Thought about this recently, and was a bit stuck.
Is all mathematics based on the concept that $1+1=2$?
For example, if $1+1\ne2$, then all arithmetic won't work, right?
Sometimes $1+1=0$, when you're working in a unital ring with characteristic $2$!
But those are the only possibilities. $2$ is defined as $1+1$. (So actually in the characteristic $2$ case, $1+1=2$ still- it's just that $0=2$.)
If $1+1=5$ just declare $5$ as $2$ and move on with it. You are mixing the idea of $2$ with its symbol.
We usually define the meaning of the symbol $2$ as a shorthand for $1+1$, but it might as well be any other symbol. It should be noted that if we assume the "usual" axioms of mathematics and we manage to prove that $1+1\neq2$ then our axioms have an inherent contradiction and we need to find it. But so far we're good.