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In about 445 B.C., the Greek philosopher Zeno of Elea offered several arguments that led to conclusions contradicting what we all know from our physical experience. The paradoxes had a dramatic impact upon the later development of mathematics, science, and philosophy. His most familiar paradox, the paradox of Achilles and the Tortoise, involves the fast-running Achilles and the slow-crawling tortoise. The tortoise has a head start. If Achilles hopes to overtake the tortoise, he must at least run to where the tortoise is, but by the time he arrives there, the tortoise has crawled to a new place. So, Achilles must run to the new place; but of course the tortoise isn't there, having crawled on to yet another place, and so on forever. Therefore, Zeno argues, good reasoning shows that fast runners never can catch slow ones. So much the worse for good reasoning. Notice that Zeno's reasoning rests on the assumption that time is continuous, that is, that time can be divided into infinitely many parts. We assume this continuity of time when we assume that a basektball dropped onto the court will bounce an infinite number of times before stopping.
In his Progressive Dichotomy Paradox, Zeno argued that a runner will never reach the goal line because he first must have time to reach the halfway point to the goal, but after arriving there he will need time to get to the 3/4 point, then the 7/8 point, and so forth. If the distance to the goal is, say, 1 meter, then the runner must cover a distance of 1/2 + 1/4 + 1/8 + ... meters. Zeno believed this sum is infinite and concluded that the runner will never have the infinite time it takes to reach this infinitely distant goal. Because at any time there is always more time needed, motion can never be completed. Worse yet, argued Zeno in his Regressive Dichotomy Paradox, the runner can't even take a first step. Any first step may be divided into a first half and a second half. Before taking a full step, the runner must have time to take a 1/2 step, but before that a 1/4 step, and so forth. The runner will need an infinite amount of time just to take a first step, and so will never get going.
Zeno's Arrow Paradox takes a different approach to challenging the coherence of the concepts of time and motion. Consider one instant of an arrow's flight. For that entire instant the arrow occupies a region of space equal to its total length, so at that instant the arrow isn't moving, he reasoned. If at every instant the arrow isn't moving, then the arrow can't move.
Yet another paradox created by Zeno attacks the notion that there are shorter and shorter times. Consider a duration of one second. It can be divided into two non-overlapping parts. They, in turn, can be divided, and so on. At the end of this infinite division we reach the elements. Here there is a problem. If these elements have zero duration, then adding an infinity of zeros yields a zero sum, and the total duration is zero seconds, which is absurd. Alternatively, if that infinite division produced elements having a finite duration, then adding an infinite number of these together will produce an infinite duration, which is also absurd. So, a second lasts either for no time at all or else for an infinite amount of time.
These paradoxes by Zeno can be considered to challenge the notion that time (and space) is continuous. Some of his other paradoxes, not discussed here, challenge the presumption that time might be discrete or discontinuous, with instants being like atoms of time.
Zeno's paradoxical arguments are valid, given his assumptions about space, time, motion and mathematics; and they reveal the underlying incoherence in ancient Greek thought, an incoherence that was not adequately resolved for 2,300 years. The way out of Zeno's paradoxes requires revising the concepts of duration, distance, instantaneous speed, and sum of a series. The relevant revisions were made by Leibniz, Newton, Cauchy, Weierstrass, Dedekind, Cantor, Einstein, and Lebesque over two centuries. The notion of infinite sums of numbers had to be revised so that an infinite series of numbers that decrease sufficiently rapidly can have a finite sum. Although 1/2 + 1/3 + 1/4 +... is infinite, the more rapidly decreasing series 1/2 + 1/4 + 1/8 +... is 1. The other key idea was to appreciate that durations and distances must be topologically like an interval of the linear continuum, a dense ordering of uncountably many points. Although individual points of the continuum have zero measure (that is, zero 'total length'), the modern notion of measure on the linear continuum does not allow the measure of a segment (continuous region) to be the sum of the measures of its individual points, as Zeno had assumed in his argument against plurality. With these contemporary concepts, we can now make sense of Achilles covering an infinite number of distances in a finite time while running at a normal, finite speed. The new concepts restore the coherence of mathematics and science with our experience of space and time, and they are behind today's declaration that Zeno's arguments are based on naive and false assumptions.
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