So far I have ever seen the definition of addition of two natural numbers is like:

where m, n are both natural numbers, and S(m) is the successor of m. Today I remembered I have never seen some books or whatever defining m+S(n)=S(m)+n. What makes m+S(n)=S(m+n) better than m+S(n)=S(m)+n?

For example, let's take 1+1=2.

If m+S(n)=S(m+n), then

S(0)+S(0)

=S(S(0)+0)

=S(S(0)).

If m+S(n)=S(m)+n, then

S(0)+S(0)

=S(S(0))+0

=S(S(0)).

Whichever you define what m+S(n) equals, it seems you get same result.

- m+0=m
- m+S(n)=S(m+n)

where m, n are both natural numbers, and S(m) is the successor of m. Today I remembered I have never seen some books or whatever defining m+S(n)=S(m)+n. What makes m+S(n)=S(m+n) better than m+S(n)=S(m)+n?

For example, let's take 1+1=2.

If m+S(n)=S(m+n), then

S(0)+S(0)

=S(S(0)+0)

=S(S(0)).

If m+S(n)=S(m)+n, then

S(0)+S(0)

=S(S(0))+0

=S(S(0)).

Whichever you define what m+S(n) equals, it seems you get same result.