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ec429 comments on How to Convince Me That 2 + 2 = 3 - Less Wrong

53 Post author: Eliezer_Yudkowsky 27 September 2007 11:00PM

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Comment author: ec429 14 September 2011 06:03:20AM 2 points [-]

I, at least, was not suggesting that you don't know the difference, merely that your article failed to take account of the difference and was therefore confusing and initially unconvincing to me because I was taking account of that difference.

However (and it took me too damn long to realise this; I can't wait for Logic and Set Theory this coming year), I wasn't talking about "models" in the sense that pebbles are a Model of the Theory PA. I was talking in the sense that PA is a model of the behaviour observed in pebbles. If PA fails to model pebbles, that doesn't mean PA is wrong, it just means that pebbles don't follow PA. If a Model of PA exists in which SS0+SS0 = SSS0, then the Theory PA materially cannot prove that SS0+SS0 ≠ SSS0, and if such a proof has been constructed from the axiomata of the Theory then either the proof is in error (exists a step not justified by the inference rules), or the combination of axiomata and inference rules contains a contradiction (which can be rephrased as "under these inference rules, the Theory is not consistent"), or the claimed Model is not in fact a Model at all (in which case one of the axiomata does not, in fact, apply to it).

I should probably write down what I think I know about the epistemic status of mathematics and why I think I know it, because I'm pretty sure I disagree quite strongly with you (and my prior probability of me being right and you being wrong is rather low).

Comment author: Oscar_Cunningham 14 September 2011 09:15:02AM 9 points [-]

Scientists and mathematicians use the word "model" in exactly opposite ways. This is occasionally confusing.