A Tribute to Zeno of Elea.

I remember back when I used to take Remedial Math classes for struggling Engineering students. I was trying to explain Zeno’s Paradoxes to someone, to illustrate a Convergent Infinite Series. I wasn’t making any progress with this kid. When I thought I had explained about half of what I was trying to explain, he fell over backwards and asked me to repeat myself.

I thought I was done a dozen times, but his comprehension just split again without warning every time.

“How is this possible?! How does one get anywhere?!” He even managed to enunciate the exclamation marks as he laughed his way through this blasphemy.

He was laughing at one of the most famous quandaries of human philosophy – Zeno’s Paradox of Dichotomy. What this paradox postulates is that a man traveling from point A to point B must first reach the halfway point between A and B. To reach this halfway point, he must first reach the halfway point to that halfway point, ad infinitum. This is where the phrase, ‘So close, yet so far,’ was born. A man doesn’t even begin to move, because he has to reach the halfway point to moving itself. However, the point is that men do move. Therein lies a problem that vexed free thinkers for over two millennia. In the face of that, this boy really wasn’t special.

A man can never even move.
Hector charges Achilles. Before he can get to Achilles, he must get halfway there. Before he can get halfway there, he must get a quarter of the way there. Before traveling a fourth, he must travel one-eighth; before an eighth, one-sixteenth; and so on. This description requires one to complete an infinite number of tasks, which Zeno maintains is a mathematical impossibility.

He didn’t seem to want to understand the subtleties of the calculus or geometric series involved in my proof, nor was he excited about the myriad Greek symbols and integrals drawn on the page. Sitting like ducks in a row, these vestigial remains of great civilizations and minds had a beauty of their own.

In my mind, the ducks were swimming for shore. No matter how hard their little feet rowed under the placid water, they never got closer. They got slower and slower and slower until it seemed like the world itself had come to a halt. Even the ripples they sailed on crested and never settled. It was a tiny surfer’s greatest dream. The trees had grown heavy with age. The bees and the birds labored with their wings until they stopped beating and yet they hung suspended in the air from invisible strings. The puppeteer was asleep, relaxed, blissful. The wind was caught in the jar of the world, and its inaudible melodies were trapped as though in amber. In this world, there is magic that exists outside Art.

No one knows what it is like to live in the moment until they live in a moment frozen in time. Lovers know this, and so do people who have almost died. The frozen moment is a reminder for everyone to slow down. A man is dry brush waiting for Life’s fire to catch, but the flames will never catch when he’s moving too quickly.

People mistake moving too quickly for living the moment. They think that Carpe Diem is about doing all the things possible in the mad blur of life. Well, it is, but only because a man takes every moment for itself. He looks at it square in the eye. He notices it for what it is and by understanding the beauty of it, truly lives it. That is Zen.

Zeno of Elea
Zeno of Elea (ca. 490 BC? – ca. 430 BC?)

I am convinced that before Carpe Diem, it was Zeno that propagated this philosophy. Every single one of his thought experiments imagines the world as it were in a single moment – with no concern for anything but the very immediate, yet still dwindling future. The end was never in his sights. His life was dedicated to the the journey to that end. He divided space and time into smaller and smaller halves until he visualized moments outside time, instants that are outside this mortal coil itself. That is why his ideas are paradoxes. They are not of this world. Zeno’s Paradoxes are a journey to a world where the things that seem so important here mean nothing.

Men can fly in that journey merely by the simple act of jumping. I would never reach the halfway point as my feet left the ground. I would stay in the air forever. I always imagined myself jumping in front of those high speed cameras on Time Warp. On playback, I’d have them slow down and stop me at the apex of flight. Then, I would see magic. At that moment, Zen would be all over my face as it was frozen on a TV screen, like animal parts in Aspic.

This journey toward Zen isn’t about the Math. It is about the point that the Math doesn’t even matter. Even though Calculus eventually explained the paradox in terms of a Convergent Infinite Series, Zeno’s Paradoxes are a reminder that life itself is the Convergent Infinite Series. It is filled with approximations and the lessons of common sense because nothing can be quantified exactly. It is the sum of infinite moments frozen in time. More importantly, it tells us that despite the infinite number of these experiences, we always get to the end of the trip. Always.

This boy was laughing at that.

Instead of trying to repeat myself the thirteenth time, I grabbed him by the hair and banged his head into the page twelve times in the hope that some of it would seep through his skull. The boys head passed the halfway point and reached the page a dozen times with force. Before Newton, this is how Zeno’s paradoxes were solved. Zeno’s Arrow Paradox was solved by men shooting arrows into other men. Even the indisputable logic of Greeks could not argue with a shaft in the torso. No matter what the arrows did at finite points of stopped time, no matter how still they were, they always found themselves in the spleens of other men.

Needless to say, by the third hit, the boy had closed his eyes and lost consciousness. It’s not my temper. It’s not my fault. I just cannot stand people who cannot appreciate beauty. It still boggles my mind that some people are about as deep as a puddle of rainwater.

Now I have red spots on my piece of paper and a sleeping man with a Sigma scarred across his bloodied forehead, chained helplessly to a pipe in the basement. He might not be able to seize the day, but he can sure seize the pipe. He has nothing but time to think about what I was trying to say. He didn’t understand Calculus, so he got common sense instead. It is a learning experience. When I finally do let him go, his grades are bound to get better.

Despite what most people think, Math makes for very interesting situations.


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