Philosophy Mathematics

ISBN-13: 9781405189927 / Angielski / Twarda / 2009 / 344 str.

David Bostock

Opis

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

Recenzje

“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)

 

 

Spis treści

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

O autorze

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).

Blurb

In this new introduction to the philosophy of mathematics, David Bostock guides the reader through the basic ideas on the nature of mathematics that have played a major part in the development of philosophy from antiquity to the present.

The chapters proceed historically, beginning with the earliest serious views on the interpretation of mathematics, due to Plato and Aristotle. They then move quickly through the middle ages and the early moderns, but continue with extended discussions of Kant, Mill, and Frege. Later chapters explore the main schools of thought at the start of the twentieth century, i.e. the movements known as logicism, formalism, intuitionism, and what may be called predicativism. These chapters also discuss more modern variations on the same basic themes. Finally the book concludes with a discussion of the most recent debates between realists and nominalists. The emphasis throughout is not simply to describe, but to offer a critical appraisal of, the views discussed. The result is an engaging, clear, and remarkably comprehensive panorama of the major issues in the field.

The book assumes no prior knowledge of mathematics, beyond what is commonly taught in schools, and only a minimal exposure to standard philosophical terminology. It is aimed at undergraduates in philosophy and mathematics, but will also provide an important new perspective on the subject for more experienced readers.