3
Achilles and Zeno
One day, as Achilles was idly picking at his toenails with the end of his spear, an old Greek philosopher sidled up to him. "Wanna race?" said the ancient. Had he not been sitting on the ground, Achilles would have looked him up and down. Since he was sitting on the ground, Achilles contented himself with merely looking him up. "You must be joking," laughed Achilles.
"No," said the old man, "I'm his brother, Fred King," and laughed much longer and much laster than Achilles. "Sorry, couldn't resist it," he finally managed to get out. "It really is true what they say about the old ones being the best. My name is really Zeno. Anyway, I propose that you and I race twice, once to see how much faster you are. Then we can work out a fair start for me and have a proper second race where it's anyone's guess who will reach the finishing line first."
"Okay," said Achilles, thinking of how he could take it easy on the first stage and so win easily on the second.
So, Achilles and Zeno lined up. On Zeno's count of three, Achilles shot away, followed at a more leisurely pace by Zeno. The latter just managed to reach the exact halfway point by the time Achilles had finished.
For the second leg, Zeno positioned himself at the halfway point. On Achilles' count of three, he set off at a much greater speed. Being a pretty good Greek philosopher, he had reasoned that Achilles would take it easy during the first race and so had done likewise. However, Achilles was himself no mean philosopher and had quite rightly reasoned that Zeno would have reasoned that he, Achilles, would take it easy during the first stage. Following the logic of this reasoning, he had taken it twice as easy. However, being only an amateur philosopher, he had failed to reason any further than that. Had he done so, he would have realised that Zeno (who was not just a pretty good Greek philosopher but that more awesome figure - a pretty good professional Greek philosopher), had worked every step out very carefully before even approaching Achilles.
The second race was obviously going to be a close finish. Although Achilles had twice the distance to cover that Zeno had, he was running twice as fast; for his part, Zeno was running at only half the speed of Achilles but, fortunately, he had but half the distance to run. The finishing line grew ever closer.
The finishing line drew ever closer.
The finishing line drew ever closer.
"Erm...," said Achilles, as the finishing line drew ever closer, "Shouldn't we have finished this race by now?"
"I think so," panted Zeno, as the finishing line drew ever closer. "In fact it's beginning to worry me." He turned to Achilles who was very nearly at his side, "You must have guessed by now that I didn't challenge you to this race just for a laugh. I'm trying to prove a philosophical point."
As the finishing line drew ever closer, Achilles, still not breathing too heavily, replied, "I thought as much. You didn't seem to be the athletic type. But what philosophical point are you attempting to prove?"
"That with a head start, you'll never catch me!" retorted Zeno as the finishing line drew ever closer.
Achilles thought about it for a few moments while the finishing line drew ever closer. "Ah!" he said, "The drachma's just dropped! By the time I'd reached your starting point - halfway for me - you'd already reached your halfway point; and when I reached that point, you'd gone halfway from there to the finishing line."
"That's right," gasped Zeno, who was now finding the exertions somewhat of a strain. "And it works like that for every stage. Every time you reach where I was, I've gone a little further. Although you're continually getting closer to me, you can never catch me."
"Like the finishing line, do you mean?" said Achilles.
"I hadn't thought of that," said Zeno, "But now you come to mention it, I can see what you mean. Since you are running twice the distance that I am but doing it twice as fast then you and I should reach the finishing line at the same time - the exact moment that you catch up with me. The problem with this is that I have just proved philosophically that you can never catch up with me."
"So," said Achilles, who was beginning to look decidedly uneasy and had found his mouth going terribly dry during Zeno's last speech, "Although the finishing line is drawing ever nearer, we can never actually reach it?"
"That seems to be about the size of it," said Zeno, somewhat incoherently for his mouth had also gone dry with fear at the prospect of eternally running towards an unreachable finishing line.
At that moment a diminutive figure strode past both of them. His nose and attention were buried in a great leather-bound Almagest apparently bookmarked by strands of a long, grey, straggly beard. From behind them he came; Past the finishing line he stepped.
"Excuse me!" Zeno shouted after him, "Do you think you could help us?"
Joachim (for it was he, our hero) looked up startled, having failed to realise that the two figures in a running pose that he'd just passed were not statues but living beings. "Certainly," Joachim replied with a precision of meaning few could match, "What is your problem?"
Zeno set about trying to explain but due to the dryness of his mouth and the fact of his shaking from sheer terror, his words made not a great deal of sense. Nevertheless, and luckily for him and Achilles, Joachim was an adept at reading entire histories solely from a simple observation of emotions. He cut Zeno's exposition short with a wave of his hand, "No need to go on, dear fellow," he smiled, "It's perfectly obvious that you've got yourselves caught up in the Einstein Dry Fear continuum. You'll be out of it in no time - relatively speaking, of course!"
"Your problem lies in the mistaken assumption that time is a fourth dimension of the same order as the three spatial dimensions. Thus, as Achilles takes increasingly shorter steps through space to reach the point where Zeno was, he does so in correspondingly shorter periods of time. If space and time are of the same order, and if both are infinitely divisible, then Achilles and Zeno will run forever, never growing old as the world ages and crumbles about them. The noticeable effects of this are dryness of the throat and fear. What you have to realise is that these causes and effects are actually relative - that the increasingly shorter spatial and temporal distances can just as readily be seen as the effects that are caused by the dryness and fear."
"Cut the lecture!" Achilles shouted, "And get us out of here!"
"Oh, I can't do that," smiled Joachim, "You have to do it for yourselves."
"Then please tell us how," Zeno pleaded.
"It's really quite easy," said Joachim, "All you need is something to wet your whistle and calm your nerves. Just keep shouting 'Einstein Dry Fear'. Eventually, your dry mouths will make the 'F' sound like 'B' and a large blond-haired girl will come over with a lager for us all."
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This is one of the Tales of Joachim |
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Ink Amera |
(C) David 1/9/2007 |
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