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Pentashagon 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.
2, +, 2, =, and 4 are just definitions. What they are definitions for depends on the underlying representation (2 might be a definition for S(S(0)) in PA, { {} , {{}} } in ZF set theory, or two apples in school) but what really matters is that there exists a homomorphism between all our representations.
Even better, there's generally an inverse homomorphism back the real world.
We can convert between any of our representations while preserving the structure of the relationships between the objects in our representations. What we have discovered is not that "2 + 2 = 4" was always true but that any possible equivalent representation is an inherent property of the universe.
"2 + 2 = 5" just lacks a homomorphism to any other useful representations of reality based on our common definitions.