Jens,
To me, Zeno's paradox is fun because it shows how thinking logically
can give wrong answers!
It is a paradox only to someone who
assumes that moving through
"an infinite number of points" (whatever that means) requires an
infinite amount of time. Or to put it another way, moving through
n points requires
n seconds. There is actually no connection
between the number of points and the amount of time.
If it takes one second to move a distance of one meter, it takes
one second to move through all those points. It doesn't matter
if the number of points is 100 or 100 googolplex.
Some time ago I wrote the following for a web page that is still
unfinished. I want critical feedback.
Quote:
Simple Answers to Eternal Questions
Zeno's Paradox: Can Achilles ever catch the tortoise?
Zeno asked: If Achilles and the tortoise run a race, and the
tortoise is given a head start, then Achilles must run for some
time before he reaches the point where the tortoise was when
they started. By then, the tortoise has moved ahead some
distance. Achilles must then run for an additional time to
reach that point. Again the tortoise will have moved further
ahead. And so on, without end. How can Achilles ever catch
up to the tortoise?
No problem.
Every increment is a smaller distance and shorter length of time
than the one before. The endless number of ever-smaller distances
add up to the distance from Achilles' starting point to the point
where he passes the tortoise. The endless number of ever-shorter
time periods add up to the time it takes Achilles to catch up to the
tortoise. Zeno's analysis simply looks at smaller and smaller pieces
of the interval remaining just before Achilles passes the tortoise,
and avoids ever looking at that point or beyond.
The process of dividing something into more and more pieces which
become ever smaller is the basis of calculus, invented by Isaac Newton
and Gottfried von Leibniz more than 2000 years after Zeno's time.
Calculus is needed to accurately describe changes in a thing when the
rate of change is itself changing. Although Zeno's paradox seems
rather silly, it introduced a very powerful mathematical idea
essential to the development of modern technology.
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Is that all correct? Is it clear? Did I leave out something
important? I don't want to get into the question of discrete
versus continuous space and time, though.
-- Jeff, in Minneapolis
__________________
http://www.FreeMars.org/jeff/
"I find astronomy very interesting, but I wouldn't if I thought we
were just going to sit here and look." -- "Van Rijn"
"The other planets? Well, they just happen to be there, but the
point of rockets is to explore them!" -- Kai Yeves