Zeno
of Elea conceived
a number
of "paradoxes".
Zeno conceived
these not
as mathematical
amusement,
but as
an attempt
to support
the doctrine
of his
teacher,
the ancient
Greek philosopher
Parmenides,
that all
evidence
of the
senses
is illusory,
particularly
the illusion
of "motion".
One
of Zeno's
most famous
paradoxes
posited
a race
between
the popular
Greek hero
Achilles,
and a tortoise.
Zeno
set out
to logically
show that,
with the
tortoise
given a
head start,
Achilles,
speedy
as he might
be, could,
in fact,
never
overtake
the plodding
reptile.
Zeno
reasoned
that when
Achilles
reached
the starting
point of
the tortoise,
the tortoise
would have
advanced
incrementally
further.
Achilles
would continually
reach a
point the
tortoise
had already
reached,
while the
tortoise
would at
the same
time have
reached
a slightly
further
point.
Thus, Zeno
reasoned,
the tortoise
could never
be overtaken
by Achilles.
Zeno's
paradox
provided
an
early entree
into the
science
and mathematics
of limits.
Zeno's
paradox
is resolved
with
the
insight
that
a sum of
infinitely
many
terms can
nevertheless
yield
a finite
result,
an insight
of calculus.
It was not until Cantor's
development of the theory of infinite
sets
in the
midnineteenth
century
that,
after more than two millennia, Zeno's Paradoxes could
be fully
resolved.
Zeno's
portrait is depicted
in the
style of
an ancient
Grecian
amphora
vase.
Overlaying
Zeno is
a graphical
depiction
of 'limits'.
