Proofs are sometimes seen as an exercise in tedium, other times as a pure form of beauty, and often as both. But from time immemorial, people have been using mathematics to demonstrate new theorems, and advance the state of the art of mathematics. However, it is only relatively recently, within the last 3,000 years, that the art of mathematical proof has been considered essential to the study of mathematics. Mathematicians constantly fight over what constitutes a proof, and even what makes a proof valid, partially because proof requires delicate insight. So what is the art of mathematical proof? How has it changed? And who can do it?

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