From my understanding, when we use metric spaces, we are trying to measure how "different" certain elements in a metric space are from one another.
We all know that a metric space $(S,d)$ satisfies:
$d(x,x) = 0$ $\forall x \in S$
$d(x,y) = d(y,x)$$ \forall x,y \in S$
$d(x,y) \leq d(x,z) + d(z,y)$$\forall x,y,z \in S$
The first two make perfect sense to me. If we are trying to measure how different two points are, then the difference between x and x should be nothing. Furthermore, the difference between x and y shouldn't depend on their order.
Now, I'm trying to make sense of why number 3 is a fundamental property of distance. My only guess is that if 3 didn't hold, then there's another "path" that can be taken between x and y that is "shorter" than our metric assigned to x and y. But why is this a fundamental property of measuring distance? Of course, you can give me the euclidian metric as an example, to why this is important, and I would agree...but why must it be fundamental for ALL metrics?
I hope this question isn't too silly. Thank you!