The so-called Eleatic challenge launched by Parmenides of Elea against traditional cosmogonies through his monism was taken up by his disciple Zeno (also of Elea). Zeno pushed the logos to its extremes by formulating paradoxes aimed at proving, for instance, the logical impossibility of motion if one failed to accept the foundational principle of being as a single, unified whole. The most famous is that of swift-footed Achilles and the tortoise (slow-footed?): in my own words, “Even if Achilles gives a head start (in space or time) to a tortoise in a race, for every interval Achilles covers, the tortoise will move ahead (even if by a smaller distance than Achilles). By imagining increasingly smaller intervals of movement for Achilles, the tortoise keeps advancing in every instance, meaning Achilles can never overtake it.”
The applied logos SEEMS consistent, but with a bit of mathematics, one can expose the flaw in this reasoning. Let us look at an example using distances.
Imagine Achilles runs 10 times faster than the tortoise and grants it a 10-meter head start.
For every advancement Achilles makes, the tortoise will have covered 1/10th of that distance. “For every” means we can sum all these (infinite) intervals together. We can thus write the formula for the total distance
D = head start + displacements covered by Achilles as:
D = 10 + 1 + 1/10 + 1/100 + 1/1000 + … to infinity, which is written as
D = 10 + ∑n=0∞ (1/10)n .
This is a series of the form ∑n=0∞ qn, which converges if |q| < 1 (as in this case, where q = 0.1), yielding the value 1/(1 − q) .
We can easily compute D = 10 + 10/9 ≈ 11.11 meters as the distance at which Achilles catches up with the tortoise.
Good old Zeno could hardly have imagined, nearly 2,500 years ago, that a sum of infinitely many terms could yield a finite number; however, as mentioned, his aim was not mathematical but philosophical (a term that was actually still anachronistic back then).
It is fascinating to note that, although a concept analogous to convergent series was tackled by Archimedes of Syracuse (in the quadrature of the parabola) just a couple of centuries later, the conceptual constraints surrounding infinity (potential vs. actual) and the absence of the concept of a limit left Zeno’s paradoxes unresolved until Augustin-Louis Cauchy in the 19th century.
