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.

Zusammenfassung
'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.

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

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