Like if someone said, "Black holes have infinite density," I would have to ask for clarification.
Actually, the way I learned calculus, allowable values of functions are real (or complex), not infinite. The value of the function 1/x at x=0 is not "infinity", but "undefined" (which is to say, there is no function at that point); similarly for derivatives of functions where the functions go vertical. Since that time, I discovered that apparently physicists have supplemented the calculus I know with infinite values. They actually did it because this was useful to them. Don't ask me why, I don't remember. But here is a case where the pure math does not have infinities, and then the practical folk over in the physics department add them in. Apparently the practical folk think that infinity can pay rent.
As for gravitational singularities, the problem here is not the concept of infinity. That is an innocent bystander. The problem is that the math breaks down. That happens even if you replace "infinite" with "undefined".
This isn't really correct. Allowable values of functions are whatever you want. If you define a function on R-{0} by "x goes to 1/x", it's not defined at 0; I explicitly excluded it from the domain. If you define a function on R by "x goes to 1/x"... you can't, there's no such thing as 1/0. If you define a function on R by "x goes to 1/x if x is nonzero, and 0 goes to infinity", this is a perfectly sensible function, which it is convenient to just abbreviate as "1/x". Though for obvious reasons I would only recomm...
[edit: sorry, the formatting of links and italics in this is all screwy. I've tried editing both the rich-text and the HTML and either way it looks ok while i'm editing it but the formatted terms either come out with no surrounding spaces or two surrounding spaces]
In the latest Rationality Quotes thread, CronoDAS quoted Paul Graham: