New MAA Book: Proof and Other Dilemmas

Proof and Other Dilemmas: Mathematics and Philosophy
Bonnie Gold and Roger Simons, Editors
320 pp., hardbound, 2008. Series: Spectrum.
ISBN: 978-0-88385-567-6

Sixteen original essays provide a sampler of current topics in the philosophy of mathematics, from the nature of mathematical objects (How can one tell when one mathematical object is really the same as another?) to the boundaries between mathematics and other disciplines (What does one relegate to "applied" mathematics?). The authors are Barry Mazur, Stewart Shapiro, Charles Chihara, Mark Balaguer, Øystein Linnebo, Michael Detlefsen, Joseph Auslander, Jonathan Borwein, Robert Thomas, Guershon Harel, Reuben Hersh, Julian Cole, Mark Steiner, Philip J. Davis, Keith Devlin, Julian Cole, and Alan Hájek.

The essays by this broad range of mathematicians and philosophers of mathematics offer the most recent thinking on emerging questions in mathematics and philosophy as well as on classical topics, from the problem of "What is mathematics?" to the nature of proof and probabilities. The material is accessible to anyone who has been thinking about or would like an introduction to the philosophy of mathematics and suitable as supplementary readings for upper-level mathematics undergraduates.

Contents:
Acknowledgments; Introduction; Part I. Proof and How it is Changing: 1. Proof: Its Nature and Significance (Michael Detlefsen); 2. Implications of Experimental Mathematics for the Philosophy of Mathematics (Jonathan Borwein); 3. On the Roles of Proof in Mathematics (Joseph Auslander); Part II. Social Constructivist Views of Mathematics: 4. When is a Problem Solved? (Philip J. Davis); 5. Mathematical Practice as a Scientific Problem (Reuben Hersh); 6. Mathematical Domains: Social Constructs? (Julian Cole); Part III. The Nature of Mathematical Objects and Mathematical Knowledge: 7. The Existence of Mathematical Objects (Charles Chihara); 8. Mathematical Objects (Stewart Shapiro); 9. Mathematical Platonism (Mark Balaguer); 10. The Nature of Mathematical Objects (Øystein Linnebo); 11. When is One Thing Equal to Some Other Thing? (Barry Mazur); Part IV. The Nature of Mathematics and its Applications: 12. Extreme Science: Mathematics as the Science of Relations as Such (R.S.D. Thomas); 13. What is Mathematics? A Pedagogical Answer to a Philosophical Question (Guershon Harel); 14. What Will Count as Mathematics in 2100? (Keith Devlin); 15. Mathematics Applied: The Case of Addition (Mark Steiner); 16. Probability—A Philosophical Overview (Alan Hájek); Glossary of Common Philosophical Terms.
           
About the Editors:
Bonnie Gold received her A.B. degree from the University of Rochester, her M.A. in mathematics from Princeton University, and her Ph.D. in mathematics from Cornell University. She taught for twenty years at Wabash College in Indiana, and since 1998 has been in the Mathematics Department at Monmouth University in New Jersey.

Roger A. Simons received his A.B. degree at the University of California, Los Angeles, and his M.A. and Ph.D. from the University of California, Berkeley. He also received a Master of Science degree for computer science from Brown University He retired after 27 years as a professor of mathematics and computer science at Rhode Island College.