I guess its more flipped. How many pieces equal the whole, not how many pieces can the whole be broken up into. That's why you can't divide by 0, any number of nothing can't make a whole.
The function "division" is just like any other function. It takes two inputs, and gives an output.
The reason it's a useful function is it has some properties:
y=1/x is continuous except where x==0. That is, as you approach 0 from either side, it grows to either +- infinity. The developers of this language want to make this function ==0 when x is 0.
You can do that, sure, but it doesn't change the fact that when you go from +0.0000001 to -0.0000001, you now have 2 non-continuous jumps instead of just one.
*disclamer: I have no math background whatso ever