Bear with me as I'm a philosophy (not math) student. First some philosophical background, and then the math question.
One philosophical view is that physical space is composed of infinitely many points of zero volume. But the question arises, how can points of zero volume add up to a non-zero finite volume? Even with an infinite number of points this seems impossible--from nothing you get nothing, no matter how many times you add nothing.
The response to this question is typically an appeal to measure theory in mathematics. While the Lebesgue measure of a set containing a single point, e.g. $[1,1]$ is $0$, the Lebesgue measure of the set of points $[0,1]$ is non-zero. Hence we have what is in some sense an addition of $0$'s equaling something nonzero. Thus if it is the case that measure theory correctly describes physical space, we have an answer to our original question.
Now here is my question to you guys. Are there any alternatives to measure theory which yield different results? That is, are there any instances of mathematics where the sum of an actually infinite amount of zeros (this rules out limits and sums defined in terms of limits) is $0$? One candidate to me seemed to be the hyperreal numbers from non-standard analysis. While I do not know how to formalize an "infinite sum" in non-standard analysis (e.g. is there a sort of measure theory equivalent in non-standard analysis?) but I can do something similar--in non-standard analysis, $0$ times infinity is always zero (right?). Whether this calculation is a better analog to physical space than measure theory is something I am skeptical of, but nonetheless it would be interesting to hear from you mathematicians about sums of zeros in non-standard analysis or other alternative mathematical tools/theories.