A Puzzle about Sums

Oxford Studies in Metaphysics (forthcoming)
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Abstract

A famous mathematical theorem says that the sum of an infinite series of numbers can depend on the order in which those numbers occur. Suppose we interpret the numbers in such a series as representing instances of some physical quantity, such as the weights of a collection of items. The mathematics seems to lead to the result that the weight of a collection of items can depend on the order in which those items are weighed. But that is very hard to believe. A puzzle then arises: How do we interpret the metaphysical significance of this mathematical theorem? I first argue that prior solutions to the puzzle lead to implausible consequences. Then I develop my own solution, where the basic idea is that the weight of a collection of items is equal to the limit of the weights of its finite subcollections contained within ever-expanding regions of space. I show how my solution is intuitively plausible and philosophically motivated, how it reveals an underexplored line of metaphysical inquiry about quantities and locations, and how it elucidates some classic puzzles concerning supertasks.

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Andrew Y. Lee
University of Toronto at Scarborough

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References found in this work

Theories of Location.Josh Parsons - 2008 - In Dean W. Zimmerman (ed.), Oxford Studies in Metaphysics. Oxford University Press. pp. 201-232.
Infinite Ethics.Nick Bostrom - 2011 - Analysis and Metaphysics 10:9–59.
Tasks and Supertasks.James Thomson - 1954 - Analysis 15 (1):1--13.

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