Thinking about Mathematics

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

Thinking about Mathematics Book Excerpt:

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.

Philosophy of Mathematics

Philosophy of Mathematics
Author: Øystein Linnebo
Publsiher: Princeton University Press
Total Pages: 216
Release: 2017-05-30
ISBN: 0691161402
Category: Philosophy
Language: EN, FR, DE, ES & NL

Philosophy of Mathematics Book Excerpt:

A sophisticated, original introduction to the philosophy of mathematics from one of its leading contemporary scholars Mathematics is one of humanity's most successful yet puzzling endeavors. It is a model of precision and objectivity, but 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 yet accessible introduction to the field that is trying to answer that question: the philosophy of mathematics. Written by Øystein Linnebo, one of the world's leading scholars on the subject, the book introduces all of the classical approaches to the field, including logicism, formalism, intuitionism, empiricism, and structuralism. It also contains accessible introductions to some more specialized issues, such as mathematical intuition, potential infinity, the iterative conception of sets, and the search for new mathematical axioms. The groundbreaking work of German mathematician and philosopher Gottlob Frege, one of the founders of analytic philosophy, figures prominently throughout the book. Other important thinkers whose work is introduced and discussed include Immanuel Kant, John Stuart Mill, David Hilbert, Kurt Gödel, W. V. Quine, Paul Benacerraf, and Hartry H. Field. Sophisticated but clear and approachable, this is an essential introduction for all students and teachers of philosophy, as well as mathematicians and others who want to understand the foundations of mathematics.

Philosophy of Mathematics

Philosophy of Mathematics
Author: Stewart Shapiro
Publsiher: Oxford University Press on Demand
Total Pages: 279
Release: 1997-08-07
ISBN: 0195094522
Category: Mathematics
Language: EN, FR, DE, ES & NL

Philosophy of Mathematics Book Excerpt:

Shapiro argues that both realist and anti-realist accounts of mathematics are problematic. To resolve this dilemma, he articulates a "structuralist" approach, arguing that the subject matter of a mathematical theory is not a fixed domain of numbers that exist independent of each other, but rather is the natural structure, the pattern common to any system of objects that has an initial object and successor relation satisfying the induction principle.

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: 308
Release: 2014-01-30
ISBN: 1107050170
Category: Mathematics
Language: EN, FR, DE, ES & NL

Why Is There Philosophy of Mathematics At All Book Excerpt:

Hacking explores how mathematics became possible for the human race, and how it ensured our status as the dominant species.

Philosophy of Mathematics

Philosophy of Mathematics
Author: Ahmet Cevik
Publsiher: CRC Press
Total Pages: 352
Release: 2021-11-09
ISBN: 1000468801
Category: Mathematics
Language: EN, FR, DE, ES & NL

Philosophy of Mathematics Book Excerpt:

The philosophy of mathematics is an exciting subject. Philosophy of Mathematics: Classic and Contemporary Studies explores the foundations of mathematical thought. The aim of this book is to encourage young mathematicians to think about the philosophical issues behind fundamental concepts and about different views on mathematical objects and mathematical knowledge. With this new approach, the author rekindles an interest in philosophical subjects surrounding the foundations of mathematics. He offers the mathematical motivations behind the topics under debate. He introduces various philosophical positions ranging from the classic views to more contemporary ones, including subjects which are more engaged with mathematical logic. Most books on philosophy of mathematics have little to no focus on the effects of philosophical views on mathematical practice, and no concern on giving crucial mathematical results and their philosophical relevance, consequences, reasons, etc. This book fills this gap. The book can be used as a textbook for a one-semester or even one-year course on philosophy of mathematics. "Other textbooks on the philosophy of mathematics are aimed at philosophers. This book is aimed at mathematicians. Since the author is a mathematician, it is a valuable addition to the literature." - Mark Balaguer, California State University, Los Angeles "There are not many such texts available for mathematics students. I applaud efforts to foster the dialogue between mathematics and philosophy." - Michele Friend, George Washington University and CNRS, Lille, France

Philosophy of Mathematics

Philosophy of Mathematics
Author: Thomas Bedürftig,Roman Murawski
Publsiher: Walter de Gruyter GmbH & Co KG
Total Pages: 474
Release: 2018-10-26
ISBN: 3110470772
Category: Mathematics
Language: EN, FR, DE, ES & NL

Philosophy of Mathematics Book Excerpt:

The present book is an introduction to the philosophy of mathematics. It asks philosophical questions concerning fundamental concepts, constructions and methods - this is done from the standpoint of mathematical research and teaching. It looks for answers both in mathematics and in the philosophy of mathematics from their beginnings till today. The reference point of the considerations is the introducing of the reals in the 19th century that marked an epochal turn in the foundations of mathematics. In the book problems connected with the concept of a number, with the infinity, the continuum and the infinitely small, with the applicability of mathematics as well as with sets, logic, provability and truth and with the axiomatic approach to mathematics are considered. In Chapter 6 the meaning of infinitesimals to mathematics and to the elements of analysis is presented. The authors of the present book are mathematicians. Their aim is to introduce mathematicians and teachers of mathematics as well as students into the philosophy of mathematics. The book is suitable also for professional philosophers as well as for students of philosophy, just because it approaches philosophy from the side of mathematics. The knowledge of mathematics needed to understand the text is elementary. Reports on historical conceptions. Thinking about today‘s mathematical doing and thinking. Recent developments. Based on the third, revised German edition. For mathematicians - students, teachers, researchers and lecturers - and readersinterested in mathematics and philosophy. Contents On the way to the reals On the history of the philosophy of mathematics On fundamental questions of the philosophy of mathematics Sets and set theories Axiomatic approach and logic Thinking and calculating infinitesimally – First nonstandard steps Retrospection

Philosophy of Mathematics

Philosophy of Mathematics
Author: James Robert Brown
Publsiher: Routledge
Total Pages: 264
Release: 2010-03-17
ISBN: 1135902399
Category: Mathematics
Language: EN, FR, DE, ES & NL

Philosophy of Mathematics Book Excerpt:

In his long-awaited new edition of Philosophy of Mathematics, James Robert Brown tackles important new as well as enduring questions in the mathematical sciences. Can pictures go beyond being merely suggestive and actually prove anything? Are mathematical results certain? Are experiments of any real value? This clear and engaging book takes a unique approach, encompassing non-standard topics such as the role of visual reasoning, the importance of notation, and the place of computers in mathematics, as well as traditional topics such as formalism, Platonism, and constructivism. The combination of topics and clarity of presentation make it suitable for beginners and experts alike. The revised and updated second edition of Philosophy of Mathematics contains more examples, suggestions for further reading, and expanded material on several topics including a novel approach to the continuum hypothesis.

Philosophy of Mathematics Today

Philosophy of Mathematics Today
Author: E. Agazzi,György Darvas
Publsiher: Springer Science & Business Media
Total Pages: 361
Release: 2012-12-06
ISBN: 9401156905
Category: Mathematics
Language: EN, FR, DE, ES & NL

Philosophy of Mathematics Today Book Excerpt:

Mathematics is often considered as a body of knowledge that is essen tially independent of linguistic formulations, in the sense that, once the content of this knowledge has been grasped, there remains only the problem of professional ability, that of clearly formulating and correctly proving it. However, the question is not so simple, and P. Weingartner's paper (Language and Coding-Dependency of Results in Logic and Mathe matics) deals with some results in logic and mathematics which reveal that certain notions are in general not invariant with respect to different choices of language and of coding processes. Five example are given: 1) The validity of axioms and rules of classical propositional logic depend on the interpretation of sentential variables; 2) The language dependency of verisimilitude; 3) The proof of the weak and strong anti inductivist theorems in Popper's theory of inductive support is not invariant with respect to limitative criteria put on classical logic; 4) The language-dependency of the concept of provability; 5) The language dependency of the existence of ungrounded and paradoxical sentences (in the sense of Kripke). The requirements of logical rigour and consistency are not the only criteria for the acceptance and appreciation of mathematical proposi tions and theories.

Philosophies of Mathematics

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

Philosophies of Mathematics Book Excerpt:

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.

Wittgenstein s Philosophy of Mathematics

Wittgenstein s Philosophy of Mathematics
Author: Pasquale Frascolla
Publsiher: Routledge
Total Pages: 200
Release: 2006-12-05
ISBN: 113497437X
Category: Philosophy
Language: EN, FR, DE, ES & NL

Wittgenstein s Philosophy of Mathematics Book Excerpt:

Wittgenstein's role was vital in establishing mathematics as one of this century's principal areas of philosophic inquiry. In this book, the three phases of Wittgenstein's reflections on mathematics are viewed as a progressive whole, rather than as separate entities. Frascolla builds up a systematic construction of Wittgenstein's representation of the role of arithmetic in the theory of logical operations. He also presents a new interpretation of Wittgenstein's rule-following considerations - the `community view of internal relations'.

An Historical Introduction to the Philosophy of Mathematics A Reader

An Historical Introduction to the Philosophy of Mathematics  A Reader
Author: Russell Marcus,Mark McEvoy
Publsiher: Bloomsbury Publishing
Total Pages: 896
Release: 2016-02-11
ISBN: 1472532910
Category: Philosophy
Language: EN, FR, DE, ES & NL

An Historical Introduction to the Philosophy of Mathematics A Reader Book Excerpt:

A comprehensive collection of historical readings in the philosophy of mathematics and a selection of influential contemporary work, this much-needed introduction reveals the rich history of the subject. An Historical Introduction to the Philosophy of Mathematics: A Reader brings together an impressive collection of primary sources from ancient and modern philosophy. Arranged chronologically and featuring introductory overviews explaining technical terms, this accessible reader is easy-to-follow and unrivaled in its historical scope. With selections from key thinkers such as Plato, Aristotle, Descartes, Hume and Kant, it connects the major ideas of the ancients with contemporary thinkers. A selection of recent texts from philosophers including Quine, Putnam, Field and Maddy offering insights into the current state of the discipline clearly illustrates the development of the subject. Presenting historical background essential to understanding contemporary trends and a survey of recent work, An Historical Introduction to the Philosophy of Mathematics: A Reader is required reading for undergraduates and graduate students studying the philosophy of mathematics and an invaluable source book for working researchers.

The Oxford Handbook of Philosophy of Mathematics and Logic

The Oxford Handbook of Philosophy of Mathematics and Logic
Author: Stewart Shapiro
Publsiher: Oxford University Press
Total Pages: 856
Release: 2005-02-10
ISBN: 0190287535
Category: Mathematics
Language: EN, FR, DE, ES & NL

The Oxford Handbook of Philosophy of Mathematics and Logic Book Excerpt:

Mathematics and logic have been central topics of concern since the dawn of philosophy. Since logic is the study of correct reasoning, it is a fundamental branch of epistemology and a priority in any philosophical system. Philosophers have focused on mathematics as a case study for general philosophical issues and for its role in overall knowledge- gathering. Today, philosophy of mathematics and logic remain central disciplines in contemporary philosophy, as evidenced by the regular appearance of articles on these topics in the best mainstream philosophical journals; in fact, the last decade has seen an explosion of scholarly work in these areas. This volume covers these disciplines in a comprehensive and accessible manner, giving the reader an overview of the major problems, positions, and battle lines. The 26 contributed chapters are by established experts in the field, and their articles contain both exposition and criticism as well as substantial development of their own positions. The essays, which are substantially self-contained, serve both to introduce the reader to the subject and to engage in it at its frontiers. Certain major positions are represented by two chapters--one supportive and one critical. The Oxford Handbook of Philosophy of Math and Logic is a ground-breaking reference like no other in its field. It is a central resource to those wishing to learn about the philosophy of mathematics and the philosophy of logic, or some aspect thereof, and to those who actively engage in the discipline, from advanced undergraduates to professional philosophers, mathematicians, and historians.

From Foundations to Philosophy of Mathematics

From Foundations to Philosophy of Mathematics
Author: Joan Roselló
Publsiher: Cambridge Scholars Publishing
Total Pages: 230
Release: 2011-10-18
ISBN: 1443834793
Category: Mathematics
Language: EN, FR, DE, ES & NL

From Foundations to Philosophy of Mathematics Book Excerpt:

From Foundations to Philosophy of Mathematics provides an historical introduction to the most exciting period in the foundations of mathematics, starting with the discovery of the paradoxes of logic and set theory at the beginning of the twentieth century and continuing with the great foundational debate that took place in the 1920s. As a result of the efforts of several mathematicians and philosophers during this period to ground mathematics and to clarify its nature from a certain philosophical standpoint, the four main schools in the philosophy of mathematics that have largely dominated the twentieth century arose, namely, logicism, intuitionism, formalism and predicativism. It was due precisely to the insufficiencies of the first three foundational programs and the objections raised against them, that interest in Platonism was renewed in the 1940s, mainly by Gödel. Not only does this book pay special attention to the foundational programs of these philosophies of mathematics, but also to some technical accomplishments that were developed in close connection with them and have largely shaped our understanding of the nature of mathematics, such as Russell’s type theory, Zermelo’s set theory and Gödel’s incompleteness theorems. Finally, it also examines some current research programs that have been pursued in the last decades and have tried, at least to some extent, to show the feasibility of the foundational programs developed in the schools mentioned above. This is the case of neologicism, constructivism, and predicativist and finitist reductionism, this last one developed closely with the research program of reverse 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: 135
Release: 2020-01-31
ISBN: 1108669530
Category: Philosophy
Language: EN, FR, DE, ES & NL

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

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.

The Philosophy of Mathematics and Natural Laws

The Philosophy of Mathematics and Natural Laws
Author: Noel Curran
Publsiher: Routledge
Total Pages: 208
Release: 2019-07-23
ISBN: 0429808089
Category: Philosophy
Language: EN, FR, DE, ES & NL

The Philosophy of Mathematics and Natural Laws Book Excerpt:

First published in 1997, this title is a sequel to Dr Noel Curran's first book The Logical Universe: The Real Universe (published by Ashgate under the Avebury imprint, 1994). The philosophy of mathematics in this book is based on ideas of Sir William Rowan Hamilton on the ordinal character of numbers, the real numbers, the measure numbers, scalar numbers and the extension to vectors. The final extension is to Hamilton’s quaternions. This algebra is interpreted as the mathematics of spin. This led to a a new theory of time and space which is Euclidian. The motion of spin is absolute, no frame of reference is required. If time is assumed to have a beginning it would be asymmetric with an arrow. This concept is applied to the laws of nature, which are symmetrical. This is another Copernican Revolution in three aspects: absolute time is restored, time has an arrow - is asymmetric, and thirdly the theory is based on the motion of spin which is absolute and more fundamental than the motion of translation. This opens the way to the final unification of physics.

Wittgenstein and the Turning Point in the Philosophy of Mathematics

Wittgenstein and the Turning Point in the Philosophy of Mathematics
Author: S.G. Shanker
Publsiher: Routledge
Total Pages: 376
Release: 2013-12-19
ISBN: 1317832035
Category: Philosophy
Language: EN, FR, DE, ES & NL

Wittgenstein and the Turning Point in the Philosophy of Mathematics Book Excerpt:

First published in 2005. Routledge is an imprint of Taylor & Francis, an informa company.

Philosophy of Mathematics

Philosophy of Mathematics
Author: John V. Canfield
Publsiher: Articles-Garlan
Total Pages: 401
Release: 1986
ISBN: 1928374650XXX
Category: Mathematics
Language: EN, FR, DE, ES & NL

Philosophy of Mathematics Book Excerpt:

Introducing Philosophy of Mathematics

Introducing Philosophy of Mathematics
Author: Michele Friend
Publsiher: Routledge
Total Pages: 240
Release: 2014-12-05
ISBN: 1317493796
Category: Philosophy
Language: EN, FR, DE, ES & NL

Introducing Philosophy of Mathematics Book Excerpt:

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.

Philosophy of Mathematics

Philosophy of Mathematics
Author: Vincent F. Hendricks,Hannes Leitgeb
Publsiher: Automatic Press / VIP
Total Pages: 342
Release: 2008
ISBN: 1928374650XXX
Category: Mathematics
Language: EN, FR, DE, ES & NL

Philosophy of Mathematics Book Excerpt:

Interviews of more than two dozen scholars about the philosophy of mathematics, their interests in the field, and their ideas about future developments.

The New Empiricism in the Philosophy of Mathematics

The New Empiricism in the Philosophy of Mathematics
Author: Margarita Rosa Levin
Publsiher: Unknown
Total Pages: 584
Release: 1986
ISBN: 1928374650XXX
Category: Electronic Book
Language: EN, FR, DE, ES & NL

The New Empiricism in the Philosophy of Mathematics Book Excerpt: