I can agree with "there are infinitely many primes" if I interpret it as something like "if I ever expect to run out of primes, that belief won't pay rent."
In this case, and in most cases in mathematics, these statements may look and operate the same - except mine might be slower and harder to work with. So why do I insist on it? I'm happy to work with infinities for regular math stuff, but there are some cases where it does matter, and these might all be outside of pure math. But in applied math there can be problems if infinity is taken seriously as a static concept rather than as a process where the expectation that it will end will never pay rent.
Like if someone said, "Black holes have infinite density," I would have to ask for clarification. Can it be put into a verbal form at least? How would it pay rent in terms of measurements? That kind of thing.
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 actual...
[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: