This article introduces three of the twentieth century's main philosophies ofmathematics andarguesthat ofthose three, one describesmathematical reality, the \reality " of the other two being merely virtual. What are mathematical objects, really? What, for example, is that thing that we call \the number one", or \the set of all positive whole numbers", or \the shortest path between two points on the surface of a sphere"? Most mathematicians (let alone most people) would ¯nd little interest in such questions, since they are totally preoccupied with the practice of their discipline rather than with questions about its meaning. In this essay I shall outline three 1 of the standard philosophical approaches to the meaning of mathematics and present a case that one of those three represents the reality of mathematics, each of the other two amounting to virtual reality. The ¯rst approach that I want to mention is known as platonism. The platonist mathematician believes that mathematical objects do exist, in perfect