Author: Charles Parsons

Publisher: Cambridge University Press

ISBN: 9781139467278

Category: Science

Page: N.A

View: 9290

Charles Parsons examines the notion of object, with the aim to navigate between nominalism, denying that distinctively mathematical objects exist, and forms of Platonism that postulate a transcendent realm of such objects. He introduces the central mathematical notion of structure and defends a version of the structuralist view of mathematical objects, according to which their existence is relative to a structure and they have no more of a 'nature' than that confers on them. Parsons also analyzes the concept of intuition and presents a conception of it distantly inspired by that of Kant, which describes a basic kind of access to abstract objects and an element of a first conception of the infinite.
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Author: Jacob Klein

Publisher: Courier Corporation

ISBN: 0486319814

Category: Mathematics

Page: 384

View: 3966

Important study focuses on the revival and assimilation of ancient Greek mathematics in the 13th–16th centuries, via Arabic science, and the 16th-century development of symbolic algebra. This brought about the crucial change in the concept of number that made possible modern science — in which the symbolic "form" of a mathematical statement is completely inseparable from its "content" of physical meaning. Includes a translation of Vieta's Introduction to the Analytical Art. 1968 edition. Bibliography.
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Author: Mark Kac,Stanislaw M. Ulam

Publisher: Courier Corporation

ISBN: 0486670856

Category: Philosophy

Page: 170

View: 4739

Fascinating study of the origin and nature of mathematical thought, including relation of mathematics and science, 20th-century developments, impact of computers, and more.Includes 34 illustrations. 1968 edition."
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Author: Mark Balaguer

Publisher: Oxford University Press on Demand

ISBN: 9780195143980

Category: Mathematics

Page: 217

View: 8327

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.
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Author: Tim Crane

Publisher: Oxford University Press

ISBN: 0199682747

Category: Philosophy

Page: 182

View: 2303

Tim Crane addresses the ancient question of how it is possible to think about what does not exist. He argues that the representation of the non-existent is a pervasive feature of our thought about the world, and that to understand thought's representational power ('intentionality') we need to understand the representation of the non-existent.
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Structure and Ontology

Author: Stewart Shapiro

Publisher: Oxford University Press

ISBN: 9780198025450

Category: Philosophy

Page: 296

View: 3950

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.
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Author: Jean-Pierre Changeux,Alain Connes

Publisher: Princeton University Press

ISBN: 9780691004051

Category: Mathematics

Page: 272

View: 8239

Do numbers and the other objects of mathematics enjoy a timeless existence independent of human minds, or are they the products of cerebral invention? Do we discover them, as Plato supposed and many others have believed since, or do we construct them? Does the physical world actually obey mathematical laws, or does it seem to conform to them simply because physicists have increasingly been able to make mathematical sense of it? Does mathematics constitute a universal language that in principle would permit human beings to communicate with extraterrestrial civilizations elsewhere in the universe, or is it merely an earthly language that owes its accidental existence to the peculiar evolution of neuronal networks in our brains? Jean-Pierre Changeux, an internationally renowned neurobiologist, and Alain Connes, one of the most eminent living mathematicians, find themselves deeply divided by these questions. Why order should exist in the world at all, and why it should be comprehensible to human beings, is the question that lies at the heart of these remarkable dialogues.
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An Introduction to the Philosophy of Mathematics

Author: E.W. Beth

Publisher: Taylor & Francis

ISBN: 9789027700704

Category: Mathematics

Page: 208

View: 980

In contributing a foreword to this book I am complying with a wish my husband expressed a few days before his death. He had completed the manuscript of this work, which may be considered a companion volume to his book Formal Methods. The task of seeing it through the press was undertaken by Mr. J. J. A. Mooij, acting director of the Institute for Research in Foundations and the Philosophy of Science (Instituut voor Grondslagenonderzoek en Filoso:fie der Exacte Wetenschappen) of the University of Amsterdam, with the help of Mrs. E. M. Barth, lecturer at the Institute. I wish to thank Mr. Mooij and Mrs. Barth most cordially for the care with which they have acquitted themselves of this delicate task and for the speed with which they have brought it to completion. I also wish to express my gratitude to Miss L. E. Minning, M. A. , for the helpful advice she has so kindly given to Mr. Mooij and Mrs. Barth during the proof reading. C. P. C. BETH-PASTOOR VII PREFACE A few years ago Mr. Horace S.
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Author: Morris Kline

Publisher: Oxford University Press

ISBN: 9780195345452

Category: Mathematics

Page: 512

View: 5639

This book gives a remarkably fine account of the influences mathematics has exerted on the development of philosophy, the physical sciences, religion, and the arts in Western life.
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Author: Morris Kline

Publisher: Courier Corporation

ISBN: 0486136310

Category: Mathematics

Page: 512

View: 8149

Stimulating account of development of mathematics from arithmetic, algebra, geometry and trigonometry, to calculus, differential equations, and non-Euclidean geometries. Also describes how math is used in optics, astronomy, and other phenomena.
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Author: Reuben Hersh

Publisher: Oxford University Press, USA

ISBN: 9780195130874

Category: Mathematics

Page: 343

View: 7598

Reflecting an insider's view of mathematical life, the author argues that mathematics must be historically evolved, and intelligible only in a social context.
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A True Story of Religious Mysticism and Mathematical Creativity

Author: Loren Graham,Jean-Michel Kantor

Publisher: Harvard University Press

ISBN: 0674032934

Category: History

Page: 239

View: 8649

Looks at the competition between French and Russian mathematicians over the nature of infinity during the twentieth century.
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Where Engineering and Mathematics Meet

Author: John Bryant,Chris Sangwin

Publisher: Princeton University Press

ISBN: 1400837952

Category: Mathematics

Page: 320

View: 9665

How do you draw a straight line? How do you determine if a circle is really round? These may sound like simple or even trivial mathematical problems, but to an engineer the answers can mean the difference between success and failure. How Round Is Your Circle? invites readers to explore many of the same fundamental questions that working engineers deal with every day--it's challenging, hands-on, and fun. John Bryant and Chris Sangwin illustrate how physical models are created from abstract mathematical ones. Using elementary geometry and trigonometry, they guide readers through paper-and-pencil reconstructions of mathematical problems and show them how to construct actual physical models themselves--directions included. It's an effective and entertaining way to explain how applied mathematics and engineering work together to solve problems, everything from keeping a piston aligned in its cylinder to ensuring that automotive driveshafts rotate smoothly. Intriguingly, checking the roundness of a manufactured object is trickier than one might think. When does the width of a saw blade affect an engineer's calculations--or, for that matter, the width of a physical line? When does a measurement need to be exact and when will an approximation suffice? Bryant and Sangwin tackle questions like these and enliven their discussions with many fascinating highlights from engineering history. Generously illustrated, How Round Is Your Circle? reveals some of the hidden complexities in everyday things.
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The Modernist Transformation of Mathematics

Author: Jeremy Gray

Publisher: Princeton University Press

ISBN: 9781400829040

Category: Mathematics

Page: 528

View: 8768

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.
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Author: Alexander George

Publisher: Oxford University Press on Demand

ISBN: 0195079299

Category: History

Page: 204

View: 2870

The essays in this volume investigate the conceptual foundations of mathematics illuminating the powers of the mind. Contributors include Alexander George, Michael Dummett, George Boolos, W.W. Tait, Wilfried Sieg, Daniel Isaacson, Charles Parsons, and Michael Hallett.
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Author: Charles Parsons

Publisher: Harvard University Press

ISBN: 0674065425

Category: Philosophy

Page: 242

View: 3991

Main description: In From Kant to Husserl, Charles Parsons examines a wide range of historical opinion on philosophical questions, from mathematics to phenomenology. Amplifying his early ideas on Kant's philosophy of arithmetic, Parsons uses Kant's lectures on metaphysics to explore how his arithmetical concepts relate to the categories. He then turns to early reactions by two immediate successors of Kant, Johann Schultz and Bernard Bolzano, to shed light on disputed questions regarding interpretation of Kant's philosophy of mathematics. Interested, as well, in what Kant meant by 0pure natural science,0 Parsons considers the relationship between the first Critique and the Metaphysical Foundations of Natural Science. His commentary on Kant's Transcendental Aesthetic departs from mathematics to engage the vexed question of what it tells about the meaning of Kant's transcendental idealism.Proceeding on to phenomenology, Parsons examines Frege's evolving idea of extensions, his attitude toward set theory, and his correspondence, particularly exchanges with Russell and Husserl. An essay on Brentano brings out, in the case of judgment, an alternative to the now standard Fregean view of negation, and, on truth, alternatives to the traditional correspondence view that are still discussed today. Ending with the question of why Husserl did not take the 0linguistic turn,0 a final essay included here marks the only article-length discussion of Husserl Parsons has ever written, despite a long-standing engagement with this philosopher.
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How Mathematical Thinking Evolved And Why Numbers Are Like Gossip

Author: Keith Devlin

Publisher: Basic Books

ISBN: 9780465016198

Category: Science

Page: 352

View: 9012

Why is math so hard? And why, despite this difficulty, are some people so good at it? If there's some inborn capacity for mathematical thinking—which there must be, otherwise no one could do it —why can't we all do it well? Keith Devlin has answers to all these difficult questions, and in giving them shows us how mathematical ability evolved, why it's a part of language ability, and how we can make better use of this innate talent.He also offers a breathtakingly new theory of language development—that language evolved in two stages, and its main purpose was not communication—to show that the ability to think mathematically arose out of the same symbol-manipulating ability that was so crucial to the emergence of true language. Why, then, can't we do math as well as we can speak? The answer, says Devlin, is that we can and do—we just don't recognize when we're using mathematical reasoning.
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Mathematical Proof of Implausible Ideas

Author: Julian Havil

Publisher: Princeton University Press

ISBN: 9781400837380

Category: Mathematics

Page: 216

View: 5230

Math--the application of reasonable logic to reasonable assumptions--usually produces reasonable results. But sometimes math generates astonishing paradoxes--conclusions that seem completely unreasonable or just plain impossible but that are nevertheless demonstrably true. Did you know that a losing sports team can become a winning one by adding worse players than its opponents? Or that the thirteenth of the month is more likely to be a Friday than any other day? Or that cones can roll unaided uphill? In Nonplussed!--a delightfully eclectic collection of paradoxes from many different areas of math--popular-math writer Julian Havil reveals the math that shows the truth of these and many other unbelievable ideas. Nonplussed! pays special attention to problems from probability and statistics, areas where intuition can easily be wrong. These problems include the vagaries of tennis scoring, what can be deduced from tossing a needle, and disadvantageous games that form winning combinations. Other chapters address everything from the historically important Torricelli's Trumpet to the mind-warping implications of objects that live on high dimensions. Readers learn about the colorful history and people associated with many of these problems in addition to their mathematical proofs. Nonplussed! will appeal to anyone with a calculus background who enjoys popular math books or puzzles.
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Author: Vladimir Tasic

Publisher: Oxford University Press

ISBN: 9780195349955

Category: Mathematics

Page: 200

View: 5745

This is a charming and insightful contribution to an understanding of the "Science Wars" between postmodernist humanism and science, driving toward a resolution of the mutual misunderstanding that has driven the controversy. It traces the root of postmodern theory to a debate on the foundations of mathematics early in the 20th century, then compares developments in mathematics to what took place in the arts and humanities, discussing issues as diverse as literary theory, arts, and artificial intelligence. This is a straightforward, easily understood presentation of what can be difficult theoretical concepts It demonstrates that a pattern of misreading mathematics can be seen both on the part of science and on the part of postmodern thinking. This is a humorous, playful yet deeply serious look at the intellectual foundations of mathematics for those in the humanities and the perfect critical introduction to the bases of modernism and postmodernism for those in the sciences.
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Author: William Aspray,Philip Kitcher

Publisher: U of Minnesota Press

ISBN: 9780816615674

Category: Mathematics

Page: 386

View: 6409

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.
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