Zeno’s Paradox

Zeno’s paradox is my favourite conversation filler. The beauty of the paradox is how universally simple it is to explain it to people from different ages and backgrounds. Here’s how it was originally told:

The Tortoise challenged Achilles to a race, claiming that he would win as long as Achilles gave him a small head start. Achilles laughed at this, for of course he was a mighty warrior and swift of foot, whereas the Tortoise was heavy and slow.

“How big a head start do you need?” he asked the Tortoise with a smile.

“Ten meters,” the latter replied.

Achilles laughed louder than ever. “You will surely lose, my friend, in that case,” he told the Tortoise, “but let us race, if you wish it.”

“On the contrary,” said the Tortoise, “I will win, and I can prove it to you by a simple argument.”

“Go on then,” Achilles replied, with less confidence than he felt before. He knew he was the superior athlete, but he also knew the Tortoise had the sharper wits, and he had lost many a bewildering argument with him before this.

“Suppose,” began the Tortoise, “that you give me a 10-meter head start. Would you say that you could cover that 10 meters between us very quickly?”

“Very quickly,” Achilles affirmed.

“And in that time, how far should I have gone, do you think?”

“Perhaps a meter—no more,” said Achilles after a moment’s thought.

“Very well,” replied the Tortoise, “so now there is a meter between us. And you would catch up that distance very quickly?”

“Very quickly indeed!”

“And yet, in that time I shall have gone a little way farther, so that now you must catch that distance up, yes?”

“Ye-es,” said Achilles slowly.

“And while you are doing so, I shall have gone a little way farther, so that you must then catch up the new distance,” the Tortoise continued smoothly.

Achilles said nothing.

“And so you see, in each moment you must be catching up the distance between us, and yet I—at the same time—will be adding a new distance, however small, for you to catch up again.”

“Indeed, it must be so,” said Achilles wearily.

“And so you can never catch up,” the Tortoise concluded sympathetically.

“You are right, as always,” said Achilles sadly—and conceded the race.

zeno_achilles_paradox

[ from  https://upload.wikimedia.org/wikipedia/commons/6/66/Zeno_Achilles_Paradox.png ]

The Tortoise claims that if he is given a head start Achilles will  never be able to catch up with him. Why? By the time Achilles reaches the point where the Tortoise started from, some time would have elapsed and in that time the Tortoise would have moved a tiny bit ahead. Now the Tortoise is again some distance ahead of Achilles and the same argument can be repeated. If this argument was true, Achilles would never be able to catch up with the Tortoise. But we know from experience that Achilles would definitely overtake the Tortoise as he is faster than the tortoise. Therein lies the paradox that has puzzled people for centuries.

I like to annoy my friends by bringing up this paradox for discussion. Most of them are stunned by how correct the argument sounds in spite of knowing there must be something wrong in the argument. Some of them have heard this before and say that the resolution of the paradox has something to do with discretization of space-time. However, in the form the above proof is presented it doesn’t really depend on discretization explicitly.

There are two informal ways how I think we can resolve the paradox (which are not necessarily new or unique):

  1. We claim we know the speed of the Tortoise and Achilles with the certainty that it is not zero. If their speeds are not zero it becomes difficult to pinpoint where each of them is in space. If we can say for sure a given object is at a given point in space, it must be at rest. If it is moving, it is impossible to say that the object is at a particular point in space. This is related to the Heisenberg’s Uncertainty Principle. If we know the speed of the Tortoise or Achilles with uncertainty that tends to zero, the uncertainty in determining the position of either one of them increases to infinity. That is where the fallacy creeps in when trying to analyze the problem by looking at points in space to mark the position of the Tortoise and Achilles.
  2. Suppose our measurements of where both Achilles and the Tortoise take some infinitesimally small but non-zero time. If you were taking a video this measurement time is determined by the frame rate. At a point in time when the positions of the Tortoise and Achilles are almost the same. The distance which Achilles would have moved by the time we finish making our measurement would be greater than the distance between Achilles and the Tortoise because the Tortoise is slower than Achilles. This makes me wonder about how we try to judge which horse won the race by looking at the video. May be if we had seen the video a fraction of a second later, the horse that came second would have won. Recent advances in camera technology using which you can capture 20 billion frames per second should be helpful.

It’s amazing how such a simple thought experiment tugs at our very naive understanding of space and time.

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