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    "segs": [ {
      "utf8": "Translator: Andrea McDonough\nReviewer: Bedirhan Cinar"
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    "segs": [ {
      "utf8": "This is Zeno of Elea,"
    } ]
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    "segs": [ {
      "utf8": "an ancient Greek philosopher"
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    "segs": [ {
      "utf8": "famous for inventing a number of paradoxes,"
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  }, {
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    "segs": [ {
      "utf8": "arguments that seem logical,"
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  }, {
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    "segs": [ {
      "utf8": "but whose conclusion is absurd or contradictory."
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    "segs": [ {
      "utf8": "For more than 2,000 years,"
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  }, {
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    "segs": [ {
      "utf8": "Zeno's mind-bending riddles have inspired"
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    "segs": [ {
      "utf8": "mathematicians and philosophers"
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  }, {
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    "dDurationMs": 2436,
    "segs": [ {
      "utf8": "to better understand the nature of infinity."
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    "segs": [ {
      "utf8": "One of the best known of Zeno's problems"
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    "tStartMs": 35525,
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    "segs": [ {
      "utf8": "is called the dichotomy paradox,"
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  }, {
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    "segs": [ {
      "utf8": "which means, \"the paradox of cutting in two\" in ancient Greek."
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    "segs": [ {
      "utf8": "It goes something like this:"
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    "segs": [ {
      "utf8": "After a long day of sitting around, thinking,"
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    "segs": [ {
      "utf8": "Zeno decides to walk from his house to the park."
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    "segs": [ {
      "utf8": "The fresh air clears his mind"
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  }, {
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    "dDurationMs": 1523,
    "segs": [ {
      "utf8": "and help him think better."
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    "segs": [ {
      "utf8": "In order to get to the park,"
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    "segs": [ {
      "utf8": "he first has to get half way to the park."
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  }, {
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    "segs": [ {
      "utf8": "This portion of his journey"
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      "utf8": "takes some finite amount of time."
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    "segs": [ {
      "utf8": "Once he gets to the halfway point,"
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  }, {
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    "segs": [ {
      "utf8": "he needs to walk half the remaining distance."
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    "segs": [ {
      "utf8": "Again, this takes a finite amount of time."
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    "segs": [ {
      "utf8": "Once he gets there, he still needs to walk"
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    "segs": [ {
      "utf8": "half the distance that's left,"
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    "segs": [ {
      "utf8": "which takes another finite amount of time."
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  }, {
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    "segs": [ {
      "utf8": "This happens again and again and again."
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    "segs": [ {
      "utf8": "You can see that we can keep going like this forever,"
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  }, {
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    "dDurationMs": 1662,
    "segs": [ {
      "utf8": "dividing whatever distance is left"
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    "segs": [ {
      "utf8": "into smaller and smaller pieces,"
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    "segs": [ {
      "utf8": "each of which takes some finite time to traverse."
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    "segs": [ {
      "utf8": "So, how long does it take Zeno to get to the park?"
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    "segs": [ {
      "utf8": "Well, to find out, you need to add the times"
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  }, {
    "tStartMs": 90317,
    "dDurationMs": 1967,
    "segs": [ {
      "utf8": "of each of the pieces of the journey."
    } ]
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    "segs": [ {
      "utf8": "The problem is, there are infinitely many of these finite-sized pieces."
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    "tStartMs": 96616,
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    "segs": [ {
      "utf8": "So, shouldn't the total time be infinity?"
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    "segs": [ {
      "utf8": "This argument, by the way, is completely general."
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    "segs": [ {
      "utf8": "It says that traveling from any location to any other location"
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    "segs": [ {
      "utf8": "should take an infinite amount of time."
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    "segs": [ {
      "utf8": "In other words, it says that all motion is impossible."
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    "segs": [ {
      "utf8": "This conclusion is clearly absurd,"
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    "segs": [ {
      "utf8": "but where is the flaw in the logic?"
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    "segs": [ {
      "utf8": "To resolve the paradox,"
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    "segs": [ {
      "utf8": "it helps to turn the story into a math problem."
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    "segs": [ {
      "utf8": "Let's supposed that Zeno's house is one mile from the park"
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      "utf8": "and that Zeno walks at one mile per hour."
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    "segs": [ {
      "utf8": "Common sense tells us that the time for the journey"
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    "segs": [ {
      "utf8": "should be one hour."
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    "segs": [ {
      "utf8": "But, let's look at things from Zeno's point of view"
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    "segs": [ {
      "utf8": "and divide up the journey into pieces."
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    "segs": [ {
      "utf8": "The first half of the journey takes half an hour,"
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  }, {
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    "dDurationMs": 2126,
    "segs": [ {
      "utf8": "the next part takes quarter of an hour,"
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    "segs": [ {
      "utf8": "the third part takes an eighth of an hour,"
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    "segs": [ {
      "utf8": "and so on."
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    "dDurationMs": 1297,
    "segs": [ {
      "utf8": "Summing up all these times,"
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    "dDurationMs": 2106,
    "segs": [ {
      "utf8": "we get a series that looks like this."
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    "dDurationMs": 1252,
    "segs": [ {
      "utf8": "\"Now\", Zeno might say,"
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    "dDurationMs": 2340,
    "segs": [ {
      "utf8": "\"since there are infinitely many of terms"
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  }, {
    "tStartMs": 147964,
    "dDurationMs": 1657,
    "segs": [ {
      "utf8": "on the right side of the equation,"
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    "dDurationMs": 2262,
    "segs": [ {
      "utf8": "and each individual term is finite,"
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  }, {
    "tStartMs": 151883,
    "dDurationMs": 2635,
    "segs": [ {
      "utf8": "the sum should equal infinity, right?\""
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  }, {
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    "dDurationMs": 2152,
    "segs": [ {
      "utf8": "This is the problem with Zeno's argument."
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    "segs": [ {
      "utf8": "As mathematicians have since realized,"
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  }, {
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    "dDurationMs": 3763,
    "segs": [ {
      "utf8": "it is possible to add up infinitely many finite-sized terms"
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  }, {
    "tStartMs": 162618,
    "dDurationMs": 2196,
    "segs": [ {
      "utf8": "and still get a finite answer."
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  }, {
    "tStartMs": 164814,
    "dDurationMs": 1175,
    "segs": [ {
      "utf8": "\"How?\" you ask."
    } ]
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    "tStartMs": 165989,
    "dDurationMs": 1497,
    "segs": [ {
      "utf8": "Well, let's think of it this way."
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  }, {
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    "dDurationMs": 2904,
    "segs": [ {
      "utf8": "Let's start with a square that has area of one meter."
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  }, {
    "tStartMs": 170390,
    "dDurationMs": 2138,
    "segs": [ {
      "utf8": "Now let's chop the square in half,"
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  }, {
    "tStartMs": 172528,
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    "segs": [ {
      "utf8": "and then chop the remaining half in half,"
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    "segs": [ {
      "utf8": "and so on."
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    "segs": [ {
      "utf8": "While we're doing this,"
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    "segs": [ {
      "utf8": "let's keep track of the areas of the pieces."
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    "segs": [ {
      "utf8": "The first slice makes two parts,"
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    "segs": [ {
      "utf8": "each with an area of one-half"
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  }, {
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    "segs": [ {
      "utf8": "The next slice divides one of those halves in half,"
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  }, {
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    "segs": [ {
      "utf8": "and so on."
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    "segs": [ {
      "utf8": "But, no matter how many times we slice up the boxes,"
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    "dDurationMs": 4587,
    "segs": [ {
      "utf8": "the total area is still the sum of the areas of all the pieces."
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  }, {
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    "dDurationMs": 2628,
    "segs": [ {
      "utf8": "Now you can see why we choose this particular way"
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  }, {
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    "dDurationMs": 1529,
    "segs": [ {
      "utf8": "of cutting up the square."
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  }, {
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    "dDurationMs": 1917,
    "segs": [ {
      "utf8": "We've obtained the same infinite series"
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  }, {
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    "dDurationMs": 2468,
    "segs": [ {
      "utf8": "as we had for the time of Zeno's journey."
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    "tStartMs": 203356,
    "dDurationMs": 2435,
    "segs": [ {
      "utf8": "As we construct more and more blue pieces,"
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  }, {
    "tStartMs": 205791,
    "dDurationMs": 1523,
    "segs": [ {
      "utf8": "to use the math jargon,"
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    "dDurationMs": 3428,
    "segs": [ {
      "utf8": "as we take the limit as n tends to infinity,"
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  }, {
    "tStartMs": 210742,
    "dDurationMs": 2614,
    "segs": [ {
      "utf8": "the entire square becomes covered with blue."
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  }, {
    "tStartMs": 213356,
    "dDurationMs": 2071,
    "segs": [ {
      "utf8": "But the area of the square is just one unit,"
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  }, {
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    "dDurationMs": 3273,
    "segs": [ {
      "utf8": "and so the infinite sum must equal one."
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    "dDurationMs": 1054,
    "segs": [ {
      "utf8": "Going back to Zeno's journey,"
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    "segs": [ {
      "utf8": "we can now see how how the paradox is resolved."
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    "segs": [ {
      "utf8": "Not only does the infinite series sum to a finite answer,"
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    "segs": [ {
      "utf8": "but that finite answer is the same one"
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    "segs": [ {
      "utf8": "that common sense tells us is true."
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    "segs": [ {
      "utf8": "Zeno's journey takes one hour."
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