(P/B) THINKING ABOUT MATHEMATICS

Παρουσίαση

This unique text by Stewart Shapiro looks at a range of philosophical issues and positions concerning mathematics in four comprehensive sections. The first describes questions and issues about mathematics that have motivated philosophers almost since the beginning of intellectual history. Part II is an historical survey, discussing the role of mathematics in such thinkers as Plato, Aristotle, Kant, and Mill. The third section covers the three major positions, and battle lines, throughout the twentieth century: that mathematics is logic (logicism), 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 looks at contemporary positions and work which brings the reader up-to-date on the discipline. Thinking about Mathematics is accessible to those with little background in either mathematics or philosophy. It is aimed at students and professionals in mathematics who have little contact with academic philosophy and at philosophy students and other philosophers who forgot much of their mathematics. (From the publisher)

Περιεχόμενα

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
Godel
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