One might define multiplication $\bullet$ on $\mathbb Z$ as follows:
$\bullet: \mathbb Z\times \mathbb N\ni (a,b) \mapsto a+\cdots+a\in \mathbb Z$ where we add $b$ times.
But suppose we are in a universe where we can only multiply. How would one define addtion, or could one even define it?
Silly approach 1: $\log(e^ae^b)=\log(e^{a+b})=a+b$, but this assumes existence of $\log$ and $e$ and is rather unsatisfying.
Approach 2: If we could find a formula for $a+1$ (where $a\in \mathbb N$), then we could successively extend this notion to get an addition function $+:\mathbb N\times\mathbb N\to \mathbb N$. But we can only define $a+1$ through multiplication and the only given parameter is $a$, so $a+1$ is be a product of $a$, hence $a\mid (a+1)$. But this is impossible for $a>1$. So it seems we cannot define an addition function using only multiplication, which is again unsatisfying.
Does anybody have an idea whether this is at all possible? What if we could use a (possibly infinite) product of real numbers to define $a+1$?