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Arhenius comments on The role of mathematical truths - Less Wrong
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Is 4 not by defintion 2+2, Is math not self proving? I mean why all this "explantion" when it is more evident to say that this thing mathematics is a complex game with rules designed to match the reality.
Typically, 4 is by definition 1+(1+(1+1))), so "2+2=4" means
(1+1)+(1+1) = 1+(1+(1+1)))
In other words, it means two particular different processes for adding up 1's will yield the same result. This is not assumed in Peano arithmetic, but proven from a selection of even more basic assumptions (which need not explicitly mention associativity), albeit a very clever selection.