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.
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 for Thinking about Mathematics: The Philosophy of 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
