The philosopher's favorite mathematician Here's an interesting thing: a mathematician who claims that most mathematicians don't really make use of Goedel's theorem. In my limited experience, this is accurate. But if you take my metalogic class, you'll hear a good deal about Goedel, and I were teaching on the history of 20th Century Analytic philosophy, and not simply 20th Centur philosophy, then you'd hear a lot more about Goedel. For mathematicians, the proof is a very interesting proof that doesn't necessarily touch on anything they were interested in to begin with. For philosophers, this was a serious jolt to the asperations of Russell and Frege, one consequence of Goedel's presentation is that the rules of logic alone are not sufficient to provide a complete encoding of all arithmetic truths. (Although the converse is true, you can translate all of your logical truths into arithmetic, which how he pulls off his "P is not provable in L" trick.) One consequence of the Goedel's methods, rather than the theorems themselves is that studying numbers allows for interesting investigations that logic alone can't investigate. Of course, most working mathematicians probably took this somewhat for granted, which is why "Gödel's theorem, for most working mathematicians, is like a sign warning us away from logical terrain we'd never visit anyway".  ¶ #

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