On conditional convergence and non-well ordering

Linnebo recently presented a puzzle about infinite sums according to which the net weight marked by a spring balance depends on the order the weights are placed on it when these weights form a conditionally convergent series. The aim here is to show that Linnebo’s analysis has an interesting limitation that has gone unnoticed: its conclusions only follow when conditional convergence is considered on the basis of a well-ordered sequence of operations. The consequences otherwise are even more puzzling. There is however a way to avoid them, but only for magnitudes (which certainly do not include weight nor the like), for which a continuity principle is available (such as the principle of continuity of motion, associated with the position of a material object). This also has interesting implications regarding the metaphysical adequacy of the standard mathematical concept of convergence for a whole class of infinite series, so-called doubly infinite series. Cases are shown here in which this concept of convergence is metaphysically adequate (useful), and cases where it is not.