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Reality and virtual reality in mathematics
"... In this talk I introduce three of the twentieth centurys main philosophies of mathematics and argue that of those three, one describes mathematical reality, the realityof the other two being merely virtual. What are mathematical objects, really? What, for example, is that thing that we call the nu ..."
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In this talk I introduce three of the twentieth centurys main philosophies of mathematics and argue that of those three, one describes mathematical reality, the realityof 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 talk I shall outline three 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. (I should add that there is a fourth standard approach, known as logicism, in which mathematics is regarded as, or reduced to, the formal, axiomatic theory of logical propositions. This philosophy,