The Bunny and Zeno’s Paradoxes

It’s a well known-fact that bunnies can never outrun a turtle in a speed race. Aesop proved that, in theory. Cousin Bugs tried to challenge this truth, and he failed. If even Bugs Bunny lost, what hope is left for us common bunnies?

But as it seems, we are not the only species slower than turtles. Zeno of Elea theorised that even a well known for his speed demigod and hero, Achilles, can’t outrun a turtle, if certain conditions are meant.

Achilles and the Turtle

According to Zeno, if Achilles runs ten times faster than a turtle, and the turtle starts running ten stadiums (unit of length) ahead of Achilles, the turtle will win the race, no matter what. He used these numbers to make maths easy, but following the same logic, no matter how much faster Achilles is, or how small of a lead the turtle has, as long as it has one, Achilles is doomed to lose.

Therefore, let’s also keep maths simple, and turn stadiums to kilometres for easier understanding.

So, the turtle and Achilles start running with Achilles ten kilometres behind the turtle, and he runs ten times faster than the turtle. According to Zeno, by the time Achilles will have covered these 10 km (reaching the turtle’s starting point), the turtle will already be ahead another kilometre. By the time Achilles gets there, the turtle will be 100 m ahead, and so on and so forth. And, as long as the distance between Achilles and the turtle can be divided by ten (which is forever) Achilles won’t even reach the turtle, let alone win the race.

It’s maths. It’s proven. Therefore, it must be the truth. Turtles can’t lose a speed race against demigods and bunnies alike. End of story!

I hear you screaming, “BUT!!!”

Some of you might say that from one point onwards the distance between the turtle and Achilles will be smaller than Achilles’ stride, or even his foot. And, therefore, Achilles will overtake the turtle.

Good argument, but it also shows you’ve fallen in Zeno’s trap. Were you to say that, Zeno would smirk at you and ask, “Then if they were point particles, Achilles wouldn’t have a chance to overtake the turtle. Right?”

Zeno’s Trick

Actually, what Zeno says is the truth and nothing but the truth.

Read the previous statement again, and you’ll see what’s wrong.

Indeed, Zeno speaks the truth. But not the whole truth. And he skilfully misleads you into thinking that his calculations investigate space (the distance between Achilles and the turtle). But what he actually does is expanding time. Or slowing down time, if you prefer. But he never mentions time, does he?

Let’s see what he doesn’t say, then. If Achilles is capable of shaming the World Record keeper for 10 km (26min 11 sec), and he can cover the distance of 10 km in 10 minutes, in those ten minutes, the turtle will have run 1 km. Then, Achilles, in 1 minute will run this 1 km. By then, the turtle will be ahead 100 m. Achilles will need 6 seconds to cover this distance. Once again, the turtle will be ahead of him 10 metres. Achilles will need 0.6 seconds to cover that. And then, 0.06 seconds to cover the 1 metre the turtle will be ahead of him, and so on and so forth.

Therefore, the time “stops” in 11 minutes and 6.666… seconds. Achilles will reach the turtle in 11 minutes, 6 and 6/9 seconds. The last fraction is zero point infinite sixes. In 11 minutes and 6.7 seconds the turtle will be eating Achilles’ dust (quite literally, too, actually), but Zeno never reaches that point.

Zeno was not a stupid person. He knew and understood maths quite well – that’s how he could come up with this paradox. He wasn’t an evil person either. He wasn’t trying to mislead people into believing that maths is wrong. He intended to challenge people’s thinking and to test whether they know what they thought they “knew”. In other words, whether they understand the information they have.

On the other hand, we can’t as easily say the same for the demagogues who hide the most important parts of the truth and try to make the people focus on the wrong parts, in order to convince them that what they say is the truth, the whole truth and the only truth. And social sciences, or the psychology of the masses are not subjects we know nearly as well as plain old maths. Therefore, it’s easier to fall for their traps. After all, we may not get that, “But Achilles will win the race, even if I don’t know why,” kind of thought.

The Stack of Wheat

Another interesting Zeno’s paradox is the stack of wheat. If we take a single grain of wheat and put it on the ground, we have one grain. Then we add another. Then another… and eventually, we have what we can accurately describe as a stack of wheat. Right?

But in this “accurate” description we lose track of the individual grains and the amount of them. Then, can that description really be called “accurate”?

The Noise of Silence

In the same group of paradoxes, Zeno said that if we take a grain of wheat and let it fall on the ground it will make no noise. But if we take a sack of wheat and empty it on the ground, it will make noise. So, how come a bunch of silences produces noise?

That’s easy to understand. Right? The “no noise” of a single grain is not true. It does make a sound, but this sound is unnoticeable, or imperceptible. Add many of those miniscule sounds together and they become a noise we all can notice.

Unlike poor little grains of wheat, we can choose the kind of “noise” we make. And no matter how insignificant this “noise” is, if enough of these “noises” gather, it will become a significant “noise”.

But! If we really want to choose our noise, we need to be a sack of individual grains, and not a stack of wheat.

The Grains That Don’t Fall on Earth

This is not a Zeno’s paradox, but I couldn’t resist the temptation. Did you know that if you put in a sack 200 octillion grains of wheat and leave it mid-air it won’t fall on Earth?

If on average a grain of wheat has a mass of 35 mg, 200 octillion have a mass of 7·10²⁴ kg. Earth has a mass of nearly 6·10²⁴ kg (or just above that, if we include the Moon). Therefore, Earth will fall on the sack, instead.

That’s all.

Be a good grain, make a nice noise, and watch out for speedy turtles!