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John_Maxwell_IV comments on Formalization is a rationality technique - Less Wrong
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Perhaps a better response to the forays of amateurs would be to define a formal model that represents your understanding of their argument, explain it to them, and see if they agree that it's accurate.
That's a lot of work to respond to an amateur's argument with. Probably at least an order of magnitude more work than went into the original argument. And the formal argument is likely to end up being very different from the original, informal one; it would be very frustrating to take someone's informal argument, formalize it, show that the formal version of the argument is incorrect, and then be told that your formalization missed some important insight.
Really? Maybe I'm just naive. Could you give me an example of an argument and its formalization?
Good examples of this include the efforts of posters on the newsgroup sci.math to make sense of arguments by math cranks who believe that they've proved the denumerability of the reals or what have you.