Why is distributivity "the only way to reconcile addition and multiplication" Today my prof called distributivity "amazing". I asked him why he thought so, and he replied "it's the only way to reconcile addition and multiplication." It was a tangential question, so I didn't ask him to elaborate, despite having no idea what he meant. 
What does reconciling addition and multiplication involve?
Thank you
-Hal

Here's a possible answer. By definition $0$ is the additive identity (say, in the integers, or any other ring). Prove that $0\cdot r=0$ for every $r$. Despite popular opinion, this is a proposition, not an axiom.
