On a puzzle about sums

Abstract While the problem of the philosophical significance of Riemann’s theorem on conditionally convergent series has been discussed in detail for some time, specific versions of it have appeared in the literature very recently, over which there have been widespread disagreements. I argue that such discrepancies can be clarified by introducing a rather conventional type of composition rule for the treatment of some infinite systems (as well as supertasks) while analysing and clarifying the role of the concept of continuity by stripping it of the excesses to which its application by the Leibnizian tradition has led. The conclusion reached is that the indeterminacy associated with conditional convergence has a clear philosophical significance, but no fundamental ontological significance.