Abstract
Three of Zeno's objections to motion are answered by utilizing a version of nonstandard analysis, internal set theory, interpreted within an empirical context. Two of the objections are without force because they rely upon infinite sets, which always contain nonstandard real numbers. These numbers are devoid of ‘numerical meaning’, and thus one cannot render the judgment that an object is, in fact, located at a point in spacetime for which they would serve as coordinates. The third objection, ‘an arrow never appears to be moving’, is answered by showing that it only applies to a finite number of instants of time. A theory of motion is also advanced; it consists of a finite series of contiguous infinitesimal steps. The theory is immune to Zeno's first two objections because the number of steps is finite and each lies outside the domain of observation. ‘Present motion’ is hypothesized to be an unobservable process taking place within each step. The fact of motion is apparent through a summing (Riemann integration) of the steps.
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References
Allen, R. E.: 1983, Plato's Parmenides, University of Minnesota Press, Minneapolis.
Bishop, E.: 1967, Foundations of Constructive Mathematics, McGraw-Hill, New York.
Borel, E.: [1950]/1963, Probability and Certainty, Walker, New York (first published as Probabilité et Certitude, Presses Universitaires de France).
Cajori, F.: 1915, ‘The History of Zeno's Arguments on Motion’, American Mathematical Monthly 22, 1–6, 38–47, 77–82, 109–15, 143–49, 179–86, 215–20, 253–58, 292–97.
Davis, M.: 1983, Review of E. Nelson's Internal Set Theory: A New Approach to Nonstandard Analysis, Journal of Symbolic Logic 48, 1203–204.
Kyburg, H. E.: 1988, Review of L. Narens' Abstract Measurement Theory, Synthese 76, 179–82.
Mandelkern, M.: 1985, ‘Constructive Mathematics’, Mathematics Magazine 58, 272–80.
Narens, L.: 1985, Abstract Measurement Theory, MIT Press, Cambridge.
Nelson, E.: 1977, ‘Internal Set Theory: A New Approach to Nonstandard Analysis’, Bulletin of the American Mathematical Society 83, 1165–98.
Robert, A.: 1988, Nonstandard Analysis, John Wiley, New York.
Robinson, A.: 1966, Non-standard Analysis, North-Holland, Amsterdam.
Salmon, W. C.: 1970, ‘Introduction’, in Zeno's Paradoxes, Bobbs-Merrill, Indianapolis and New York, pp. 5–44.
Van Dantzig, D.: 1955, ‘Is \(10^{10^{10} } \) a Finite Number?’, Dialectica 9, 273–77.
Vlastos, G.: 1966a, ‘A Note on Zeno's Arrow’, Phronesis 11, 3–18.
Vlastos, G.: 1966b, ‘Zeno's Race Course’, Journal of the History of Philosophy 4, 95–108.
Vlastos, G.: 1967, ‘Zeno of Elea’, in The Encyclopedia of Philosophy, Macmillan, New York, pp. 369–79.
Whitehead, A. N.: 1948, Essays in Science and Philosophy, Philosophical Library, New York.
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McLaughlin, W.I., Miller, S.L. An epistemological use of nonstandard analysis to answer Zeno's objections against motion. Synthese 92, 371–384 (1992). https://doi.org/10.1007/BF00414288
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DOI: https://doi.org/10.1007/BF00414288