I'm an electrical engineering major, and as a result, I've spent a lot of time sitting in math classes (my math minor is partly to blame, I guess) and I've always had this secret belief that, for all of their brilliance, many mathematicians have approximately the same understanding of physics and mathematical applications that one would expect from the average fashion major. This belief was recently confirmed for me in a particularly funny example (at least to us engineers).
My faculty advisor, an electrical engineer and a specialist in digital signal analysis, was asked to sit on the board for the oral defense of a mathematics student's doctoral dissertation, and I was invited to attend. I thought that it would be interesting to see what the ground-pounders of modern mathematics were doing, but I didn't think that I would get much out of it. Sure enough, this dissertation was far too esoteric for me to even understand what it was saying, let alone follow it. I did, however, notice that it made heavy use of the Fourier Transform, which I recognized because it forms the cornerstone, foundation, and capstone of all signal processing.
After the presentation, the members of the board asked questions while my advisor remained silent. At the very end, he asked the doctoral candidate if he were aware of any practical applications of the Fourier Transform. The student looked puzzled for a second, and then replied that he couldn't think of what it could possibly be good for. My professor smiled, and I had to suppress the urge to laugh out loud. The Fourier Series and the Fourier Transform were developed to solve the engineering problem of heat conduction through metal and only given a firm mathematical basis years later. Don't these guys have to take a math history class? However, after the doctoral candidate had shuffled out of the room, obviously relieved, another member of the board, a mathematics professor, turned to my advisor and asked "You mean that the Fourier Transform does have practical applications?"