WEBVTT 00:00:15.096 --> 00:00:16.871 This is Zeno of Elea, 00:00:16.871 --> 00:00:18.377 an ancient Greek philosopher 00:00:18.377 --> 00:00:21.042 famous for inventing a number of paradoxes, 00:00:21.042 --> 00:00:22.560 arguments that seem logical, 00:00:22.560 --> 00:00:25.779 but whose conclusion is absurd or contradictory. 00:00:25.779 --> 00:00:27.183 For more than 2,000 years, 00:00:27.183 --> 00:00:29.694 Zeno's mind-bending riddles have inspired 00:00:29.694 --> 00:00:31.310 mathematicians and philosophers 00:00:31.310 --> 00:00:33.746 to better understand the nature of infinity. 00:00:33.746 --> 00:00:35.525 One of the best known of Zeno's problems 00:00:35.525 --> 00:00:37.741 is called the dichotomy paradox, 00:00:37.741 --> 00:00:41.527 which means, "the paradox of cutting in two" in ancient Greek. 00:00:41.527 --> 00:00:43.315 It goes something like this: 00:00:43.315 --> 00:00:46.154 After a long day of sitting around, thinking, 00:00:46.154 --> 00:00:48.950 Zeno decides to walk from his house to the park. 00:00:48.950 --> 00:00:50.397 The fresh air clears his mind 00:00:50.397 --> 00:00:51.920 and help him think better. 00:00:51.920 --> 00:00:53.075 In order to get to the park, 00:00:53.075 --> 00:00:55.428 he first has to get half way to the park. 00:00:55.428 --> 00:00:56.601 This portion of his journey 00:00:56.601 --> 00:00:58.443 takes some finite amount of time. 00:00:58.443 --> 00:01:00.452 Once he gets to the halfway point, 00:01:00.452 --> 00:01:02.841 he needs to walk half the remaining distance. 00:01:02.841 --> 00:01:05.868 Again, this takes a finite amount of time. 00:01:05.868 --> 00:01:08.140 Once he gets there, he still needs to walk 00:01:08.140 --> 00:01:09.882 half the distance that's left, 00:01:09.882 --> 00:01:12.371 which takes another finite amount of time. 00:01:12.371 --> 00:01:15.522 This happens again and again and again. 00:01:15.522 --> 00:01:18.195 You can see that we can keep going like this forever, 00:01:18.195 --> 00:01:19.857 dividing whatever distance is left 00:01:19.857 --> 00:01:21.772 into smaller and smaller pieces, 00:01:21.772 --> 00:01:25.278 each of which takes some finite time to traverse. 00:01:25.278 --> 00:01:27.958 So, how long does it take Zeno to get to the park? 00:01:27.958 --> 00:01:30.317 Well, to find out, you need to add the times 00:01:30.317 --> 00:01:32.284 of each of the pieces of the journey. 00:01:32.284 --> 00:01:36.616 The problem is, there are infinitely many of these finite-sized pieces. 00:01:36.616 --> 00:01:39.750 So, shouldn't the total time be infinity? 00:01:39.750 --> 00:01:42.548 This argument, by the way, is completely general. 00:01:42.548 --> 00:01:45.092 It says that traveling from any location to any other location 00:01:45.092 --> 00:01:47.254 should take an infinite amount of time. 00:01:47.254 --> 00:01:51.006 In other words, it says that all motion is impossible. 00:01:51.006 --> 00:01:52.785 This conclusion is clearly absurd, 00:01:52.785 --> 00:01:54.784 but where is the flaw in the logic? 00:01:54.784 --> 00:01:55.966 To resolve the paradox, 00:01:55.966 --> 00:01:58.731 it helps to turn the story into a math problem. 00:01:58.731 --> 00:02:01.618 Let's supposed that Zeno's house is one mile from the park 00:02:01.618 --> 00:02:04.341 and that Zeno walks at one mile per hour. 00:02:04.341 --> 00:02:06.692 Common sense tells us that the time for the journey 00:02:06.692 --> 00:02:08.205 should be one hour. 00:02:08.205 --> 00:02:10.867 But, let's look at things from Zeno's point of view 00:02:10.867 --> 00:02:13.196 and divide up the journey into pieces. 00:02:13.196 --> 00:02:15.656 The first half of the journey takes half an hour, 00:02:15.656 --> 00:02:17.782 the next part takes quarter of an hour, 00:02:17.782 --> 00:02:20.064 the third part takes an eighth of an hour, 00:02:20.064 --> 00:02:20.969 and so on. 00:02:20.969 --> 00:02:22.266 Summing up all these times, 00:02:22.266 --> 00:02:24.372 we get a series that looks like this. 00:02:24.372 --> 00:02:25.624 "Now", Zeno might say, 00:02:25.624 --> 00:02:27.964 "since there are infinitely many of terms 00:02:27.964 --> 00:02:29.621 on the right side of the equation, 00:02:29.621 --> 00:02:31.883 and each individual term is finite, 00:02:31.883 --> 00:02:34.518 the sum should equal infinity, right?" 00:02:34.518 --> 00:02:36.670 This is the problem with Zeno's argument. 00:02:36.670 --> 00:02:38.855 As mathematicians have since realized, 00:02:38.855 --> 00:02:42.618 it is possible to add up infinitely many finite-sized terms 00:02:42.618 --> 00:02:44.814 and still get a finite answer. 00:02:44.814 --> 00:02:45.989 "How?" you ask. 00:02:45.989 --> 00:02:47.486 Well, let's think of it this way. 00:02:47.486 --> 00:02:50.390 Let's start with a square that has area of one meter. 00:02:50.390 --> 00:02:52.528 Now let's chop the square in half, 00:02:52.528 --> 00:02:54.909 and then chop the remaining half in half, 00:02:54.909 --> 00:02:56.172 and so on. 00:02:56.172 --> 00:02:57.239 While we're doing this, 00:02:57.239 --> 00:03:00.380 let's keep track of the areas of the pieces. 00:03:00.380 --> 00:03:02.169 The first slice makes two parts, 00:03:02.169 --> 00:03:04.028 each with an area of one-half 00:03:04.028 --> 00:03:06.545 The next slice divides one of those halves in half, 00:03:06.545 --> 00:03:07.796 and so on. 00:03:07.796 --> 00:03:10.227 But, no matter how many times we slice up the boxes, 00:03:10.227 --> 00:03:14.814 the total area is still the sum of the areas of all the pieces. 00:03:14.814 --> 00:03:17.442 Now you can see why we choose this particular way 00:03:17.442 --> 00:03:18.971 of cutting up the square. 00:03:18.971 --> 00:03:20.888 We've obtained the same infinite series 00:03:20.888 --> 00:03:23.356 as we had for the time of Zeno's journey. 00:03:23.356 --> 00:03:25.791 As we construct more and more blue pieces, 00:03:25.791 --> 00:03:27.314 to use the math jargon, 00:03:27.314 --> 00:03:30.742 as we take the limit as n tends to infinity, 00:03:30.742 --> 00:03:33.356 the entire square becomes covered with blue. 00:03:33.356 --> 00:03:35.427 But the area of the square is just one unit, 00:03:35.427 --> 00:03:38.700 and so the infinite sum must equal one. 00:03:38.700 --> 00:03:39.754 Going back to Zeno's journey, 00:03:39.754 --> 00:03:42.370 we can now see how how the paradox is resolved. 00:03:42.370 --> 00:03:45.713 Not only does the infinite series sum to a finite answer, 00:03:45.713 --> 00:03:47.745 but that finite answer is the same one 00:03:47.745 --> 00:03:50.172 that common sense tells us is true. 00:03:50.172 --> 00:03:52.877 Zeno's journey takes one hour.