![]() ![]() The reason is simple: the paradox isn't simply about dividing a finite thing up into an infinite number of parts, but rather about the inherently physical concept of a rate.Īlthough the paradox is usually posed in terms of distances alone, the paradox is really about motion, which is about the amount of distance covered in a specific amount of time. It might seem counterintuitive, but pure mathematics alone cannot provide a satisfactory solution to the paradox. It is not enough to contend that time jumps get shorter as distance jumps get shorter a quantitative relationship is necessary. smaller than the previous term, but the total series still diverges: i.e., has a sum that tends towards infinity. The harmonic series, as shown here, is a classic example of a series where each and every term is. As it turns out, the limit does not exist: this is a diverging series. You can check this for yourself by trying to find what the series sums to. It's eminently possible that the time it takes to finish each step will still go down: half the original time, a third of the original time, a quarter of the original time, a fifth, etc., but that the total journey will take an infinite amount of time. Only, this line of thinking is flawed, too. And therefore, if that's true, Atalanta can finally reach her destination and complete her journey. Specifically, as asserted by Archimedes, it must take less time to complete a smaller distance jump than it does to complete a larger distance jump, and therefore if you travel a finite distance, it must take you only a finite amount of time. Many thinkers, both ancient and contemporary, tried to resolve this paradox by invoking the idea of time. It was only through a physical understanding of distance, time, and their relationship that this paradox was resolved. One of the many representations (and formulations) of Zeno of Elea's paradox relating to the. Under this line of thinking, it may still be impossible for Atalanta to reach her destination. If each jump took the same amount of time, for example, regardless of the distance traveled, it would take an infinite amount of time to cover whatever tiny fraction-of-the-journey remains. How could time come into play to ruin this mathematically elegant and compelling "solution" to Zeno's paradox?īecause there's no guarantee that each of the infinite number of jumps you need to take - even to cover a finite distance - occurs in a finite amount of time. It doesn't tell you anything about how long it takes you to reach your destination, and that's the tricky part of the paradox. ![]() This mathematical line of reasoning is only good enough to show that the total distance you must travel converges to a finite value. convergent series: one entire "thing" can be obtained by summing up one half plus one fourth plus one eighth, etc. By continuously halving a quantity, you can show that the sum of each successive half leads to a. ![]()
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