Addition does not require numbers. It turns out, no math requires numbers. Even the math we normally use numbers for.

For instance, here is associativity defined on addition over non-numbers a and b:

a + b = b + a

What if you add a twice?

a + a + b

To do that without numbers, you just leave it there. Given associativity, you probably want to normalize (or standardize) expressions so that equal expressions end up looking identical. For instance, moving references of the same elements together, ordering different elements in a standard way (a before b):

i.e. a + b + a => a + a + b

Here I use => to mean "equal, and preferred/simplified/normalized".

Now we can easily see that (a + b + a => a + a + b) is equal to (b + a + a => a + a + b).

You can go on, and prove anything about non-numbers without numbers, even if you normally would use numbers to simplify the relations and proofs.

Numbers are just a shortcut for dealing with repetitions, by taking into account the commonality of say a + a + a, and b + b + b. But if you do non-number math with those expressions, they still work. Less efficiently than if you can unify triples with a number 3, i.e. 3a and 3b, but by definition those expressions are respectively equal (a + a + a = 3, etc.) and so still work. The answer will be the same, just more verbose.