The ABC's of Point Count Systems

When I have taught the unit on Social Choice, one of the questions that sometimes arises concerns the numbers used in the Borda Count. Everyone agrees that first place deserves more points than second, second more than third, and so forth.

We shall call any such a monotone point count system. But, except for purely historical reasons, why do the numbers have to be 5, 4, 3, 2, 1 (or 4, 3, 2, 1, 0 as in more recent editions)?

A natural question arises: How do you characterize a collection of ballots so that the social choice result will be the same under all monotone point count systems? The question has a nice answer, and an interesting connection to the calculus, which is, on the surface, surprising in an inherently discrete branch of mathematics applications.