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 pelihosphor famuos 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 rdidels have isneiprd 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 ctnitug in two" in ancient Greek. It goes something like this: After a long day of sttniig 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 potiron 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 anuomt 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 smaellr and smaller pieces, each of which takes some fiitne time to tsaverre. 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 ttaol time be infinity? This amuregnt, by the way, is completely general. It says that tlervanig 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 cnuolsocin is clearly abrusd, but where is the flaw in the logic? To roevlse the paradox, it helps to turn the story into a math problem. Let's supposed that Zeno's hsuoe 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 pniot of view and divide up the journey into pieces. The first half of the journey takes half an hour, the next part tekas qaeurtr of an hour, the third part takes an ethgih of an hour, and so on. Summing up all these times, we get a sieers 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 iiitfnny, 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 rnneimaig half in half, and so on. While we're doing this, let's keep trcak of the areas of the peeics. The first sclie 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 sqarue. 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 rlsevoed. 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.

Open Cloze

This is Zeno of Elea, an ancient Greek ___________ ______ 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 ________ 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 _______ in two" in ancient Greek. It goes something like this: After a long day of _______ 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 _______ 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 ______ 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 _______ and smaller pieces, each of which takes some ______ time to ________. 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 _____ time be infinity? This ________, by the way, is completely general. It says that _________ 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 __________ is clearly ______, but where is the flaw in the logic? To _______ the paradox, it helps to turn the story into a math problem. Let's supposed that Zeno's _____ 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 _____ of view and divide up the journey into pieces. The first half of the journey takes half an hour, the next part _____ _______ of an hour, the third part takes an ______ of an hour, and so on. Summing up all these times, we get a ______ 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 ________, 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 _________ half in half, and so on. While we're doing this, let's keep _____ of the areas of the ______. The first _____ 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 ______. 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 ________. 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.

Solution

absurd
series
famous
finite
total
pieces
track
house
takes
conclusion
traverse
philosopher
square
riddles
resolve
quarter
resolved
infinity
cutting
portion
remaining
inspired
amount
smaller
slice
eighth
traveling
argument
point
sitting

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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