Interesting. at first glance, I’m not sure how to explain addition without using addition as a concept. I’ll give it a go.
I seem to remember from Godel, Esher, Bach that numbers were thought of as having successors. So for each number there is a successor that is a unit higher than itself. This forms an infinite number line.
Addition is the the stepping up through a succession of numbers, from one number to new number, by a certain amount of steps.
So 4 + 5 could be thought of the 5th successor of 4, or the 4th successor of 5.
I’m not confident about all of this, but I think number lines concept is related.