" I'm not nearly as mathematics-versed as you are, so you probably know better than I do why these things are called 'infinities' in a field where infinities are a no-no (as in the Feynman example Richard cited earlier). "
Thanks for the compliment, although I don't think that high of myself. There are far more knowledgeable people here, I'll only comment because I found the issue interesting.
Paul made an interesting comment a few weeks ago regarding the way physicists use math. Being one of them, I knew engineers care more about getting things done than about formalities, but I was surprised to hear that physicists do the same. But it does make sense after all.
I can tell you one thing. If you ask an engineer what happens to some electric circuits when you apply a pulse to them, they will tell you that you may get infinite current or voltage during the moment of transition. That's what our mathematical models tell us, and that's the impression you get when you see a hydro transformer being blown up during a storm. Now engineers are sensible people and won't get into silly philosophical debates on whether infinite currents or voltages really exist, we know full well how to prevent them and that's all we care about. Whether the voltage on your neighborhood hydro transformer is "infinite" or "too damn high" doesn't change the fact that it blows up just the same.
As a sidenote, I cringed when I read Dick arguing with Yanniru that the integral over a function that is infinite on a singularity yields a finite number. I thought it was funny because it's the kind of mathematically unacceptable thing engineers use because it gets things done. I was surprised to learn that physicists are closer to engineers than I used to think.
Well, that's enough nonsense for a day. Back to work...