During the first 75 years of the twentieth century almost all work in the philosophy of mathematics concerned foundational questions. In the last quarter of the century, philosophers of mathematics began to return to basic questions concerning the philosophy of mathematics such as, what is the nature of mathematical knowledge and of mathematical objects, and how is mathematics related to science? Two new schools of philosophy of mathematics, social constructivism and structuralism, were added to the four traditional views (formalism, intuitionalism, logicism, and platonism). The advent of the computer led to proofs and the development of mathematics assisted by computer, and to questions of the role of the computer in mathematics. This book of 16 essays, all written specifically for this volume, is the first to explore this range of new developments in a language accessible to mathematicians. Approximately half the essays were written by mathematicians, and consider questions that philosophers are not yet discussing. The other half, written by philsophers of mathematics, summarize the discussion in that community during the last 35 years. In each case, a connection is made to issues relevant to the teach of mathematics.

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This collection of sixteen original essays is the first to explore a range of new developments in the philosophy of mathematics, in a way mathematicians will understand. Coverage includes emerging questions in the field as well as recent thinking on classical ideas, all relevant to the teaching of mathematics.

Bonnie Gold 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 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. 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. 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, 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 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.

Roger A. Simons received his A.B. degree in Education at the University of California (Los Angeles), his Masters in Science from Brown University and his Ph.D. from UCLA (Berkeley). He was the past Chair and founding member of POMSIGMAA, Philosophy of Mathematics Special Interest Group of the MAA.

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