Abstract
We explore the better known paradoxes of Zeno including modern variants based on infinite processes, from the point of view of standard, classical analysis, from which there is still much to learn (especially concerning the paradox of division), and then from the viewpoints of non-standard and non-classical analysis (the logic of the latter being intuitionist).
The standard, classical or “Cantorian” notion of the continuum, modeled on the real number line, is well known, as is the definition of motion as the time derivative of distance (we are not concerned with position and motion in more than one dimension, since Zeno wasn't). The real number line consists of its points, the distance between distinct points being positive and finite. The standard, classical derivative relies on the classical notion of limit, which does not use infinitesimals.
In non-standard analysis, the real line is again the set of its points, but infinitesimal distances between points are allowed, while the derivative is defined in terms of the ratio of infinitesimals. This has the surprising consequence that there is a function of time giving positions, which is not constant, but whose derivative is everywhere zero, so that a particle whose position is given by this function can move a finite distance in a finite time, while being at rest all along. Such a function, of course is “external” — it can't be defined in the formal language. More suprising still, a model of the non-standard real line can be found which is “internally finite” — that is there is an injection from an initial segment of non-negative integers to the non-standard interval [0,1) whose range includes all the standard real numbers, and the ratio of the number of points in the range in any subset S of the interval to the number of those in the entire interval differs by an infinitesimal from the Lebesgue measure of the set of standard points in S. Of course, the formalism can't tell the difference between standard and non-standard integers or points — that is an external concept. Still, this allows a discrete model of the line with points an infinitesimal distance apart, without sacrificing any of the results of standard analysis, including measure theory.
Non-classical analysis is based on topos theory in an intuitionist setting. All curves are piecewise linear, or “straight” over infinitesimal distances, and the non-classical derivative is defined as the slope in such “intervals”. So whereas the direction of a non-standard smooth curve changes infinitesimally over infinitesimal distances, the direction of a non-classical curve (all of them are smooth), doesn't change at all over infitesimal distances. And while a standard or non-standard line can be identified with the set of its points, the points on a non-classical line might be said not to occupy all “positions”, and the line is an object which is not the set of its points.
We explore the implications of all this for the various paradoxes of Zeno, and some modern variants.
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Harrison, C. The three arrows of Zeno. Synthese 107, 271–292 (1996). https://doi.org/10.1007/BF00413609
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DOI: https://doi.org/10.1007/BF00413609