This is the first book on the philosophy of mathematics to explore the breadth of contemporary issues that interest mathematicians, using language accessible to non-philosophers. It should interest users and teachers of mathematics. Most authors of the 16 original essays are either prominent mathematicians or prominent mathematicians with philosophical interests. It can serve as a non-technical primer on current thinking in the philosophy of mathematics. It includes comments on applications to teaching mathematics and could be a text for a philosophy of mathematics course for mathematics majors.

### Table of Contents

Acknowledgments

Introduction

Proof and How it is Changing

Social constructivism Views of Mathematics

The Nature of Mathematical Objects and Mathematical Knowledge

The Nature of Mathematics and its Applications

Glossary of Common Philosophical Terms

About the Editors

### About the Editors

**Bonnie Gold** was born in New York city and attended high school at the Bronx High School of Science, received her A.B. degree from the University of Rochester with highest honors in mathematics, her M.A. in mathematics from Princeton University, and her PhD in mathematics from Cornell University. She taught for 20 years at Wabash College in Indiana, and since 1998 has been in the mathematics department at Monmouth University in New Jersey. While at Wabash College, she received the McLain-Turner-Arnold Award for Excellence in Teaching, as well as a Lilly Open Faculty Fellowship to study and begin research in the philosophy of mathematics. In 2006, she received the Distinguished Teaching Award from the New Jersey section of the Mathematical Association of America. At Monmouth University, she was co-director of the 21st Century Science Teachers’ Skills Project. She is a member of the American Mathematical Society, the Mathematical Association of America (MAA), the Canadian Society for the History and Philosophy of Mathematics, is a founding member of the Special Interest Group of the MAA for the Philosophy of Mathematics (POMSIGMAA) (and was its first Chair and is current Public Information Officer), and is currently Vice-Chair for Speakers of the New Jersey section of the MAA. She was co-editor (with Sandra Keith and William Marion) of Assessment Practices in Undergraduate Mathematics. She is the editor of MAA Online’s Innovative Teaching Exchange, and is the director of NJ-NExT, a state version of a national project for new college faculty in mathematics. She has written one article in the philosophy of mathematics, “What is the Philosophy of Mathematics, and What Should it Be?” *Mathematical Intelligencer*, 1994.

**Roger A. Simons** was born in Detroit, Michigan. When he was three years old, his family moved to Los Angeles, where he received all his early education. He graduated from University High School and received an A.B. degree with honors from UCLA as a mathematics major. He received an M.A. and PhD from the University of California, Berkeley in mathematics and also an Sc.M. from Brown University in computer science. During his PhD program, he worked two summers at Aerospace Corporation on NASA’s Gemini and Apollo projects. He was on the faculty of the University of Wisconsin, Green Bay for ten years and is just retiring after 27 years as a professor of mathematics and computer science at Rhode Island College. He has been a visiting faculty member in mathematics at the University of Hawaii six times, an adjunct professor of computer science at the University of Rhode Island, and has consulted to three different companies, including Chemical Bank, helping with their in-house software development. He has refereed for the *ACM Symposium on Theory of Computing, Algorithmica, Communications of the ACM, IEEE Computer Society Transactions on Knowledge and Data Engineering, Journal of Computer System Sciences, Theoretical Computer Science*, the Consortium for Computing in Small Colleges, and the Association for Symbolic Logic. He has also evaluated grant proposals for the NSF CCLI program in computer science. He appeared in *Who’s Who Among America’s Teachers 2005*, a recognition resulting only by the nomination of an honor student as the one teacher in her academic career who made a difference in her life. He is a member of the Mathematical Association of America, American Mathematical Society, Association for Symbolic Logic, Association for Computing Machinery, Sigma Xi, and Pi Mu Epsilon. He served one term as president of the Rhode Island College Chapter of Sigma Xi and has been its treasurer for 15 years. His PhD studies in logic and foundations aroused his curiosity about several issues not explained or not addressed by logic or set theory. Thirty years later some reading on the philosophy of mathematics brought back his former thoughts and raised his interests enough to become active in this field. He is a founding member of the Special Interest Group of the MAA for the Philosophy of Mathematics (POMSIGMAA) and was its second Chair.