In a recent BBC interview one of my colleagues, ``Dr. Z.'', suggested that there are only finitely many integers. When the perplexed interviewer pressed him, Dr. Z. answered something like this: `I know this sounds nonsensical, but it makes more sense than the mess you get yourself into when you allow infinitely many integers.' I am not sure I would share his radical sentiment, but infinity is a tricky business, indeed. The Lorentz force on a point charge can even be infinite in all directions, and this has gotten people into a fine mess. I will explain how physicists computed solutions to electromagnetic equations which don't have any solutions, and how on other occasions they proved the absence of solutions which do exist. Underlying this mess is sometimes a (little too) careless handling of mathematics, but more often a confusion about important conceptual notions in physics. I conclude with an uncensored inquiry into the life of naked singularities in general relativity.