TedTest

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 prdoxaeas, arguments that seem lcigoal, but whose conclusion is absurd or contradictory. For more than 2,000 years, Zeno's mind-bending riddles have inspired maciiahtametns and peihporohsls to better understand the ntaure of iiitnfny. One of the best known of Zeno's problems is caleld 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, tniikhng, 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 dntacsie is left into smaller and smaller pieces, each of which tkaes 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 jonreuy. The plerbom 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 lcoitaon to any other location should take an infinite amount of time. In other wdros, it says that all miootn is impossible. This conclusion is clearly absrud, but where is the flaw in the logic? To resolve the poadrax, 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 qaretur of an hour, the third part takes an eighth of an hour, and so on. smnmuig up all these times, we get a sreies that looks like this. "Now", Zeno might say, "since there are iiinfltney 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 relazied, 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 meetr. 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 temis we slice up the boxes, the total area is still the sum of the areas of all the peiecs. 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 etnrie srquae becomes creoved 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 fntiie answer is the same one that common 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 _________, arguments that seem _______, but whose conclusion is absurd or contradictory. For more than 2,000 years, Zeno's mind-bending riddles have inspired ______________ and ____________ to better understand the ______ of ________. One of the best known of Zeno's problems is ______ 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, ________, 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 ________ is left into smaller and smaller pieces, each of which _____ 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 _______. The _______ 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 ________ to any other location should take an infinite amount of time. In other _____, it says that all ______ is impossible. This conclusion is clearly ______, but where is the flaw in the logic? To resolve the _______, 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 _______ of an hour, the third part takes an eighth of an hour, and so on. _______ up all these times, we get a ______ that looks like this. "Now", Zeno might say, "since there are __________ 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 ________, 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 _____. 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 _____ we slice up the boxes, the total area is still the sum of the areas of all the ______. 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 ______ ______ becomes _______ 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 common sense tells us is true. Zeno's journey takes one hour.

Solution

mathematicians
takes
infinitely
summing
thinking
absurd
series
problem
motion
paradoxes
philosophers
logical
distance
meter
covered
words
pieces
called
entire
paradox
finite
square
times
journey
location
infinity
quarter
realized
nature

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

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