islamic terrorists spurn the modern world & pine for a culture based on unquestioning faith; where scientists write bestsellers that passionately make the case for atheism; where others struggle to find a balance between faith & reason. Descartes’ Bonesis a historical detective story about the creation of the modern mind, with twists & turns leading up to the present day—to the science museum in Paris where the philosopher’s skull now resides & to the church a few kilometers away where, not long ago, a philosopher-priest said a mass for his bones.

    What happened during those years and what has happened since to refine the understanding of this phenomenon is the fascinating story told here.We move from a coffee shop in Zurich, where Einstein and Max von Laue discuss the madness of quantum theory, to a bar in Brazil, as David Bohm and Richard Feynman chat over cervejas. We travel to the campuses of American universities—from J. Robert Oppenheimer’s Berkeley to the Princeton of Einstein and Bohm to Bell’s Stanford sabbatical—and we visit centers of European physics: Copenhagen, home to Bohr’s famous institute, and Munich, where Werner Heisenberg and Wolfgang Pauli picnic on cheese and heady discussions of electron orbits.Drawing on the papers, letters, and memoirs of the twentieth century’s greatest physicists, Louisa Gilder both humanizes and dramatizes the story by employing their own words in imagined face-to-face dialogues. Here are Bohr and Einstein clashing, and Heisenberg and Pauli deciding which mysteries to pursue. We see Schrödinger and Louis de Broglie pave the way for Bell, whose work is here given a long-overdue revisiting. And with his characteristic matter-of-fact eloquence, Richard Feynman challenges his contemporaries to make something of this entanglement.

    At its heart is the discovery of the order that lies deep within the most complex of systems, from the origin of life, to the workings of giant corporations, to the rise and fall of greatcivilizations. And more than anyone else, this revolution is the work of one man, Stuart Kauffman, a MacArthur Fellow and visionary pioneer of the new science of complexity. Now, in At Home in the Universe, Kauffman brilliantly weaves together the excitement of intellectual discovery and a fertilemix of insights to give the general reader a fascinating look at this new science--and at the forces for order that lie at the edge of chaos. We all know of instances of spontaneous order in nature--an oil droplet in water forms a sphere, snowflakes have a six-fold symmetry. What we are only now discovering, Kauffman says, is that the range of spontaneous order is enormously greater than we had supposed. Indeed, self-organization is agreat undiscovered principle of nature. But how does this spontaneous order arise? Kauffman contends that complexity itself triggers self-organization, or what he calls order for free, that if enough different molecules pass a certain threshold of complexity, they begin to self-organize into a newentity--a living cell. Kauffman uses the analogy of a thousand buttons on a rug--join two buttons randomly with thread, then another two, and so on. At first, you have isolated pairs; later, small clusters; but suddenly at around the 500th repetition, a remarkable transformation occurs--much likethe phase transition when water abruptly turns to ice--and the buttons link up in one giant network. Likewise, life may have originated when the mix of different molecules in the primordial soup passed a certain level of complexity and self-organized into living entities (if so, then life is not ahighly improbable chance event, but almost inevitable). Kauffman uses the basic insight of order for free to illuminate a staggering range of phenomena. We see how a single-celled embryo can grow to a highly complex organism with over two hundred different cell types. We learn how the science ofcomplexity extends Darwin's theory of evolution by natural selection: that self-organization, selection, and chance are the engines of the biosphere. And we gain insights into biotechnology, the stunning magic of the new frontier of genetic engineering--generating trillions of novel molecules tofind new drugs, vaccines, enzymes, biosensors, and more. Indeed, Kauffman shows that ecosystems, economic systems, and even cultural systems may all evolve according to similar general laws, that tissues and terra cotta evolve in similar ways. And finally, there is a profoundly spiritual element toKauffman's thought. If, as he argues, life were bound to arise, not as an incalculably improbable accident, but as an expected fulfillment of the natural order, then we truly are at home in the universe. Kauffman's earlier volume, The Origins of Order, written for specialists, received lavish praise. Stephen Jay Gould called it a landmark and a classic. And Nobel Laureate Philip Anderson wrote that there are few people in this world who ever ask the right questions of science, and they are theones who affect its future most profoundly. Stuart Kauffman is one of these. In At Home in the Universe, this visionary thinker takes you along as he explores new insights into the nature of life.

    . . . Readers will never again look into the night sky the same way.” —MICHAEL SHERMER, author of The Believing Brain on Parallax: The Race to Measure the CosmosIn 1930, Edwin Hubble announced the greatest discovery in the history of astronomy since Galileo first turned a telescope to the heavens. The galaxies, previously believed to float serenely in the void, are in fact hurtling apart at an incredible speed; the universe is expanding. This stunning discovery was the culmination of a decades-long arc of scientific and technical advancement. In its shadow lies an untold, yet equally fascinating, backstory whose cast of characters illuminates the gritty, hard-won nature of scientific progress.The path to a broader mode of cosmic observation was blazed by a cadre of 19th-century amateur astronomers and inventors, galvanized by the advent of photography, spectral analysis, and innovative technology to create the entirely new field of astrophysics. From William Bond, who turned his home into a functional observatory, to John and Henry Draper, a father and son team who were trailblazers of astrophotography and spectroscopy, to geniuses of invention such as Léon Foucault, and George Hale, who founded the Mount Wilson Observatory, Hirshfeld reveals the incredible stories—and the ambitious dreamers—behind the birth of modern astronomy.Alan Hirshfeld, Professor of Physics at the University of Massachusetts Dartmouth and an Associate of the Harvard College Observatory, is the author of Parallax: The Race to Measure the Cosmos, The Electric Life of Michael Faraday, and Eureka Man: The Life and Legacy of Archimedes.

    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.