Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Zeno’s Paradoxes by James D. Nickel Copyright 2007 www.biblicalchristianworldview.net Zeno of Elea ca. 490 BC – 430 BC. A pre-Socratic Greek philosopher. Zeno's paradoxes have puzzled, challenged, influenced, inspired, infuriated, and amused philosophers, mathematicians, physicists and school children for over two millennia. The most famous are the so-called "arguments against motion" described by Aristotle (384-322 BC), in his Physics. Dichotomy Aristotle “That which is in locomotion must arrive at the half-way stage before it arrives at the goal.” In other words, to reach a goal, you first must arrive at the half-way mark. Before you get to the half-way mark, you must arrive at the quarter mark. Before you get to the quarter mark, you must arrive at the eighth mark. Before you get the the eighth mark, you must arrive at the sixteenth mark. Conclusion Carried ad infinitum, you cannot even start! Another perspective Every time I take a step, I get halfway closer to my goal. I make it to the half-way mark with my first step. With my second step I make it to the quarter mark. With my third I make it to the eighth mark. Will I ever reach my goal? My Starting Point My Goal Step 1 1/2 My Starting Point My Goal Step 2 1/2 + 1/4 = 3/4 My Starting Point My Goal Step 3 1/2 + 1/4 + 1/8 = 7/8 My Starting Point My Goal Step 4 1/2 + 1/4 + 1/8 + 1/16 = 15/16 My Starting Point My Goal Step 5 1/2 + 1/4 + 1/8 + 1/16 + 1/32 = 31/32 My Starting Point My Goal Step 6 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 = 63/64 My Starting Point My Goal Step 7 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + 1/128 = 127/128 Will I Ever Reach My Goal? According to Zeno, it would take an infinite number of steps to reach my goal. To him, this was impossible. What do you think? Consider this Infinite Sum 1 1 1 1 1 ... ? 2 4 8 16 32 Draw a square 8 by 8 on graph paper. Let the area of the square be 1 square unit. Now, color in half of it (1/2 - the first number in the series). Area of the first term Sum of the first two terms Next, color an additional one-fourth (1/4 is the second number in the series). What is the total area? 3 1 1 42 6 3 2 4 8 8 4 4 Area of the first and second terms Sum of the first three terms Next, color an additional one-eighth (1/8 - the third number in the series). What is the total area? 7 3 1 24 4 28 7 4 8 32 32 8 8 Area of the first three terms Sum of the first four terms Next, color an additional onesixteenth (1/16 - the fourth number in the series). What is the total area? 15 7 1 112 8 120 15 8 16 128 128 16 16 Area of the first four terms Continue the Process Go on and color in 1/16 and 1/32, the 5th and 6th terms. Questions What is happening to the partial sums? Are the sums getting bigger, smaller, or staying the same size? Answer: Getting bigger (ever so slightly). Questions Are the numbers you are adding each time getting bigger or smaller? Answer: Getting smaller (ever so slightly). Questions Will the square ever get filled as we keep on going? Answer: As long as you stop at some point, there will always be a tiny bit unfilled. Questions What is the sum getting close to (as a threshold)? Answer: 1 Questions What is the smallest number that each term in the series gets closer and closer to (as a threshold)? Answer: 0 Mathematical Analysis 1 1 1 1 1 ... 1 2 4 8 16 32 by James D. Nickel Copyright 2007 www.biblicalchristianworldview.net