This won't be a very precise answer, but might still be useful. I have occasionally been able to convince someone that a precise definition is a useful thing because _you can know for sure when you've checked it._ For example, it's surprisingly involved to define whether a graph is connected, under people's usual intuition: _for all_ pairs of vertices, _there exists_ a finite number $n$, such that _there exists_ a sequence of vertices, such that _for all_ vertices $v_i$ in that sequence, $(v_i,v_{i+1})$ is in your edge set. $\bf But$: once you've gone to the bother of making that precise, it's often pretty easy to show that one or another reasonably defined graph is connected. (Then there's the exercise to show that this connectedness is iff there doesn't exist a separating function onto {0,1}.)