I've been trying to learn naive set theory through a YouTube series by 'The Bright Side of Mathematics'. So far, I've been able to understand successor maps and the definition of $\mathbb{N}_{0}$. I have now gotten to the point where he defines addition. His definition is the following:
$$\text{Addition is a map } \mathbb{N}_{0} \times \mathbb{N}_{0} \rightarrow \mathbb{N}_{0}$$ $$(m,n) \mapsto m + n$$ He also gives the following: $$m+0:=m$$ $$m + s(n) := s(m+n)$$
I am able to understand how he gets these statements, as he works through it quite well on the video. However, to me this seems like circular reasoning. How can you define addition using the $+$ symbol, isn't that against the purpose of a definition?
Whilst doing further research I managed to find the following article: How is addition defined?. The answers on it however don't make much sense to me as a complete beginner, and I am wondering whether anyone has a definition of addition without using $+$ (or if that's even possible)?