Zeno of Elea was born in Elea, Italy, in 490 B.C. He died there in 430 B.C., in an
attempt to oust the city's tyrant. He was a noted pupil of Parmenides, from whom he
learned most of his doctrines and political ideas. He believed that what exists is one,
permanent, and unchanging. Zeno argued against multiplicity and motion. He did so by
showing the contradictions that result from assuming that they were real. His argument
against multiplicity stated that if the many exists, it must be both infinitely large
and
infinitely small, and it must be both limited and unlimited in number. His argument
against motion is characterized by two famous illustrations: the flying arrow, and the
runner in the race. It is the illustration with the runner that is associated the first
part of
the assignment. In this illustration, Zeno argued that a runner can never reach the end
of a
race course. He stated that the runner first completes half of the race course, and then
half
of the remaining distance, and will continue to do so for infinity. In this way, the
runner
can never reach the end of the course, as it would be infinitely long, much as the
semester
would be infinitely long if we completed half, and then half the remainder, ad infinitum.
This interval will shrink infinitely, but never quite disappear. This type of argument
may
be called the antinonomy of infinite divisibilty, and was part of the dialectic which
Zeno
invented.
These are only a small part of Zeno's arguments, however. He is believed to have
devised at least forty arguments, eight of which have survived until the present. While
these arguments seems simple, they have managed to raise a number of profound
philosophical and scientific questions about space, time, and infinity, throughout
history.
These issues still interest philosophers and scientists today.
The problem with both Zeno's argument and yours is that neither of you deal with
adding the infinite. Your argument suggests that if one adds the infinite, the sum will
be
infinity, which is not the case. If the numbers are shrinking infinitely at the same
rate,
then eventually they will equal a certain number, not infinity as both Zeno's argument
and
yours suggest. A simpler way to explain this would be to say that if the first half of
the
semester takes a certain amount of time, and time always passes at the same rate, then
the
second half of the semester will also take a certain amount of time, which can be
measured.
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