Author: Charles Parsons

Publisher: Cambridge University Press

ISBN: 9781139467278

Category: Science

Page: N.A

View: 3221

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: Mark Kac,Stanislaw M. Ulam

Publisher: Courier Corporation

ISBN: 0486670856

Category: Philosophy

Page: 170

View: 7079

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: Jacob Klein

Publisher: Courier Corporation

ISBN: 0486319814

Category: Mathematics

Page: 384

View: 8181

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: Reuben Hersh

Publisher: Oxford University Press, USA

ISBN: 9780195130874

Category: Medical

Page: 343

View: 8636

Most philosophers of mathematics treat it as isolated, timeless, ahistorical, inhuman. Reuben Hersh argues the contrary, that mathematics must be understood as a human activity, a social phenomenon, part of human culture, historically evolved, and intelligible only in a social context. Hersh pulls the screen back to reveal mathematics as seen by professionals, debunking many mathematical myths, and demonstrating how the "humanist" idea of the nature of mathematics more closely resembles how mathematicians actually work. At the heart of his book is a fascinating historical account of the mainstream of philosophy--ranging from Pythagoras, Descartes, and Spinoza, to Bertrand Russell, David Hilbert, and Rudolph Carnap--followed by the mavericks who saw mathematics as a human artifact, including Aristotle, Locke, Hume, Mill, and Lakatos. What is Mathematics, Really? reflects an insider's view of mathematical life, and will be hotly debated by anyone with an interest in mathematics or the philosophy of science.
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Author: Tim Crane

Publisher: OUP Oxford

ISBN: 019150520X

Category: Philosophy

Page: 208

View: 2380

The Objects of Thought addresses the ancient question of how it is possible to think about what does not exist. Tim Crane argues that the representation of the non-existent is a pervasive feature of our thought about the world, and that we will not adequately understand thought's representational power ('intentionality') unless we have understood the representation of the non-existent. Intentionality is conceived by Crane in terms of the direction of the mind upon an object of thought, or an intentional object. Intentional objects are what we think about. Some intentional objects exist and some do not. Non-existence poses a problem because there seem to be truths about non-existent intentional objects, but truths are answerable to reality, and reality contains only what exists. The proposed solution is to accept that there are some genuine truths about non-existent intentional objects, but to hold that they must be reductively explained in terms of truths about what does exist. The Objects of Thought offers both an original account of the nature of intentionality and a solution to the problem of thought about the non-existent.
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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: 5036

Looks at the competition between French and Russian mathematicians over the nature of infinity during the twentieth century.
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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: 7545

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: G. Duke

Publisher: Springer

ISBN: 0230378439

Category: Philosophy

Page: 212

View: 8998

This historically-informed critical assessment of Dummett's account of abstract objects, examines in detail some of the Fregean presuppositions of Dummett's account whilst also engaging with phenomenological approaches and recent work on the problem of abstract entities.
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Author: John P. Burgess

Publisher: OUP Oxford

ISBN: 019103360X

Category: Philosophy

Page: 224

View: 9811

While we are commonly told that the distinctive method of mathematics is rigorous proof, and that the special topic of mathematics is abstract structure, there has been no agreement among mathematicians, logicians, or philosophers as to just what either of these assertions means. John P. Burgess clarifies the nature of mathematical rigor and of mathematical structure, and above all of the relation between the two, taking into account some of the latest developments in mathematics, including the rise of experimental mathematics on the one hand and computerized formal proofs on the other hand. The main theses of Rigor and Structure are that the features of mathematical practice that a large group of philosophers of mathematics, the structuralists, have attributed to the peculiar nature of mathematical objects are better explained in a different way, as artefacts of the manner in which the ancient ideal of rigor is realized in modern mathematics. Notably, the mathematician must be very careful in deriving new results from the previous literature, but may remain largely indifferent to just how the results in the previous literature were obtained from first principles. Indeed, the working mathematician may remain largely indifferent to just what the first principles are supposed to be, and whether they are set-theoretic or category-theoretic or something else. Along the way to these conclusions, a great many historical developments in mathematics, philosophy, and logic are surveyed. Yet very little in the way of background knowledge on the part of the reader is presupposed.
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Author: Charles Parsons

Publisher: Harvard University Press

ISBN: 0674419502

Category: Philosophy

Page: N.A

View: 4985

In these selected essays, Charles Parsons surveys the contributions of philosophers and mathematicians who shaped the philosophy of mathematics over the past century: Brouwer, Hilbert, Bernays, Weyl, Gödel, Russell, Quine, Putnam, Wang, and Tait.
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Essays on Mathematics, Science and Philosophy

Author: Mark Kac,Gian-Carlo Rota,Jacob T. Schwartz

Publisher: Springer Science & Business Media

ISBN: 9780817647742

Category: Mathematics

Page: 266

View: 5376

This is a volume of essays and reviews that delightfully explores mathematics in all its moods — from the light and the witty, and humorous to serious, rational, and cerebral. These beautifully written articles from three great modern mathematicians will provide a source for supplemental reading for almost any math class. Topics include: logic, combinatorics, statistics, economics, artificial intelligence, computer science, and broad applications of mathematics. Readers will also find coverage of history and philosophy, including discussion of the work of Ulam, Kant, and Heidegger, among others.
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Mathematical Proof of Implausible Ideas

Author: Julian Havil

Publisher: Princeton University Press

ISBN: 9781400837380

Category: Mathematics

Page: 216

View: 7929

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: Alexander George

Publisher: Oxford University Press on Demand

ISBN: 0195079299

Category: History

Page: 204

View: 2840

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: David West

Publisher: Pearson Education

ISBN: 0735619654

Category: Computers

Page: 334

View: 8781

In OBJECT THINKING, esteemed object technologist David West contends that the mindset makes the programmer--not the tools and techniques. Delving into the history, philosophy, and even politics of object-oriented programming, West reveals how the best programmers rely on analysis and conceptualization--on thinking--rather than formal process and methods. Both provocative and pragmatic, this book gives form to what's primarily been an oral tradition among the field's revolutionary thinkers--and it illustrates specific object-behavior practices that you can adopt for true object design and superior results. Gain an in-depth understanding of: Prerequisites and principles of object thinking. Object knowledge implicit in eXtreme Programming (XP) and Agile software development. Object conceptualization and modeling. Metaphors, vocabulary, and design for object development. Learn viable techniques for: Decomposing complex domains in terms of objects. Identifying object relationships, interactions, and constraints. Relating object behavior to internal structure and implementation design. Incorporating object thinking into XP and Agile practice.
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A New Rationalist Manifesto

Author: A. Chapman,A. Ellis,R. Hanna,T. Hildebrand,H. Pickford

Publisher: Springer

ISBN: 1137347953

Category: Philosophy

Page: 427

View: 3472

A reply to contemporary skepticism about intuitions and a priori knowledge, and a defense of neo-rationalism from a contemporary Kantian standpoint, focusing on the theory of rational intuitions and on solving the two core problems of justifying and explaining them.
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Author: William Aspray,Philip Kitcher

Publisher: U of Minnesota Press

ISBN: 9780816615674

Category: Mathematics

Page: 386

View: 4482

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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Author: Charles Parsons

Publisher: Harvard University Press

ISBN: 0674065425

Category: Philosophy

Page: 242

View: 1335

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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A Short History of Mathematical Notation and Its Hidden Powers

Author: Joseph Mazur

Publisher: Princeton University Press

ISBN: 1400850118

Category: Mathematics

Page: 312

View: 5143

While all of us regularly use basic math symbols such as those for plus, minus, and equals, few of us know that many of these symbols weren't available before the sixteenth century. What did mathematicians rely on for their work before then? And how did mathematical notations evolve into what we know today? In Enlightening Symbols, popular math writer Joseph Mazur explains the fascinating history behind the development of our mathematical notation system. He shows how symbols were used initially, how one symbol replaced another over time, and how written math was conveyed before and after symbols became widely adopted. Traversing mathematical history and the foundations of numerals in different cultures, Mazur looks at how historians have disagreed over the origins of the numerical system for the past two centuries. He follows the transfigurations of algebra from a rhetorical style to a symbolic one, demonstrating that most algebra before the sixteenth century was written in prose or in verse employing the written names of numerals. Mazur also investigates the subconscious and psychological effects that mathematical symbols have had on mathematical thought, moods, meaning, communication, and comprehension. He considers how these symbols influence us (through similarity, association, identity, resemblance, and repeated imagery), how they lead to new ideas by subconscious associations, how they make connections between experience and the unknown, and how they contribute to the communication of basic mathematics. From words to abbreviations to symbols, this book shows how math evolved to the familiar forms we use today.
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Author: Vladimir Tasic

Publisher: Oxford University Press

ISBN: 9780195349955

Category: Mathematics

Page: 200

View: 7924

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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