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

Let $\mathbf{N}$ denote the structure $(\mathbb{N},+,\times,0,1).$
Then $(\mathbb{N},+,0)$ satisfies the following identities.


*

*$+$ is associative

*$+$ is commutative

*$0$ is left-neutral for $+$

*$0$ is right-neutral for $+$


In fact, it turns out that all the identities that hold for $(\mathbb{N},+,0)$ can be proven from the above four. (I'm not claiming the list minimal, of course.)
Also, $(\mathbb{N},\times,0,1)$ satisfies the following identities.


*

*$\times$ is associative

*$\times$ is commutative

*$1$ is left-neutral for $\times$

*$1$ is right-neutral for $\times$

*$0$ is left-absorptive for $\times$

*$0$ is right-absorptive for $\times$


And yep, you guessed it; all the identities that hold for $(\mathbb{N},\times,0,1)$ can be proven from the above six. (Once again, no claims to minimality).
Despite all this, some identities for $\mathbf{N}$ cannot be proven from the above $10.$ For example, the identity $$(x+y)^2 = x^2+2xy+y^2$$ cannot be proven. Intuitively, this is because we have not specified any constraints on how $+$ and $\times$ interact. Amazingly, we only need one more identity before our list of axioms for the equational theory of $\mathbf{N}$ is complete. Namely,
$$a(x+y) = ax+ay.$$
So distributivity alone "generates" all the relationships (expressible as identities) that hold between $+$ and $\times$ (in the presence of the previous $10$ axioms, of course). Frankly, this amazing. The miracle is not so much that "it is the only way to reconcile addition and multiplication" (what does that even mean?), rather the miracle is that "distributivity suffices to reconcile addition and multiplication."
