My granddaughter Katelyn is 8.

Today, I had lunch with her at her school. The mascot of the school is a Bronco, a wild half-tamed horse. A Bronco is very fast. A tortoise is a turtle. A turtle is not fast. In a race, who should win ?

Before you answer, let’s look at that number 8 again.

Eight is a very interesting number. If you turn 8 on its side, it’s the symbol for infinity – this symbol ∞. See, it’s a side-wise 8.

Infinity and its symbol, ∞, refer to and look like something without any limit. Start anywhere on the side-wise 8 or infinity symbol ∞ and start going, and you’ll come back to where you started, and just keep going, and going, and going, forever. That’s infinity, no end. Our word infinity comes from the Greek word apeiros, which mean endless.

Back to that race between the tortoise and the Bronco, do you think it could go on forever, be endless, be infinite?

Zeno thought so.

Zeno of Elea was a Greek philosopher who lived in southern Italy about 2,500 hundred years ago. He’s best known for his paradoxes, Zeno’s paradoxes.

A paradox is a statement that seems absurd (untrue), but it may express a possible truth. The word paradox is composed of “para” for contrary and “dox” for opinions. Where contrary opinions both seem to be true, you have a paradox.

Zeno was very good at paradoxes. Perhaps his most famous is “Achilles and the Tortoise.” Now, Achilles is a famous Greek hero of the Trojan wars, and he was very strong and a very quick runner. Here is what Zeno said about a very quick runner like Achilles:

“In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point where the pursued started, so that the slower will always hold the lead.”

That’s absurd, we say loudly. It’s simply untrue. The faster runner always wins the race. I saw it on TV. How can you say that, Mr. Zeno? It’s a paradox. I hold a contrary opinion.

And, here is what Zeno would say to you.

I’ll give you Achilles for your team’s runner – we’ll call your team the Bronco Team. From the Trojan Wars, you know Achilles is fast. I’ll take the tortoise for my team – the Zeno Team. All I ask, for the Zeno Team, is that you, the Bronco Team, give our turtle, the tortoise, a head start of 100 yards. Agreed?

Sure, why not, this will be over in a blink, a cake walk.

The gun goes off. The turtle takes his time and finally reaches the 100-yard mark. Our Achilles takes off like a hare, a very fast rabbit, and reaches that 100-yard mark in no time at all. But wait, the slow tortoise has now advanced a little farther down the race track. No problem, our hare-like Achilles zips to the spot. But wait, the turtle is now a little farther ahead. “I’ll catch him,” our rabbit-fast Achilles shouts and races to where the tortoise just was. But, the turtle is not there. He’s moved a little more ahead. And so it goes on and on to infinity. There are an infinite number of points Achilles must reach where the tortoise has already been. In this very logical and ordered way of looking at this race, Achilles never overtakes the tortoise.

“In real life, it is not so!” you yell. “It’s not true.”

Maybe there is more than one truth here, and they just appear to be contradictory.

In real life, Achilles and the Bronco Team win. High fives all around to the Bronco Team members!!! We won!!!!

But wait, in the real life of mathematics (numbers) and perhaps philosophy (fundamentals), the Zeno Team also wins, because, in mathematics, once you start at Point A and start dividing the distance in half from Point A to Point B, you can keep dividing the next half in half forever, for infinity. Those halves get infinitesimally smaller as you go, but, in concept, at least, you never stop dividing, and you never reach Point B – mathematically speaking.

I think that was Zeno’s point: If you look at the same race differently, you can get different results, both of which are true. Of course, the real life hare beats the tortoise, unless he takes a nap, which is what happened in Aesop’s Fable of “The Tortoise and the Hare” – that’s the real life race perspective. But, just as “of course,” in mathematical parlance, the tortoise with the lead can never be overtaken by hare – the tortoise wins.

Zeno was a smart guy. He figured out that sometimes you have to confuse people with two truths to help them see that both are true. That’s a true paradox.

And that’s what happens when you have lunch with a bunch of 8-year old Broncos.

Will wonders never cease?

Hope not,

Grandpa Jim