Relativity gives every bit of matter
throughout infinite space a unique address. Scientists agree that two or
more bits of matter cannot
take up the same space at the same time, and since matter is infinite,
then every bit of matter is related to all other bits of matter with
regard to their distances from all other bits of matter. Note: some
might argue that there are alternate worlds that exist in the same space
as another world. The problem with this argument is they are now talking
fourth dimensional reality. The solution of Zeno's motion paradoxes
exists in the third dimension.
Zeno was a Greek philosopher who lived between the
years of 490 B.C. and 430 B.C. His birth predated Socrates' birth (470
B.C.), Plato's (427 B.C.), and Aristotle's (384 B.C.). These three
famous philosophers, who were born after Zeno, tried to solve Zeno's
paradoxes without success.
There is very little known about Zeno's book or books
and his paradoxes. (It is not known if he had written more than one
book.) Proclus, an ancient Greek philosopher, born in the year 412, is
one of the philosophers who has brought forth Zeno's paradoxes to the
twenty-first century. Proclus stated that in Zeno's book he listed 40
paradoxes, and of those forty there are only ten that have continued on
to modern times. Zeno's book has long since been lost. Nevertheless,
those of Zeno's paradoxes that still exist have created much controversy
with scientists, mathematicians, and philosophers since their inception
nearly twenty-five hundred years ago.
During this time of high technology and vast amounts of
knowledge there is still a question of whether or not Zeno's paradoxes
have been solved. Some believe that mathematics using convergence of an
infinite series has offered a solution for several of the paradoxes, but
this essay will prove that this is not true. Furthermore, in this essay
it will be shown how to solve two of the more famous paradoxes: the
Achilles Paradox and the Dichotomy Paradox.
The following is a description of the Achilles Paradox.
Achilles is in a footrace with a tortoise. He gives the
tortoise a head start of 100 meters. Now, Achilles will have to run
the 100 meters, bringing him to the tortoise's original starting
point. During this time the tortoise has advanced a distance of 10
meters. Now, Achilles will have to run the 10 meters, but the
tortoise has advanced another 5 meters. Now, Achilles will have to
run the 5 meters, but the tortoise has advanced another 2.5 meters.
Now Achilles will have to run the 2.5 meters, . . . , but the
tortoise has advanced one tenth of a meter, . . . , but the tortoise
has advanced one millionth of a meter . . . . This division of
mathematical units of distances grows smaller and smaller and
continues on into infinity proving that Achilles will never catch up
with the tortoise.
The Dichotomy Paradox also incorporates an infinite
distance to be traveled: