Thinking about Mathematics

Description

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.

• The only introductory guide to the philosophy of mathematics aimed at the general reader/undergraduate student
• Uniquely guides reader through the 'history' of the subject providing an overview of major philosophical positions and debates, as well as the major philosophers of the 20th Century
• Only basic prior knowledge of either philosophy or mathematics needed, making it ideal for the student or reader coming to the subject for the first time

About the Author(s)

Stewart Shapiro, Professor of Philosophy, Ohio State University at Newark, USA

Table of Contents

Part I. Perspective
Chapter 1. What is so interesting about mathematics (for philosopher)?
Attraction - of opposites?
Philosophy and mathematics: chicken or egg?
Naturalism and mathematics
Chapter 2. A Potpourri of questions and attempted answers
Necessity and a priori knowledge
Global matters: objects and objectivity
The mathematical and the physical
Local maters: theorems, theories, and concepts
Part II. History
Chapter 3. Plato's Rationalism, and Aristotle
The world of Being
Plato on mathematics
Mathematics on Plato
Aristotle, the worthy opponent
Further reading
Chapter 4. Near opposites: Kant and Mill
Reorientation
Kant
Mill
Further reading
Part III. The big three
Chapter 5. Logicism: Is mathematics (just) logic?
Frege
Russell
Carnap and logical positivism
Contemporary views
Further reading
Chapter 6. Formalism: Do mathematical statements mean anything?
Basic views: Freg's onslaught
Deductivism: Hilbert's Grundlagen der Geometrie
Finitism: the Hilbert program
Incompleteness
Curry
Further reading
Chapter 7. Intuitionism: is something wrong with our logic?
1. Revising classical logic
2. The teacher, Brouwer
3. The student, Heyting
4. Dummett
5. Further reading
Part IV. The contemporary scene
Chapter 8. Numbers exist
Gödel
The web of belief
Set-theoretic realism
Further reading
Chapter 9. No they don't
Fictionalism
Modal construction
What should we make of all this?
Addendum: Young Turks
Further reading
Chapter 10. Structuralism
The underlying idea
Ante rem structures, and objects
Structuralism without structures
Knowledge of structures
Further reading
References
Index

Reviews

""Stewart Shapiro admirably provides an accessible introduction to contemporary thinking in mathematics, while avoiding caricature of the technicalities. His ease with the subject and lucid style makes this book a succinct introduction to a fascinating intellectual discipline." Times Literary Supplement"