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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}.)