Thinking about Mathematics

The Philosophy of Mathematics

Author: Stewart Shapiro

Publisher: OUP Oxford

ISBN: 9780192893062

Category: Philosophy

Page: 328

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Thinking about Mathematics covers the range of philosophical issues and positions concerning mathematics. The text describes the questions about mathematics that motivated philosophers throughout history and covers historical figures such as Plato, Aristotle, Kant, and Mill. It also presents the major positions and arguments concerning mathematics throughout the twentieth century, bringing the reader up to the present positions and battle lines.

An Introduction to the Philosophy of Mathematics

Author: Mark Colyvan

Publisher: Cambridge University Press

ISBN: 0521826020

Category: Mathematics

Page: 188

View: 8763

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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

Selected Readings

Author: Paul Benacerraf,Hilary Putnam

Publisher: Cambridge University Press

ISBN: 1107268133

Category: Science

Page: N.A

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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.

New Directions in the Philosophy of Mathematics

An Anthology

Author: Thomas Tymoczko

Publisher: Princeton University Press

ISBN: 9780691034980

Category: Mathematics

Page: 436

View: 1961

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The traditional debate among philosophers of mathematics is whether there is an external mathematical reality, something out there to be discovered, or whether mathematics is the product of the human mind. This provocative book, now available in a revised and expanded paperback edition, goes beyond foundationalist questions to offer what has been called a "postmodern" assessment of the philosophy of mathematics--one that addresses issues of theoretical importance in terms of mathematical experience. By bringing together essays of leading philosophers, mathematicians, logicians, and computer scientists, Thomas Tymoczko reveals an evolving effort to account for the nature of mathematics in relation to other human activities. These accounts include such topics as the history of mathematics as a field of study, predictions about how computers will influence the future organization of mathematics, and what processes a proof undergoes before it reaches publishable form. This expanded edition now contains essays by Penelope Maddy, Michael D. Resnik, and William P. Thurston that address the nature of mathematical proofs. The editor has provided a new afterword and a supplemental bibliography of recent work.

Philosophy of Mathematics

Structure and Ontology

Author: Stewart Shapiro

Publisher: Oxford University Press

ISBN: 9780198025450

Category: Philosophy

Page: 296

View: 7385

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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.

The Philosophy of Mathematical Practice

Author: Paolo Mancosu

Publisher: Oxford University Press on Demand

ISBN: 0199296456

Category: Philosophy

Page: 447

View: 1903

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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.

Towards a Philosophy of Real Mathematics

Author: David Corfield

Publisher: Cambridge University Press

ISBN: 9781139436397

Category: Philosophy

Page: N.A

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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.

Platonism and Anti-Platonism in Mathematics

Author: Mark Balaguer

Publisher: Oxford University Press on Demand

ISBN: 9780195143980

Category: Mathematics

Page: 217

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In this book, Balaguer demonstrates that there are no good arguments for or against mathematical platonism. He establishes that both platonism and anti-platonism are defensible views and introduces a form of platonism ("full-blooded platonism") that solves all problems traditionally associated with the view, proceeding to defend anti-platonism (in particular, mathematical fictionalism) against various attacks--most notably the Quine-Putnam indispensability attack.

How Mathematicians Think

Using Ambiguity, Contradiction, and Paradox to Create Mathematics

Author: William Byers

Publisher: Princeton University Press

ISBN: 9780691145990

Category: Mathematics

Page: 424

View: 5124

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To many outsiders, mathematicians appear to think like computers, grimly grinding away with a strict formal logic and moving methodically--even algorithmically--from one black-and-white deduction to another. Yet mathematicians often describe their most important breakthroughs as creative, intuitive responses to ambiguity, contradiction, and paradox. A unique examination of this less-familiar aspect of mathematics, How Mathematicians Think reveals that mathematics is a profoundly creative activity and not just a body of formalized rules and results. Nonlogical qualities, William Byers shows, play an essential role in mathematics. Ambiguities, contradictions, and paradoxes can arise when ideas developed in different contexts come into contact. Uncertainties and conflicts do not impede but rather spur the development of mathematics. Creativity often means bringing apparently incompatible perspectives together as complementary aspects of a new, more subtle theory. The secret of mathematics is not to be found only in its logical structure. The creative dimensions of mathematical work have great implications for our notions of mathematical and scientific truth, and How Mathematicians Think provides a novel approach to many fundamental questions. Is mathematics objectively true? Is it discovered or invented? And is there such a thing as a "final" scientific theory? Ultimately, How Mathematicians Think shows that the nature of mathematical thinking can teach us a great deal about the human condition itself.

Introducing Philosophy of Mathematics

Author: Michele Friend

Publisher: Routledge

ISBN: 1317493788

Category: Philosophy

Page: 240

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What is mathematics about? Does the subject-matter of mathematics exist independently of the mind or are they mental constructions? How do we know mathematics? Is mathematical knowledge logical knowledge? And how is mathematics applied to the material world? In this introduction to the philosophy of mathematics, Michele Friend examines these and other ontological and epistemological problems raised by the content and practice of mathematics. Aimed at a readership with limited proficiency in mathematics but with some experience of formal logic it seeks to strike a balance between conceptual accessibility and correct representation of the issues. Friend examines the standard theories of mathematics - Platonism, realism, logicism, formalism, constructivism and structuralism - as well as some less standard theories such as psychologism, fictionalism and Meinongian philosophy of mathematics. In each case Friend explains what characterises the position and where the divisions between them lie, including some of the arguments in favour and against each. This book also explores particular questions that occupy present-day philosophers and mathematicians such as the problem of infinity, mathematical intuition and the relationship, if any, between the philosophy of mathematics and the practice of mathematics. Taking in the canonical ideas of Aristotle, Kant, Frege and Whitehead and Russell as well as the challenging and innovative work of recent philosophers like Benacerraf, Hellman, Maddy and Shapiro, Friend provides a balanced and accessible introduction suitable for upper-level undergraduate courses and the non-specialist.

Plato's Ghost

The Modernist Transformation of Mathematics

Author: Jeremy Gray

Publisher: Princeton University Press

ISBN: 9781400829040

Category: Mathematics

Page: 528

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Plato's Ghost is the first book to examine the development of mathematics from 1880 to 1920 as a modernist transformation similar to those in art, literature, and music. Jeremy Gray traces the growth of mathematical modernism from its roots in problem solving and theory to its interactions with physics, philosophy, theology, psychology, and ideas about real and artificial languages. He shows how mathematics was popularized, and explains how mathematical modernism not only gave expression to the work of mathematicians and the professional image they sought to create for themselves, but how modernism also introduced deeper and ultimately unanswerable questions. Plato's Ghost evokes Yeats's lament that any claim to worldly perfection inevitably is proven wrong by the philosopher's ghost; Gray demonstrates how modernist mathematicians believed they had advanced further than anyone before them, only to make more profound mistakes. He tells for the first time the story of these ambitious and brilliant mathematicians, including Richard Dedekind, Henri Lebesgue, Henri Poincaré, and many others. He describes the lively debates surrounding novel objects, definitions, and proofs in mathematics arising from the use of naïve set theory and the revived axiomatic method--debates that spilled over into contemporary arguments in philosophy and the sciences and drove an upsurge of popular writing on mathematics. And he looks at mathematics after World War I, including the foundational crisis and mathematical Platonism. Plato's Ghost is essential reading for mathematicians and historians, and will appeal to anyone interested in the development of modern mathematics.

Philosophies of Mathematics

Author: Alexander L. George,Daniel Velleman

Publisher: Wiley-Blackwell

ISBN: 9780631195443

Category: Science

Page: 244

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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 and Mathematical Practice in the Seventeenth Century

Author: Paolo Mancosu

Publisher: Oxford University Press on Demand

ISBN: 0195132440

Category: Drama

Page: 275

View: 7328

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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.

Philosophy of Mathematics

An Introduction

Author: David Bostock

Publisher: John Wiley & Sons

ISBN: 1405189924

Category: Mathematics

Page: 332

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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

Realism in Mathematics

Author: Penelope Maddy

Publisher: Oxford University Press

ISBN: 019824035X

Category: Political Science

Page: 204

View: 5581

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When engaged in mathematics, most people tend to think of themselves as scientists investigating the features of real mathematical things, and the wildly successful application of mathematics in the physical sciences reinforces this picture of mathematics as an objective study. For philosophers, however, this realism about mathematics raises serious questions: What are mathematical things? Where are they? How do we know about them? Penelope Maddy delineates and defends a novel versionof mathematical realism that answers the traditional questions and refocuses philosophical attention on the pressing foundational issues of contemporary mathematics.

Mathematics

The Loss of Certainty

Author: Morris Kline

Publisher: Oxford University Press, USA

ISBN: 9780195030853

Category: Mathematics

Page: 366

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Refuting the accepted belief that mathematics is exact and infallible, the author examines the development of conflicting concepts of mathematics and their implications for the physical, applied, social, and computer sciences

More Precisely

The Math You Need to Do Philosophy

Author: Eric Steinhart

Publisher: Broadview Press

ISBN: 9781770480094

Category: Philosophy

Page: 208

View: 9709

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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.

How Not to Be Wrong

The Power of Mathematical Thinking

Author: Jordan Ellenberg

Publisher: Penguin

ISBN: 0698163842

Category: Mathematics

Page: 480

View: 7822

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The Freakonomics of math—a math-world superstar unveils the hidden beauty and logic of the world and puts its power in our hands The math we learn in school can seem like a dull set of rules, laid down by the ancients and not to be questioned. In How Not to Be Wrong, Jordan Ellenberg shows us how terribly limiting this view is: Math isn’t confined to abstract incidents that never occur in real life, but rather touches everything we do—the whole world is shot through with it. Math allows us to see the hidden structures underneath the messy and chaotic surface of our world. It’s a science of not being wrong, hammered out by centuries of hard work and argument. Armed with the tools of mathematics, we can see through to the true meaning of information we take for granted: How early should you get to the airport? What does “public opinion” really represent? Why do tall parents have shorter children? Who really won Florida in 2000? And how likely are you, really, to develop cancer? How Not to Be Wrong presents the surprising revelations behind all of these questions and many more, using the mathematician’s method of analyzing life and exposing the hard-won insights of the academic community to the layman—minus the jargon. Ellenberg chases mathematical threads through a vast range of time and space, from the everyday to the cosmic, encountering, among other things, baseball, Reaganomics, daring lottery schemes, Voltaire, the replicability crisis in psychology, Italian Renaissance painting, artificial languages, the development of non-Euclidean geometry, the coming obesity apocalypse, Antonin Scalia’s views on crime and punishment, the psychology of slime molds, what Facebook can and can’t figure out about you, and the existence of God. Ellenberg pulls from history as well as from the latest theoretical developments to provide those not trained in math with the knowledge they need. Math, as Ellenberg says, is “an atomic-powered prosthesis that you attach to your common sense, vastly multiplying its reach and strength.” With the tools of mathematics in hand, you can understand the world in a deeper, more meaningful way. How Not to Be Wrong will show you how.

The Philosophy of Schopenhauer

Author: Dale Jacquette

Publisher: Routledge

ISBN: 1317494482

Category: Philosophy

Page: 320

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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.

Principia Mathematica

Author: Alfred North Whitehead,Bertrand Russell

Publisher: N.A

ISBN: N.A

Category: Logic, Symbolic and mathematical

Page: N.A

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