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.

    His investigations shed light on what we can ultimately know about the universe and the very nature of life. In an infectious and enthusiastic narrative, Chaitin delineates the specific intellectual and intuitive steps he took toward the discovery. He takes us to the very frontiers of scientific thinking, and helps us to appreciate the art—and the sheer beauty—in the science of math.

    

    This book contains my answer to that question. The purpose of the book is to tell the beginning student of advanced mathematics the basic set- theoretic facts of life, and to do so with the minimum of philosophical discourse and logical formalism. The point of view throughout is that of a prospective mathematician anxious to study groups, or integrals, or manifolds. From this point of view the concepts and methods of this book are merely some of the standard mathematical tools; the expert specialist will find nothing new here. Scholarly bibliographical credits and references are out of place in a purely expository book such as this one. The student who gets interested in set theory for its own sake should know, however, that there is much more to the subject than there is in this book. One of the most beautiful sources of set-theoretic wisdom is still Hausdorff's Set theory. A recent and highly readable addition to the literature, with an extensive and up-to-date bibliography, is Axiomatic set theory by Suppes.

    

    Offering inspiration and intellectual delight, the problems throughout the book encourage students to express their ideas in writing to explain how they conceive problems, what conjectures they make, and what conclusions they reach. Applying specific techniques and strategies, readers will acquire a solid understanding of the fundamental concepts and ideas of number theory.

    Topics: elimination of metaphysics, function of philosophy, nature of philosophical analysis, the a priori, truth & probability, critique of ethics & theology, self & the common world etc.IntroductionThe elimination of metaphysicsThe function of philosophy The nature of philosophical analysisThe a priori Truth & probabilityCritique of ethics & theologyThe self & the common worldSolutions of outstanding philosophical disputesIndex

    E. Moore, and Ludwig Wittgenstein in the four decades around the turn of the twentieth century, analytic philosophy established itself in various forms in the 1930s. After the Second World War, it developed further in North America, in the rest of Europe, and is now growing in influence as the dominant philosophical tradition right across the world, from Latin America to East Asia.In this Very Short Introduction Michael Beaney introduces some of the key ideas of the founders of analytic philosophy by exploring certain fundamental philosophical questions and showing how those ideas can be used in offering answers. Considering the work of Susan Stebbing, he also explores the application of analytic philosophy to critical thinking, and emphasizes the conceptual creativity that lies at the heart of fruitful analysis. Throughout, Beaney illustrates why clarity of thinking, precision of expression, and rigour of argumentation are rightly seen as virtues of analytic philosophy.

    Harry Gensler engages students with the basics of logic through practical examples and important arguments both in the history of philosophy and from contemporary philosophy. Using simple and manageable methods for testing arguments, students are led step-by-step to master the complexities of logic.The companion LogiCola instructional program and various teaching aids (including a teacher's manual) are available from the book's website: www.routledge.com/textbooks/gensler_l...

    The Clay Mathematics Institute is offering a prize of $1 million to anyone who can discover a general solution to the problem. In this book, Avner Ash and Robert Gross guide readers through the mathematics they need to understand this captivating problem.The key to the conjecture lies in elliptic curves, which are cubic equations in two variables. These equations may appear simple, yet they arise from some very deep--and often very mystifying--mathematical ideas. Using only basic algebra and calculus while presenting numerous eye-opening examples, Ash and Gross make these ideas accessible to general readers, and in the process venture to the very frontiers of modern mathematics. Along the way, they give an informative and entertaining introduction to some of the most profound discoveries of the last three centuries in algebraic geometry, abstract algebra, and number theory. They demonstrate how mathematics grows more abstract to tackle ever more challenging problems, and how each new generation of mathematicians builds on the accomplishments of those who preceded them. Ash and Gross fully explain how the Birch and Swinnerton-Dyer Conjecture sheds light on the number theory of elliptic curves, and how it provides a beautiful and startling connection between two very different objects arising from an elliptic curve, one based on calculus, the other on algebra.

    More than two centuries after Euler's death, it is still regarded as a conceptual diamond of unsurpassed beauty. Called Euler's identity or God's equation, it includes just five numbers but represents an astonishing revelation of hidden connections. It ties together everything from basic arithmetic to compound interest, the circumference of a circle, trigonometry, calculus, and even infinity. In David Stipp's hands, Euler's identity formula becomes a contemplative stroll through the glories of mathematics. The result is an ode to this magical field.

    In Out of Our Heads, he restates and reexamines the problem of consciousness, and then proposes a startling solution: Do away with the two hundred-year-old paradigm that places consciousness within the confines of the brain. Our culture is obsessed with the brain—how it perceives; how it remembers; how it determines our intelligence, our morality, our likes and our dislikes. It’s widely believed that consciousness itself, that Holy Grail of science and philosophy, will soon be given a neural explanation. And yet, after decades of research, only one proposition about how the brain makes us conscious—how it gives rise to sensation, feeling, and subjectivity—has emerged unchallenged: We don’t have a clue. In this inventive work, Noë suggests that rather than being something that happens inside us, consciousness is something we do. Debunking an outmoded philosophy that holds the scientific study of consciousness captive, Out of Our Heads is a fresh attempt at understanding our minds and how we interact with the world around us.

    Composed of such luminaries as Kurt Gödel and Rudolf Carnap, and stimulated by the works of Ludwig Wittgenstein and Karl Popper, the Vienna Circle left an indelible mark on science.Exact Thinking in Demented Times tells the often outrageous, sometimes tragic, and never boring stories of the men who transformed scientific thought. A revealing work of history, this landmark book pays tribute to those who dared to reinvent knowledge from the ground up.

    Barrow's Pi in the Sky is a profound -- and profoundly different -- exploration of the world of mathematics: where it comes from, what it is, and where it's going to take us if we follow it to the limit in our search for the ultimate meaning of the universe. Barrow begins by investigating whether math is a purely human invention inspired by our practical needs. Or is it something inherent in nature waiting to be discovered?In answering these questions, Barrow provides a bridge between the usually irreconcilable worlds of mathematics and theology. Along the way, he treats us to a history of counting all over the world, from Egyptian hieroglyphics to logical friction, from number mysticism to Marxist mathematics. And he introduces us to a host of peculiar individuals who have thought some of the deepest and strangest thoughts that human minds have ever thought, from Lao-Tse to Robert Pirsig, Charles Darwin, and Umberto Eco. Barrow thus provides the historical framework and the intellectual tools necessary to an understanding of some of today's weightiest mathematical concepts.