Herewith a culling from my personal archive - an illustrative fable, borrowing heavily but unwittingly from Lewis Carroll's What the Tortoise Said to Achilles, which I didn't actually encounter until later in a linear-time paradigm. The tortoise in my story advances three of the "immeasurably subtle and profound" (thank you, Bertrand Russell) paradoxes of Zeno of Elea, an ancient Greek philosopher hailed by Aristotle as the inventor of the dialectic. It is perhaps appropriate that this fable takes a dialectical form.
"Let us race," declaimed the tortoise with a loftiness belying his low center of gravity.
The hare left off appreciating the feel of the wind cool against his long ears and regarded the tortoise with leporine indulgence.
"You want... us... to race?" clarified the hare, not unkindly he thought.
The tortoise nodded, an exercise that took long enough that the hare had time to watch a small cloud scud across the sky overhead. It looked like a nice head of lettuce, he thought. His stomach rumbled: his was an active metabolism.
"Each other?" The hare was a thorough if not a very bright animal.
Again the tortoise nodded, and the hare occupied himself scratching at his flank.
"But I'm much faster than you," explained the hare as sensitively as he could. "You racing me would be just pointless."
"Oh, it doesn't matter how fast we go, you know," remarked the tortoise pleasantly. "It's a question of how far."
This seemed a rather odd view of racing to the hare, who felt that he knew a thing or two about racing after his serial adventures with the farmer's hounds; but he kept his counsel, because he felt rather sorry for the tortoise.
"However," the latter was now saying, "you do make a good point. Would you agree to give me a head start?"
"Hang on," said the hare. "A head start to where? A race has to have a start and a finish."
"It does, does it?" The tortoise seemed amused. "Let's suppose a race goes from some place to some other place that's different. Then to move between those places, we can surely agree that at any moment one would have to be moving either from a spot or to a spot between them. Correct?"
The hare thought about this. His nose twitched as he worked it out, and this time the tortoise had the luxury of observing a cloud scud by overhead.
"I think so," said the hare at length.
"And," continued the tortoise carelessly, "at any moment one would have to be in a spot between the start and finish of the race, yes?"
The hare chewed pensively. His feet were much happier with racing than his brain.
"That seems right," he ventured.
"Well then," said the tortoise, "in that case, no matter where the start and finish are, or when we consider one's position, one has to either be in the place where one is and the place one just came from, or the place where one is and the place one is headed."
The hare's ears drooped.
"That can't be right," he said miserably.
"Oh it can," said the tortoise with placid contentment. "Unless of course movement is continuous and it's the instant that's an illusion. But then there's no reason to suppose that it means anything to say that at any moment anything's in any particular place. One might as well say one starts the race at the finish, since no moment's any more real than any other."
The hare hopped about unhappily in a circle. "You're confusing me," he said.
"Don't worry about it," enjoined the tortoise earnestly. "Let's pretend that it does matter and we actually do occupy some coordinate in spacetime."
The hare's eyes were glazing.
"I'm going to walk over here," said the tortoise, starting out on that journey, "and you can wait there."
He sauntered off, while the hare tried to work out if he had time for a snack and whether he might actually already be having one. The tortoise had upset his delicate sensibilities, and he was sorely in need of a radish.
Some minutes later, the tortoise hailed him, from a distance of some ten feet away.
"That should be enough of a headstart, don't you think?"
The hare looked doubtfully at the space between them.
"It doesn't seem very far," he suggested timidly.
"Oh but quite far enough," returned the tortoise. "Because now to win the race, you'll have to pass me. And to pass me you'll have to reach me."
"Yes..." said the hare, with the uneasy feeling that he was about to get a migraine.
"Of course, to reach me, you'll first have to get halfway here."
The hare thought about this. "Of course," he said, but he didn't sound certain.
"And naturally, to get there you'll have to go halfway first."
"... Naturally ..."
"And to get there you'll have to go halfway as well."
"I suppose so..."
"And in fact no matter how many times we divide that distance up, you'll always be able to divide it up one more time. For you to reach me, you'll have to first travel an infinitely short distance."
The hare extended a paw towards the tortoise, and hesitated, and withdrew it.
"Can't I just -?"
"Not logically," said the tortoise firmly. "In any case, even if you could start out, and even if having started out you could move from place to place... there's still the problem that you'll never catch up with me, let alone get past me."
The hare felt sure there was something badly wrong with this, but he couldn't put his paw on it.
"Because," explained the tortoise patiently, "even though I travel much slower than you, I am travelling at the same time as you are and on the same journey, albeit from a different start point. We're going in the same direction, is what I mean to say."
"But," blurted out the hare, "you're only ten feet away! I can cover that distance in no time!"
"On the contrary," sniffed the tortoise. "You can only cover that distance - or any distance - in some time. And in that time, I can cover some distance too. Oh, not nearly as much of course... but enough."
"You'd only go..." the hare struggled with mental arithmetic, "one foot."
"Correct!" said the tortoise delightedly. "And of course you'd cover that distance easily as well... but I'd have moved on..."
"... a tenth of a foot..." whispered the hare miserably.
"Quite so!" went on the tortoise. "And I know you'd cover that distance even quicker... but I'd have moved on another hundredth of a foot. Oh, you'd get very close," he smiled, "but you'd never catch me."
The hare scratched his other side. The smell of radishes was overwhelming.
"You win," he sighed, and hopped past the tortoise to get some food.
The hare left off appreciating the feel of the wind cool against his long ears and regarded the tortoise with leporine indulgence.
"You want... us... to race?" clarified the hare, not unkindly he thought.
The tortoise nodded, an exercise that took long enough that the hare had time to watch a small cloud scud across the sky overhead. It looked like a nice head of lettuce, he thought. His stomach rumbled: his was an active metabolism.
"Each other?" The hare was a thorough if not a very bright animal.
Again the tortoise nodded, and the hare occupied himself scratching at his flank.
"But I'm much faster than you," explained the hare as sensitively as he could. "You racing me would be just pointless."
"Oh, it doesn't matter how fast we go, you know," remarked the tortoise pleasantly. "It's a question of how far."
This seemed a rather odd view of racing to the hare, who felt that he knew a thing or two about racing after his serial adventures with the farmer's hounds; but he kept his counsel, because he felt rather sorry for the tortoise.
"However," the latter was now saying, "you do make a good point. Would you agree to give me a head start?"
"Hang on," said the hare. "A head start to where? A race has to have a start and a finish."
"It does, does it?" The tortoise seemed amused. "Let's suppose a race goes from some place to some other place that's different. Then to move between those places, we can surely agree that at any moment one would have to be moving either from a spot or to a spot between them. Correct?"
The hare thought about this. His nose twitched as he worked it out, and this time the tortoise had the luxury of observing a cloud scud by overhead.
"I think so," said the hare at length.
"And," continued the tortoise carelessly, "at any moment one would have to be in a spot between the start and finish of the race, yes?"
The hare chewed pensively. His feet were much happier with racing than his brain.
"That seems right," he ventured.
"Well then," said the tortoise, "in that case, no matter where the start and finish are, or when we consider one's position, one has to either be in the place where one is and the place one just came from, or the place where one is and the place one is headed."
The hare's ears drooped.
"That can't be right," he said miserably.
"Oh it can," said the tortoise with placid contentment. "Unless of course movement is continuous and it's the instant that's an illusion. But then there's no reason to suppose that it means anything to say that at any moment anything's in any particular place. One might as well say one starts the race at the finish, since no moment's any more real than any other."
The hare hopped about unhappily in a circle. "You're confusing me," he said.
"Don't worry about it," enjoined the tortoise earnestly. "Let's pretend that it does matter and we actually do occupy some coordinate in spacetime."
The hare's eyes were glazing.
"I'm going to walk over here," said the tortoise, starting out on that journey, "and you can wait there."
He sauntered off, while the hare tried to work out if he had time for a snack and whether he might actually already be having one. The tortoise had upset his delicate sensibilities, and he was sorely in need of a radish.
Some minutes later, the tortoise hailed him, from a distance of some ten feet away.
"That should be enough of a headstart, don't you think?"
The hare looked doubtfully at the space between them.
"It doesn't seem very far," he suggested timidly.
"Oh but quite far enough," returned the tortoise. "Because now to win the race, you'll have to pass me. And to pass me you'll have to reach me."
"Yes..." said the hare, with the uneasy feeling that he was about to get a migraine.
"Of course, to reach me, you'll first have to get halfway here."
The hare thought about this. "Of course," he said, but he didn't sound certain.
"And naturally, to get there you'll have to go halfway first."
"... Naturally ..."
"And to get there you'll have to go halfway as well."
"I suppose so..."
"And in fact no matter how many times we divide that distance up, you'll always be able to divide it up one more time. For you to reach me, you'll have to first travel an infinitely short distance."
The hare extended a paw towards the tortoise, and hesitated, and withdrew it.
"Can't I just -?"
"Not logically," said the tortoise firmly. "In any case, even if you could start out, and even if having started out you could move from place to place... there's still the problem that you'll never catch up with me, let alone get past me."
The hare felt sure there was something badly wrong with this, but he couldn't put his paw on it.
"Because," explained the tortoise patiently, "even though I travel much slower than you, I am travelling at the same time as you are and on the same journey, albeit from a different start point. We're going in the same direction, is what I mean to say."
"But," blurted out the hare, "you're only ten feet away! I can cover that distance in no time!"
"On the contrary," sniffed the tortoise. "You can only cover that distance - or any distance - in some time. And in that time, I can cover some distance too. Oh, not nearly as much of course... but enough."
"You'd only go..." the hare struggled with mental arithmetic, "one foot."
"Correct!" said the tortoise delightedly. "And of course you'd cover that distance easily as well... but I'd have moved on..."
"... a tenth of a foot..." whispered the hare miserably.
"Quite so!" went on the tortoise. "And I know you'd cover that distance even quicker... but I'd have moved on another hundredth of a foot. Oh, you'd get very close," he smiled, "but you'd never catch me."
The hare scratched his other side. The smell of radishes was overwhelming.
"You win," he sighed, and hopped past the tortoise to get some food.
This concludes, only slightly late depending on how one measures these things, my personal A to Z challenge. I want to reiterate thanks to everyone who helped me along the way, and apologies to all of them, and more besides, for not getting out there and commenting more in return. It is my earnest hope that I'll be redressing that balance in coming days.
I also have a mailbox to compile, which I'll be getting around to hopefully tomorrow. I've been honored with an award, from a blogger I appreciate greatly but haven't done much to show that lately, so that needs to go up as well. Busy, busy, busy! PLUS the inaugural Mojo's Monthly Mindbender (which for May will have a musical and mathematical bent, since I might as well milk the alliteration for all it's worth) and perhaps the first tentative steps towards the Mojofesto, in preparation for November 2012. I eagerly await the inspirations and communications the blogosphere has in store for me over the coming thirty days - thanks for the ride so far, guys and gals!
I also have a mailbox to compile, which I'll be getting around to hopefully tomorrow. I've been honored with an award, from a blogger I appreciate greatly but haven't done much to show that lately, so that needs to go up as well. Busy, busy, busy! PLUS the inaugural Mojo's Monthly Mindbender (which for May will have a musical and mathematical bent, since I might as well milk the alliteration for all it's worth) and perhaps the first tentative steps towards the Mojofesto, in preparation for November 2012. I eagerly await the inspirations and communications the blogosphere has in store for me over the coming thirty days - thanks for the ride so far, guys and gals!
