Proofs are important not just for developing critical reasoning, and not simply for avoiding errors, but for progress in mathematics itself. This talk will explain why, and Mark Ronan will present a fascinating array of mathematical examples. The moral of the lecture is two-fold. We need to question our proofs, however good they look, and we must question our assumptions. If we don't, we may miss something vitally important, and non-Euclidean geometry is a glorious example. It has become extremely useful to modern mathematics, in areas surprisingly far removed from geometry.

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