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MixedNuts comments on Proofs, Implications, and Models - Less Wrong
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Meditation:
It has been claimed that logic and mathematics is the study of which conclusions follow from which premises. But when we say that 2 + 2 = 4, are we really just assuming that? It seems like 2 + 2 = 4 was true well before anyone was around to assume it, that two apples equalled two apples before there was anyone to count them, and that we couldn't make it 5 just by assuming differently.
My response, before reading the other responses, is that this is not a matter of assumption but of definition; the symbols '2', '+', '=' and '4' have been defined in such a way that 2+2=4 is a true statement. (The important symbol, here, is + in my view: 2, 4 and = are such basic operations that it's near certain that there would have been some symbol with those meanings. + is pretty basic, but to my mind less basic - it's not the only way to combine two quantities).
This could be seen as taking the definitions of 2, 4, = and + as premises and the truth of the statement 2+2=4 as a conclusion. The conclusion (2+2=4) follows from the premises whether anyone's around to postulate the premises or not; that remains true of any logical statement. So, similarly, if all kittens are little, and if all little things are innocent, then it remains true that all kittens are innocent whether anyone has ever considered that chain of logic before or not.