Zeno’s Paradoxes

by James D. Nickel Copyright 

Zeno of Elea ca. 490 BC – 430 BC. ca. 490 BC – 430 BC. A pre-Socratic Greek philosopher. 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. 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 ( BC), in his Physics. The most famous are the so-called "arguments against motion" described by Aristotle ( BC), in his Physics.

Dichotomy Aristotle “That which is in locomotion must arrive at the half-way stage before it arrives at the goal.” 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. 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 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 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. Before you get the the eighth mark, you must arrive at the sixteenth mark.

Conclusion Carried ad infinitum, you cannot even start! Carried ad infinitum, you cannot even start!

Another perspective Every time I take a step, I get half- way closer to my goal. Every time I take a step, I get half- way closer to my goal. I make it to the half-way mark with my first step. 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 second step I make it to the quarter mark. With my third I make it to the eighth mark. With my third I make it to the eighth mark. Will I ever reach my goal? Will I ever reach my goal?

My Goal My Starting Point Step 1 1/2

My Goal My Starting Point Step 2 1/2 + 1/4 = 3/4

My Goal My Starting Point Step 3 1/2 + 1/4 + 1/8 = 7/8

My Goal My Starting Point Step 4 1/2 + 1/4 + 1/8 + 1/16 = 15/16

My Goal My Starting Point Step 5 1/2 + 1/4 + 1/8 + 1/16 + 1/32 = 31/32

My Goal My Starting Point Step 6 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 = 63/64

My Goal My Starting Point 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. According to Zeno, it would take an infinite number of steps to reach my goal. To him, this was impossible. To him, this was impossible. What do you think? What do you think?

Consider this Infinite Sum Draw a square 8 by 8 on graph paper. Draw a square 8 by 8 on graph paper. Let the area of the square be 1 square unit. Let the area of the square be 1 square unit. Now, color in half of it (1/2 - the first number in the series). 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). Next, color an additional one-fourth (1/4 is the second number in the series). What is the total area? What is the total area?

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). Next, color an additional one-eighth (1/8 - the third number in the series). What is the total area? What is the total area?

Area of the first three terms

Sum of the first four terms Next, color an additional one- sixteenth (1/16 - the fourth number in the series). Next, color an additional one- sixteenth (1/16 - the fourth number in the series). What is the total area? What is the total area?

Area of the first four terms

Continue the Process Go on and color in 1/16 and 1/32, the 5 th and 6 th terms. Go on and color in 1/16 and 1/32, the 5 th and 6 th terms.

Questions What is happening to the partial sums? What is happening to the partial sums? Are the sums getting bigger, smaller, or staying the same size? 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? 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? 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)? 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)? What is the smallest number that each term in the series gets closer and closer to (as a threshold)? Answer: 0

Mathematical Analysis

by James D. Nickel Copyright 