TITLE: How would one be able to prove mathematically that $1+1 = 2$?
QUESTION [48 upvotes]: Is it possible to prove that $1+1 = 2$? Or rather, how would one prove this algebraically or mathematically?

REPLY [6 votes]: 1 is the convention name of 0++, 2 is the convention name of (0++)++, so what you need to prove is 0++ + 0++ = (0++)++.
$+$ is defined as:

$0+m := m$
$(n++)+m := (n+m)++$

Just apply this definition to the left of the equation and you will get the right. You don't even need to know what '0' or '++' is, all you do is shuffling some sequences of symbols according to some rules.