Intersection Theory – Integral Notation for Degree Homomorphism on Algebraic Cycles

Last night, Peter asked the following question on math.stackexchange:

In Fulton’s Intersection Theory, he develops the notation {\int_X} for the degree homomorphism from {A^*(X)} to {\mathbb{Z}}, and I was wondering if there was a reason for the notation. Is this in any sense a kind of integration?

This resulted in a quite interesting discussion over Facebook chat (that perhaps lasted longer than it needed to), which I had screencapped and posted as an answer on math.stackexchange.

The folks at math.stackexchange were less than amused by this answer (which is reasonable on their part, given the rules of the site), as seen by their discussion here on meta.math.stackexchange. Not surprisingly, the answer was deleted.