"The Catholic clergy seldom bother to make their arguments plausible; it is plain that they have little respect for human intelligence, and indeed little belief in its existence."

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The goal of a definition is to introduce a mathematical object. The goal of a theorem is to state some of its properties, or interrelations between various objects. The goal of a proof is to make such a statement convincing by presenting a reasoning subdivided into small steps each of which is justified as an "elementary" convincing argument.

— IU?. I. Manin