The reason that every major university maintains a department of mathematics is that it is cheaper to do this than to institutionalize all those people. The engineer calculated the average area required per sheep, multiplied by the number of sheep, and proceeded to build a circular fence enclosing the proper area. The physicist gathered all of the sheep together as closely as possible, and built a circular fence around them. The Mathematician just laughed at them. Engineers think that their equations approximate reality.
Physicists think reality approximates their equations.
A mathematician walks into a room and finds a burning trash can sitting on a table. He quickly pours his coffee into the can extinguishing the flames. The next day, he walks in to again find the trash can ablaze, but this time it sits on the floor. He promptly places the trash can on the table and leaves the room. Prove that you're the Pope.