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1. Zeno's Paradox Slides prepared by: Pamela Leutwyler,    Professor of Mathematics   Bucks County Community College

2. The hare and the tortoise decide to race

3. Since I run twice as fast as you do, I will give you a half mile head start. Thanks!

4. ½ ¼

5. ½ ¼ The hare quickly reaches the turtle’s starting point – but in that same time The turtle moves ¼ mile ahead.

6. ½ ¼ By the time the rabbit reaches the turtle’s new position, the turtle has had time to move ahead.

7. ½ ¼ No matter how quickly the hare covers the distance between himself and the turtle, the turtle uses that time to move ahead.

8. ½ ¼ Can the hare ever catch the turtle???

9. How can I ever catch the turtle. If it takes me 1 second to reach his current position, in that 1 second, he will have moved ahead again!

10. This is a paradox because common sense tells us that eventually the much swifter hare must overtake the plodding tortoise!

11. If the rabbit runs twice as fast as the turtle, then the rabbit runs 2 miles in the same time the turtle runs 1 mile. 1 mile 2 miles

12. HOW DO WE MODEL TIME AND SPACE? A unit of time( hour, minute, second ) or a unit of space(mile, foot, inch) can be divided in half, and then divided in half again, and again. Can we continue to break it into smaller and smaller pieces ad infinitum, or do we eventually reach some unit so small it can no longer be divided?

13. TWENTIETH CENTURY PHILOSOPHERS ON “ZENO”

14. “Zeno’s arguments in some form, have afforded grounds for almost all the theories of space and time and infinity which have been constructed from his day to our own.” B. Russell

15. “The kernel of the paradoxes … lies in the fact that it is paradoxical to describe a finite time or distance as an infinite series of diminishing magnitudes.”E.TeHennepe

16. “If I literally thought of a line as consisting of an assemblage of points of zero length and of an interval of time as the sum of moments without duration, paradox would then present itself.”P.W. Bridgman

17. In classical physics, time and space are modeled as mathematically continuous - able to be subdivided into smaller and smaller pieces, ad infinitum. Quantum theory posits a minimal unit of time - called a chronon - and a minimal unit of space- called a hodon . These units are discrete and indivisible. OPPOSING MODELS

18. With the race between the turtle and the rabbit, Zeno argues against a model of space and time that allows units to be divided into smaller and smaller pieces to infinity. • Zeno has another paradox, called “the stadium” that argues against the existence of indivisible units of space and time!

19. The paradox of the stadium is about soldiers marching in formation - turtles will play the rolls of soldiers.

20. If the bottom two rows march in the directions indicated, will turtle 5 in row 2 pass turtle 9 in row 3? 5 9

21. Now, suppose the turtles are 1 hodon apart, and marching at a rate of 1 hodon per chronon. The 2 bottom rows move simultaneously. One instant they are here: 5 9

22. The next instant, (one chronon later) they are here. At no point in time was turtle 5 in row 2 opposite the turtle 9 in row three. The red faced turtles do not pass! 5 9

23. In one indivisible instant (chronon) , turtles move from top position to bottom position, and the red faced turtles do not pass!