What paradoxes taught me.

Perhaps each time a paradox is encountered in mathematics, there is a useful distinction or mathematical tool hiding in plain sight, one that hasn't be discovered or utilized. For cursory evidence I give you: division by zero, the speed of an arrow at any point in flight, and calculus.

Maybe this isn't true for some paradoxes, or even most, but as time goes on I suspect people will discover it is more true than they might have thought.

Undefined behavior and results aren't nonsense: They look to me like golden seams to be explored for possible utility when approached from uncommon angles with uncommon problems.