[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:15.10,0:00:16.87,Default,,0000,0000,0000,,This is Zeno of Elea,
Dialogue: 0,0:00:16.87,0:00:18.38,Default,,0000,0000,0000,,an ancient Greek philosopher
Dialogue: 0,0:00:18.38,0:00:21.04,Default,,0000,0000,0000,,famous for inventing a number of paradoxes,
Dialogue: 0,0:00:21.04,0:00:22.56,Default,,0000,0000,0000,,arguments that seem logical,
Dialogue: 0,0:00:22.56,0:00:25.78,Default,,0000,0000,0000,,but whose conclusion is absurd or contradictory.
Dialogue: 0,0:00:25.78,0:00:27.18,Default,,0000,0000,0000,,For more than 2,000 years,
Dialogue: 0,0:00:27.18,0:00:29.69,Default,,0000,0000,0000,,Zeno's mind-bending riddles have inspired
Dialogue: 0,0:00:29.69,0:00:31.31,Default,,0000,0000,0000,,mathematicians and philosophers
Dialogue: 0,0:00:31.31,0:00:33.75,Default,,0000,0000,0000,,to better understand the nature of infinity.
Dialogue: 0,0:00:33.75,0:00:35.52,Default,,0000,0000,0000,,One of the best known of Zeno's problems
Dialogue: 0,0:00:35.52,0:00:37.74,Default,,0000,0000,0000,,is called the dichotomy paradox,
Dialogue: 0,0:00:37.74,0:00:41.53,Default,,0000,0000,0000,,which means, "the paradox of cutting in two" in ancient Greek.
Dialogue: 0,0:00:41.53,0:00:43.32,Default,,0000,0000,0000,,It goes something like this:
Dialogue: 0,0:00:43.32,0:00:46.15,Default,,0000,0000,0000,,After a long day of sitting around, thinking,
Dialogue: 0,0:00:46.15,0:00:48.95,Default,,0000,0000,0000,,Zeno decides to walk from his house to the park.
Dialogue: 0,0:00:48.95,0:00:50.40,Default,,0000,0000,0000,,The fresh air clears his mind
Dialogue: 0,0:00:50.40,0:00:51.92,Default,,0000,0000,0000,,and help him think better.
Dialogue: 0,0:00:51.92,0:00:53.08,Default,,0000,0000,0000,,In order to get to the park,
Dialogue: 0,0:00:53.08,0:00:55.43,Default,,0000,0000,0000,,he first has to get half way to the park.
Dialogue: 0,0:00:55.43,0:00:56.60,Default,,0000,0000,0000,,This portion of his journey
Dialogue: 0,0:00:56.60,0:00:58.44,Default,,0000,0000,0000,,takes some finite amount of time.
Dialogue: 0,0:00:58.44,0:01:00.45,Default,,0000,0000,0000,,Once he gets to the halfway point,
Dialogue: 0,0:01:00.45,0:01:02.84,Default,,0000,0000,0000,,he needs to walk half the remaining distance.
Dialogue: 0,0:01:02.84,0:01:05.87,Default,,0000,0000,0000,,Again, this takes a finite amount of time.
Dialogue: 0,0:01:05.87,0:01:08.14,Default,,0000,0000,0000,,Once he gets there, he still needs to walk
Dialogue: 0,0:01:08.14,0:01:09.88,Default,,0000,0000,0000,,half the distance that's left,
Dialogue: 0,0:01:09.88,0:01:12.37,Default,,0000,0000,0000,,which takes another finite amount of time.
Dialogue: 0,0:01:12.37,0:01:15.52,Default,,0000,0000,0000,,This happens again and again and again.
Dialogue: 0,0:01:15.52,0:01:18.20,Default,,0000,0000,0000,,You can see that we can keep going like this forever,
Dialogue: 0,0:01:18.20,0:01:19.86,Default,,0000,0000,0000,,dividing whatever distance is left
Dialogue: 0,0:01:19.86,0:01:21.77,Default,,0000,0000,0000,,into smaller and smaller pieces,
Dialogue: 0,0:01:21.77,0:01:25.28,Default,,0000,0000,0000,,each of which takes some finite time to traverse.
Dialogue: 0,0:01:25.28,0:01:27.96,Default,,0000,0000,0000,,So, how long does it take Zeno to get to the park?
Dialogue: 0,0:01:27.96,0:01:30.32,Default,,0000,0000,0000,,Well, to find out, you need to add the times
Dialogue: 0,0:01:30.32,0:01:32.28,Default,,0000,0000,0000,,of each of the pieces of the journey.
Dialogue: 0,0:01:32.28,0:01:36.62,Default,,0000,0000,0000,,The problem is, there are infinitely many of these finite-sized pieces.
Dialogue: 0,0:01:36.62,0:01:39.75,Default,,0000,0000,0000,,So, shouldn't the total time be infinity?
Dialogue: 0,0:01:39.75,0:01:42.55,Default,,0000,0000,0000,,This argument, by the way, is completely general.
Dialogue: 0,0:01:42.55,0:01:45.09,Default,,0000,0000,0000,,It says that traveling from any location to any other location
Dialogue: 0,0:01:45.09,0:01:47.25,Default,,0000,0000,0000,,should take an infinite amount of time.
Dialogue: 0,0:01:47.25,0:01:51.01,Default,,0000,0000,0000,,In other words, it says that all motion is impossible.
Dialogue: 0,0:01:51.01,0:01:52.78,Default,,0000,0000,0000,,This conclusion is clearly absurd,
Dialogue: 0,0:01:52.78,0:01:54.78,Default,,0000,0000,0000,,but where is the flaw in the logic?
Dialogue: 0,0:01:54.78,0:01:55.97,Default,,0000,0000,0000,,To resolve the paradox,
Dialogue: 0,0:01:55.97,0:01:58.73,Default,,0000,0000,0000,,it helps to turn the story into a math problem.
Dialogue: 0,0:01:58.73,0:02:01.62,Default,,0000,0000,0000,,Let's supposed that Zeno's house is one mile from the park
Dialogue: 0,0:02:01.62,0:02:04.34,Default,,0000,0000,0000,,and that Zeno walks at one mile per hour.
Dialogue: 0,0:02:04.34,0:02:06.69,Default,,0000,0000,0000,,Common sense tells us that the time for the journey
Dialogue: 0,0:02:06.69,0:02:08.20,Default,,0000,0000,0000,,should be one hour.
Dialogue: 0,0:02:08.20,0:02:10.87,Default,,0000,0000,0000,,But, let's look at things from Zeno's point of view
Dialogue: 0,0:02:10.87,0:02:13.20,Default,,0000,0000,0000,,and divide up the journey into pieces.
Dialogue: 0,0:02:13.20,0:02:15.66,Default,,0000,0000,0000,,The first half of the journey takes half an hour,
Dialogue: 0,0:02:15.66,0:02:17.78,Default,,0000,0000,0000,,the next part takes quarter of an hour,
Dialogue: 0,0:02:17.78,0:02:20.06,Default,,0000,0000,0000,,the third part takes an eighth of an hour,
Dialogue: 0,0:02:20.06,0:02:20.97,Default,,0000,0000,0000,,and so on.
Dialogue: 0,0:02:20.97,0:02:22.27,Default,,0000,0000,0000,,Summing up all these times,
Dialogue: 0,0:02:22.27,0:02:24.37,Default,,0000,0000,0000,,we get a series that looks like this.
Dialogue: 0,0:02:24.37,0:02:25.62,Default,,0000,0000,0000,,"Now", Zeno might say,
Dialogue: 0,0:02:25.62,0:02:27.96,Default,,0000,0000,0000,,"since there are infinitely many of terms
Dialogue: 0,0:02:27.96,0:02:29.62,Default,,0000,0000,0000,,on the right side of the equation,
Dialogue: 0,0:02:29.62,0:02:31.88,Default,,0000,0000,0000,,and each individual term is finite,
Dialogue: 0,0:02:31.88,0:02:34.52,Default,,0000,0000,0000,,the sum should equal infinity, right?"
Dialogue: 0,0:02:34.52,0:02:36.67,Default,,0000,0000,0000,,This is the problem with Zeno's argument.
Dialogue: 0,0:02:36.67,0:02:38.86,Default,,0000,0000,0000,,As mathematicians have since realized,
Dialogue: 0,0:02:38.86,0:02:42.62,Default,,0000,0000,0000,,it is possible to add up infinitely many finite-sized terms
Dialogue: 0,0:02:42.62,0:02:44.81,Default,,0000,0000,0000,,and still get a finite answer.
Dialogue: 0,0:02:44.81,0:02:45.99,Default,,0000,0000,0000,,"How?" you ask.
Dialogue: 0,0:02:45.99,0:02:47.49,Default,,0000,0000,0000,,Well, let's think of it this way.
Dialogue: 0,0:02:47.49,0:02:50.39,Default,,0000,0000,0000,,Let's start with a square that has area of one meter.
Dialogue: 0,0:02:50.39,0:02:52.53,Default,,0000,0000,0000,,Now let's chop the square in half,
Dialogue: 0,0:02:52.53,0:02:54.91,Default,,0000,0000,0000,,and then chop the remaining half in half,
Dialogue: 0,0:02:54.91,0:02:56.17,Default,,0000,0000,0000,,and so on.
Dialogue: 0,0:02:56.17,0:02:57.24,Default,,0000,0000,0000,,While we're doing this,
Dialogue: 0,0:02:57.24,0:03:00.38,Default,,0000,0000,0000,,let's keep track of the areas of the pieces.
Dialogue: 0,0:03:00.38,0:03:02.17,Default,,0000,0000,0000,,The first slice makes two parts,
Dialogue: 0,0:03:02.17,0:03:04.03,Default,,0000,0000,0000,,each with an area of one-half
Dialogue: 0,0:03:04.03,0:03:06.54,Default,,0000,0000,0000,,The next slice divides one of those halves in half,
Dialogue: 0,0:03:06.54,0:03:07.80,Default,,0000,0000,0000,,and so on.
Dialogue: 0,0:03:07.80,0:03:10.23,Default,,0000,0000,0000,,But, no matter how many times we slice up the boxes,
Dialogue: 0,0:03:10.23,0:03:14.81,Default,,0000,0000,0000,,the total area is still the sum of the areas of all the pieces.
Dialogue: 0,0:03:14.81,0:03:17.44,Default,,0000,0000,0000,,Now you can see why we choose this particular way
Dialogue: 0,0:03:17.44,0:03:18.97,Default,,0000,0000,0000,,of cutting up the square.
Dialogue: 0,0:03:18.97,0:03:20.89,Default,,0000,0000,0000,,We've obtained the same infinite series
Dialogue: 0,0:03:20.89,0:03:23.36,Default,,0000,0000,0000,,as we had for the time of Zeno's journey.
Dialogue: 0,0:03:23.36,0:03:25.79,Default,,0000,0000,0000,,As we construct more and more blue pieces,
Dialogue: 0,0:03:25.79,0:03:27.31,Default,,0000,0000,0000,,to use the math jargon,
Dialogue: 0,0:03:27.31,0:03:30.74,Default,,0000,0000,0000,,as we take the limit as n tends to infinity,
Dialogue: 0,0:03:30.74,0:03:33.36,Default,,0000,0000,0000,,the entire square becomes covered with blue.
Dialogue: 0,0:03:33.36,0:03:35.43,Default,,0000,0000,0000,,But the area of the square is just one unit,
Dialogue: 0,0:03:35.43,0:03:38.70,Default,,0000,0000,0000,,and so the infinite sum must equal one.
Dialogue: 0,0:03:38.70,0:03:39.75,Default,,0000,0000,0000,,Going back to Zeno's journey,
Dialogue: 0,0:03:39.75,0:03:42.37,Default,,0000,0000,0000,,we can now see how how the paradox is resolved.
Dialogue: 0,0:03:42.37,0:03:45.71,Default,,0000,0000,0000,,Not only does the infinite series sum to a finite answer,
Dialogue: 0,0:03:45.71,0:03:47.74,Default,,0000,0000,0000,,but that finite answer is the same one
Dialogue: 0,0:03:47.74,0:03:50.17,Default,,0000,0000,0000,,that common sense tells us is true.
Dialogue: 0,0:03:50.17,0:03:52.88,Default,,0000,0000,0000,,Zeno's journey takes one hour.