Zeno's Paradox. 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 .
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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.
The hare quickly reaches the turtle’s
starting point – but in that same time
The turtle moves ¼ mile ahead.
By the time the rabbit reaches the
turtle’s new position, the turtle
has had time to move ahead.
No matter how quickly the hare
covers the distance between himself
and the turtle, the turtle uses that
time to move ahead.
Can the hare ever catch the
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?
“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
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.
If the bottom two rows march in the directions indicated, will blue in row 2 pass yellow in row 3?
If the motion is continuous, yes!
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:
The next instant, (one chronon later) they are here. At no point in time was the blue turtle in row 2 opposite the yellow turtle in row three. The red faced turtles do not pass!
In one indivisible instant (chronon) , turtles move from top position to bottom position, and the red faced turtles do not pass!