full transcript

From the Ted Talk by Colm Kelleher: What is Zeno's Dichotomy Paradox?

Unscramble the Blue Letters

This is Zeno of Elea, an ancient Greek philosopher famous for inventing a number of paradoxes, arguments that seem logical, but whose conclusion is absurd or contradictory. For more than 2,000 years, Zeno's mind-bending ridelds have inspired mathematicians and philosophers to better udsearnntd the nature of infinity. One of the best known of Zeno's problems is clelad the doicmtohy paradox, which means, "the paradox of ctunitg in two" in ancient Greek. It goes something like this: After a long day of sitting around, thinking, Zeno decides to walk from his hosue to the park. The fresh air clears his mind and help him think better. In order to get to the park, he first has to get half way to the park. This portion of his jrouney takes some finite aunomt of time. Once he gets to the halfway point, he needs to walk half the remaining distance. Again, this tekas a finite amount of time. Once he gets there, he still needs to walk half the distance that's left, which takes another finite amount of time. This happens again and again and again. You can see that we can keep going like this forever, ddnviiig whatever distance is left into smaller and smaller pieces, each of which takes some finite time to traverse. So, how long does it take Zeno to get to the park? Well, to find out, you need to add the times of each of the pecies of the journey. The problem is, there are infinitely many of these finite-sized pieces. So, shouldn't the total time be innftiiy? This argument, by the way, is cpmleetloy general. It says that tavrinelg from any location to any other location should take an iinfinte amount of time. In other wrods, it says that all motion is impossible. This conclusion is clearly absurd, but where is the flaw in the logic? To resolve the paradox, it helps to turn the story into a math problem. Let's supposed that Zeno's house is one mile from the park and that Zeno wklas at one mile per hour. Common ssnee tells us that the time for the journey should be one hour. But, let's look at things from Zeno's point of view and divide up the journey into pieces. The first half of the journey takes half an hour, the next part takes quarter of an hour, the third part takes an eighth of an hour, and so on. smiumng up all these times, we get a series that looks like this. "Now", Zeno might say, "since there are infinitely many of terms on the right side of the eqioatun, and each individual term is finite, the sum should equal infinity, right?" This is the problem with Zeno's argument. As mathematicians have since realized, it is possible to add up ilnfiteniy many finite-sized terms and still get a finite answer. "How?" you ask. Well, let's think of it this way. Let's start with a square that has area of one meter. Now let's chop the square in half, and then chop the rinmnaeig half in half, and so on. While we're doing this, let's keep track of the araes of the pieces. The first slice makes two parts, each with an area of one-half The next sicle divides one of those halves in half, and so on. But, no maettr how many tiems we slice up the boxes, the ttaol area is still the sum of the areas of all the pieces. Now you can see why we choose this particular way of cutting up the square. We've otenaibd the same infinite series as we had for the time of Zeno's journey. As we construct more and more blue pieces, to use the math jargon, as we take the limit as n tends to infinity, the entire square becomes covered with blue. But the area of the square is just one unit, and so the infinite sum must equal one. Going back to Zeno's journey, we can now see how how the paradox is resolved. Not only does the infinite series sum to a finite answer, but that ftinie answer is the same one that coommn sense tells us is true. Zeno's journey takes one hour.

Open Cloze

This is Zeno of Elea, an ancient Greek philosopher famous for inventing a number of paradoxes, arguments that seem logical, but whose conclusion is absurd or contradictory. For more than 2,000 years, Zeno's mind-bending _______ have inspired mathematicians and philosophers to better __________ the nature of infinity. One of the best known of Zeno's problems is ______ the _________ paradox, which means, "the paradox of _______ in two" in ancient Greek. It goes something like this: After a long day of sitting around, thinking, Zeno decides to walk from his _____ to the park. The fresh air clears his mind and help him think better. In order to get to the park, he first has to get half way to the park. This portion of his _______ takes some finite ______ of time. Once he gets to the halfway point, he needs to walk half the remaining distance. Again, this _____ a finite amount of time. Once he gets there, he still needs to walk half the distance that's left, which takes another finite amount of time. This happens again and again and again. You can see that we can keep going like this forever, ________ whatever distance is left into smaller and smaller pieces, each of which takes some finite time to traverse. So, how long does it take Zeno to get to the park? Well, to find out, you need to add the times of each of the ______ of the journey. The problem is, there are infinitely many of these finite-sized pieces. So, shouldn't the total time be ________? This argument, by the way, is __________ general. It says that _________ from any location to any other location should take an ________ amount of time. In other _____, it says that all motion is impossible. This conclusion is clearly absurd, but where is the flaw in the logic? To resolve the paradox, it helps to turn the story into a math problem. Let's supposed that Zeno's house is one mile from the park and that Zeno _____ at one mile per hour. Common _____ tells us that the time for the journey should be one hour. But, let's look at things from Zeno's point of view and divide up the journey into pieces. The first half of the journey takes half an hour, the next part takes quarter of an hour, the third part takes an eighth of an hour, and so on. _______ up all these times, we get a series that looks like this. "Now", Zeno might say, "since there are infinitely many of terms on the right side of the ________, and each individual term is finite, the sum should equal infinity, right?" This is the problem with Zeno's argument. As mathematicians have since realized, it is possible to add up __________ many finite-sized terms and still get a finite answer. "How?" you ask. Well, let's think of it this way. Let's start with a square that has area of one meter. Now let's chop the square in half, and then chop the _________ half in half, and so on. While we're doing this, let's keep track of the _____ of the pieces. The first slice makes two parts, each with an area of one-half The next _____ divides one of those halves in half, and so on. But, no ______ how many _____ we slice up the boxes, the _____ area is still the sum of the areas of all the pieces. Now you can see why we choose this particular way of cutting up the square. We've ________ the same infinite series as we had for the time of Zeno's journey. As we construct more and more blue pieces, to use the math jargon, as we take the limit as n tends to infinity, the entire square becomes covered with blue. But the area of the square is just one unit, and so the infinite sum must equal one. Going back to Zeno's journey, we can now see how how the paradox is resolved. Not only does the infinite series sum to a finite answer, but that ______ answer is the same one that ______ sense tells us is true. Zeno's journey takes one hour.

Solution

  1. pieces
  2. summing
  3. infinite
  4. dichotomy
  5. common
  6. equation
  7. total
  8. house
  9. finite
  10. infinitely
  11. words
  12. walks
  13. remaining
  14. riddles
  15. obtained
  16. sense
  17. journey
  18. cutting
  19. amount
  20. traveling
  21. matter
  22. slice
  23. infinity
  24. understand
  25. areas
  26. called
  27. takes
  28. times
  29. dividing
  30. completely

Original Text

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

Frequently Occurring Word Combinations

ngrams of length 2

collocation frequency
journey takes 3
finite amount 3
ancient greek 2
common sense 2
sense tells 2
part takes 2
finite answer 2
infinite series 2

ngrams of length 3

collocation frequency
common sense tells 2

Important Words

  1. absurd
  2. add
  3. air
  4. amount
  5. ancient
  6. answer
  7. area
  8. areas
  9. argument
  10. arguments
  11. blue
  12. boxes
  13. called
  14. choose
  15. chop
  16. clears
  17. common
  18. completely
  19. conclusion
  20. construct
  21. contradictory
  22. covered
  23. cutting
  24. day
  25. decides
  26. dichotomy
  27. distance
  28. divide
  29. divides
  30. dividing
  31. eighth
  32. elea
  33. entire
  34. equal
  35. equation
  36. famous
  37. find
  38. finite
  39. flaw
  40. fresh
  41. general
  42. greek
  43. halfway
  44. halves
  45. helps
  46. hour
  47. house
  48. impossible
  49. individual
  50. infinite
  51. infinitely
  52. infinity
  53. inspired
  54. inventing
  55. jargon
  56. journey
  57. left
  58. limit
  59. location
  60. logic
  61. logical
  62. long
  63. math
  64. mathematicians
  65. matter
  66. means
  67. meter
  68. mile
  69. mind
  70. motion
  71. nature
  72. number
  73. obtained
  74. order
  75. paradox
  76. paradoxes
  77. park
  78. part
  79. parts
  80. philosopher
  81. philosophers
  82. pieces
  83. point
  84. portion
  85. problem
  86. problems
  87. quarter
  88. realized
  89. remaining
  90. resolve
  91. resolved
  92. riddles
  93. sense
  94. series
  95. side
  96. sitting
  97. slice
  98. smaller
  99. square
  100. start
  101. story
  102. sum
  103. summing
  104. supposed
  105. takes
  106. tells
  107. term
  108. terms
  109. thinking
  110. time
  111. times
  112. total
  113. track
  114. traveling
  115. traverse
  116. true
  117. turn
  118. understand
  119. unit
  120. view
  121. walk
  122. walks
  123. words
  124. years
  125. zeno