Surprisingly, nobody has yet given the mathematical explanation as to why the "paradox" of Achilles and the tortoise (or the runner and the stadium) is no
paradox at all. People have pointed out that with today's understanding of mathematics, we can easily show that the infinite series have a finite sum. Zeno was born somewhere around 495-480 BCE.
Newton and
Leibniz are credited with the discovery of
The Calculus not until the 17th century of the common era.
Let's take a look at the geometric series 1/2 + 1/4 + 1/8 + ..., which describes the segments that the runner must move around the stadium.
This geometric series has ratio r=1/2. We will prove that a geometric series with a common ratio whose absolute value is less than 1 converges, and that its sum is described as a/(1-r), where a is its initial term and r its ratio. In the case of the runner, a=1/2 and r=1/2 . This series converges, and its sum is a/(1-r) = (1/2)/(1-1/2) = (1/2)/(1/2) = 1.
Each term of a geometric series is obtained by multiplying the previous term by the common ratio, r. The series can thus be described as a + ar + ar2 + ar3 + ... + arn-1 + ..., or Σ(n=1, ∞, arn-1).
We can define the nth partial sum sn = Σ(i=1, n, ari-1) =
a + ar + ar2 + ... + arn-1.
We know that when r=1, the series is a + a + a + ... + a. In this case, sn = na, which goes to infinity as n grows to infinity. In this case, the series diverges.
When r≠1, sn = a + ar + ar2 + ... + an-1.
rsn = ar + ar2 + ... + arn-1 + arn.
Subtracting these equations, sn-rsn = a - arn = sn(1-r).
So sn = (a(1-rn))/(1-r).
when -1 < r < 1, rn goes to 0 as n goes to infinity. So
lim(n→∞, sn) = lim(n→∞,(a(1-rn)/(1-r)) = (a/(1-r))(1 - lim(n→∞,rn)) = (a/(1-r))(1-0) = a/(1-r).
So there we have it. A geometric series with |r|<1 converges, and its sum is a/(1-r). The length of each successively smaller segment that the runner must run around the stadium is a term of the geometric series 1/2 + 1/4 + 1/8 + ..., in which r = 1/2. This series' sum is (1/2)/(1-(1/2)) = (1/2)/(1/2) = 1. So the runner does run each half of each segment, and eventually runs around the whole circumference of the stadium.