look at "the foundations of mathematics": set theory.
Per definition, 0 is the empty set
Per definition, 1={0}
Per definition, 2={0,1}={0,{0}}
Per definition, x+1=xU{x} (that's a union)
Per definition, x+y is defined recursively from x+1 (I won't go into the details).
Now, the depends on whether you see 1+1 as the function x+1 with parameter x=1 (this function is often written as S(x)), or whether you see 1+1 as the function x+y with parameters x=y=1. In the former case, 1+1=2 is pretty much the definition of the symbol "2". In the latter case, 1+1=2 is a theorem, though one which is so easy that you have to be really pedantic to see why it is not per definition.
In algrebra, you sort of assume that you already have the set of integers and the addition operation on them. Algebra doesn't really deal with where they came from. |
