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RichardKennaway comments on Why CFAR's Mission? - Less Wrong
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I learned math with the Peano axioms and we considered the symbol
2to refer to the1+1, 3 to(1+1)+1and so on. However even if you consider it to be more complicated it still stays an analytic statement and isn't a synthetic one.If you define 2 differently what's the definition of 2?
One popular definition (at least, among that small class of people who need to define 2) is { { }, { { } } }.
Another, less used nowadays, is { z : ∃x,y. x∈z ∧ y∈z ∧ x ≠ y ∧ ∀w∈z.(w=x ∨ w=y) }.
In surreal numbers, 2 is { { { | } | } | }.