Starting from what basis? In Peano's axioms for the natural number system, in which "successor" is taken as a primitive notion, "2" is defined as the "successor" of 1 and then "a+ 1" is defined to be the successor of a. From that, immediately, 1+ 1 is the successor of 1, 2.

Peano's axioms: The natural numbers is a set of objects, N, (called "numbers") together with a "successor function", s(n), such that:

1) There is a unique member of the set (called "1") such that the successor function maps N one to one and onto N-{1}.

2) If A is a subset of N such that 1 is in A and, whenever x is in A, the successor of x, s(x), is in A, then A= N.

Once we have that we define "2" to be the successor of "1", "3" to be the successor of "2", "4" to be the successor of "3", etc.

We define "a+ b" by:

1) a+ 1 is s(a).

2) if b is not 1, then there exist c such that b= s(c). In that case, a+ b= s(a+ c).

From (1), 1+ 1= s(1)= 2.

A little more interesting is that 2+ 2= 4. Since 2 is NOT 1, 2= s(1) so 2+ 2= s(2+ 1). 2+ 1= s(2)= 3 so 2+ 2= s(3)= 4.