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.

    Discovered by an 18th century mathematician and preacher, Bayes' rule is a cornerstone of modern probability theory. In this richly illustrated book, intuitive visual representations of real-world examples are used to show how Bayes' rule is actually a form of commonsense reasoning. The tutorial style of writing, combined with a comprehensive glossary, makes this an ideal primer for novices who wish to gain an intuitive understanding of Bayesian analysis. As an aid to understanding, online computer code (in MatLab, Python and R) reproduces key numerical results and diagrams.Stone's book is renowned for its visually engaging style of presentation, which stems from teaching Bayes' rule to psychology students for over 10 years as a university lecturer.

    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. Inspired by the myth of a man condemned to ceaselessly push a rock up a mountain and watch it roll back to the valley below, The Myth of Sisyphus transformed twentieth-century philosophy with its impassioned argument for the value of life in a world without religious meaning.

    Part I describes questions and issues about mathematics that have motivated philosophers since the beginning of intellectual history. Part II is an historical survey, discussing the role of mathematics in the thought of such philosophers as Plato, Aristotle, Kant, and Mill. Part III covers the three major positions held throughout the twentieth century: the idea that mathematics is logic (logicism), the view that the essence of mathematics is the rule-governed manipulation of characters (formalism), and a revisionist philosophy that focuses on the mental activity of mathematics (intuitionism). Finally, Part IV brings the reader up-to-date with a look at contemporary developments within the discipline.This sweeping introductory guide to the philosophy of mathematics makes these fascinating concepts accessible to those with little background in either mathematics or philosophy.

    It is something we do. In Action in Perception, Noë argues that perception and perceptual consciousness depend on capacities for action and thought--that perception is a kind of thoughtful activity. Touch, not vision, should be our model for perception. Perception is not a process in the brain, but a kind of skillful activity of the body as a whole. We enact our perceptual experience. To perceive, according to this enactive approach to perception, is not merely to have sensations; it is to have sensations that we understand. In Action in Perception, Noë investigates the forms this understanding can take. He begins by arguing, on both phenomenological and empirical grounds, that the content of perception is not like the content of a picture; the world is not given to consciousness all at once but is gained gradually by active inquiry and exploration. Noë then argues that perceptual experience acquires content thanks to our possession and exercise of practical bodily knowledge, and examines, among other topics, the problems posed by spatial content and the experience of color. He considers the perspectival aspect of the representational content of experience and assesses the place of thought and understanding in experience. Finally, he explores the implications of the enactive approach for our understanding of the neuroscience of perception.

    Polanyi, originally a chemist and chemical physicist, is widely acclaimed for his epistemology which opposes the prevailing positivist approaches. His discussion of tacit knowledge has proved to be influential in many fields from theology to artificial intelligence. This text represnts a contribution to 20th-century thought, and continues to make valuable insights to our understanding of how knowledge functions.

    Throughout, the author provides a fair and balanced blending of the standard positions on epistemology with his own carefully reasoned positions or stances into the analysis of each concept. KEY TOPICS: Epistemological Questions. The Traditional Analysis of Knowledge. Modifying the Traditional Analysis of Knowledge. Evidentialist Theories of Justification. Non-evidentialist Theories of Knowledge and Justification. Skepticism. Epistemology and Science. Relativism. MARKET: For anyone interested in the philosophy of knowledge and rationality.

    Also, to open a method for rendering the sciences more easy, useful, and compendious.

    Discourses and Selected Writings is a transcribed collection of informal lectures given by the philosopher around AD 108. A gateway into the life and mind of a great intellectual, it is also an important example of the usage of Koine or ?common? Greek, an ancestor to Standard Modern Greek.

    From Aristotle, Plato, Epicurus, Confucius, Cicero and Heraclitus in ancient times to 17th century rationalists Descartes, Leibniz and Spinoza, from 20th-century greats Jean-Paul Sartre, Jean Baudrillard and Simone de Beauvoir to contemporary thinkers Michael Sandel, Peter Singer and Slavoj Zizek, 50 Philosophy Classics explores key writings that have shaped the discipline and had an impact on the real world. Philosophy can no longer be confined to academia, and 50 Philosophy Classics shows how powerful it can be as a tool for opening our minds and helping us think. Whether you are fascinated or daunted by the big questions of how to think, how to be, how to act and how to see, this is the perfect introduction to some of humanity's greatest minds and their landmark books.

    The author's work can serve either as a textbook in a university course on metaphysics or as an introduction to metaphysical thinking for the interested reader. This second edition, revised though not fundamentally changed, includes the basis of the first edition with a new chapter on the nature of time.

    This volume provides basic writings of Dostoevsky, Kierkegaard, Nietzsche, Rilke, Kafka, Ortega, Jaspers, Heidegger, Sartre, and Camus, including some not previously translated, along with an invaluable introductory essay by Walter Kaufmann.

    It aim is nothing less than insight into insight itself, a comprehensive view of knowledge and understanding, and to state what one needs to understand and how one proceeds to understand it.In Lonergan's own words: 'Thoroughly understand what it is to understand, and not only will you understand the broad lines of all there is to be understood but also you will possess a fixed base, and invariant pattern, opening upon all further developments of understanding.'The editors of the Collected Works of Bernard Lonergan have established the definitive text for Insight after examining all the variant forms in Lonergan's manuscripts and papers. The volume includes introductory material and annotation to enable the reader to appreciate more fully this challenging work.

    How can it be possible that the world we see is not objective reality? And how can our senses be useful if they are not communicating the truth? Hoffman grapples with these questions and more over the course of this eye-opening work.Ever since Homo sapiens has walked the earth, natural selection has favored perception that hides the truth and guides us toward useful action, shaping our senses to keep us alive and reproducing. We observe a speeding car and do not walk in front of it; we see mold growing on bread and do not eat it. These impressions, though, are not objective reality. Just like a file icon on a desktop screen is a useful symbol rather than a genuine representation of what a computer file looks like, the objects we see every day are merely icons, allowing us to navigate the world safely and with ease.The real-world implications for this discovery are huge. From examining why fashion designers create clothes that give the illusion of a more “attractive” body shape to studying how companies use color to elicit specific emotions in consumers, and even dismantling the very notion that spacetime is objective reality, The Case Against Reality dares us to question everything we thought we knew about the world we see.

    This Penguin Classics edition of The Complete Essays is translated from the French and edited with an introduction and notes by M.A. Screech.In 1572 Montaigne retired to his estates in order to devote himself to leisure, reading and reflection. There he wrote his constantly expanding 'assays', inspired by the ideas he found in books contained in his library and from his own experience. He discusses subjects as diverse as war-horses and cannibals, poetry and politics, sex and religion, love and friendship, ecstasy and experience. But, above all, Montaigne studied himself as a way of drawing out his own inner nature and that of men and women in general. The Essays are among the most idiosyncratic and personal works in all literature and provide an engaging insight into a wise Renaissance mind, continuing to give pleasure and enlightenment to modern readers.With its extensive introduction and notes, M.A. Screech's edition of Montaigne is widely regarded as the most distinguished of recent times.Michel de Montaigne (1533-1586) studied law and spent a number of years working as a counsellor before devoting his life to reading, writing and reflection.If you enjoyed The Complete Essays, you might like Francois Rabelais's Gargantua and Pantagruel, also available in Penguin Classics.'Screech's fine version ... must surely serve as the definitive English Montaigne'A.C. Grayling, Financial Times'A superb edition'Nicholas Wollaston, Observer