Zeno's Paradoxes Zeno of Elea (c. 490 BC - 430 BC)

Zeno of Elea was a philosopher Zeno of Elea was a philosopher. Unfortunately, we know little of his life, and none of his publications have survived.  We know of his work only from other references.  Zeno did produce a book that contained 40 paradoxes trying to argue about the impossibility of motion. Motion is an illusion and in reality it cannot even start. Four of the paradoxes had significant effect on mathematics. Aristotle's work refers to Zeno's four paradoxes of "Dichotomy," "Achilles," "Arrow," and "Stadium."  Aristotle was not taken in by Zeno's paradoxes and called them fallacies.  However, it wasn't until more modern times and through the development of the calculus that mathematics developed the notation and results to adequately handle Zeno's challenging contradictions.

Other proposed solutions to Zeno's paradoxes have included: Zeno's paradoxes were a major problem for ancient and medieval philosophers. In modern times, calculus provides at least a practical solution (as we will see). Other proposed solutions to Zeno's paradoxes have included: the denial that space and time are themselves infinitely divisible, the denial that the terms space and time refer to any entity with any innate properties at all.

The Arrow and The Target

Then, the arrow must first reach the point M3. To reach the target, the arrow must first reach the midpoint M1. Then, the arrow must first reach the point M2. BEGIN M1 M2 M3 TARGET And so on…ad infinitum. Therefore, if the space is infinitely divisible, the arrow can never reach the target. Even worse, the motion cannot even start since one can apply the same reasoning in the very first interval.

HARE Achilleas and Turtle

The hare and the tortoise decide to race

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

½ ¼

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

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

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

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

How can I ever catch the turtle 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!

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

TWENTIETH CENTURY PHILOSOPHERS ON “ZENO”

“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

“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

“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

OPPOSING MODELS 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.