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An accessible and fascinating exploration of how Alan Turing’s mathematical theory gave rise to modern computer science and applications—from the desktops to cell phonesIn 1936, when he was just 24 years old, Alan Turing wrote a remarkable paper in which he outlined the theory of computation, laying out the ideas that underlie all modern computers. This groundbreaking and powerful theory now forms the basis of computer science. In Turing’s Vision, Chris Bernhardt explains the theory for the general reader, beginning with its foundations and systematically building to its surprising conclusions. He also views Turing’s theory in the context of mathematical history, other views of computation (including those of Alonzo Church), Turing’s later work, and the birth of the modern computer.Turing wanted to show that there were problems that were beyond any computer’s ability to solve; in particular, he wanted to find a decision problem that he could prove was undecidable. To explain Turing’s ideas, Bernhardt examines 3 well-known decision problems to explore the concept of undecidability; investigates theoretical computing machines, including Turing machines; explains universal machines; and proves that certain problems are undecidable, including Turing’s problem concerning computable numbers.

Throughout history, some books have changed the world. They have transformed the way we see ourselves - and each other. They have inspired debate, dissent, war and revolution. They have enlightened, outraged, provoked and comforted. They have enriched lives - and destroyed them. Now Penguin brings you the works of the great thinkers, pioneers, radicals and visionaries whose ideas shook civilization and helped make us who we are. The Father of Existentialism, Kierkegaard transformed philosophy with his conviction that we must all create our own nature; in this great work of religious anxiety, he argues that a true understanding of God can only be attained by making a personal 'leap of faith'.

Like masterpieces of art, music, and literature, great mathematical theorems are creative milestones, works of genius destined to last forever. Now William Dunham gives them the attention they deserve.Dunham places each theorem within its historical context and explores the very human and often turbulent life of the creator — from Archimedes, the absentminded theoretician whose absorption in his work often precluded eating or bathing, to Gerolamo Cardano, the sixteenth-century mathematician whose accomplishments flourished despite a bizarre array of misadventures, to the paranoid genius of modern times, Georg Cantor. He also provides step-by-step proofs for the theorems, each easily accessible to readers with no more than a knowledge of high school mathematics. A rare combination of the historical, biographical, and mathematical, Journey Through Genius is a fascinating introduction to a neglected field of human creativity.“It is mathematics presented as a series of works of art; a fascinating lingering over individual examples of ingenuity and insight. It is mathematics by lightning flash.” —Isaac Asimov

One of the major philosophical texts of the 20th century, Process and Reality is based on Alfred North Whitehead’s influential lectures that he delivered at the University of Edinburgh in the 1920s on process philosophy.Whitehead’s master work in philsophy, Process and Reality propounds a system of speculative philosophy, known as process philosophy, in which the various elements of reality into a consistent relation to each other. It is also an exploration of some of the preeminent thinkers of the seventeenth and eighteenth centuries, such as Descartes, Newton, Locke, and Kant.The ultimate edition of Whitehead’s magnum opus, Process and Reality is a standard reference for scholars of all backgrounds.

The story of philosophy is an epic tale: an exploration of the ideas, views and teachings of some of the most creative minds known to humanity. But since the long-popular classic Bertrand Russell's History of Western Philosophy, first published in 1945, there has been no comprehensive and entertaining, single-volume history of this great intellectual journey.With his characteristic clarity and elegance A. C. Grayling takes the reader from the world-views and moralities before the age of the Buddha, Confucius and Socrates, through Christianity's dominance of the European mind to the Renaissance and Enlightenment, and on to Mill, Nietzsche, Sartre, and philosophy today. And, since the story of philosophy is incomplete without mention of the great philosophical traditions of India, China and the Persian-Arabic world, he gives a comparative survey of them too.Intelligible for students and eye-opening for philosophy readers, he covers epistemology, metaphysics, ethics, aesthetics, logic, the philosophy of mind, the philosophy of language, political philosophy and the history of debates in these areas of enquiry, through the ideas of the celebrated philosophers as well as less well-known influential thinkers. He also asks what we have learnt from this body of thought, and what progress is still to be made.The first authoritative and accessible single-volume history of philosophy for decades, remarkable for its range and clarity, this is a landmark work.

It has only been a couple of decades since Benoit Mandelbrot published his famous picture of what is now called the Mandelbrot set. That picture, now seeming graphically primitive, has changed our view of the mathematical and physical universe. The properties and circumstances of the discovery of the Mandelbrot Set continue to generate much interest in the research community and beyond. This book contains the hard-to-obtain original papers, many unpublished illustrations dating back to 1979 and extensive documented historical context showing how Mandelbrot helped change our way of looking at the world.

Kurt Godel, the greatest logician of our time, startled the world of mathematics in 1931 with his Theorem of Undecidability, which showed that some statements in mathematics are inherently "undecidable." His work on the completeness of logic, the incompleteness of number theory, and the consistency of the axiom of choice and the continuum theory brought him further worldwide fame. In this introductory volume, Raymond Smullyan, himself a well-known logician, guides the reader through the fascinating world of Godel's incompleteness theorems. The level of presentation is suitable for anyone with a basic acquaintance with mathematical logic. As a clear, concise introduction to a difficult but essential subject, the book will appeal to mathematicians, philosophers, and computer scientists.

When is it right to go to war? The most persuasive answer to this question has always been 'in self-defense'. In a penetrating new analysis, bringing together moral philosophy, political science, and law, David Rodin shows what's wrong with this answer. He proposes a comprehensive new theory of the right of self-defense which resolves many of the perplexing questions that have dogged both jurists and moral philosophers. By applying the theory of self-defense to international relations, Rodin produces a far-reaching critique of the canonical Just War theory. The simple analogy between self-defense and national defense - between the individual and the state - needs to be fundamentally rethought, and with it many of the basic elements of international law and the ethics of international relations.

"[Mythologies] illustrates the beautiful generosity of Barthes's progressive interest in the meaning (his word is signification) of practically everything around him, not only the books and paintings of high art, but also the slogans, trivia, toys, food, and popular rituals (cruises, striptease, eating, wrestling matches) of contemporary life . . . For Barthes, words and objects have in common the organized capacity to say something; at the same time, since they are signs, words and objects have the bad faith always to appear natural to their consumer, as if what they say is eternal, true, necessary, instead of arbitrary, made, contingent. Mythologies finds Barthes revealing the fashioned systems of ideas that make it possible, for example, for 'Einstein's brain' to stand for, be the myth of, 'a genius so lacking in magic that one speaks about his thought as a functional labor analogous to the mechanical making of sausages.' Each of the little essays in this book wrenches a definition out of a common but constructed object, making the object speak its hidden, but ever-so-present, reservoir of manufactured sense."--Edward W. Said

Who ought to govern? Why should I obey the law? How should conflict be controlled? What is the proper education for a citizen and a statesman? These questions probe some of the deepest and most enduring problems that every society confronts, regardless of time and place. Today we ask the same crucial questions about law, authority, justice, and freedom that Plato, Aristotle, Machiavelli, Hobbes, Locke, Rousseau, and Tocqueville faced in previous centuries.In this lively and enlightening book, Professor Steven B. Smith introduces the wide terrain of political philosophy through the classic texts of the discipline. Works by the greatest thinkers illuminate the permanent problems of political life, Smith shows, and while we may not accept all their conclusions, it would be a mistake to overlook the relevance of their insights.