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Philosophy of Mathematics

Philosophy of Mathematics

  • Author: Øystein Linnebo
  • Publisher: Princeton University Press
  • ISBN: 1400885248
  • Category: Philosophy
  • Page: 216
  • View: 5029
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A sophisticated, original introduction to the philosophy of mathematics from one of its leading contemporary scholars Mathematics is one of humanity's most successful yet puzzling endeavors. It is a model of precision and objectivity, but appears distinct from the empirical sciences because it seems to deliver nonexperiential knowledge of a nonphysical reality of numbers, sets, and functions. How can these two aspects of mathematics be reconciled? This concise book provides a systematic yet accessible introduction to the field that is trying to answer that question: the philosophy of mathematics. Written by Øystein Linnebo, one of the world's leading scholars on the subject, the book introduces all of the classical approaches to the field, including logicism, formalism, intuitionism, empiricism, and structuralism. It also contains accessible introductions to some more specialized issues, such as mathematical intuition, potential infinity, the iterative conception of sets, and the search for new mathematical axioms. The groundbreaking work of German mathematician and philosopher Gottlob Frege, one of the founders of analytic philosophy, figures prominently throughout the book. Other important thinkers whose work is introduced and discussed include Immanuel Kant, John Stuart Mill, David Hilbert, Kurt Gödel, W. V. Quine, Paul Benacerraf, and Hartry H. Field. Sophisticated but clear and approachable, this is an essential introduction for all students and teachers of philosophy, as well as mathematicians and others who want to understand the foundations of mathematics.

Philosophical Logic

Philosophical Logic

  • Author: John P. Burgess
  • Publisher: Princeton University Press
  • ISBN: 9780691137896
  • Category: Philosophy
  • Page: 153
  • View: 2493
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Philosophical Logic is a clear and concise critical survey of nonclassical logics of philosophical interest written by one of the world's leading authorities on the subject. After giving an overview of classical logic, John Burgess introduces five central branches of nonclassical logic (temporal, modal, conditional, relevantistic, and intuitionistic), focusing on the sometimes problematic relationship between formal apparatus and intuitive motivation. Requiring minimal background and arranged to make the more technical material optional, the book offers a choice between an overview and in-depth study, and it balances the philosophical and technical aspects of the subject. The book emphasizes the relationship between models and the traditional goal of logic, the evaluation of arguments, and critically examines apparatus and assumptions that often are taken for granted. Philosophical Logic provides an unusually thorough treatment of conditional logic, unifying probabilistic and model-theoretic approaches. It underscores the variety of approaches that have been taken to relevantistic and related logics, and it stresses the problem of connecting formal systems to the motivating ideas behind intuitionistic mathematics. Each chapter ends with a brief guide to further reading. Philosophical Logic addresses students new to logic, philosophers working in other areas, and specialists in logic, providing both a sophisticated introduction and a new synthesis.

Philosophy of Language

Philosophy of Language

  • Author: Scott Soames
  • Publisher: Princeton University Press
  • ISBN: 9781400833931
  • Category: Philosophy
  • Page: 200
  • View: 6711
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In this book one of the world's foremost philosophers of language presents his unifying vision of the field--its principal achievements, its most pressing current questions, and its most promising future directions. In addition to explaining the progress philosophers have made toward creating a theoretical framework for the study of language, Scott Soames investigates foundational concepts--such as truth, reference, and meaning--that are central to the philosophy of language and important to philosophy as a whole. The first part of the book describes how philosophers from Frege, Russell, Tarski, and Carnap to Kripke, Kaplan, and Montague developed precise techniques for understanding the languages of logic and mathematics, and how these techniques have been refined and extended to the study of natural human languages. The book then builds on this account, exploring new thinking about propositions, possibility, and the relationship between meaning, assertion, and other aspects of language use. An invaluable overview of the philosophy of language by one of its most important practitioners, this book will be essential reading for all serious students of philosophy.

Philosophy of Physics

Philosophy of Physics

Space and Time

  • Author: Tim Maudlin
  • Publisher: Princeton University Press
  • ISBN: 0691143099
  • Category: Philosophy
  • Page: 183
  • View: 2783
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Introduces non-physicists to core philosophical issues surrounding the nature & structure of space & time, & is also an ideal resource for physicists interested in the conceptual foundations of space-time theory. Provides a broad historical overview, from Aristotle to Einstein, & covers the Twins Paradox, Galilean relativity, time travel, & more.

An Introduction to the Philosophy of Mathematics

An Introduction to the Philosophy of Mathematics

  • Author: Mark Colyvan
  • Publisher: Cambridge University Press
  • ISBN: 0521826020
  • Category: Mathematics
  • Page: 188
  • View: 3129
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This introduction to the philosophy of mathematics focuses on contemporary debates in an important and central area of philosophy. The reader is taken on a fascinating and entertaining journey through some intriguing mathematical and philosophical territory, including such topics as the realism/anti-realism debate in mathematics, mathematical explanation, the limits of mathematics, the significance of mathematical notation, inconsistent mathematics and the applications of mathematics. Each chapter has a number of discussion questions and recommended further reading from both the contemporary literature and older sources. Very little mathematical background is assumed and all of the mathematics encountered is clearly introduced and explained using a wide variety of examples. The book is suitable for an undergraduate course in philosophy of mathematics and, more widely, for anyone interested in philosophy and mathematics.

Philosophy of Mathematics

Philosophy of Mathematics

Selected Readings

  • Author: Paul Benacerraf,Hilary Putnam
  • Publisher: Cambridge University Press
  • ISBN: 1107268133
  • Category: Science
  • Page: N.A
  • View: 7051
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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.

Truth

Truth

  • Author: Alexis G. Burgess,John P. Burgess
  • Publisher: Princeton University Press
  • ISBN: 9781400838691
  • Category: Philosophy
  • Page: 176
  • View: 5450
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This is a concise introduction to current philosophical debates about truth. Combining philosophical and technical material, the book is organized around, but not limited to, the view known as deflationism. In clear language, Burgess and Burgess cover a wide range of issues, including the nature of truth, the status of truth-value gaps, the relationship between truth and meaning, relativism and pluralism about truth, and semantic paradoxes from Alfred Tarski to Saul Kripke and beyond. The book provides a rich picture of contemporary philosophical theorizing about truth, one that will be essential reading for philosophy students as well as philosophers specializing in other areas.

The Philosophy of Mathematical Practice

The Philosophy of Mathematical Practice

  • Author: Paolo Mancosu
  • Publisher: Oxford University Press on Demand
  • ISBN: 0199296456
  • Category: Philosophy
  • Page: 447
  • View: 5021
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This book gives a coherent and unified presentation of a new direction of work in philosophy of mathematics. This new approach in philosophy of mathematics requires extensive attention to mathematical practice and provides philosophical analyses of important novel characteristics of contemporary (twentieth century) mathematics and of many aspects of mathematical activity-such as visualization, explanation, understanding etc.-- which escape purely formal logicaltreatment.The book consists of a lengthy introduction by the editor and of eight chapters written by some of the very best scholars in this area. Each chapter consists of a short introduction to the general topic of the chapter and of a longer research article in the very same area. Theeight topics selected represent a broad spectrum of the contemporary philosophical reflection on different aspects of mathematical practice: Diagrammatic reasoning and representational systems; Visualization; Mathematical Explanation; Purity of Methods; Mathematical Concepts; Philosophical relevance of category theory; Philosophical aspects of computer science in mathematics; Philosophical impact of recent developments in mathematical physics.

Philosophy of Mathematics

Philosophy of Mathematics

Structure and Ontology

  • Author: Stewart Shapiro
  • Publisher: Oxford University Press
  • ISBN: 9780198025450
  • Category: Philosophy
  • Page: 296
  • View: 7782
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Do numbers, sets, and so forth, exist? What do mathematical statements mean? Are they literally true or false, or do they lack truth values altogether? Addressing questions that have attracted lively debate in recent years, Stewart Shapiro contends that standard realist and antirealist accounts of mathematics are both problematic. As Benacerraf first noted, we are confronted with the following powerful dilemma. The desired continuity between mathematical and, say, scientific language suggests realism, but realism in this context suggests seemingly intractable epistemic problems. As a way out of this dilemma, Shapiro articulates a structuralist approach. On this view, the subject matter of arithmetic, for example, is not a fixed domain of numbers independent of each other, but rather is the natural number structure, the pattern common to any system of objects that has an initial object and successor relation satisfying the induction principle. Using this framework, realism in mathematics can be preserved without troublesome epistemic consequences. Shapiro concludes by showing how a structuralist approach can be applied to wider philosophical questions such as the nature of an "object" and the Quinean nature of ontological commitment. Clear, compelling, and tautly argued, Shapiro's work, noteworthy both in its attempt to develop a full-length structuralist approach to mathematics and to trace its emergence in the history of mathematics, will be of deep interest to both philosophers and mathematicians.

Defending the Axioms

Defending the Axioms

On the Philosophical Foundations of Set Theory

  • Author: Penelope Maddy
  • Publisher: Oxford University Press
  • ISBN: 0199596182
  • Category: Mathematics
  • Page: 150
  • View: 1729
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Mathematics depends on proofs, and proofs must begin somewhere, from some fundamental assumptions. The axioms of set theory have long played this role, so the question of how they are properly judged is of central importance. Maddy discusses the appropriate methods for such evaluations and the philosophical backdrop that makes them appropriate.

Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century

Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century

  • Author: Paolo Mancosu
  • Publisher: Oxford University Press on Demand
  • ISBN: 0195132440
  • Category: Drama
  • Page: 275
  • View: 995
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The seventeenth century saw dramatic advances in mathematical theory and practice than any era before or since. With the recovery of many of the classical Greek mathematical texts, new techniques were introduced, and within 100 years, analytic geometry, the geometry of indivisibles, the arithmetic of infinites, and the calculus had been developed. Although many technical studies have been devoted to these innovations, Paolo Mancosu provides the first comprehensive account of the relationship between mathematical advances of the seventeenth century and the philosophy of mathematics of the period. Beginning with the Renaissance debates on the certainty of mathematics, Mancosu leads the reader through the foundational issues raised by the emergence of these new mathematical techniques, including the influence of the Aristotelian conception of science in Cavalieri and Guldin, the foundational relevance of Descartes' Geometrie, the relationship between empiricist epistemology and infinitistic theorems in geometry, and the debates concerning the foundations of the Leibnizian calculus In the process Mancosu draws a sophisticated picture of the subtle dependencies between technical development and philosophical reflection in seventeenth century mathematics.

Fixing Frege

Fixing Frege

  • Author: John P. Burgess
  • Publisher: Princeton University Press
  • ISBN: 0691187061
  • Category: Philosophy
  • Page: N.A
  • View: 8450
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Philosophy of Biology

Philosophy of Biology

  • Author: Peter Godfrey-Smith
  • Publisher: Princeton University Press
  • ISBN: 1400850444
  • Category: Science
  • Page: 200
  • View: 2357
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This is a concise, comprehensive, and accessible introduction to the philosophy of biology written by a leading authority on the subject. Geared to philosophers, biologists, and students of both, the book provides sophisticated and innovative coverage of the central topics and many of the latest developments in the field. Emphasizing connections between biological theories and other areas of philosophy, and carefully explaining both philosophical and biological terms, Peter Godfrey-Smith discusses the relation between philosophy and science; examines the role of laws, mechanistic explanation, and idealized models in biological theories; describes evolution by natural selection; and assesses attempts to extend Darwin's mechanism to explain changes in ideas, culture, and other phenomena. Further topics include functions and teleology, individuality and organisms, species, the tree of life, and human nature. The book closes with detailed, cutting-edge treatments of the evolution of cooperation, of information in biology, and of the role of communication in living systems at all scales. Authoritative and up-to-date, this is an essential guide for anyone interested in the important philosophical issues raised by the biological sciences.

The Philosophy of Schopenhauer

The Philosophy of Schopenhauer

  • Author: Dale Jacquette
  • Publisher: Routledge
  • ISBN: 1317494482
  • Category: Philosophy
  • Page: 320
  • View: 7034
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Dale Jacquette charts the development of Schopenhauer's ideas from the time of his early dissertation on The Fourfold Root of the Principle of Sufficient Reason through the two editions of his magnum opus The World as Will and Representation to his later collections of philosophical aphorisms and competition essays. Jacquette explores the central topics in Schopenhauer's philosophy including his metaphysics of the world as representation and Will, his so-called pessimistic philosophical appraisal of the human condition, his examination of the concept of death, his dualistic analysis of free will, and his simplified non-Kantian theory of morality. Jacquette shows how these many complex themes fit together in a unified portrait of Schopenhauer's philosophy. The synthesis of Plato, Kant and Buddhist and Hindu ideas is given particular attention as is his influence on Nietzsche, first a follower and then arch opponent of Schopenhauer's thought, and the early Wittgenstein. The book provides a comprehensive and in-depth historical and philosophical introduction to Schopenhauer's distinctive contribution to philosophy.

Set Theory and its Philosophy

Set Theory and its Philosophy

A Critical Introduction

  • Author: Michael Potter
  • Publisher: Clarendon Press
  • ISBN: 0191556432
  • Category: Philosophy
  • Page: 360
  • View: 2617
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Michael Potter presents a comprehensive new philosophical introduction to set theory. Anyone wishing to work on the logical foundations of mathematics must understand set theory, which lies at its heart. Potter offers a thorough account of cardinal and ordinal arithmetic, and the various axiom candidates. He discusses in detail the project of set-theoretic reduction, which aims to interpret the rest of mathematics in terms of set theory. The key question here is how to deal with the paradoxes that bedevil set theory. Potter offers a strikingly simple version of the most widely accepted response to the paradoxes, which classifies sets by means of a hierarchy of levels. What makes the book unique is that it interweaves a careful presentation of the technical material with a penetrating philosophical critique. Potter does not merely expound the theory dogmatically but at every stage discusses in detail the reasons that can be offered for believing it to be true. Set Theory and its Philosophy is a key text for philosophy, mathematical logic, and computer science.

Philosophies of Mathematics

Philosophies of Mathematics

  • Author: Alexander L. George,Daniel Velleman
  • Publisher: Wiley-Blackwell
  • ISBN: 9780631195443
  • Category: Science
  • Page: 244
  • View: 6847
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This book provides an accessible, critical introduction to the three main approaches that dominated work in the philosophy of mathematics during the twentieth century: logicism, intuitionism and formalism.

Philosophy of Mathematics

Philosophy of Mathematics

An Introduction

  • Author: David Bostock
  • Publisher: John Wiley & Sons
  • ISBN: 1405189924
  • Category: Mathematics
  • Page: 332
  • View: 6828
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Philosophy of Mathematics: An Introduction provides a critical analysis of the major philosophical issues and viewpoints in the concepts and methods of mathematics - from antiquity to the modern era. Offers beginning readers a critical appraisal of philosophical viewpoints throughout history Gives a separate chapter to predicativism, which is often (but wrongly) treated as if it were a part of logicism Provides readers with a non-partisan discussion until the final chapter, which gives the author?s personal opinion on where the truth lies Designed to be accessible to both undergraduates and graduate students, and at the same time to be of interest to professionals

Epistemology

Epistemology

  • Author: Ernest Sosa
  • Publisher: Princeton University Press
  • ISBN: 1400883059
  • Category: Philosophy
  • Page: 256
  • View: 9504
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In this concise book, one of the world's leading epistemologists provides a sophisticated, revisionist introduction to the problem of knowledge in Western philosophy. Modern and contemporary accounts of epistemology tend to focus on limited questions of knowledge and skepticism, such as how we can know the external world, other minds, the past through memory, the future through induction, or the world’s depth and structure through inference. This book steps back for a better view of the more general issues posed by the ancient Greek Pyrrhonists. Returning to and illuminating this older, broader epistemological tradition, Ernest Sosa develops an original account of the subject, giving it substance not with Cartesian theology but with science and common sense. Descartes is a part of this ancient tradition, but he goes beyond it by considering not just whether knowledge is possible at all but also how we can properly attain it. In Cartesian epistemology, Sosa finds a virtue-theoretic account, one that he extends beyond the Cartesian context. Once epistemology is viewed in this light, many of its problems can be solved or fall away. The result is an important reevaluation of epistemology that will be essential reading for students and teachers.

Abstractionism

Abstractionism

Essays in Philosophy of Mathematics

  • Author: Philip A. Ebert,Marcus Rossberg
  • Publisher: Oxford University Press
  • ISBN: 0199645264
  • Category:
  • Page: 368
  • View: 465
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Abstractionism, which is a development of Frege's original Logicism, is a recent and much debated position in the philosophy of mathematics. This volume contains 16 original papers by leading scholars on the philosophical and mathematical aspects of Abstractionism. After an extensive editors' introduction to the topic of abstractionism, five contributions deal with the semantics and meta-ontology of Abstractionism, as well as the so-called Caesar Problem. Four papers then discuss abstractionist epistemology, focusing on the idea of implicit definitions and non-evidential warrants (entitlements) to account for a priori mathematical knowledge. This is followed by four chapters concerning the mathematics of Abstractionism, in particular the issue of impredicativity, the Bad Company objection, and the question of abstractionist set theory. Finally, the last section of the book contains three contributions that discuss Frege's application constraint within an abstractionist setting.

History and Philosophy of Modern Mathematics

History and Philosophy of Modern Mathematics

  • Author: William Aspray,Philip Kitcher
  • Publisher: U of Minnesota Press
  • ISBN: 9780816615674
  • Category: Mathematics
  • Page: 386
  • View: 3166
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History and Philosophy of Modern Mathematics was first published in 1988. Minnesota Archive Editions uses digital technology to make long-unavailable books once again accessible, and are published unaltered from the original University of Minnesota Press editions. The fourteen essays in this volume build on the pioneering effort of Garrett Birkhoff, professor of mathematics at Harvard University, who in 1974 organized a conference of mathematicians and historians of modern mathematics to examine how the two disciplines approach the history of mathematics. In History and Philosophy of Modern Mathematics, William Aspray and Philip Kitcher bring together distinguished scholars from mathematics, history, and philosophy to assess the current state of the field. Their essays, which grow out of a 1985 conference at the University of Minnesota, develop the basic premise that mathematical thought needs to be studied from an interdisciplinary perspective. The opening essays study issues arising within logic and the foundations of mathematics, a traditional area of interest to historians and philosophers. The second section examines issues in the history of mathematics within the framework of established historical periods and questions. Next come case studies that illustrate the power of an interdisciplinary approach to the study of mathematics. The collection closes with a look at mathematics from a sociohistorical perspective, including the way institutions affect what constitutes mathematical knowledge.