The Search for Certainty A Philosophical Account of Foundations of Mathematics

Author: Marcus Giaquinto
Publisher: Clarendon Press
ISBN: 0191588172
Format: PDF, Mobi
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The nineteenth century saw a movement to make higher mathematics rigorous. This seemed to be on the brink of success when it was thrown into confusion by the discovery of the class paradoxes. That initiated a period of intense research into the foundations of mathematics, and with it the birth of mathematical logic and a new, sharper debate in the philosophy of mathematics. The Search for Certainty examines this foundational endeavour from the discovery of the paradoxes to the present. Focusing on Russell's logicist programme and Hilbert's finitist programme, Giaquinto investigates how successful they were and how successful they could be. These questions are set in the context of a clear, non-technical exposition and assessment of the most important discoveries in mathematical logic, above all G--ouml--;del's underivability theorems. More than six decades after those discoveries, Giaquinto asks what our present perspective should be on the question of certainty in mathematics. Taking recent developments into account, he gives reasons for a surprisingly positive response.

Introducing Philosophy of Mathematics

Author: Michele Friend
Publisher: Routledge
ISBN: 1317493796
Format: PDF, Mobi
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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.

An Introduction to the Philosophy of Mathematics

Author: Mark Colyvan
Publisher: Cambridge University Press
ISBN: 0521826020
Format: PDF, Mobi
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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.

Set Theory and its Philosophy

Author: Michael Potter
Publisher: Clarendon Press
ISBN: 0191556432
Format: PDF, Docs
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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.

Mathematics

Author: Morris Kline
Publisher: Oxford University Press, USA
ISBN: 9780195030853
Format: PDF, Docs
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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

The Princeton Companion to Mathematics

Author: Timothy Gowers
Publisher: Princeton University Press
ISBN: 9781400830398
Format: PDF
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This is a one-of-a-kind reference for anyone with a serious interest in mathematics. Edited by Timothy Gowers, a recipient of the Fields Medal, it presents nearly two hundred entries, written especially for this book by some of the world's leading mathematicians, that introduce basic mathematical tools and vocabulary; trace the development of modern mathematics; explain essential terms and concepts; examine core ideas in major areas of mathematics; describe the achievements of scores of famous mathematicians; explore the impact of mathematics on other disciplines such as biology, finance, and music--and much, much more. Unparalleled in its depth of coverage, The Princeton Companion to Mathematics surveys the most active and exciting branches of pure mathematics. Accessible in style, this is an indispensable resource for undergraduate and graduate students in mathematics as well as for researchers and scholars seeking to understand areas outside their specialties. Features nearly 200 entries, organized thematically and written by an international team of distinguished contributors Presents major ideas and branches of pure mathematics in a clear, accessible style Defines and explains important mathematical concepts, methods, theorems, and open problems Introduces the language of mathematics and the goals of mathematical research Covers number theory, algebra, analysis, geometry, logic, probability, and more Traces the history and development of modern mathematics Profiles more than ninety-five mathematicians who influenced those working today Explores the influence of mathematics on other disciplines Includes bibliographies, cross-references, and a comprehensive index Contributors incude: Graham Allan, Noga Alon, George Andrews, Tom Archibald, Sir Michael Atiyah, David Aubin, Joan Bagaria, Keith Ball, June Barrow-Green, Alan Beardon, David D. Ben-Zvi, Vitaly Bergelson, Nicholas Bingham, Béla Bollobás, Henk Bos, Bodil Branner, Martin R. Bridson, John P. Burgess, Kevin Buzzard, Peter J. Cameron, Jean-Luc Chabert, Eugenia Cheng, Clifford C. Cocks, Alain Connes, Leo Corry, Wolfgang Coy, Tony Crilly, Serafina Cuomo, Mihalis Dafermos, Partha Dasgupta, Ingrid Daubechies, Joseph W. Dauben, John W. Dawson Jr., Francois de Gandt, Persi Diaconis, Jordan S. Ellenberg, Lawrence C. Evans, Florence Fasanelli, Anita Burdman Feferman, Solomon Feferman, Charles Fefferman, Della Fenster, José Ferreirós, David Fisher, Terry Gannon, A. Gardiner, Charles C. Gillispie, Oded Goldreich, Catherine Goldstein, Fernando Q. Gouvêa, Timothy Gowers, Andrew Granville, Ivor Grattan-Guinness, Jeremy Gray, Ben Green, Ian Grojnowski, Niccolò Guicciardini, Michael Harris, Ulf Hashagen, Nigel Higson, Andrew Hodges, F. E. A. Johnson, Mark Joshi, Kiran S. Kedlaya, Frank Kelly, Sergiu Klainerman, Jon Kleinberg, Israel Kleiner, Jacek Klinowski, Eberhard Knobloch, János Kollár, T. W. Körner, Michael Krivelevich, Peter D. Lax, Imre Leader, Jean-François Le Gall, W. B. R. Lickorish, Martin W. Liebeck, Jesper Lützen, Des MacHale, Alan L. Mackay, Shahn Majid, Lech Maligranda, David Marker, Jean Mawhin, Barry Mazur, Dusa McDuff, Colin McLarty, Bojan Mohar, Peter M. Neumann, Catherine Nolan, James Norris, Brian Osserman, Richard S. Palais, Marco Panza, Karen Hunger Parshall, Gabriel P. Paternain, Jeanne Peiffer, Carl Pomerance, Helmut Pulte, Bruce Reed, Michael C. Reed, Adrian Rice, Eleanor Robson, Igor Rodnianski, John Roe, Mark Ronan, Edward Sandifer, Tilman Sauer, Norbert Schappacher, Andrzej Schinzel, Erhard Scholz, Reinhard Siegmund-Schultze, Gordon Slade, David J. Spiegelhalter, Jacqueline Stedall, Arild Stubhaug, Madhu Sudan, Terence Tao, Jamie Tappenden, C. H. Taubes, Rüdiger Thiele, Burt Totaro, Lloyd N. Trefethen, Dirk van Dalen, Richard Weber, Dominic Welsh, Avi Wigderson, Herbert Wilf, David Wilkins, B. Yandell, Eric Zaslow, Doron Zeilberger

Derrida Badiou and the Formal Imperative

Author: Christopher Norris
Publisher: A&C Black
ISBN: 1441193979
Format: PDF, ePub, Mobi
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In this path-breaking study Christopher Norris proposes a transformed understanding of the much-exaggerated differences between analytic and continental philosophy. While keeping the analytic tradition squarely in view his book focuses on the work of Jacques Derrida and Alain Badiou, two of the most original and significant figures in the recent history of ideas. Norris argues that these thinkers have decisively reconfigured the terrain of contemporary philosophy and, between them, pointed a way beyond some of those seemingly intractable issues that have polarised debate on both sides of the notional rift between the analytic and continental traditions. In particular his book sets out to show - against the received analytic wisdom - that continental philosophy has its own analytic resources and is capable of bringing some much-needed fresh insight to bear on problems in philosophy of language, logic and mathematics. Norris provides not only a unique comparative account of Derrida's and Badiou's work but also a remarkably wide-ranging assessment of their joint contribution to philosophy's current - if widely resisted - potential for self-transformation.

Wittgenstein s Lectures on the Foundations of Mathematics Cambridge 1939

Author: Ludwig Wittgenstein
Publisher: University of Chicago Press
ISBN: 022630860X
Format: PDF
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For several terms at Cambridge in 1939, Ludwig Wittgenstein lectured on the philosophical foundations of mathematics. A lecture class taught by Wittgenstein, however, hardly resembled a lecture. He sat on a chair in the middle of the room, with some of the class sitting in chairs, some on the floor. He never used notes. He paused frequently, sometimes for several minutes, while he puzzled out a problem. He often asked his listeners questions and reacted to their replies. Many meetings were largely conversation. These lectures were attended by, among others, D. A. T. Gasking, J. N. Findlay, Stephen Toulmin, Alan Turing, G. H. von Wright, R. G. Bosanquet, Norman Malcolm, Rush Rhees, and Yorick Smythies. Notes taken by these last four are the basis for the thirty-one lectures in this book. The lectures covered such topics as the nature of mathematics, the distinctions between mathematical and everyday languages, the truth of mathematical propositions, consistency and contradiction in formal systems, the logicism of Frege and Russell, Platonism, identity, negation, and necessary truth. The mathematical examples used are nearly always elementary.

Companion to the History of Modern Science

Author: G N Cantor
Publisher: Routledge
ISBN: 1134977514
Format: PDF, ePub, Mobi
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* A descriptive and analytical guide to the development of Western science from AD 1500, and to the diversity and course of that development first in Europe and later across the world * Presented in clear, non-technical language * Extensive indexes of Subjects and Names `Indeed a companion volume whose 67 essays give pleasure and instruction ... an ambitious and successful work.' - Times Literary Supplement `This work is an essential resource for libraries everywhere. For specialist science libraries willing to keep just one encyclopaedic guide to history, for undergraduate libraries seeking to provide easily accessible information, for the devisers of university curricula, for the modern social historian or even the eclectic scientist taking a break from simply making history, this is the book for you.' - Times Higher Education Supplement `A pleasure to read with a carefully chosen typeface, well organized pages and ample margins ... it is very easy to find one's way around. This is a book which will be consulted widely.' - Technovation `This is a commendably easy book to use.' - British Journal of the History of Science `Scholars from other areas entering this field, students taking the vertical approach and teachers coming from any direction cannot fail to find this an invaluable text.' - History of Science Journal

The Search for Certainty

Author: Krzysztof Burdzy
Publisher: World Scientific
ISBN: 9814273708
Format: PDF, Docs
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This volume represents a radical departure from the current philosophical duopoly in the area of foundations of probability, that is, the frequency and subjective theories. One of the main new ideas is a set of scientific laws of probability. The new laws are simple, intuitive and, last but not least, they agree well with the contents of current textbooks on probability. Another major new claim is that the ?frequency statistics? has nothing in common with the ?frequency philosophy of probability,? contrary to popular belief. Similarly, contrary to the general perception, the ?Bayesian statistics? shares nothing in common with the ?subjective philosophy of probability.? This volume is non-partisan on the scientific side ? that it is supportive of both frequency statistics and Bayesian statistics. On the other hand, it contains well-documented and thoroughly-explained criticisms of the frequency and subjective philosophies of probability. Short reviews of other philosophical theories of probability and basic mathematical methods of probability and statistics are incorporated. This volume includes essential chapters on decision theory and teaching probability, and it is easily accessible to the general audience.