Achilles and the Tortoise
How will Achilles ever overtake the tortoise?
W. S. Robinson
Zeno of Elea (c. 450 BCE) is credited with creating several famous paradoxes, and perhaps the best known is the paradox of the Tortoise and Achilles. For those who are unaware, below are the basics of Zeno’s logic puzzle.
Achilles, the fleet-footed hero of the Trojan War, is engaged in a race with a lowly tortoise, which has been granted a head start. The task initially seems simple and Achilles is confident, but he faces a problem. While he is covering the gap between him and the point at which the tortoise existed when he began the race, the tortoise has created a new gap. This new gap is smaller than the first but is still a finite distance that Achilles must cover before he can catch up with the animal. Achilles races across the new gap. To his frustration, while he was scampering across the second gap, the tortoise was establishing a third. The cycle repeats infinitely; Achilles reaches the point the tortoise was at, and in the time that it takes him to do so, the tortoise has covered a new distance. In this lies the crux of the problem – how will Achilles ever overtake the tortoise? No matter how quickly he closes each gap, the slow-but-steady tortoise will open new, smaller ones, and remain just ahead of the Greek hero.
It is tempting to dismiss the problem as sophistry, out of fear or laziness. Laziness, because thinking about the paradox gives the feeling that you’re perpetually on the verge of solving it without ever actually doing so – similarly to how Achilles felt about catching the tortoise. Fear, because being defeated by a man who died before humans conceived the number zero delivers a hefty blow to one’s self-image.
Yes, the paradox is a trick. Zeno created the paradox to support the argument that motion and change are not real. Nick Huggett, a philosopher of physics at the University of Illinois, says “sure it’s crazy to deny motion, but to accept it is worse.” The paradox reveals a fundamental mismatch between the way that we think about the world and the way that the world actually is, a trick in making you think about space, time and motion the wrong way. The question then arises; what is precisely wrong with our thinking? We can observe that motion is possible, and that a fast human runner can beat a tortoise in a race. The problem lies in our conception of infinity.
There is more than one kind of infinity. The task seems impossible, as Achilles would have to cover an infinite number of spaces in a finite amount of time. However, not all infinities are created the same. Consider the series:
1
2
+
1
4
+
1
8
+
1
16
+
1
32
12+14+18+116+132
… Although the number goes on forever, the series converges, and the solution is one. As long as Achilles is making the gaps smaller at a sufficiently fast rate (so that their distances look like the above equation), he will complete the series in a measurable amount of time, and therefore catch the tortoise.
The convergence of infinite series explains countless things that we observe in the world – not just the fact that a fast runner can overtake a tortoise in a race. Any distance, time or force that exists in our world can be broken into an infinite number of pieces (just like the distance that Achilles must cover), but decades of physics and mathematics has proved that they can be treated as finite.
This answer might not completely satisfy ancient Greece philosophers, many of whom felt that their logic was more powerful than the observed reality. However, the way that philosophers and mathematicians have answered the challenge, using observation to create a durable theory, is a testament to the role that research and experimentation plays in advancing understanding.