Good question, I guess I should have made that more clear. Terence Tao has a nice lecture published... trying to find it, brb. In any case, the gist of it was

Good question, I guess I should have made that more clear. Terence Tao has a nice lecture published... trying to find it, brb. In any case, the gist of it was: long as we treat the numbers in the sum as integers (ie discrete), we will get ill-defined sums, contradictions, infinities, and so on. Once we switch to real numbers (and there is an underlying function that is differentiable), things "can" be made to work and converge. To -1/12 always. So that's it. In particular, if we decided that our smallest unit of measure is... the size of atom, or whatever finite value, many of these techniques will just fail to work, and they won't match what we measured. Which means something is wrong. Maybe things really are continuous. Or maybe they are not, it is just that our math is not sophisticated enough. Obviously this is an open question.