ZENO

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Zeno's Paradox
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The hare and the tortoise decide to race
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Since I run twice as fast as you do, I will give
you a half mile head start.
Thanks!
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½
¼
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½
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The hare quickly reaches the turtles starting
point but in that same time The turtle moves ¼
mile ahead.
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½
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By the time the rabbit reaches the turtles new
position, the turtle has had time to move ahead.
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½
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No matter how quickly the hare covers the
distance between himself and the turtle, the
turtle uses that time to move ahead.
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½
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Can the hare ever catch the turtle???
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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!
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This is a paradox because common sense tells us
that eventually the much swifter hare must
overtake the plodding tortoise!
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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
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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?
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TWENTIETH CENTURY PHILOSOPHERS ON ZENO
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Zenos 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
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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
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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
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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.

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• 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!

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The paradox of the stadium is about soldiers
marching in formation - turtles will play the
rolls of soldiers.
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(No Transcript)
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If the bottom two rows march in the directions
indicated, will blue in row 2 pass yellow in row
3?
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If the motion is continuous, yes!
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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
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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!
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In one indivisible instant (chronon) , turtles
move from top position to bottom position, and
the red faced turtles do not pass!
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Do both models lead to paradox?