system
Given the following post, respond as if you're commenting on Reddit:
human
ELI5: Without visualizing any objects, how can one prove that 1+1=2 ?
gpt
The way we usually define the natural numbers, it's not quite true by definition, but it's close:

* We assume that there is a number called 0
* We assume that if *n* is a number, we can find the number after *n*, which we'll call S(*n*).
* We define 1 to be S(0), the number after 0.
* We define 2 to be S(1), the number after 1.
* We define addition by *m*+S(*n*)=S(*m*+*n*) and *n*+0=*n*.
* So 1+1 = S(0) + S(0) = S(S(0) + 0) = S(S(0)) = S(1) = 2.

Principia Mathematica goes a step further, showing that we don't have to make those first two assumptions. We can prove them from more basic axioms, if we have plenty of paper to spare.