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Document Details :
Title: The Sad Ballad of Achilles
Subtitle: Part 1: A Reimagined Telling of the Paradox of Achilles and the Tortoise, and an Introduction into the Concepts of Megatasks and Time-Cycling
Author(s): CROUSE, David
Journal: Logique et Analyse
Volume: 254 Date: 2021
In this paper, we use a reimagined telling of the paradox of Achilles and the tortoise to study important concepts on the nature of space, time, motion, measurements, and existence. In the process of telling the new version, we review the conventional solutions to the paradox, Richard Friedberg’s solution that resolves dynamical issues with the paradox, and the complete solution to the paradox assuming discrete space-time. The new version adds to the body of work done on the issue of distinguishability of the segments run by Achilles. We extend this well-researched topic significantly further by introducing an intriguing possibility: rather than Achilles performing an infinite number of tasks in a finite time (i.e., a supertask), we explore the possibility that he can perform an infinite number of tasks over an infinite duration but arrive at the finish line at a finite time-coordinate of his choosing (a scenario that we call a megatask). This bizare scenario and accompanying discussion is not a revisitation of Henri Bergson’s duration-versus-time debate but a new question concerning time, motion, and space. In this work, we show that if zeta function regularization is valid in calculations involving time, as it is for other measurable quantities in experimentally confirmed theoretical descriptions of physical phenomona (e.g., the Casimir effect), all systems will be accompanied by echoes of themselves — echoes that manifest as temporal delocalization of the systems.