A proof is a representation of a claim that is interesting but not trivial as a chain of claims that are trivial but not interesting. The four-color proof illustrates the difference between 'trivial' and 'easy'.
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(A claim that is trivial and interesting is a lemma, a claim that is not trivial and not interesting is ignored.)
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I've been explicitly told that I shouldn't put the detailed rigorous proof in my paper, because it is too complicated and long and distracting and mostly book keeping, and I should put the short hand wavy one in instead, and the reader can figure out the rigorous one if they want
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Liked this a lot! Coming from the logic end I reciprocate; once I have a base assumption, like that there's only true or false, or some such, I can build contraptions of exquisite complication. But not without! Russell meets Godel, and it all falls down.
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This is absolutely fascinating. The weaknesses of other rational systems are reasonably disseminated, but pure math still maintains an air of popular immaculate infallibility...Except when punctured by folk conceptions of Godel's work
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I always thought proofs were just ways of showing that some system is self consistent. This lets you build even more complex self consistent systems on top. Self consistent systems allow you to enter a frame of reasoning that prevents you from contradicting yourself.
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