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Philosophy of Mathematics: An Introduction

ISBN: 978-1-4051-8992-7
344 pages
March 2009, Wiley-Blackwell
Philosophy of Mathematics: An Introduction (1405189924) cover image
Philosophy of Mathematics: An Introduction provides a critical analysis of the major philosophical issues and viewpoints in the concepts and methods of mathematics - from antiquity to the modern era.
  • Offers beginning readers a critical appraisal of philosophical viewpoints throughout history
  • Gives a separate chapter to predicativism, which is often (but wrongly) treated as if it were a part of logicism
  • Provides readers with a non-partisan discussion until the final chapter, which gives the author’s personal opinion on where the truth lies
  • Designed to be accessible to both undergraduates and graduate students, and at the same time to be of interest to professionals
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Introduction.

Part I: Plato versus Aristotle:.

A. Plato.

1. The Socratic Background.

2. The Theory of Recollection.

3. Platonism in Mathematics.

4. Retractions: the Divided Line in Republic VI (509d−511e).

B. Aristotle.

5. The Overall Position.

6. Idealizations.

7. Complications.

8. Problems with Infinity.

C. Prospects.

Part II: From Aristotle to Kant:.

1. Medieval Times.

2. Descartes.

3. Locke, Berkeley, Hume.

4. A Remark on Conceptualism.

5. Kant: the Problem.

6. Kant: the Solution.

Part III: Reactions to Kant:.

1. Mill on Geometry.

2. Mill versus Frege on Arithmetic.

3. Analytic Truths.

4. Concluding Remarks.

Part IV: Mathematics and its Foundations:.

1. Geometry.

2. Different Kinds of Number.

3. The Calculus.

4. Return to Foundations.

5. Infinite Numbers.

6. Foundations Again.

Part V: Logicism:.

1. Frege.

2. Russell.

3. Borkowski/Bostock.

4. Set Theory.

5. Logic.

6. Definition.

Part VI: Formalism:.

1. Hilbert.

2. Gödel.

3. Pure Formalism.

4. Structuralism.

5. Some Comments.

Part VII: Intuitionism:.

1. Brouwer.

2. Intuitionist Logic.

3. The Irrelevance of Ontology.

4. The Attack on Classical Logic.

Part VIII: Predicativism:.

1. Russell and the VCP.

2. Russell’s Ramified Theory and the Axiom of Reducibility.

3. Predicative Theories after Russell.

4. Concluding Remarks.

Part IX: Realism versus Nominalism:.

A. Realism.

1. Gödel.

2. Neo-Fregeans.

3. Quine and Putnam.

B. Nominalism.

4. Reductive Nominalism.

5. Fictionalism.

6. Concluding Remarks.

References.

Index

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David Bostock has been a Fellow and Tutor in Philosophy at Merton College, and Lecturer in Philosophy at the University of Oxford. His recent publications include Intermediate Logic (1997), Aristotle’s Ethics (2000), and Space, Time, Matter, and Form: Essays on Aristotle's Physics (2006).
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  • Offers beginning readers a critical appraisal of philosophical viewpoints throughout history

  • Gives a separate chapter to predicativism, which is often (but wrongly) treated as if it were a part of logicism

  • Provides readers with a non-partisan discussion until the final chapter, which gives the author’s personal opinion on where the truth lies

  • Designed to be accessible to both undergraduates and graduate students, and at the same time to be of interest to professionals

See More

“Given this caveat, Bostock’s new book is highly recommendable as a text for undergraduate seminars in the philosophy of mathematics and also for individual study. It covers all the essentials and more. It should appeal not only to students who have already developed a preference for the general approach and style of contemporary analytic philosophy, but also to a broader audience of students and to people with a non-professional interest in philosophy and mathematics.”  (Erkenn, 2011)

"This is a concise as well as comprehensive presentation of core topics in the philosophy of mathematics, written in a clear and engaged manner, hence well readable." (Zentralblatt MATH, 2011)

 

 

"The best textbook on the philosophy of mathematics bar none" –Alexander Paseau, University of Oxford

"Bostock's 'Philosophy of Mathematics' is remarkably comprehensive compared to other surveys of philosophy of mathematics. The writing is engaging and clear, and it treats a wide range of issues in considerable depth, including issues that are often ignored or downplayed in more general discussions." –Alan Baker, Swarthmore College

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