Includes bibliographical references.

Proof and how it is changing. Proof : its nature and significance / Michael Detlefsen -- Implications of experimental mathematics for the philosophy of mathematics / Jonathan Borwein -- On the roles of proof in mathematics / Joseph Auslander -- Social constructivist views of mathematics. When is a problem solved? / Philip J. Davis -- Mathematical practice as a scientific problem / Reuben Hersh -- Mathematical domains : social constructs? / Julian Cole -- The nature of mathematical objects and mathematical knowledge. The existence of mathematical objects / Charles Chihara -- Mathematical objects / Stewart Shapiro -- Mathematical platonism / Mark Balaguer -- The nature of mathematical objects / �ystein Linnebo -- When is one thing equal to some other thing? / Barry Mazur -- The nature of mathematics and its applications. Extreme science : mathematics as the science of relations as such / R.S.D. Thomas -- What is mathematics? A pedagogical answer to a philosophical question / Guershon Harel -- What will count as mathematics in 2100? / Keith Devlin -- Mathematics applied : the case of addition / Mark Steiner -- Probability : a philosophical overview / Alan H�ajek.