Philosophy of Mathematics

Philosophy of Mathematics
Author: Paul Benacerraf,Hilary Putnam
Publsiher: Cambridge University Press
Total Pages: 329
Release: 1984-01-27
ISBN 10: 1107268133
ISBN 13: 9781107268135
Language: EN, FR, DE, ES & NL

Philosophy of Mathematics Book Review:

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.

Introduction to Mathematical Philosophy

Introduction to Mathematical Philosophy
Author: Bertrand Russell
Publsiher: Cosimo, Inc.
Total Pages: 220
Release: 2007-04-01
ISBN 10: 1602062919
ISBN 13: 9781602062917
Language: EN, FR, DE, ES & NL

Introduction to Mathematical Philosophy Book Review:

Not to be confused with the philosophy of mathematics, mathematical philosophy is the structured set of rules that govern all existence. Or, in a word: logic. While this branch of philosophy threatens to be an intimidating and abstract subject, it is one that is surprisingly simple and necessarily sensible, particularly at the pen of writer Bertrand Russell, who infuses this work, first published in 1919, with a palpable and genuine desire to assist the reader in understanding the principles he illustrates. Anyone interested in logic and its development and application here will find a comprehensive and accessible account of mathematical philosophy, from the idea of what numbers actually are, through the principles of order, limits, and deduction, and on to infinity. British philosopher and mathematician BERTRAND ARTHUR WILLIAM RUSSELL (1872-1970) won the Nobel Prize for Literature in 1950. Among his many works are Why I Am Not a Christian (1927), Power: A New Social Analysis (1938), and My Philosophical Development (1959).

Why Is There Philosophy of Mathematics At All

Why Is There Philosophy of Mathematics At All
Author: Ian Hacking
Publsiher: Cambridge University Press
Total Pages: 212
Release: 2014-01-30
ISBN 10: 1107729823
ISBN 13: 9781107729827
Language: EN, FR, DE, ES & NL

Why Is There Philosophy of Mathematics At All Book Review:

This truly philosophical book takes us back to fundamentals - the sheer experience of proof, and the enigmatic relation of mathematics to nature. It asks unexpected questions, such as 'what makes mathematics mathematics?', 'where did proof come from and how did it evolve?', and 'how did the distinction between pure and applied mathematics come into being?' In a wide-ranging discussion that is both immersed in the past and unusually attuned to the competing philosophical ideas of contemporary mathematicians, it shows that proof and other forms of mathematical exploration continue to be living, evolving practices - responsive to new technologies, yet embedded in permanent (and astonishing) facts about human beings. It distinguishes several distinct types of application of mathematics, and shows how each leads to a different philosophical conundrum. Here is a remarkable body of new philosophical thinking about proofs, applications, and other mathematical activities.

Philosophy of Mathematics

Philosophy of Mathematics
Author: Anonim
Publsiher: Elsevier
Total Pages: 733
Release: 2009-07-08
ISBN 10: 9780080930589
ISBN 13: 0080930581
Language: EN, FR, DE, ES & NL

Philosophy of Mathematics Book Review:

One of the most striking features of mathematics is the fact that we are much more certain about the mathematical knowledge we have than about what mathematical knowledge is knowledge of. Are numbers, sets, functions and groups physical entities of some kind? Are they objectively existing objects in some non-physical, mathematical realm? Are they ideas that are present only in the mind? Or do mathematical truths not involve referents of any kind? It is these kinds of questions that have encouraged philosophers and mathematicians alike to focus their attention on issues in the philosophy of mathematics. Over the centuries a number of reasonably well-defined positions about the nature of mathematics have been developed and it is these positions (both historical and current) that are surveyed in the current volume. Traditional theories (Platonism, Aristotelianism, Kantianism), as well as dominant modern theories (logicism, formalism, constructivism, fictionalism, etc.), are all analyzed and evaluated. Leading-edge research in related fields (set theory, computability theory, probability theory, paraconsistency) is also discussed. The result is a handbook that not only provides a comprehensive overview of recent developments but that also serves as an indispensable resource for anyone wanting to learn about current developments in the philosophy of mathematics. -Comprehensive coverage of all main theories in the philosophy of mathematics -Clearly written expositions of fundamental ideas and concepts -Definitive discussions by leading researchers in the field -Summaries of leading-edge research in related fields (set theory, computability theory, probability theory, paraconsistency) are also included

Lectures on the Philosophy of Mathematics

Lectures on the Philosophy of Mathematics
Author: Joel David Hamkins
Publsiher: MIT Press
Total Pages: 352
Release: 2021-02-02
ISBN 10: 0262362651
ISBN 13: 9780262362658
Language: EN, FR, DE, ES & NL

Lectures on the Philosophy of Mathematics Book Review:

An introduction to the philosophy of mathematics grounded in mathematics and motivated by mathematical inquiry and practice. In this book, Joel David Hamkins offers an introduction to the philosophy of mathematics that is grounded in mathematics and motivated by mathematical inquiry and practice. He treats philosophical issues as they arise organically in mathematics, discussing such topics as platonism, realism, logicism, structuralism, formalism, infinity, and intuitionism in mathematical contexts. He organizes the book by mathematical themes--numbers, rigor, geometry, proof, computability, incompleteness, and set theory--that give rise again and again to philosophical considerations.

Kant s Philosophy of Mathematics

Kant   s Philosophy of Mathematics
Author: C.J. Posy
Publsiher: Springer Science & Business Media
Total Pages: 380
Release: 2013-03-09
ISBN 10: 9401580464
ISBN 13: 9789401580465
Language: EN, FR, DE, ES & NL

Kant s Philosophy of Mathematics Book Review:

Kant's views about mathematics were controversial in his own time, and they have inspired or infuriated thinkers ever since. Though specific Kantian doctrines fell into disrepute earlier in this century, the past twenty-five years have seen a surge of interest in and respect for Kant's philosophy of mathematics among both Kant scholars and philosophers of mathematics. The present volume includes the classic papers from the 1960s and 1970s which spared this renaissance of interest, together with updated postscripts by their authors. It also includes the most important recent work on Kant's philosophy of mathematics. The essays bring to bear a wealth of detailed Kantian scholarship, together with powerful new interpretative tools drawn from modern mathematics, logic and philosophy. The cumulative effect of this collection upon the reader will be a deeper understanding of the centrality of mathematics in all aspects of Kant's thought and a renewed respect for the power of Kant's thinking about mathematics. The essays contained in this volume will set the agenda for further work on Kant's philosophy of mathematics for some time to come.

Philosophies of Mathematics

Philosophies of Mathematics
Author: Alexander L. George,Daniel Velleman
Publsiher: Wiley-Blackwell
Total Pages: 244
Release: 2001-12-03
ISBN 10: 9780631195443
ISBN 13: 0631195440
Language: EN, FR, DE, ES & NL

Philosophies of Mathematics Book Review:

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.

The Philosophy of Mathematics Today

The Philosophy of Mathematics Today
Author: Matthias Schirn
Publsiher: Oxford University Press
Total Pages: 638
Release: 2003
ISBN 10: 9780199262625
ISBN 13: 0199262624
Language: EN, FR, DE, ES & NL

The Philosophy of Mathematics Today Book Review:

The Philosophy of Mathematics Today gives a panorama of the best current work in this lively field, through twenty essays specially written for this collection by leading figures. The topics include indeterminacy, logical consequence, mathematical methodology, abstraction, and both Hilbert'sand Frege's foundational programmes. The collection will be an important source for research in the philosophy of mathematics for years to come.

Thinking about Mathematics

Thinking about Mathematics
Author: Stewart Shapiro
Publsiher: OUP Oxford
Total Pages: 328
Release: 2000-07-13
ISBN 10: 0192893068
ISBN 13: 9780192893062
Language: EN, FR, DE, ES & NL

Thinking about Mathematics Book Review:

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

An Introduction to the Philosophy of Mathematics
Author: Mark Colyvan
Publsiher: Cambridge University Press
Total Pages: 188
Release: 2012-06-14
ISBN 10: 0521826020
ISBN 13: 9780521826020
Language: EN, FR, DE, ES & NL

An Introduction to the Philosophy of Mathematics Book Review:

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

Philosophy of Mathematics
Author: James Robert Brown
Publsiher: Routledge
Total Pages: 232
Release: 2005-08-09
ISBN 10: 1134806434
ISBN 13: 9781134806430
Language: EN, FR, DE, ES & NL

Philosophy of Mathematics Book Review:

Philosophy of Mathematics is an excellent introductory text. This student friendly book discusses the great philosophers and the importance of mathematics to their thought. It includes the following topics: * the mathematical image * platonism * picture-proofs * applied mathematics * Hilbert and Godel * knots and nations * definitions * picture-proofs and Wittgenstein * computation, proof and conjecture. The book is ideal for courses on philosophy of mathematics and logic.

Mathematics and Philosophy

Mathematics and Philosophy
Author: Daniel Parrochia
Publsiher: John Wiley & Sons
Total Pages: 352
Release: 2018-05-24
ISBN 10: 1119527791
ISBN 13: 9781119527794
Language: EN, FR, DE, ES & NL

Mathematics and Philosophy Book Review:

This book, which studies the links between mathematics and philosophy, highlights a reversal. Initially, the (Greek) philosophers were also mathematicians (geometers). Their vision of the world stemmed from their research in this field (rational and irrational numbers, problem of duplicating the cube, trisection of the angle...). Subsequently, mathematicians freed themselves from philosophy (with Analysis, differential Calculus, Algebra, Topology, etc.), but their researches continued to inspire philosophers (Descartes, Leibniz, Hegel, Husserl, etc.). However, from a certain level of complexity, the mathematicians themselves became philosophers (a movement that begins with Wronsky and Clifford, and continues until Grothendieck).

Philosophy of Mathematics

Philosophy of Mathematics
Author: Øystein Linnebo
Publsiher: Princeton University Press
Total Pages: 216
Release: 2020-03-24
ISBN 10: 069120229X
ISBN 13: 9780691202297
Language: EN, FR, DE, ES & NL

Philosophy of Mathematics Book Review:

A sophisticated, original introduction to the philosophy of mathematics from one of its leading thinkers Mathematics is a model of precision and objectivity, but it appears distinct from the empirical sciences because it seems to deliver nonexperiential knowledge of a nonphysical reality of numbers, sets, and functions. How can these two aspects of mathematics be reconciled? This concise book provides a systematic, accessible introduction to the field that is trying to answer that question: the philosophy of mathematics. Øystein Linnebo, one of the world's leading scholars on the subject, introduces all of the classical approaches to the field as well as more specialized issues, including mathematical intuition, potential infinity, and the search for new mathematical axioms. Sophisticated but clear and approachable, this is an essential book for all students and teachers of philosophy and of mathematics.

The Philosophy of Mathematical Practice

The Philosophy of Mathematical Practice
Author: Paolo Mancosu
Publsiher: OUP Oxford
Total Pages: 460
Release: 2008-06-19
ISBN 10: 9780191559099
ISBN 13: 0191559091
Language: EN, FR, DE, ES & NL

The Philosophy of Mathematical Practice Book Review:

Contemporary philosophy of mathematics offers us an embarrassment of riches. Among the major areas of work one could list developments of the classical foundational programs, analytic approaches to epistemology and ontology of mathematics, and developments at the intersection of history and philosophy of mathematics. But anyone familiar with contemporary philosophy of mathematics will be aware of the need for new approaches that pay closer attention to mathematical practice. This book is the first attempt to give a coherent and unified presentation of this new wave of work in philosophy of mathematics. The new approach is innovative at least in two ways. First, it holds that there are important novel characteristics of contemporary mathematics that are just as worthy of philosophical attention as the distinction between constructive and non-constructive mathematics at the time of the foundational debates. Secondly, it holds that many topics which escape purely formal logical treatment - such as visualization, explanation, and understanding - can nonetheless be subjected to philosophical analysis. The Philosophy of Mathematical Practice comprises an introduction by the editor and eight chapters written by some of the leading scholars in the field. Each chapter consists of short introduction to the general topic of the chapter followed by a longer research article in the area. The eight topics selected represent a broad spectrum of contemporary philosophical reflection on different aspects of mathematical practice: diagrammatic reasoning and representation systems; visualization; mathematical explanation; purity of methods; mathematical concepts; the philosophical relevance of category theory; philosophical aspects of computer science in mathematics; the philosophical impact of recent developments in mathematical physics.

The Philosophy of Mathematics Education Today

The Philosophy of Mathematics Education Today
Author: Paul Ernest
Publsiher: Springer
Total Pages: 367
Release: 2018-06-19
ISBN 10: 9783319777597
ISBN 13: 3319777599
Language: EN, FR, DE, ES & NL

The Philosophy of Mathematics Education Today Book Review:

This book offers an up-to-date overview of the research on philosophy of mathematics education, one of the most important and relevant areas of theory. The contributions analyse, question, challenge, and critique the claims of mathematics education practice, policy, theory and research, offering ways forward for new and better solutions. The book poses basic questions, including: What are our aims of teaching and learning mathematics? What is mathematics anyway? How is mathematics related to society in the 21st century? How do students learn mathematics? What have we learnt about mathematics teaching? Applied philosophy can help to answer these and other fundamental questions, and only through an in-depth analysis can the practice of the teaching and learning of mathematics be improved. The book addresses important themes, such as critical mathematics education, the traditional role of mathematics in schools during the current unprecedented political, social, and environmental crises, and the way in which the teaching and learning of mathematics can better serve social justice and make the world a better place for the future.

Research in History and Philosophy of Mathematics

Research in History and Philosophy of Mathematics
Author: Maria Zack,Dirk Schlimm
Publsiher: Birkhäuser
Total Pages: 172
Release: 2020-01-03
ISBN 10: 9783030311964
ISBN 13: 3030311961
Language: EN, FR, DE, ES & NL

Research in History and Philosophy of Mathematics Book Review:

This volume contains ten papers that have been collected by the Canadian Society for History and Philosophy of Mathematics/Société canadienne d’histoire et de philosophie des mathématiques. It showcases rigorously-reviewed contemporary scholarship on an interesting variety of topics in the history and philosophy of mathematics from the seventeenth century to the modern era. The volume begins with an exposition of the life and work of Professor Bolesław Sobociński. It then moves on to cover a collection of topics about twentieth-century philosophy of mathematics, including Fred Sommers’s creation of Traditional Formal Logic and Alexander Grothendieck’s work as a starting point for discussing analogies between commutative algebra and algebraic geometry. Continuing the focus on the philosophy of mathematics, the next selections discuss the mathematization of biology and address the study of numerical cognition. The volume then moves to discussing various aspects of mathematics education, including Charles Davies’s early book on the teaching of mathematics and the use of Gaussian Lemniscates in the classroom. A collection of papers on the history of mathematics in the nineteenth century closes out the volume, presenting a discussion of Gauss’s “Allgemeine Theorie des Erdmagnetismus” and a comparison of the geometric works of Desargues and La Hire. Written by leading scholars in the field, these papers are accessible not only to mathematicians and students of the history and philosophy of mathematics, but also to anyone with a general interest in mathematics.

Kant s Philosophy of Mathematics Volume 1 The Critical Philosophy and its Roots

Kant s Philosophy of Mathematics  Volume 1  The Critical Philosophy and its Roots
Author: Carl Posy,Ofra Rechter
Publsiher: Cambridge University Press
Total Pages: 329
Release: 2020-01-31
ISBN 10: 1108669530
ISBN 13: 9781108669535
Language: EN, FR, DE, ES & NL

Kant s Philosophy of Mathematics Volume 1 The Critical Philosophy and its Roots Book Review:

The late 1960s saw the emergence of new philosophical interest in Kant's philosophy of mathematics, and since then this interest has developed into a major and dynamic field of study. In this state-of-the-art survey of contemporary scholarship on Kant's mathematical thinking, Carl Posy and Ofra Rechter gather leading authors who approach it from multiple perspectives, engaging with topics including geometry, arithmetic, logic, and metaphysics. Their essays offer fine-grained analysis of Kant's philosophy of mathematics in the context of his Critical philosophy, and also show sensitivity to its historical background. The volume will be important for readers seeking a comprehensive picture of the current scholarship about the development of Kant's philosophy of mathematics, its place in his overall philosophy, and the Kantian themes that influenced mathematics and its philosophy after Kant.

Introducing Philosophy of Mathematics

Introducing Philosophy of Mathematics
Author: Michele Friend
Publsiher: Routledge
Total Pages: 240
Release: 2014-12-05
ISBN 10: 1317493788
ISBN 13: 9781317493785
Language: EN, FR, DE, ES & NL

Introducing Philosophy of Mathematics Book Review:

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.

The Philosophy of Mathematics Education

The Philosophy of Mathematics Education
Author: Paul Ernest,Ole Skovsmose,Jean Paul van Bendegem,Maria Bicudo,Roger Miarka,Ladislav Kvasz,Regina Moeller
Publsiher: Springer
Total Pages: 26
Release: 2016-07-15
ISBN 10: 3319405691
ISBN 13: 9783319405698
Language: EN, FR, DE, ES & NL

The Philosophy of Mathematics Education Book Review:

This survey provides a brief and selective overview of research in the philosophy of mathematics education. It asks what makes up the philosophy of mathematics education, what it means, what questions it asks and answers, and what is its overall importance and use? It provides overviews of critical mathematics education, and the most relevant modern movements in the philosophy of mathematics. A case study is provided of an emerging research tradition in one country. This is the Hermeneutic strand of research in the philosophy of mathematics education in Brazil. This illustrates one orientation towards research inquiry in the philosophy of mathematics education. It is part of a broader practice of ‘philosophical archaeology’: the uncovering of hidden assumptions and buried ideologies within the concepts and methods of research and practice in mathematics education. An extensive bibliography is also included.

Pluralism in Mathematics A New Position in Philosophy of Mathematics

Pluralism in Mathematics  A New Position in Philosophy of Mathematics
Author: Michèle Friend
Publsiher: Springer Science & Business Media
Total Pages: 291
Release: 2013-11-20
ISBN 10: 9400770588
ISBN 13: 9789400770584
Language: EN, FR, DE, ES & NL

Pluralism in Mathematics A New Position in Philosophy of Mathematics Book Review:

This book is about philosophy, mathematics and logic, giving a philosophical account of Pluralism which is a family of positions in the philosophy of mathematics. There are four parts to this book, beginning with a look at motivations for Pluralism by way of Realism, Maddy’s Naturalism, Shapiro’s Structuralism and Formalism. In the second part of this book the author covers: the philosophical presentation of Pluralism; using a formal theory of logic metaphorically; rigour and proof for the Pluralist; and mathematical fixtures. In the third part the author goes on to focus on the transcendental presentation of Pluralism, and in part four looks at applications of Pluralism, such as a Pluralist approach to proof in mathematics and how Pluralism works in regard to together-inconsistent philosophies of mathematics. The book finishes with suggestions for further Pluralist enquiry. In this work the author takes a deeply radical approach in developing a new position that will either convert readers, or act as a strong warning to treat the word ‘pluralism’ with care.