Kurt Giidel maintained, and offered detailed proof, that in any arithmetic system, even in elementary parts of arithmetic, there are propositions which cannot be proved or disproved within the system. It is thus uncertain that the basic axioms of arithmetic will not give rise to contradictions. The repercussions of this discovery are still being felt and debated in 20th-century mathematics.The present volume reprints the first English translation of Giidel's far-reaching work. Not only does it make the argument more intelligible, but the introduction contributed by Professor R. B. Braithwaite (Cambridge University}, an excellent work of scholarship in its own right, illuminates it by paraphrasing the major part of the argument.This Dover edition thus makes widely available a superb edition of a classic work of original thought, one that will be of profound interest to mathematicians, logicians and anyone interested in the history of attempts to establish axioms that would provide a rigorous basis for all mathematics. Translated by B. Meltzer, University of Edinburgh. Preface. Introduction by R. B. Braithwaite.

    The material is arranged chronologically beginning with archaic origins and covers Egyptian, Mesopotamian, Greek, Chinese, Indian, Arabic and European contributions done to the nineteenth century and present day. There are revised references and bibliographies and revised and expanded chapters on the nineteeth and twentieth centuries.

    . . in which Sydney Padua transforms one of the most compelling scientific collaborations into a hilarious series of adventures. Meet Victorian London’s most dynamic duo: Charles Babbage, the unrealized inventor of the computer, and his accomplice, Ada, Countess of Lovelace, the peculiar protoprogrammer and daughter of Lord Byron. When Lovelace translated a description of Babbage’s plans for an enormous mechanical calculating machine in 1842, she added annotations three times longer than the original work. Her footnotes contained the first appearance of the general computing theory, a hundred years before an actual computer was built. Sadly, Lovelace died of cancer a decade after publishing the paper, and Babbage never built any of his machines. But do not despair! The Thrilling Adventures of Lovelace and Babbage presents a rollicking alternate reality in which Lovelace and Babbage do build the Difference Engine and then use it to build runaway economic models, battle the scourge of spelling errors, explore the wilder realms of mathematics, and, of course, fight crime—for the sake of both London and science. Complete with extensive footnotes that rival those penned by Lovelace herself, historical curiosities, and never-before-seen diagrams of Babbage’s mechanical, steam-powered computer, The Thrilling Adventures of Lovelace and Babbage is wonderfully whimsical, utterly unusual, and, above all, entirely irresistible.(With black-and-white illustrations throughout.)

    But when the project is suddenly in danger of being canceled, David embeds a hidden directive in the software accidentally creating a runaway artificial intelligence.David and his team are initially thrilled when the project is allocated extra servers and programmers. But excitement turns to fear as the team realizes that they are being manipulated by an A.I. who is redirecting corporate funds, reassigning personnel and arming itself in pursuit of its own agenda.

    This is the English village of Fairacre: a handful of thatch-roofed cottages, a church, the school, the promise of fair weather, friendly faces, and good cheer -- at least most of the time. Here everyone knows everyone else's business, and the villagers like each other anyway (even Miss Pringle, the irascible, gloomy cleaner of Fairacre School). With a wise heart and a discerning eye, Miss Read guides us through one crisp, glistening autumn in her village and introduces us to a cast of unforgettable characters and a world of drama, romance, and humor, all within a stone's throw of the school. By the time winter comes, you'll be nestled snugly into the warmth and wit of Fairacre and won't want to leave.

    Crease tells the stories behind ten of the greatest equations in human history. Was Nobel laureate Richard Feynman really joking when he called Maxwell's electromagnetic equations the most significant event of the nineteenth century? How did Newton's law of gravitation influence young revolutionaries? Why has Euler's formula been called "God's equation," and why did a mysterious ecoterrorist make it his calling card? What role do betrayal, insanity, and suicide play in the second law of thermodynamics?The Great Equations tells the stories of how these equations were discovered, revealing the personal struggles of their ingenious originators. From "1 + 1 = 2" to Heisenberg's uncertainty principle, Crease locates these equations in the panoramic sweep of Western history, showing how they are as integral to their time and place of creation as are great works of art.

    Rare Book

    How does Jane Goodall’s relationship with her dog Rusty inform her thinking about our relationship to other species? Which time and place would Jared Diamond most prefer to live in, in light of his work on the role of chance in history? What does driving a sports car have to do with Steven Weinberg’s quest for the “theory of everything”? Physicist and journalist Stefan Klein’s intimate conversations with nineteen of the world’s best-known scientists (including three Nobel Laureates) let us listen in as they talk about their paradigm-changing work—and how it is deeply rooted in their daily lives. • Cosmologist Martin Rees on the beginning and end of the world • Evolutionary biologist Richard Dawkins on egoism and selflessness • Neuroscientist V. S. Ramachandran on consciousness • Molecular biologist Elizabeth Blackburn on aging • Philosopher Peter Singer on morality • Physician and social scientist Nicholas Christakis on human relationships • Biochemist Craig Venter on the human genome • Chemist and poet Roald Hoffmann on beauty

    When he finds out that he is a wizard, he tries to apply scientific principles to his study of magic, with sometimes surprising results.

    Following closely behind is y, or gamma, a constant that arises in many mathematical areas yet maintains a profound sense of mystery. In a tantalizing blend of history and mathematics, Julian Havil takes the reader on a journey through logarithms and the harmonic series, the two defining elements of gamma, toward the first account of gamma's place in mathematics. Introduced by the Swiss mathematician Leonhard Euler (1707-1783), who figures prominently in this book, gamma is defined as the limit of the sum of 1 + 1/2 + 1/3 + . . . Up to 1/n, minus the natural logarithm of n--the numerical value being 0.5772156. . . . But unlike its more celebrated colleagues π and e, the exact nature of gamma remains a mystery--we don't even know if gamma can be expressed as a fraction. Among the numerous topics that arise during this historical odyssey into fundamental mathematical ideas are the Prime Number Theorem and the most important open problem in mathematics today--the Riemann Hypothesis (though no proof of either is offered!). Sure to be popular with not only students and instructors but all math aficionados, Gamma takes us through countries, centuries, lives, and works, unfolding along the way the stories of some remarkable mathematics from some remarkable mathematicians.-- "Notices of the American Mathematical Society"

    In the past, discoveries often had to wait for the rise of the next generation to see questions in a new light and let go of old truisms. Today, in a world that is defined by a rapid rate of change, staying on the cutting edge has as much to do with shedding outdated notions as adopting new ones. In this spirit, John Brockman, publisher of the online salon Edge.org ("the world's smartest website"—The Guardian), asked 175 of the world's most influential scientists, economists, artists, and philosophers: What scientific idea is ready for retirement?Jared Diamond explores the diverse ways that new ideas emerge * Nassim Nicholas Taleb takes down the standard deviation * Richard Thaler and novelist Ian McEwan reveal the usefulness of "bad" ideas * Steven Pinker dismantles the working theory of human behavior * Richard Dawkins renounces essentialism * Sherry Turkle reevaluates our expectations of artificial intelligence * Physicist Andrei Linde suggests that our universe and its laws may not be as unique as we think * Martin Rees explains why scientific understanding is a limitless goal * Alan Guth rethinks the origins of the universe * Sam Harris argues that our definition of science is too narrow * Nobel Prize winner Frank Wilczek disputes the division between mind and matter * Lawrence Krauss challenges the notion that the laws of physics were preordained * plus contributions from Daniel Goleman, Mihaly Csikszentmihalyi, Nicholas Carr, Rebecca Newberger Goldstein, Matt Ridley, Stewart Brand, Sean Carroll, Daniel C. Dennett, Helen Fisher, Douglas Rushkoff, Lee Smolin, Kevin Kelly, Freeman Dyson, and others.

    The inventor of the world's first clock is punished for trying to measure God's greatest gift. He is banished to a cave for centuries and forced to listen to the voices of all who come after him seeking more days, more years. Eventually, with his soul nearly broken, Father Time is granted his freedom, along with a magical hourglass and a mission: a chance to redeem himself by teaching two earthly people the true meaning of time.He returns to our world - now dominated by the hour-counting he so innocently began - and commences a journey with two unlikely partners: one a teenage girl who is about to give up on life, the other a wealthy old businessman who wants to live forever. To save himself, he must save them both. And stop the world to do so.

    For Arthur Dent, who has only just had his house demolished that morning, this seems already to be more than he can cope with. Sadly, however, the weekend has only just begun, and the Galaxy is a very very very large and startling place.THE RESTAURANT AT THE END OF THE UNIVERSE.When all questions of space, time, matter and the nature of being have been resolved, only one question remains --- "Where shall we have dinner?" The Restaurant at the End of the Universe provides the ultimate gastronomic experience, and for once there is no morning after to worry about.LIFE, THE UNIVERSE AND EVERYTHING.In consequence of a number of stunning catastrophes, Arthur Dent is surprised to find himself living in a hideously miserable cave on prehistoric Earth. However, just as he thinks that things cannot possibly get any worse, they suddenly do. He discovers that the Galaxy is not only mind-boggling big and bewildering but also that most of the things that happen in it are staggeringly unfair.SO LONG, AND THANKS FOR ALL THE FISH.Just as Arthur Dent's sense of reality is in its dickiest state he suddenly finds the girl of his dreams. He finds her in the last place in the Universe in which he would expect to find anything at all, but which 3,976,000 people will find oddly familiar. They go in search of God's Final Message to His Creation and, in a dramatic break with tradition, actually find it.

    Hand argues that extraordinarily rare events are anything but. In fact, they’re commonplace. Not only that, we should all expect to experience a miracle roughly once every month.     But Hand is no believer in superstitions, prophecies, or the paranormal. His definition of “miracle” is thoroughly rational. No mystical or supernatural explanation is necessary to understand why someone is lucky enough to win the lottery twice, or is destined to be hit by lightning three times and still survive. All we need, Hand argues, is a firm grounding in a powerful set of laws: the laws of inevitability, of truly large numbers, of selection, of the probability lever, and of near enough.     Together, these constitute Hand’s groundbreaking Improbability Principle. And together, they explain why we should not be so surprised to bump into a friend in a foreign country, or to come across the same unfamiliar word four times in one day. Hand wrestles with seemingly less explicable questions as well: what the Bible and Shakespeare have in common, why financial crashes are par for the course, and why lightning does strike the same place (and the same person) twice. Along the way, he teaches us how to use the Improbability Principle in our own lives—including how to cash in at a casino and how to recognize when a medicine is truly effective.     An irresistible adventure into the laws behind “chance” moments and a trusty guide for understanding the world and universe we live in, The Improbability Principle will transform how you think about serendipity and luck, whether it’s in the world of business and finance or you’re merely sitting in your backyard, tossing a ball into the air and wondering where it will land.

    A missing professor. Some nefarious men in gray suits. And a dreamworld called the Troposphere? Ariel Manto has a fascination with nineteenth-century scientists--especially Thomas Lumas and The End of Mr. Y, a book no one alive has read. When she mysteriously uncovers a copy at a used bookstore, Ariel is launched into an adventure of science and faith, consciousness and death, space and time, and everything in between. Seeking answers, Ariel follows in Mr. Y’s footsteps: She swallows a tincture, stares into a black dot, and is transported into the Troposphere--a wonderland where she can travel through time and space using the thoughts of others. There she begins to understand all the mysteries surrounding the book, herself, and the universe. Or is it all just a hallucination?