Philosophies of Mathematics

During the first few decades of the twentieth century, philosophers and mathematicians mounted a sustained effort to clarify the nature of mathematics. This led to considerable discord, even enmity, and yielded fascinating and fruitful work of both a mathematical and a philosophical nature. It was one of the most exhilarating intellectual adventures of the century, pursued at an extraordinarily high level of acuity and imagination. Its legacy principally consists of three original and finely articulated programs that seek to view mathematics in the proper light: logicism, intuitionism, and finitism. Each is notable for its symbiotic melding together of philosophical vision and mathematical work: the philosophical ideas are given their substance by specific mathematical developments, which are in turn given their point by philosophical reflection.

This book provides an accessible, critical introduction to these three projects as it describes and investigates both their philosophical and their mathematical components.

Solutions manual is available upon request.

Preface.

1. Introduction.

2. Logicism.

3. Set Theory.

4. Intuitionism.

5. Intuitionistic Mathematics.

6. Finitism.

7. The Incompleteness Theorems.

8. Coda.

References.

Index.

Alexander George is Associate Professor of Philosophy at Amherst College. He is editor of Reflections on Chomsky (1989) Western State Terrorism (1991) and Mathematics and Mind (1994).

Daniel J. Velleman is Professor of Mathematics at Amherst College. He is author of How to Prove It: A Structured Approach (1994) and co–author of Which Way Did the Bicycle Go? And Other Intriguing Mathematical Mysteries (with Joseph Konhauser and Stan Wagon, 1996).

"George and Velleman manage to accomplish a difficult feat: on the one hand, they explain, clearly and rigorously, a number of highly technical accomplishments of twentieth–century mathematical logic, making plain the relevance of the mathematical work for philosophy; yet, on the other, they presuppose little more from their readers than a first course in basic logic. The examples they choose to explicate their points are carefully selected and illuminating. This is a splendid book." William Ewald, University of Pennsylvania <!––end––>

"This book includes just the right mix of helpful historical exposition and clear, tight philosophical argument. It is extremely well written and does an excellent job of making difficult material accessible. There is nothing else currently available that discusses in a single volume such a wide range of important material. The authors are to be commended for a job well done." Andrew Irvine, University of British Columbia

"This is a well–written, informative and innovative introduction to philosophies of mathematics. It is a very valuable addition to the existing literature." Wilfried Sieg, Carnegie Mellon University