That's a distinction in goals though, not in concepts. If we want to know how many people are on a bus "the number of people you would count if you bothered" is

That's a distinction in goals though, not in concepts. If we want to know how many people are on a bus "the number of people you would count if you bothered" is a perfectly true answer. It's an accurate representation. You perhaps can't do anything more than talk vaguely about abstract bus populations with it, but that is a reasonable thing to do. On the other hand, maybe "17" is the answer you want, because you need to compare it to some other integer or plug it into a function. Also useful things to do, but not always. And not even more often than the trivial alternatives. Even though we're significantly biased toward wanting the latter form, it still means the same thing as the former. And the former is a description of a process, while the latter is a number. Or the former is a number, while the latter is a particular encoding of a process. Same difference in the end. We even define numbers by describing the processes we use to generate them.