From the title and the original question in the article I expected something like a proof in a system of formal arithmetic, maybe a discussion of the Peano axioms or a discussion of different possible representations of the natural numbers using sets (like von Neumann ordinal numbers) and the really interesting question if numbers are sets.
Well, the post wasn't very well written probably, but you can argue that they are talking about two concepts: Axioms and Notation. If you used base 2, or base 3 (or base -2, -3) math, then that particular notation would work out differently than, say, base 8 math, etc. For Axioms, here you go: http://en.wikipedia.org/wiki/Axiom
A short answer that sidesteps the question of what numbers "are" is that if two is 1+1 and four is 1+1+1+1 then it follows from the associativity of addition:
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From the title and the original question in the article I expected something like a proof in a system of formal arithmetic, maybe a discussion of the Peano axioms or a discussion of different possible representations of the natural numbers using sets (like von Neumann ordinal numbers) and the really interesting question if numbers are sets.
http://en.wikipedia.org/wiki/Peano_axioms
Well, the post wasn't very well written probably, but you can argue that they are talking about two concepts: Axioms and Notation. If you used base 2, or base 3 (or base -2, -3) math, then that particular notation would work out differently than, say, base 8 math, etc. For Axioms, here you go: http://en.wikipedia.org/wiki/Axiom
A short answer that sidesteps the question of what numbers "are" is that if two is 1+1 and four is 1+1+1+1 then it follows from the associativity of addition:
2 + 2 = (1 + 1) + (1 + 1) = ((1 + 1) + 1) + 1 = 4