Author: Charles Parsons

Publisher: Cambridge University Press

ISBN: 9781139467278

Category: Science

Page: N.A

View: 2636

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: 9780486272894

Category: Mathematics

Page: 360

View: 329

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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An Introduction to the Philosophy of Mathematics

Author: E.W. Beth

Publisher: Springer Science & Business Media

ISBN: 9401722072

Category: Science

Page: 208

View: 4018

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: 7917

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: 3468

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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A New Rationalist Manifesto

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

Publisher: Springer

ISBN: 1137347953

Category: Philosophy

Page: 427

View: 1986

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: Alexander George

Publisher: Oxford University Press on Demand

ISBN: 0195079299

Category: History

Page: 204

View: 9190

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: 5554

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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Author: Mark Kac,Stanislaw M. Ulam

Publisher: Courier Corporation

ISBN: 0486670856

Category: Philosophy

Page: 170

View: 4833

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: Vladimir Tasic

Publisher: Oxford University Press

ISBN: 9780195349955

Category: Mathematics

Page: 200

View: 6197

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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A Critical Exposition of Arguments for Intuitionism

Author: Tomasz Placek

Publisher: Springer Science & Business Media

ISBN: 9401593159

Category: Science

Page: 220

View: 623

In 1907 Luitzen Egbertus Jan Brouwer defended his doctoral dissertation on the foundations of mathematics and with this event the modem version of mathematical intuitionism came into being. Brouwer attacked the main currents of the philosophy of mathematics: the formalists and the Platonists. In tum, both these schools began viewing intuitionism as the most harmful party among all known philosophies of mathematics. That was the origin of the now-90-year-old debate over intuitionism. As both sides have appealed in their arguments to philosophical propositions, the discussions have attracted the attention of philosophers as well. One might ask here what role a philosopher can play in controversies over mathematical intuitionism. Can he reasonably enter into disputes among mathematicians? I believe that these disputes call for intervention by a philo sopher. The three best-known arguments for intuitionism, those of Brouwer, Heyting and Dummett, are based on ontological and epistemological claims, or appeal to theses that properly belong to a theory of meaning. Those lines of argument should be investigated in order to find what their assumptions are, whether intuitionistic consequences really follow from those assumptions, and finally, whether the premises are sound and not absurd. The intention of this book is thus to consider seriously the arguments of mathematicians, even if philosophy was not their main field of interest. There is little sense in disputing whether what mathematicians said about the objectivity and reality of mathematical facts belongs to philosophy, or not.
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Author: David Tall

Publisher: Springer Science & Business Media

ISBN: 0306472031

Category: Education

Page: 292

View: 3938

This book is the first major study of advanced mathematical thinking as performed by mathematicians and taught to students in senior high school and university. Topics covered include the psychology of advanced mathematical thinking, the processes involved, mathematical creativity, proof, the role of definitions, symbols, and reflective abstraction. It is highly appropriate for the college professor in mathematics or the general mathematics educator.
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Author: Morris Kline

Publisher: Oxford University Press

ISBN: 0199770484

Category: Mathematics

Page: 448

View: 6082

This comprehensive history traces the development of mathematical ideas and the careers of the men responsible for them. Volume 1 looks at the disciplines origins in Babylon and Egypt, the creation of geometry and trigonometry by the Greeks, and the role of mathematics in the medieval and early modern periods. Volume 2 focuses on calculus, the rise of analysis in the 19th century, and the number theories of Dedekind and Dirichlet. The concluding volume covers the revival of projective geometry, the emergence of abstract algebra, the beginnings of topology, and the influence of Godel on recent mathematical study.
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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: 9533

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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Author: David Webb

Publisher: Edinburgh University Press

ISBN: 0748675442

Category: Philosophy

Page: 192

View: 873

Reveals the extent to which Foucault's approach to language in The Archaeology of Knowledge was influenced by the mathematical sciences, adopting a mode of thought indebted to thinkers in the scientific and epistemological traditions such as Cavailles and
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Grounding Mathematics Education

Author: Michael H.G. Hoffmann,Johannes Lenhard

Publisher: Springer Science & Business Media

ISBN: 9780387242699

Category: Education

Page: 389

View: 5446

The advancement of a scientific discipline depends not only on the "big heroes" of a discipline, but also on a community’s ability to reflect on what has been done in the past and what should be done in the future. This volume combines perspectives on both. It celebrates the merits of Michael Otte as one of the most important founding fathers of mathematics education by bringing together all the new and fascinating perspectives, created through his career as a bridge builder in the field of interdisciplinary research and cooperation. The perspectives elaborated here are for the greatest part motivated by the impressing variety of Otte’s thoughts; however, the idea is not to look back, but to find out where the research agenda might lead us in the future. This volume provides new sources of knowledge based on Michael Otte’s fundamental insight that understanding the problems of mathematics education – how to teach, how to learn, how to communicate, how to do, and how to represent mathematics – depends on means, mainly philosophical and semiotic, that have to be created first of all, and to be reflected from the perspectives of a multitude of diverse disciplines.
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Letters, Essays and Thoughts on Studies and Conduct

Author: Henry Barnard

Publisher: N.A

ISBN: N.A

Category: Education

Page: 552

View: 6763

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Aporetic Method in Cosmology and Metaphysics

Author: John J. Cleary

Publisher: BRILL

ISBN: 9789004101593

Category: Philosophy

Page: 558

View: 9623

This book examines Aristotle's critical reaction to the mathematical cosmology of Plato's Academy, and traces the aporetic method by which he developed his own cosmological and metaphysical views, which underpin his philosophy of mathematics.
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