An original poem on some of Zeno’s paradoxes:
Traverse a line? Don’t make me laugh!
Each segment’s segment’s cut in half.
One cannot simply walk the line,
‘cross infinite halves in finite time.
Swooping down from high above
The eagle catches fleeing dove,
Yet swift Achilles, to this day,
Cannot o’ertake his tortoise prey.
We see the arrow fly through air,
But surely there’s no motion there.
The arrow, ne’er before it’s caught,
Moves where it is, nor where it’s not.
In the hippodrome they run their course,
Speeds are measured in length of horse.
Opposing directions they fly past,
Each to the next seems doubly fast.
At the risk of over-explaining, a quick note on the four paradoxes mentioned in the poem:
The first verse deals with a paradox known as the “dichotomy”. This is probably the most well known of Zeno’s paradoxes. For a runner to reach the finish line, he must first reach the midpoint. In order to reach the midpoint, he must first go half way to the midpoint. And so on. As a consequence, before any distance can be traversed, an infinite number of smaller distances must be covered. And to take an infinite number of steps must take an infinite time. Therefore, the runner cannot possibly run even a short distance.
The second verse deals with the paradox of Achilles and the tortoise. It is actually quite similar to the first. Although Achilles is much faster than a tortoise, he can never catch it because in the time that it takes him to reach where the tortoise was, the tortoise will have moved a little further away. Therefore, before Achilles can ever catch up to the tortoise, he must first reach all of the infinite points where the tortoise had already been.
The third verse is the paradox of the arrow. An arrow, just like any other physical body, always takes up a space equal to itself. So at any given instant the arrow cannot move inside that space because the space is exactly the size and shape of the arrow. But it also cannot move outside of that space because it is perfectly contained. Therefore, motion is impossible.
The final verse treats the paradox of the stadium. If two teams of horses, four horses long each, pass each other in opposite directions, an observer will notice that in the time that the lead horse has covered a total distance of two horse-lengths, it will have actually passed all four of the other team’s horses. Therefore the chariot, despite its constant speed, travels two different distances in the same amount of time, which is absurd.
Beer of the week: Cerveza Monterrey – The term “Corona knock-off” gets thrown around a bit, and this isn’t the first beer I’ve reviewed that might merit that description. This extremely pale Guatemalan lager is very carbonated. On its own, it is fairly watery and tastes of corn. The addition of lime and sea salt, however, makes this a reasonably palatable hot-weather drink. With a lot of lime, it is rather refreshing.
Reading for the week: Lives of the Eminent Philosophers by Diogenes Laërtius, Parmenides, Melissus, and Zeno of Elea – Zeno’s own writings are lost to us. However, Diogenes Laërtius (among others) preserved some of his ideas. Diogenes also relates the gruesome details of Zeno’s death, including the part where Zeno bites off the ear (or nose) of a tyrant. To help give context to Zeno’s life, this reading includes the lives of his teacher (and lover?) Parmenides and his contemporary Melissus.
Question for the week: Are the paradoxes mere logic tricks, or do they point to some more profound truth?