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Eliezer_Yudkowsky comments on Causal Reference - Less Wrong
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PA proves "2 + 2 = 4" using the associative property. PA does not prove "2 + 2 = 3". "2 + 2 = 4" is actually shorthand for "((1+1) + (1+1)) = (((1+1)+1)+1)". Moving stuff next to other stuff in our universe happens to follow the associative property; this is why the belief is useful.
(I like that! The idea that it follows just from the associative property and no other features of PA is quite elegant.)