|An unusual and strange discussion of Zeno's paradox can be found in chapters 6, 20, and 22 of Reality Inspector, a novel about chess and computer-hacking. The appropriate excerpts are presented below. If you wish to read the story context surrounding the excerpts, go to the three linked chapters. If you have comments or questions about the ideas, please contact John Caris.|
|from chapter 6:
Then he sees a human figure close to the edge of the woods. Walking over, he notices that the person is painting, no doubt a landscape scene.
The painter turns around, brush in one hand, palette in the other. "Oh, hi." He is not too enthusiastic, but a little disconcerted about the interruption.
"I'm John Ocean. And I seem to be lost. Can you tell me what place this is?"
"I've heard of you. You're a reality inspector, aren't you?"
"Yes." John feels flustered and confused, not so much by the response of the painter but by the overall strangeness of the situation.
"How many are there?"
"The Greek who fought in the Trojan war?"
"And you've taken up painting in your retirement, I see."
"No, I'm not retired yet--not by any stretch of the imagination. I have very important work to do. This painting is part of it."
John steps up to the easel and scrutinizes the painting. It is in a realistic style. He doesn't think the brushstrokes are too good, but then not everyone must be a Rembrandt. Looking closer, he sees that it is a painting of a painting; the artist is painting himself in this meadow painting. It is a little like some of the studio paintings of Velazquez, Vermeer, and others. In the center of the canvas stands the artist painting. To the left is a tortoise and to the right is a line on the ground. The woods are in the background.
Looking away from the painting and to his left, John sees a large tortoise slowly crawling toward them. To his right John sees a long line cut in the earth. What is happening here? The painting seems to be a mirror of the actual setting.
"You have arrived at a most significant moment. I have solved Zeno's paradox. I am now making an artistic event which will allow me to beat the tortoise."
John thinks about Zeno's paradox. If the tortoise starts out one hundred yards ahead of Achilles and if he covers half the distance between himself and the tortoise each minute, will he ever catch the creature?
"Yes, I've finally figured out the solution. I've tried many approaches, but they all failed. Ha, I even tried riding on the tortoise's back. I figured that when it was about to reach the finish line I would just reach out my hand a few inches in front of the tortoise's nose. But that did not work either. The blasted animal stretched its neck farther than I thought possible. Anyway, now I have discovered an infallible system for winning."
"What is it?" John is intrigued. Zeno's paradox is on par with squaring the circle. Both are mind problems that lead to intriguing ideas but never a solution.
"First, let me set the record straight. During the whole history of this event I have never lost the race. True, I have never won it either. But the tortoise has never yet beaten me across the finish line."
"I thought you lost because you never reached the tortoise, only halving the distance down to infinity."
"Ah, ha! That's the slanderous propaganda which has ruined my good name. All the world applauded me for defeating Hector outside the walls of Troy. But along comes a Mr. Nobody who gains immortality by smearing my good name. Who would remember Zeno if it wasn't me? What if he had used Jane and the tortoise instead? Who would remember? So, I must erase the evil deed thrust upon me and my tribe."
John is familiar with Achilles' background and knows that he is given to extended emotional outbursts. "What is the truth?"
"About me and the tortoise? The opinion shapers said that I tried to catch the tortoise. Now, why would I, a Greek warrior born of a goddess, want a tortoise? You see how they belittle me. Horses are my thing. Back home my excellent herd of horses is well-known."
"So you never raced with a tortoise?"
"See how the truth is distorted! A race is something else. I am also renowned for my running, bouncing legs. Did I not thank the gods, after killing Hector, for the spring in my knees?"
"The racing part is true, then?"
"Yes, certainly. Racing is my weakness--not my heel as some would have it."
"You did race the tortoise?"
"Of course. The prize was a golden trophy. The gods devised the race, and of course I couldn't refuse. Look at what happened to poor Paris when he was asked to judge a beauty contest. When the gods ask, you can't refuse. I knew I was in a pickle, that there was some trap. But now I will cleanse the dirt from my good name. Achilles will shine again."
"Well, why hasn't the tortoise won yet, after all those years?"
Achilles draws himself up, looking scornfully at the slowly crawling creature. "I have not lost and the tortoise has not won because it has not crossed the finish line."
"After all those years? Looks like it is getting pretty close right now."
"Ha! It will never cross the finish line! Never in a red moon!"
The strangeness of the place has obviously affected John. He is caught by Achilles' story and feels sympathetic toward him even with all his arrogance. The tortoise is only about one hundred feet from the finish line and should cross it in an hour or so. "Why won't the tortoise ever cross the finish line?"
"Elementary, elementary. Because the finish line is just that--a line. It has only one dimension, its length; it has no width. The tortoise, which is moving towards its non-dimensional side, will never cross it."
"I don't think I follow."
"Simple. A line has only one dimension. But to win the race one of us must cross the line. (One of us must actually move from point A through an infinite number of subpoints to point B.) To reach the other side of the line means crossing a something without a spatial dimension. Now there's a paradox, if you want one. How can one ever reach the other side?"
"I think I see. The tortoise can reach the line but never cross over it to the other side. For it would have to move through a nonspace."
"Right. So once I proved that the tortoise would never win the race, I took heart. I now have an infinity of time to work out a solution. And I have. I found the solution. The tarnish will be wiped clean from my good name."
"What is the solution?"
"I have worked out a system for crossing non-dimensional space. The basic idea is subjectivity. We humans all have a subject, a consciousness that experiences the world of objects. It is this subjectivity that shapes the basic equation. Let me give you an example. Let me show you directly how this subjectivity works." (for Achilles' example of subjectivity)
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