Introducing Philosophy of Mathematics

Author: Michele Friend

Publisher: Routledge

ISBN: 1317493796

Category: Philosophy

Page: 240

View: 2623

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

Author: Thomas Bedürftig,Roman Murawski

Publisher: Walter de Gruyter GmbH & Co KG

ISBN: 3110468336

Category: Mathematics

Page: 474

View: 8891

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

MasterClass in Mathematics Education

International Perspectives on Teaching and Learning

Author: Paul Andrews,Tim Rowland

Publisher: Bloomsbury Publishing

ISBN: 1441103333

Category: Education

Page: 240

View: 698

MasterClass in Mathematics Education provides accessible links between theory and practice and encourages readers to reflect on their own understanding of their teaching context. Each chapter, written by an internationally respected authority, explores the key concepts within the selected area of the field, drawing directly on published research to encourage readers to reflect on the content, ideas and ongoing debates. Using international case studies, each chapter will encourage readers to think about ways that the teaching and learning of mathematics reflect different cultural traditions and expectations and enable them to evaluate effective strategies for their own contexts.

Introducing Philosophy of Religion

Author: Chad Meister

Publisher: Routledge

ISBN: 1134141785

Category: Philosophy

Page: 256

View: 551

Does God exist? What about evil and suffering? How does faith relate to science? Is there life after death? These questions fascinate everyone and lie at the heart of philosophy of religion. Chad Meister offers an up-to-date introduction to the field, focussing not only on traditional debates but also on contemporary concepts such as the intelligent creator. Key topics, such as divine reality and the self and religious experience, are discussed in relation to different faiths. Introducing Philosophy of Religion: • offers a lucid overview of contemporary philosophy of religion • introduces the key figures in the history of philosophy of religion • explores the impact of religious diversity and pluralism • examines the main arguments for and against the existence of God and the nature of the divine • looks at science and issues of faith and reason • explores how the different religions approach the concept of life after death. The wealth of textbook features, including tables of essential information, questions for reflection, summaries, glossary and recommendations for further reading make the book ideal for student use. Along with its accompanying Reader, this is the perfect introductory package for undergraduate philosophy of religion courses. Visit the book's companion website at www.routledge.com/textbooks/9780415403276. Features include: an interactive glossary a timeline powerpoint slides on all the chapters chapter outlines lists of objectives for study.

An Introduction to the Philosophy of Science

Author: Rudolf Carnap

Publisher: Courier Corporation

ISBN: 0486140865

Category: Science

Page: 336

View: 5869

Stimulating, thought-provoking text by one of the 20th century's most creative philosophers makes accessible such topics as probability, measurement and quantitative language, causality and determinism, theoretical laws and concepts, more.

Introducing Philosophy

God, Mind, World, and Logic

Author: Neil Tennant

Publisher: Routledge

ISBN: 1317560876

Category: Philosophy

Page: 434

View: 6525

Written for any readers interested in better harnessing philosophy’s real value, this book covers a broad range of fundamental philosophical problems and certain intellectual techniques for addressing those problems. In Introducing Philosophy: God, Mind, World, and Logic, Neil Tennant helps any student in pursuit of a ‘big picture’ to think independently, question received dogma, and analyse problems incisively. It also connects philosophy to other areas of study at the university, enabling all students to employ the concepts and techniques of this millennia-old discipline throughout their college careers – and beyond. KEY FEATURES AND BENEFITS: -- Investigates the philosophy of various subjects (psychology, language, biology, math), helping students contextualize philosophy and view it as an interdisciplinary pursuit; also helps students with majors outside of philosophy to see the relationship between philosophy and their own focused academic pursuits -- Author comes from a distinguished background in Logic and Philosophy of Language, which gives the book a level of rigor, balance, and analytic focus sometimes missing from primers to philosophy -- Introduces students to various important philosophical distinctions (e.g. fact vs. value, descriptive vs. prescriptive, norms vs. laws of nature, analytic vs. synthetic, inductive vs. deductive, a priori vs. a posteriori) providing skills that are important for undergraduates to develop in order to inform their study at higher levels. They are essential for further work in philosophy but they are also very beneficial for students pursuing most other disciplines -- Is much more methodologically comprehensive than competing introductions, giving the student the ability to address a wide range of philosophical problems – and not just the ones reviewed in the book -- Offers a companion website with links to apt primary sources, organized chapter-by-chapter, making unnecessary a separate Reader/Anthology of primary sources – thus providing students with all reading material necessary for the course -- Provides five to ten discussion questions for each chapter, helping instructors and students better interact with the ideas and concepts in the text

Introduction to the Foundations of Mathematics

Second Edition

Author: Raymond L. Wilder

Publisher: Courier Corporation

ISBN: 0486276201

Category: Mathematics

Page: 352

View: 6152

Classic undergraduate text acquaints students with fundamental concepts and methods of mathematics. Topics include axiomatic method, set theory, infinite sets, groups, intuitionism, formal systems, mathematical logic, and much more. 1965 second edition.

18 Unconventional Essays on the Nature of Mathematics

Author: Reuben Hersh

Publisher: Springer Science & Business Media

ISBN: 0387298312

Category: Mathematics

Page: 326

View: 9662

Collection of the most interesting recent writings on the philosophy of mathematics written by highly respected researchers from philosophy, mathematics, physics, and chemistry Interdisciplinary book that will be useful in several fields—with a cross-disciplinary subject area, and contributions from researchers of various disciplines

Popular Culture, Educational Discourse, and Mathematics

Author: Peter M. Appelbaum

Publisher: SUNY Press

ISBN: 9780791422700

Category: Mathematics

Page: 309

View: 2896

This ground-breaking book analyzes contemporary education discourse in the light of curriculum politics and popular culture, using sources ranging from academic scholarship to popular magazines, music video, film and television game shows. Mathematics is used as an “extreme case,” since it is a discipline so easily accepted as separable from politics, ethics or the social construction of knowledge. Appelbaum’s juxtaposition of popular culture, public debate and professional practice enables an examination of the production and mediation of “common sense” distinctions between school mathematics and the world outside of schools. Terrain ordinarily displaced or excluded by traditional education literature becomes the pendulum for a new conversation which merges research and practice while discarding pre-conceived categories of understanding The book also serves as an entertaining introduction to emerging theories in cultural studies, progressively illustrating the uses of discourse analysis for comprehending ideology, the implications of power/knowledge links, professional practice as a technology of power, and curriculum as at once commodities and cultural resources. In this way, Appelbaum effectively reveals a direction for teachers, students and researchers to cooperatively form a community attentive to the politics of curriculum and popular culture

Mecca

The Sacred City

Author: Ziauddin Sardar

Publisher: Bloomsbury Publishing

ISBN: 1408835592

Category: History

Page: 448

View: 1156

Mecca is the heart of Islam. It is the birthplace of Muhammad, the direction towards which Muslims turn when they pray and the site of pilgrimage which annually draws some three million Muslims from all corners of the world. Yet Mecca's importance goes beyond religion. What happens in Mecca and how Muslims think about the political and cultural history of Mecca has had and continues to have a profound influence on world events to this day. In this captivating book, Ziauddin Sardar unravels the significance of Mecca. Tracing its history, from its origins as a 'barren valley' in the desert to its evolution as a trading town and sudden emergence as the religious centre of a world empire, Sardar examines the religious struggles and rebellions in Mecca that have powerfully shaped Muslim culture. Interweaving stories of his own pilgrimages to Mecca with those of others, Sardar offers a unique insight into not just the spiritual aspects of Mecca – the passion, ecstasy and longing it evokes – but also the conflict between heritage and modernity that has characterised its history. He unpeels the physical, social and cultural dimensions that have helped transform the city and also, though accounts of such Orientalist travellers as Richard Burton and Charles Doughty, the strange fascination that Mecca has long inspired in the Western imagination. And, ultimately, he explores what this tension could mean for Mecca's future. An illuminative, lyrical and witty blend of history, reportage and memoir, this outstanding book reflects all that is profound, enlightening and curious about one of the most important religious sites in the world.

Why Is There Philosophy of Mathematics At All?

Author: Ian Hacking

Publisher: Cambridge University Press

ISBN: 1107729823

Category: Science

Page: 212

View: 8485

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.

The Concept of Model

An Introduction to the Materialist Epistemology of Mathematics

Author: Alain Badiou,Zachary Luke Fraser,Tzuchien Tho

Publisher: re.press

ISBN: 0980666570

Category: Electronic books

Page: 113

View: 9724

In "The Concept of Model" Alain Badiou establishes a new logical 'concept of model'. Translated for the first time into English, the work is accompanied by an exclusive interview with Badiou in which he elaborates on the connections between his early and most recent work--for which the concept of model remains seminal.

Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century

Author: Paolo Mancosu

Publisher: Oxford University Press

ISBN: 9780195359152

Category: Philosophy

Page: 288

View: 6140

The seventeenth century saw dramatic advances in mathematical theory and practice. With the recovery of many of the classical Greek mathematical texts, new techniques were introduced, and within 100 years, the rules of analytic geometry, geometry of indivisibles, arithmetic of infinites, and calculus were developed. Although many technical studies have been devoted to these innovations, Mancosu provides the first comprehensive account of the relationship between mathematical advances of the seventeenth century and the philosophy of mathematics of the period. Starting with the Renaissance debates on the certainty of mathematics, Mancosu leads the reader through the foundational issues raised by the emergence of these new mathematical techniques, including the influence of the Aristotelian conception of science in Cavalieri and Guldin, the foundational relevance of Descartes' Geometrie, the relation between geometrical and epistemological theories of the infinite, and the Leibnizian calculus and the opposition to infinitesimalist procedures. In the process Mancosu draws a sophisticated picture of the subtle dependencies between technical development and philosophical reflection in seventeenth century mathematics.

Platonism and Anti-Platonism in Mathematics

Author: Mark Balaguer

Publisher: Oxford University Press on Demand

ISBN: 9780195143980

Category: Mathematics

Page: 217

View: 2202

In this book, Balaguer demonstrates that there are no good arguments for or against mathematical platonism. He establishes that both platonism and anti-platonism are defensible views and introduces a form of platonism ("full-blooded platonism") that solves all problems traditionally associated with the view, proceeding to defend anti-platonism (in particular, mathematical fictionalism) against various attacks--most notably the Quine-Putnam indispensability attack.

The Mathematical Experience, Study Edition

Author: Philip Davis,Reuben Hersh,Elena Anne Marchisotto

Publisher: Springer Science & Business Media

ISBN: 0817682953

Category: Mathematics

Page: 500

View: 2606

Winner of the 1983 National Book Award! "...a perfectly marvelous book about the Queen of Sciences, from which one will get a real feeling for what mathematicians do and who they are. The exposition is clear and full of wit and humor..." - The New Yorker (1983 National Book Award edition) Mathematics has been a human activity for thousands of years. Yet only a few people from the vast population of users are professional mathematicians, who create, teach, foster, and apply it in a variety of situations. The authors of this book believe that it should be possible for these professional mathematicians to explain to non-professionals what they do, what they say they are doing, and why the world should support them at it. They also believe that mathematics should be taught to non-mathematics majors in such a way as to instill an appreciation of the power and beauty of mathematics. Many people from around the world have told the authors that they have done precisely that with the first edition and they have encouraged publication of this revised edition complete with exercises for helping students to demonstrate their understanding. This edition of the book should find a new generation of general readers and students who would like to know what mathematics is all about. It will prove invaluable as a course text for a general mathematics appreciation course, one in which the student can combine an appreciation for the esthetics with some satisfying and revealing applications. The text is ideal for 1) a GE course for Liberal Arts students 2) a Capstone course for perspective teachers 3) a writing course for mathematics teachers. A wealth of customizable online course materials for the book can be obtained from Elena Anne Marchisotto ([email protected]) upon request.

Philosophy of Mathematics

Structure and Ontology

Author: Stewart Shapiro

Publisher: Oxford University Press

ISBN: 9780198025450

Category: Philosophy

Page: 296

View: 657

Do numbers, sets, and so forth, exist? What do mathematical statements mean? Are they literally true or false, or do they lack truth values altogether? Addressing questions that have attracted lively debate in recent years, Stewart Shapiro contends that standard realist and antirealist accounts of mathematics are both problematic. As Benacerraf first noted, we are confronted with the following powerful dilemma. The desired continuity between mathematical and, say, scientific language suggests realism, but realism in this context suggests seemingly intractable epistemic problems. As a way out of this dilemma, Shapiro articulates a structuralist approach. On this view, the subject matter of arithmetic, for example, is not a fixed domain of numbers independent of each other, but rather is the natural number structure, the pattern common to any system of objects that has an initial object and successor relation satisfying the induction principle. Using this framework, realism in mathematics can be preserved without troublesome epistemic consequences. Shapiro concludes by showing how a structuralist approach can be applied to wider philosophical questions such as the nature of an "object" and the Quinean nature of ontological commitment. Clear, compelling, and tautly argued, Shapiro's work, noteworthy both in its attempt to develop a full-length structuralist approach to mathematics and to trace its emergence in the history of mathematics, will be of deep interest to both philosophers and mathematicians.