Added one more point to my manifesto:
“N” is always finite. Probability theory gives us many powerful limit theorems. These tell us what to expect, mathematically, as the number of trials goes to infinity. That fine, but we don’t live our lives at the limit. In the real world, asymptotics are much more important, as are non-parametric inference and tools like bootstrap to maximize the information extracted from small sample sizes. Even in those real life situations when N could increase without limit, the growth of the sample size itself can change the problem. Toss a die enough times and the edges will start to wear off, changing the frequency that each side comes up.
Had to add it after reading the phrase “if we let N go to infinity” one to many times.