Mathematical propositions are just tautologies. The reason they are necessarily true is exactly the same as the reason that all bachelors are unmarried is necessarily true. They're true by definition . . . Russel towards the end of his life said that "if there was a God", notoriously he didn't believe there was a God, "the truths of mathematics would have exactly the same profundity as the truth that a four-legged animal is an animal." --

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