The Achilles paradox in modern discussion. (Die Achilles-Paradoxie in der modernen Diskussion.) (German) Zbl 0838.03005

Rheinwald, Rosemarie

Czermak, Johannes (ed.), Philosophy of mathematics. Proceedings of the 15th international Wittgenstein-Symposium, 16-23 August 1992, Kirchberg am Wechsel, Austria. Part I. Wien: Hölder-Pichler-Tempsky. Schriftenreihe der Wittgenstein-Gesellschaft. 20/I, 383-392 (1993).

The author analyzes the modern discussion on Zeno’s paradox and its modern variants: the paradox of dichotomy, Thompson’s lamp, the \(\pi\)-machine, Achilles’s legato run, and his staccato run. She focuses on the question whether it is possible to master a “super-task”, i.e. doing something which consists of an actual infinite number of parts. She especially analyzes Achilles’s staccato run, i.e. Achilles running a minute, then resting half a minute, then running again a quarter of a minute, ad infinitum. After two minutes he has completed an infinite number of partial runs with an actual infinite number of moves. Such run is logically possible and, according to Adolf Grünbaum [“Modern science and Zeno’s paradoxes of motion”, in: W. C. Salmon (ed.), Zeno’s paradoxes (Bobbs-Merrill, Indianapolis/Boston), 200-250 (1984)], a special version of it with decreasing speed in physically possible as well. The author criticizes Grünbaum’s result that it is physically possible to do an actual infinite number of things in a finite period of time by arguing that this result does not fit with a reasonable notion of doing as action.
For the entire collection see [Zbl 0836.00022].

Reviewer: V.Peckhaus (Erlangen)

MSC:

03A05	Philosophical and critical aspects of logic and foundations
00A30	Philosophy of mathematics

Keywords:

paradoxes of motion; Zeno’s paradox; paradox of dichotomy; Thompson’s lamp; the \(\pi\)-machine; actual infinite; Achilles’s staccato run

Citations:

Zbl 0264.02004