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Sewing-Machine comments on How to Convince Me That 2 + 2 = 3 - Less Wrong
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Exactly. This is one of Eliezer's few genuine philosophical mistakes, one which, four years later, he's still making.
I know very well the difference between a collection of axioms and a collection of models of which those axioms are true, thank you.
A lot of people seem to have trouble imagining what it means to consider the hypothesis that SS0+SS0 = SSS0 is true in all models of arithmetic, for purposes of deriving predictions which distinguish it from what we should see given the alternative hypothesis that SS0+SS0=SSSS0 is true in all models of arithmetic, thereby allowing internal or external experience to advise you on which of these alternative hypotheses is true.
Reading your essay I wondered whether it would have been more effective if you had chosen bigger numbers than 2, 2, and 3. e.g. "How to convince me that 67+41 = 112."
That would have been a damn nuisance, because throughout the rest of this comment thread we'd have been writing unhelpfully long strings of Ss. ;)
I was proud of this comment and I comfort myself with your explanation for why it got the response it did.