Thinking about Mathematics

"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

Autorentext

Professor of Philosophy, Department of Philosophy, Ohio State University at Newark and Professorial Fellow, Department of Logic and Metaphysics at the University of St Andrews, Scotland.

Klappentext

This unique book by Stewart Shapiro looks at a range of philosophical issues and positions concerning mathematics in four comprehensive sections. Part I describes questions and issues about mathematics that have motivated philosophers since the beginning of intellectual history. Part II is an
historical survey, discussing the role of mathematics in the thought of such philosophers as Plato, Aristotle, Kant, and Mill. Part III covers the three major positions held throughout the twentieth century: the idea that mathematics is logic (logicism), the view that the essence of mathematics is
the rule-governed manipulation of characters (formalism), and a revisionist philosophy that focuses on the mental activity of mathematics (intuitionism). Finally, Part IV brings the reader up-to-date with a look at contemporary developments within the discipline.
This sweeping introductory guide to the philosophy of mathematics makes these fascinating concepts accessible to those with little background in either mathematics or philosophy.

Inhalt

• 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