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
point. Again the tortoise will have moved further
ahead. And so on, without end. How can Achilles ever catch
up to the tortoise?
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.