How would one be able to prove mathematically that $1+1 = 2$?

In a very "raw" sense the symbol $2$ is just a shorthand for $1+1$. There is really not much to prove there.

If we want to talk about proof we need axioms to derive the wanted conclusion from. Let us take the "usual" axioms of the natural numbers here, namely Peano Axioms. These axioms give very basic rules which describe the natural numbers, and from them we will derive $1+1=2$.

In those axioms the numbers $1$ and $2$ don't exist. We have $0$ and we have $S(n)$, which can be thought of as a "successor function" which generates the next number, so to speak. In this system $1$ is a symbol for $S(0)$ and $2$ is a symbol for $S(S(0))$.

Addition is defined inductively, that is $x+0=x$, $x+S(y)=S(x+y)$. From this we can derive:

$$1+1 = 1+S(0) = S(1+0) = S(1)$$ Now replace $1$ with its "full form" of $S(0)$ and we have: $$S(0)+S(0) = S(S(0)+0) = S(S(0))$$

Which is what we wanted.

In a more general setting, one needs to remember that $0,1,2,3,\ldots$ are just symbols. They are devoid of meaning until we give them such, and when we write $1$ we often think of the multiplicative identity. However, as I wrote in the first part, this is often dependent on the axioms - our "ground rules".

If we consider, instead of the natural numbers, the binary digits $0,1$ with addition $\bmod 2$, then we have that $1+1=0$. Now you can argue that of course that $0\neq 2$, however in this set of axioms (which I have not expressed explicitly here) we can prove that $0=2$, where $2$ is the shorthand for $1+1$ and $0$ is the additive neutral element.

Actually, just writing $1+1=0$ is a proof of that.

I can't really stress that enough, because this is a very important part of mathematics. We often use some natural notion, such as the natural numbers, before we define it. Later we define it "to work as we want it to work" and only then we have a formal framework to work with.

These axioms, these frameworks, those often remain "in the shadows" and if you don't know where to look for them then you are less likely to find them.

This is why the question "Why $1+1=2$?" is nearly meaningless - since you don't have a formal framework, and the interpretation (while assumed to be the natural one) is ill-defined.

On that same note, this question is also very important when starting with mathematics. It helps to you understand what there is to prove, and how to do it. Of course this too lack of context because one would have to define what is a proof, and all the other things first.

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.