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In this ambitious study, David Corfield attacks the widely held view that it is the nature of mathematical knowledge which has shaped the way in which mathematics is treated philosophically and claims that contingent factors have brought us to the present thematically limited discipline. Illustrating his discussion with a wealth of examples, he sets out a variety of approaches to new thinking about the philosophy of mathematics, ranging from an exploration of whether computers producing mathematical proofs or conjectures are doing real mathematics, to the use of analogy, the prospects for a Bayesian confirmation theory, the notion of a mathematical research programme and the ways in which new concepts are justified. His inspiring book challenges both philosophers and mathematicians to develop the broadest and richest philosophical resources for work in their disciplines and points clearly to the ways in which this can be done.
Dem Versuch, die These zu stützen, daß Logik und Mathematik eins seien, hat Russell mehrere Bücher gewidmet, unter anderem das dreibändige, gemeinsam mit A. N. Whitehead verfaßte Werk "Principia Mathematica" (1910-1913). Die "Einführung in die mathematische Philosophie" faßt die Ergebnisse dieser Untersuchungen zusammen, ohne Kenntnisse der mathematischen Symbolik vorauszusetzen. Sie ist zuweilen und mit Recht "eine bewundernswerte Exposition des Monumentalwerks Principia Mathematica" genannt worden; und sie ist zugleich etwas anderes, insofern sie eine relativ eigenständige Einführung in die Grundlagen der Mathematik und der Erkenntnistheorie darstellt.Das Buch entstand 1918 im Gefängnis von Brixton, wo Russell eine sechsmonatige Haftstrafe für seine pazifistische Tätigkeit während des 1. Weltkrieges absaß. Es ist sehr anregend zu lesen, wie beinahe alles, was Bertrand Russell geschrieben hat, und es ist ein Buch von der Art, wie es nur jemand wie Russell schreiben kann, wenn er im Gefängnis sitzt und keine Hilfsmittel hat und sich daher entschließt, allen technischen Ballast abzustreifen. Anders als die heute üblichen Texte im Bereich der Philosophie der Mathematik läßt Russell seine Leser immer an seinem Denken teilhaben, an seinen Vermutungen und Irrtümern und an der Begeisterung, die er bei der Beschäftigung mit seinem Gegenstand empfindet. Da er einer der herausragenden Protagonisten des modernen wissenschaftlichen Empirismus und einer der Begründer der heute dominierenden Philosophie der Mathematik ist, gewinnt man auf diese Weise aus seinen Schriften einen einzigartigen Einblick in die Wechselfälle und Ideen der erkenntnistheoretischen und logischen Diskussionen dieses Jahrhunderts.Die Ausgabe bietet eine revidierte Fassung der deutschen Übersetzung des in den 20er Jahren prominenten Mathematikers E. J. Gumbel sowie W. Gordon.
In the four decades since Imre Lakatos declared mathematics a "quasi-empirical science," increasing attention has been paid to the process of proof and argumentation in the field -- a development paralleled by the rise of computer technology and the mounting interest in the logical underpinnings of mathematics. Explanantion and Proof in Mathematics assembles perspectives from mathematics education and from the philosophy and history of mathematics to strengthen mutual awareness and share recent findings and advances in their interrelated fields. With examples ranging from the geometrists of the 17th century and ancient Chinese algorithms to cognitive psychology and current educational practice, contributors explore the role of refutation in generating proofs, the varied links between experiment and deduction, the use of diagrammatic thinking in addition to pure logic, and the uses of proof in mathematics education (including a critique of "authoritative" versus "authoritarian" teaching styles). A sampling of the coverage: The conjoint origins of proof and theoretical physics in ancient Greece. Proof as bearers of mathematical knowledge. Bridging knowing and proving in mathematical reasoning. The role of mathematics in long-term cognitive development of reasoning. Proof as experiment in the work of Wittgenstein. Relationships between mathematical proof, problem-solving, and explanation. Explanation and Proof in Mathematics is certain to attract a wide range of readers, including mathematicians, mathematics education professionals, researchers, students, and philosophers and historians of mathematics.
For the majority of the twentieth century, philosophers of mathematics focused their attention on foundational questions. However, in the last quarter of the century they began to return to basics, and two new schools of thought were created: social constructivism and structuralism. The advent of the computer also led to proofs and development of mathematics assisted by computer, and to questions concerning the role of the computer in mathematics. This book of sixteen original essays is the first to explore this range of new developments in the philosophy of mathematics, in a language accessible to mathematicians. Approximately half the essays were written by mathematicians, and consider questions that philosophers have not yet discussed. The other half, written by philosophers of mathematics, summarise the discussion in that community during the last 35 years. A connection is made in each case to issues relevant to the teaching of mathematics.
The twentieth century has witnessed an unprecedented 'crisis in the foundations of mathematics', featuring a world-famous paradox (Russell's Paradox), a challenge to 'classical' mathematics from a world-famous mathematician (the 'mathematical intuitionism' of Brouwer), a new foundational school (Hilbert's Formalism), and the profound incompleteness results of Kurt Gödel. In the same period, the cross-fertilization of mathematics and philosophy resulted in a new sort of 'mathematical philosophy', associated most notably (but in different ways) with Bertrand Russell, W. V. Quine, and Gödel himself, and which remains at the focus of Anglo-Saxon philosophical discussion. The present collection brings together in a convenient form the seminal articles in the philosophy of mathematics by these and other major thinkers. It is a substantially revised version of the edition first published in 1964 and includes a revised bibliography. The volume will be welcomed as a major work of reference at this level in the field.
More Precisely provides a rigorous and engaging introduction to the mathematics necessary to do philosophy. It is impossible to fully understand much of the most important work in contemporary philosophy without a basic grasp of set theory, functions, probability, modality and infinity. Until now, this knowledge was difficult to acquire. Professors had to provide custom handouts to their classes, while students struggled through math texts searching for insight. More Precisely fills this key gap. Eric Steinhart provides lucid explanations of the basic mathematical concepts and sets out most commonly used notational conventions. Furthermore, he demonstrates how mathematics applies to many fundamental issues in branches of philosophy such as metaphysics, philosophy of language, epistemology, and ethics.

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