Why does 2+2 equal to 4?

Question

The question is in the title. I am very appreciative of any time and concern put into belaboring this relatively little problem.

Answer 1

The definition of 2 is $1+1$. The definition of 4 is $1+1+1+1$. In general any $n$ is defined as $\sum_{i=1}^n1$. You can think of that as a purely abstract symbol. And the sum $n+m$ is defined to be $\sum_{i=1}^{n+m}1$. It is easy to show that this set and addition make this a well defined monoid. Then you must simply show that $(1+1)+(1+1)=1+1+1+1$. This is true because addition, as defined, is associative.