Enrico Bombieri, IBM von Neumann Professor, School of Mathematics. This lecture, accessible to a wide audience, explores how mathematics has arrived at its present pragmatic view of infinity and some of the counterintuitive paradoxes, as well as some of the positive results, deriving from its acceptance. It concludes with a view of how computer science is leading today to a new precise concept, namely the impossibly large in the realm of the finite. 

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