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.

    Presenting to the world his remarkable observations using the recently invented telescope--the craters of the moon, the satellites of Jupiter--Galileo dramatically challenged our idea of the perfection of the heavens and the centrality of the Earth in the universe. Indeed, the appearance of the little book is regarded as one of the great moments in the history of science. Planned to coincide with the 400th anniversary of the publication of the Starry Messenger, this is a major new biography of Galileo, a fresh and much more rounded view of the great scientist than found in earlier works. Unlike previous biographers, Heilbron shows us that Galileo was far more than a mathematician: he was deeply knowledgeable in the arts, an expert on the epic poet Ariosto, a fine lutenist. More important, Heilbron notes that years of reading the poets and experimenting with literary forms were not mere sidebars--they enabled Galileo to write clearly and plausibly about the most implausible things. Indeed, Galileo changed the world not simply because he revolutionized astronomy, but because he conveyed his discoveries so clearly and crisply that they could not be avoided or denied. If ever a discoverer was perfectly prepared to make and exploit his discovery, it was the dexterous humanist Galileo aiming his first telescope at the sky. In Galileo, John Heilbron captures not only the great scientist, but also the creative, artistic younger man who would ultimately become the champion of Copernicus, the bête-noire of the Jesuits, and the best-known of all martyrs to academic freedom.

    QFT is the only physics theory that makes sense and that dispels or resolves the paradoxes of relativity and quantum mechanics that have confused and mystified so many people.

    The Fourth Edition of volumes 1 and 2 is concerned with mechanics and E&M/Optics. New features include: expanded coverage of classic physics topics, substantial increases in the number of in-text examples which reinforce text exposition, the latest pedagogical and technical advances in the field, numerical analysis, computer-generated graphics, computer projects and much more.

    Did humans emerge through evolutionary processes from an ape-like ancestor or are we God's creation? A comparison of scientific models provides stunning insights and a framework to test emerging evidence.

    With his usual argumentative rigor, he elucidates the distinction between science fiction, a genre in which science remains possible in spite of all the upheavals that may attend the world in which the tale takes place, and fiction outside-science, the literary concept he fashions in this book, a fiction in which science becomes impossible. With its investigations of the philosophies of Hume, Kant, and Popper, Science Fiction and Fiction of Worlds Outside-Science broadens the inquiry that Meillassoux began in After Finitude, thinking through the concrete possibilities and consequences of a chaotic world in which human beings can no longer resort to science to ground their existence. It is a significant milestone in the work of an emerging philosopher, which will appeal to readers of both philosophy and literature. The text is followed by Isaac Asimov’s essay “The Billiard Ball.”

    But information is not the same as meaning, nor is knowledge identical with wisdom. Many people feel engulfed by a tsunami of facts in which they can find no meaning. In thirteen short, accessible chapters McGrath, author of the bestselling The Dawkins Delusion, leads the reader through a nontechnical discussion of science and faith. How do we make sense of the world around us? Are belief in science and the Christian faith compatible? Does the structure of the universe point toward the existence of God?McGrath's goal is to help readers see that science is neither anathema to faith, nor does it supersede faith. Both science and faith help with the overriding human desire to make sense of things. Faith is a complex idea. It is not a blind leap into the dark but a joyful discovery of a bigger picture of wondrous things of which we are all a part.

    Quantum paradoxes and the eventful life of Schroedinger's Cat are explained, along with the Many Universe explanation of quantum measurement in this newly revised edition. Updated throughout, the book also looks ahead to the nanotechnology revolution and describes quantum cryptography, computing and teleportation. Including an account of quantum mechanics and science fiction, this accessible book is geared to the general reader. Anthony Hey teaches at the University of Southampton, UK, and is the co-author of several books, including two with Patrick Walters, The Quantum Universe (Cambridge, 1987), and Einstein's Mirror (Cambridge, 1997). Patrick Walters is a Lecturer in Continuing Education at the University of Wales at Swansea. He co-ordinates the Physical Science Programme in DACE which includes the Astronomy Programme. His research interests include science education, and he also writes non-technical books on science for the general reader and beginning undergraduates. First Edition Pb (1987): 0-521-31845-9

    The era covered by this book, 1950 to 1990, was surely one of the golden ages of science as well as the American university.Cherished myths are debunked along the way as Gian-Carlo Rota takes pleasure in portraying, warts and all, some of the great scientific personalities of the period Stanislav Ulam (who, together with Edward Teller, signed the patent application for the hydrogen bomb), Solomon Lefschetz (Chairman in the 50s of the Princeton mathematics department), William Feller (one of the founders of modern probability theory), Jack Schwartz (one of the founders of computer science), and many others.Rota is not afraid of controversy. Some readers may even consider these essays indiscreet. After the publication of the essay "The Pernicious Influence of Mathematics upon Philosophy" (reprinted six times in five languages) the author was blacklisted in analytical philosophy circles. Indiscrete Thoughts should become an instant classic and the subject of debate for decades to come."Read Indiscrete Thoughts for its account of the way we were and what we have become; for its sensible advice and its exuberant rhetoric."--The Mathematical Intelligencer"Learned, thought-provoking, politically incorrect, delighting in paradox, and likely to offend but everywhere readable and entertaining."--The American Mathematical Monthly"It is about mathematicians, the way they think, and the world in which the live. It is 260 pages of Rota calling it like he sees it... Readers are bound to find his observations amusing if not insightful. Gian-Carlo Rota has written the sort of book that few mathematicians could write. What will appeal immediately to anyone with an interest in research mathematics are the stories he tells about the practice of modern mathematics."--MAA Reviews"

    Carroll at his very best.--Bill Bryson, author of The Body: A Guide for Occupants From acclaimed writer and biologist Sean B. Carroll, a rollicking, awe-inspiring story of the surprising power of chance in our lives and the worldWhy is the world the way it is? How did we get here? Does everything happen for a reason or are some things left to chance? Philosophers and theologians have pondered these questions for millennia, but startling scientific discoveries over the past half century are revealing that we live in a world driven by chance. A Series of Fortunate Events tells the story of the awesome power of chance and how it is the surprising source of all the beauty and diversity in the living world.Like every other species, we humans are here by accident. But it is shocking just how many things--any of which might never have occurred--had to happen in certain ways for any of us to exist. From an extremely improbable asteroid impact, to the wild gyrations of the Ice Age, to invisible accidents in our parents' gonads, we are all here through an astonishing series of fortunate events. And chance continues to reign every day over the razor-thin line between our life and death.This is a relatively small book about a really big idea. It is also a spirited tale. Drawing inspiration from Monty Python, Kurt Vonnegut, and other great thinkers, and crafted by one of today's most accomplished science storytellers, A Series of Fortunate Events is an irresistibly entertaining and thought-provoking account of one of the most important but least appreciated facts of life.

    A meld of history and science, this book is a group portrait of some of the greatest minds who ever lived as they wrestled with nature’s most sweeping mysteries. The answers they uncovered still hold the key to how we understand the world.At the end of the seventeenth century—an age of religious wars, plague, and the Great Fire of London—when most people saw the world as falling apart, these earliest scientists saw a world of perfect order. They declared that, chaotic as it looked, the universe was in fact as intricate and perfectly regulated as a clock. This was the tail end of Shakespeare’s century, when the natural land the supernatural still twined around each other. Disease was a punishment ordained by God, astronomy had not yet broken free from astrology, and the sky was filled with omens. It was a time when little was known and everything was new. These brilliant, ambitious, curious men believed in angels, alchemy, and the devil, and they also believed that the universe followed precise, mathematical laws—-a contradiction that tormented them and changed the course of history.The Clockwork Universe is the fascinating and compelling story of the bewildered geniuses of the Royal Society, the men who made the modern world.

    This is such a book. Max Born is a Nobel Laureate (1955) and one of the world's great physicists: in this book he analyzes and interprets the theory of Einsteinian relativity. The result is undoubtedly the most lucid and insightful of all the books that have been written to explain the revolutionary theory that marked the end of the classical and the beginning of the modern era of physics.The author follows a quasi-historical method of presentation. The book begins with a review of the classical physics, covering such topics as origins of space and time measurements, geometric axioms, Ptolemaic and Copernican astronomy, concepts of equilibrium and force, laws of motion, inertia, mass, momentum and energy, Newtonian world system (absolute space and absolute time, gravitation, celestial mechanics, centrifugal forces, and absolute space), laws of optics (the corpuscular and undulatory theories, speed of light, wave theory, Doppler effect, convection of light by matter), electrodynamics (including magnetic induction, electromagnetic theory of light, electromagnetic ether, electromagnetic laws of moving bodies, electromagnetic mass, and the contraction hypothesis). Born then takes up his exposition of Einstein's special and general theories of relativity, discussing the concept of simultaneity, kinematics, Einstein's mechanics and dynamics, relativity of arbitrary motions, the principle of equivalence, the geometry of curved surfaces, and the space-time continuum, among other topics. Born then points out some predictions of the theory of relativity and its implications for cosmology, and indicates what is being sought in the unified field theory.This account steers a middle course between vague popularizations and complex scientific presentations. This is a careful discussion of principles stated in thoroughly acceptable scientific form, yet in a manner that makes it possible for the reader who has no scientific training to understand it. Only high school algebra has been used in explaining the nature of classical physics and relativity, and simple experiments and diagrams are used to illustrate each step. The layman and the beginning student in physics will find this an immensely valuable and usable introduction to relativity. This Dover 1962 edition was greatly revised and enlarged by Dr. Born.

    However, according to Hofstadter, the formal system that underlies all mental activity transcends the system that supports it. If life can grow out of the formal chemical substrate of the cell, if consciousness can emerge out of a formal system of firing neurons, then so too will computers attain human intelligence. Gödel, Escher, Bach is a wonderful exploration of fascinating ideas at the heart of cognitive science: meaning, reduction, recursion, and much more.

    Capturing the tone of students exchanging ideas among themselves, this unique guide also explains how calculus is taught, how to get the best teachers, what to study, and what is likely to be on exams—all the tricks of the trade that will make learning the material of first-semester calculus a piece of cake. Funny, irreverent, and flexible, How to Ace Calculus shows why learning calculus can be not only a mind-expanding experience but also fantastic fun.

    Would you believe that these five pieces can be reassembled in such a fashion so as to create two apples equal in shape and size to the original? Would you believe that you could make something as large as the sun by breaking a pea into a finite number of pieces and putting it back together again? Neither did Leonard Wapner, author of The Pea and the Sun, when he was first introduced to the Banach-Tarski paradox, which asserts exactly such a notion. Written in an engaging style, The Pea and the Sun catalogues the people, events, and mathematics that contributed to the discovery of Banach and Tarski's magical paradox. Wapner makes one of the most interesting problems of advanced mathematics accessible to the non-mathematician.