ZenoÂ¡Â¯s Paradox

Take Billy Joe Bim-Bob. HeÂ¡Â¯s your typical American country bumpkin, sitting in his beat up, plastered with duct tape, lay-Z-boy that he picked up on the side of the road, watching the Super Bowl not understanding what a dancing monkey has to do with E-trade, and what the hell E-trade is for that matter, drinking some Bud thinking how funny he would be if he said Â¡Â°I wish I had some bud to go with my Bud,Â¡Â± all while hangin with his inbred cousins, Billy Ray, Bobby Joe, and Thomas Enquivst III (rumor has it that Thomas is adopted, but we all know that Bobby JoeÂ¡Â¯s mom, Billie Jean, got with Billy Jo Bim-BobÂ¡Â¯s dad) who are all still in their boxers and a white t-shirt which is splattered with stains of beer, dropped bratwurst, and, of course, their own drool.

Finally that dancing monkey goes away and a Budweiser commercial comes on and Billy Joe breathes a sigh of relief as he sees something he can comprehend. Â¡Â°MmmmmÂ¡Â¦beer,Â¡Â± Billy Joe thinks to himself while adding another drool stain to the collection. Suddenly, with the speed of a 28.8K modem, a thought weasels its way through the thickets of bong resin in his head; triggered by that catchy ad, he realizes that Billy Joe wants another Bud (there was much rejoicing). With the keen eyesight that only an eagle can replicate, he spies the room searching for the nearest unopened can. But alas, the nearest receptacle of beer is all the way across the room! Keeping in mind that he is a lazy beast, Billy Joe contemplates the fetching of the beer. He thinks, Â¡Â°I reckon Billy Joe (yes, he sometimes likes to refer to himself in the third person) canÂ¡Â¯t make it all the way over there without passing out of exhaustion. How the hell am I going to get that Bud? Billy Joe wants Bud! Ah ha!Â¡Â± he exclaimed in sheer satisfaction of his brilliant self, Â¡Â°weÂ¡Â¯ll just do this in phases. First, IÂ¡Â¯ll walk half way there and take a break. Then IÂ¡Â¯ll walk half of the remaining distance and take another break. Once I regain my breath, IÂ¡Â¯ll go the next half, and so on until I get my coveted Bud!Â¡Â± As the half time show came to a close, Billy Joe prepared himself for the trek across the room, but then stopped, and processed the first intelligent thought of his life. Â¡Â°If I go half way every time, IÂ¡Â¯ll never get my Bud!Â¡Â±

Having not been so confused since the last Super Bowl when he drank a Bud that tasted remarkably similar to urine (compliments of Billy Ray), Billy Joe is in dire straits. Â¡Â°What to do, what to do!?Â¡Â± As Billy Joe watches the Rams finally score a touchdown, the animal instinct in him takes over and he charges the defenseless Bud. Defying all laws of nature and logic, not only did Billy Joe survive the journey across the room, he also got his Bud (and there was much rejoicing). As Billy Joe chugged the remaining amount of Bud and Vinatieri kicked the game winning field goal, he collected his thoughts and said to himself, Â¡Â°Billie Joe, donÂ¡Â¯t let your brain get in the way next time.Â¡Â±

Billy Joe has just demonstrated (albeit, in a feeble manner) ZenoÂ¡Â¯s paradox of Achilles and the Tortoise (786, textbook), where in a race, Achilles gives the Tortoise a head start. The argument attempts to show that even though Achilles runs faster than the Tortoise, he will never catch the tortoise. The argument goes like this: when Achilles reaches the point at which the Tortoise started, the Tortoise is no longer there, having advanced some distance; when Achilles arrives at the point where the Tortoise was when Achilles arrived at the point where the Tortoise started, the Tortoise is no longer there, having advanced some distance; and so on. Hence Achilles can never catch the Tortoise, no matter how much faster he may run.

The inverse of this paradox can be easily summarized by another paradox of ZenoÂ¡Â¯s called, the Racecourse (787, textbook). The Racecourse argues that an athlete in a race will never be able to start. The reason for this is that before the runner can complete the whole course they have to complete half the course; and before they can complete half the course they have to complete a quarter; and before they can complete a quarter they have to complete an eighth; and so on. Therefore the runner has to complete an infinite amount of events in a finite amount of time - assuming that the race is to be run in a finite amount of time. It is impossible to do an infinite amount of things in a finite amount of time, therefore, the race can never be started and motion is impossible.

Based on his arguments, Zeno fully believed that motion did not exist. Nevertheless, as Billy Bob has just demonstrated, motion is possible, and people are able to reach any given point. According to the assumptions of his time, ZenoÂ¡Â¯s paradox was mathematically correct, however, with the evolution of the geometric series, modern math can disprove them.

First of all, what is a paradox? By definition, a paradox is a statement or concept that contains conflicting ideas. In logic, a paradox is a statement that contradicts itself; for example, the statement "I never tell the truth" is a paradox because if the statement is true (T), it must be false (F) and if it is false (F), it must be true (T). In everyday language, a paradox is a concept that seems absurd or contradictory, yet is true. In a Windows environment, for instance, it is a paradox that when a user wants to shut down their computer, it is necessary to click "start" (http://whatis.techtarget.com/).

What is paradoxical about Achilles and the Tortoise and the Racecourse? In Achilles and the Tortoise, it is obvious that a faster moving object can pass a slower moving object, but ZenoÂ¡Â¯s logic is the perfect argument saying that you can never really pass the slower object, given a head start. In the Racecourse, we all know that motion is possible, but due to ZenoÂ¡Â¯s impeccable arguments one canÂ¡Â¯t deny that motion is impossible.

To see how Billy Bob fits into Achilles and the Tortoise, we should add some numbers. Lets say that Achilles can run twice as fast as the Tortoise and the Tortoise can run at 5 yards/second. The Tortoise also gets a 10-yard head start and the track is 40 yards long. Next we need to define some points, these will be: the start of the race, the point where Achilles passes the Tortoise, and the end of the race. The start of the race is analogous to Billy BobÂ¡Â¯s beat up lay-Z-boy and the point where Achilles passes the Tortoise is analogous to where the Bud resides.

Using simple math, one can find where Achilles would pass the Tortoise assuming that faster objects can pass slower objects given a head start. After 2 seconds, both Achilles and the Tortoise will be at the 20 yard marker, otherwise knows as the Bud (Achilles = 10yds/sec * 2 seconds = 20 yards; Tortoise = 10 yard head start + {5 yds/sec * 2 seconds = 20 yards).

Using ZenoÂ¡Â¯s logic, Achilles will never pass the Tortoise, just as Billy Bob will never reach his Bud. One second after the start of the race, Achilles will have reached the 10-yard marker where the Tortoise originally started, but the Tortoise will have advanced to the 15-yard marker. After another Â¨Ã¶ second, Achilles will be at the 15-yard marker, but the Tortoise will have advanced to the 17.5-yard marker, and so on, and so on, until infinity. Therefore, Achilles will never pass the Tortoise because he can only catch up half the distance between him and the Tortoise every time they both move. Put another way, in order to reach the point where he would pass the Tortoise, Achilles first has to travel half of that distance. From halfway, he must travel half of the remaining distance, from Â¨Ãº of the way, he must travel half of the remaining distance, etc., etc. until infinity. Substitute in Billy Bob for Achilles and the Bud for the 20-yard marker (the point where Achilles passes the Tortoise) and we see why Billy Bob came to the conclusion that if he proceeded with his plan, he would never reach his Bud.

The soft underbelly of Zeno's argument is the assumption that the sum of an infinite number of numbers is always infinite. While this seems intuitively logical, it is in fact wrong. For example, the infinite sum 1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ... is equal to 2. This type of series is known as a geometric series. A geometric series is a series that begins with one and then each successive term is found by multiplying the previous term by some fixed amount, say X (485, Calculus). For the above series, X is equal to Â¨Ã¶ (the proportion of distance that Achilles can make up).

Infinite geometric series' converge (sum to a finite number) when X is less than one. Both the distance that Achilles travels and the time that elapses before he reaches the Tortoise can be expressed as an infinite geometric series with X less than one. So, Achilles passes an infinite number of "distance intervals" before catching the Tortoise, but because the "distance intervals" are decreasing exponentially in size, the total distance that he travels before catching the tortoise is not infinite. Similarly, it takes an infinite number of time intervals for Achilles to catch the Tortoise, but the sum of these time intervals is a finite amount of time (1 sec + Â¨Ã¶ sec + Â¨Ã¹ sec + 1/8 secÂ¡Â¦Â¡Ã„ = 2 sec).

The Racecourse deals with this same issue. In it, Zeno argues that a race cannot start because in order to finish Achilles would have to accomplish an infinite amount of tasks in a finite amount of time, assuming that the race is to be run with some maximum amount of time allotted. But, as we have just proven, it is possible in this case because Achilles always has to travel Â¨Ã¶ way there first (X=1/2).

What the mathematics is telling us is that there is some incredibly small, indivisible last step that Achilles takes to pass the slower Tortoise, there is some incredibly small, indivisible first step that Achilles has to take to start the race, and, thankfully, our eyes are not deceiving us. In the case of Billy Bob, that last, indivisible step was the size of his foot. In ZenoÂ¡Â¯s defense, his arguments are impeccable, but he did not have Geometric Series at his disposal. It is, however, relieving to know that faster moving objects can pass slower moving objects, and the Olympics will not have a dead tie at Starting Line.

Take Billy Joe Bim-Bob. HeÂ¡Â¯s your typical American country bumpkin, sitting in his beat up, plastered with duct tape, lay-Z-boy that he picked up on the side of the road, watching the Super Bowl not understanding what a dancing monkey has to do with E-trade, and what the hell E-trade is for that matter, drinking some Bud thinking how funny he would be if he said Â¡Â°I wish I had some bud to go with my Bud,Â¡Â± all while hangin with his inbred cousins, Billy Ray, Bobby Joe, and Thomas Enquivst III (rumor has it that Thomas is adopted, but we all know that Bobby JoeÂ¡Â¯s mom, Billie Jean, got with Billy Jo Bim-BobÂ¡Â¯s dad) who are all still in their boxers and a white t-shirt which is splattered with stains of beer, dropped bratwurst, and, of course, their own drool.

Finally that dancing monkey goes away and a Budweiser commercial comes on and Billy Joe breathes a sigh of relief as he sees something he can comprehend. Â¡Â°MmmmmÂ¡Â¦beer,Â¡Â± Billy Joe thinks to himself while adding another drool stain to the collection. Suddenly, with the speed of a 28.8K modem, a thought weasels its way through the thickets of bong resin in his head; triggered by that catchy ad, he realizes that Billy Joe wants another Bud (there was much rejoicing). With the keen eyesight that only an eagle can replicate, he spies the room searching for the nearest unopened can. But alas, the nearest receptacle of beer is all the way across the room! Keeping in mind that he is a lazy beast, Billy Joe contemplates the fetching of the beer. He thinks, Â¡Â°I reckon Billy Joe (yes, he sometimes likes to refer to himself in the third person) canÂ¡Â¯t make it all the way over there without passing out of exhaustion. How the hell am I going to get that Bud? Billy Joe wants Bud! Ah ha!Â¡Â± he exclaimed in sheer satisfaction of his brilliant self, Â¡Â°weÂ¡Â¯ll just do this in phases. First, IÂ¡Â¯ll walk half way there and take a break. Then IÂ¡Â¯ll walk half of the remaining distance and take another break. Once I regain my breath, IÂ¡Â¯ll go the next half, and so on until I get my coveted Bud!Â¡Â± As the half time show came to a close, Billy Joe prepared himself for the trek across the room, but then stopped, and processed the first intelligent thought of his life. Â¡Â°If I go half way every time, IÂ¡Â¯ll never get my Bud!Â¡Â±

Having not been so confused since the last Super Bowl when he drank a Bud that tasted remarkably similar to urine (compliments of Billy Ray), Billy Joe is in dire straits. Â¡Â°What to do, what to do!?Â¡Â± As Billy Joe watches the Rams finally score a touchdown, the animal instinct in him takes over and he charges the defenseless Bud. Defying all laws of nature and logic, not only did Billy Joe survive the journey across the room, he also got his Bud (and there was much rejoicing). As Billy Joe chugged the remaining amount of Bud and Vinatieri kicked the game winning field goal, he collected his thoughts and said to himself, Â¡Â°Billie Joe, donÂ¡Â¯t let your brain get in the way next time.Â¡Â±

Billy Joe has just demonstrated (albeit, in a feeble manner) ZenoÂ¡Â¯s paradox of Achilles and the Tortoise (786, textbook), where in a race, Achilles gives the Tortoise a head start. The argument attempts to show that even though Achilles runs faster than the Tortoise, he will never catch the tortoise. The argument goes like this: when Achilles reaches the point at which the Tortoise started, the Tortoise is no longer there, having advanced some distance; when Achilles arrives at the point where the Tortoise was when Achilles arrived at the point where the Tortoise started, the Tortoise is no longer there, having advanced some distance; and so on. Hence Achilles can never catch the Tortoise, no matter how much faster he may run.

The inverse of this paradox can be easily summarized by another paradox of ZenoÂ¡Â¯s called, the Racecourse (787, textbook). The Racecourse argues that an athlete in a race will never be able to start. The reason for this is that before the runner can complete the whole course they have to complete half the course; and before they can complete half the course they have to complete a quarter; and before they can complete a quarter they have to complete an eighth; and so on. Therefore the runner has to complete an infinite amount of events in a finite amount of time - assuming that the race is to be run in a finite amount of time. It is impossible to do an infinite amount of things in a finite amount of time, therefore, the race can never be started and motion is impossible.

Based on his arguments, Zeno fully believed that motion did not exist. Nevertheless, as Billy Bob has just demonstrated, motion is possible, and people are able to reach any given point. According to the assumptions of his time, ZenoÂ¡Â¯s paradox was mathematically correct, however, with the evolution of the geometric series, modern math can disprove them.

First of all, what is a paradox? By definition, a paradox is a statement or concept that contains conflicting ideas. In logic, a paradox is a statement that contradicts itself; for example, the statement "I never tell the truth" is a paradox because if the statement is true (T), it must be false (F) and if it is false (F), it must be true (T). In everyday language, a paradox is a concept that seems absurd or contradictory, yet is true. In a Windows environment, for instance, it is a paradox that when a user wants to shut down their computer, it is necessary to click "start" (http://whatis.techtarget.com/).

What is paradoxical about Achilles and the Tortoise and the Racecourse? In Achilles and the Tortoise, it is obvious that a faster moving object can pass a slower moving object, but ZenoÂ¡Â¯s logic is the perfect argument saying that you can never really pass the slower object, given a head start. In the Racecourse, we all know that motion is possible, but due to ZenoÂ¡Â¯s impeccable arguments one canÂ¡Â¯t deny that motion is impossible.

To see how Billy Bob fits into Achilles and the Tortoise, we should add some numbers. Lets say that Achilles can run twice as fast as the Tortoise and the Tortoise can run at 5 yards/second. The Tortoise also gets a 10-yard head start and the track is 40 yards long. Next we need to define some points, these will be: the start of the race, the point where Achilles passes the Tortoise, and the end of the race. The start of the race is analogous to Billy BobÂ¡Â¯s beat up lay-Z-boy and the point where Achilles passes the Tortoise is analogous to where the Bud resides.

Using simple math, one can find where Achilles would pass the Tortoise assuming that faster objects can pass slower objects given a head start. After 2 seconds, both Achilles and the Tortoise will be at the 20 yard marker, otherwise knows as the Bud (Achilles = 10yds/sec * 2 seconds = 20 yards; Tortoise = 10 yard head start + {5 yds/sec * 2 seconds = 20 yards).

Using ZenoÂ¡Â¯s logic, Achilles will never pass the Tortoise, just as Billy Bob will never reach his Bud. One second after the start of the race, Achilles will have reached the 10-yard marker where the Tortoise originally started, but the Tortoise will have advanced to the 15-yard marker. After another Â¨Ã¶ second, Achilles will be at the 15-yard marker, but the Tortoise will have advanced to the 17.5-yard marker, and so on, and so on, until infinity. Therefore, Achilles will never pass the Tortoise because he can only catch up half the distance between him and the Tortoise every time they both move. Put another way, in order to reach the point where he would pass the Tortoise, Achilles first has to travel half of that distance. From halfway, he must travel half of the remaining distance, from Â¨Ãº of the way, he must travel half of the remaining distance, etc., etc. until infinity. Substitute in Billy Bob for Achilles and the Bud for the 20-yard marker (the point where Achilles passes the Tortoise) and we see why Billy Bob came to the conclusion that if he proceeded with his plan, he would never reach his Bud.

The soft underbelly of Zeno's argument is the assumption that the sum of an infinite number of numbers is always infinite. While this seems intuitively logical, it is in fact wrong. For example, the infinite sum 1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ... is equal to 2. This type of series is known as a geometric series. A geometric series is a series that begins with one and then each successive term is found by multiplying the previous term by some fixed amount, say X (485, Calculus). For the above series, X is equal to Â¨Ã¶ (the proportion of distance that Achilles can make up).

Infinite geometric series' converge (sum to a finite number) when X is less than one. Both the distance that Achilles travels and the time that elapses before he reaches the Tortoise can be expressed as an infinite geometric series with X less than one. So, Achilles passes an infinite number of "distance intervals" before catching the Tortoise, but because the "distance intervals" are decreasing exponentially in size, the total distance that he travels before catching the tortoise is not infinite. Similarly, it takes an infinite number of time intervals for Achilles to catch the Tortoise, but the sum of these time intervals is a finite amount of time (1 sec + Â¨Ã¶ sec + Â¨Ã¹ sec + 1/8 secÂ¡Â¦Â¡Ã„ = 2 sec).

The Racecourse deals with this same issue. In it, Zeno argues that a race cannot start because in order to finish Achilles would have to accomplish an infinite amount of tasks in a finite amount of time, assuming that the race is to be run with some maximum amount of time allotted. But, as we have just proven, it is possible in this case because Achilles always has to travel Â¨Ã¶ way there first (X=1/2).

What the mathematics is telling us is that there is some incredibly small, indivisible last step that Achilles takes to pass the slower Tortoise, there is some incredibly small, indivisible first step that Achilles has to take to start the race, and, thankfully, our eyes are not deceiving us. In the case of Billy Bob, that last, indivisible step was the size of his foot. In ZenoÂ¡Â¯s defense, his arguments are impeccable, but he did not have Geometric Series at his disposal. It is, however, relieving to know that faster moving objects can pass slower moving objects, and the Olympics will not have a dead tie at Starting Line.