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Zenos-Paradox-of-the-Tortoise-and-Achilles

 
 
Reply Sat 8 Apr, 2017 06:41 pm
http://platonicrealms.com/encyclopedia/Zenos-Paradox-of-the-Tortoise-and-Achilles

Zeno’s Paradox of the Tortoise and Achilles

Zeno of Elea (c. 450 BCE) is credited with creating several famous paradoxes, and perhaps the best known is the paradox of the Tortoise and Achilles. (Achilles was the great Greek hero of Homer’s The Iliad.) It has inspired many writers and thinkers through the ages, notably Lewis Carroll (see Carroll’s Paradox) and Douglas Hofstadter, both of whom wrote expository dialogues involving the Tortoise and Achilles.

The original goes something like this:

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’s Paradox may be rephrased as follows.

Suppose I wish to cross the room. First, of course, I must cover half the distance. Then, I must cover half the remaining distance. Then, I must cover half the remaining distance. Then I must cover half the remaining distance…and so on forever. The consequence is that I can never get to the other side of the room.

What this actually does is to make all motion impossible, for before I can cover half the distance I must cover half of half the distance, and before I can do that I must cover half of half of half of the distance, and so on, so that in reality I can never move any distance at all, because doing so involves moving an infinite number of small intermediate distances first.

Now, since motion obviously is possible, the question arises, what is wrong with Zeno? What is the "flaw in the logic?" If you are giving the matter your full attention, it should begin to make you squirm a bit, for on its face the logic of the situation seems unassailable. You shouldn’t be able to cross the room, and the Tortoise should win the race! Yet we know better. Hmm.

Rather than tackle Zeno head-on, let us pause to notice something remarkable. Suppose we take Zeno’s Paradox at face value for the moment, and agree with him that before I can walk a mile I must first walk a half-mile. And before I can walk the remaining half-mile I must first cover half of it, that is, a quarter-mile, and then an eighth-mile, and then a sixteenth-mile, and then a thirty-secondth-mile, and so on. Well, suppose I could cover all these infinite number of small distances, how far should I have walked? One mile! In other words.

What is wrong with the logic used above? (My comment Alan McDougall)
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layman
 
  0  
Reply Sat 8 Apr, 2017 06:54 pm
Infinite division has nothing to do with it. If I take one step, I have already traversed an "infinite" number of points. And I can do that again. And again....

The problem is with thinking that "infinity," as either a mathematical or intuitive concept, has any "real" meaning.

Take a yardstick, for example. In theory, I could subdivide that into an "infinite" number of sub-pieces. If I do that, and then add them all up, what do I get? An infinite number? No, I get one yard.

Mental conceptualizations have no physical effect on the yardstick whatsoever.
layman
 
  -1  
Reply Sat 8 Apr, 2017 07:10 pm
@layman,
Edit: I should have said "infinite distance," not "infinite number," but I'm sure the point is clear enough, either way.
0 Replies
 
chai2
 
  0  
Reply Sat 8 Apr, 2017 08:14 pm
bait and switch making the unwary to concentrate on the meaningless aspect, and not the end result.

Zeno was a bit of a dunder head apparantly.

I'll bet he was the one that came up with the 3 hotel guests and the 25 dollar hotel bill.

Next case.
layman
 
  0  
Reply Sat 8 Apr, 2017 10:33 pm
@layman,
Quote:
Mental conceptualizations have no physical effect on the yardstick whatsoever.


But don't NEVER run around tryna tell Max that, eh?
0 Replies
 
ekename
 
  0  
Reply Sun 9 Apr, 2017 03:00 am
@Alan McDougall,
Once upon a time there was a believer, let's call him Alan.

He knew that half of all that he thought about the existence of a particular imaginary super being was infantile drivel so he dismissed it as absolute rubbish.

He knew that half of what was left of what he thought was also infantile drivel so he dismissed it as absolute rubbish.

He knew that half of what was left of what he thought was also infantile drivel so he dismissed it as absolute rubbish.

This went on for what seemed like an eternity until the infantile drivel that was dismissed as absolute rubbish was piled so high that the imaginary super being couldn't jump over it.

Did Alan resile from this rodomontade for his own back?

No he did not, instead he redoubled his proselytising.
0 Replies
 
farmerman
 
  1  
Reply Sun 9 Apr, 2017 05:15 am
@chai2,
Thats why philosophy is such a grand waste of an otherwise useful mind. It is mostly masturbatory self delusion posing as eunoia.

This "half step to the wall" is born of the same crapola as the "Irreducible complexity" BS that IDers use as their hand to demonstrate that they are presenting Creationism as real "science"
chai2
 
  1  
Reply Sun 9 Apr, 2017 09:30 am
@farmerman,
This is the weirdest thing farmer.

Last night I settled down to watch Nymphomaniac Volume 2, the directors cut (which if anyone thinks watching Volume 1 was a fun romp, a walk in the park, you're wrong), and the very first thing Seligman brings up is this Zenos Paradox.

I was like "I know this man" like Tommy Chong from Cheech and Chong.

I'll bet Alan was watching Nymphomaniac Volume 2 before he posted this.

That is one dark mo-fo of a movie man.

0 Replies
 
chai2
 
  1  
Reply Sun 9 Apr, 2017 09:35 am
@farmerman,
I had to look up that word eunoia

"Eunoia is the shortest English word containing all five main vowel graphemes."

Fun experiemnt. Ask someone to give you the name of a tree with all 5 vowels in it.

I guaran-damn-tee you the first thing they will say is "elm"

Like, wtf?

Fil Albuquerque
 
  1  
Reply Sun 9 Apr, 2017 12:48 pm
@farmerman,
This is a pretty simple topic. Either you have reasons to take the discrete approach to space or the continuum, what is yours and what is your argument for it? Don't forget to provide a mechanically comprehensive explanation to avoid meta "mathurbation".
By the way when you give a step how much is it, eh? How much space, energy, time?
farmerman
 
  1  
Reply Sun 9 Apr, 2017 01:00 pm
@Fil Albuquerque,
It keeps guys like you busy and sober I suppose, Is your name Bruce from the Ewnavehsetee a Walamaleoo??
0 Replies
 
farmerman
 
  1  
Reply Sun 9 Apr, 2017 01:02 pm
@chai2,
is that a suthern joke?


edgarblythe
 
  1  
Reply Sun 9 Apr, 2017 02:35 pm
This reminds me of a routine I used to run with my kids. "When you were born I was 100% older than you." You know the rest. Just idle time spent.
0 Replies
 
chai2
 
  1  
Reply Sun 9 Apr, 2017 05:03 pm
@farmerman,
farmerman wrote:

is that a suthern joke?





I guess. The only people I've ever asked were people here.

I don't know what it is though. I can remember when I first read that question, the first thing I thought was "elm?" Then I mentally bitch slapped myself and came up with the answer.
farmerman
 
  1  
Reply Sun 9 Apr, 2017 06:12 pm
@chai2,
you know Im an honorary suthrner, livin in NAwlins for a spell, so Too whit too wee.
Alan McDougall
 
  1  
Reply Mon 10 Apr, 2017 03:15 pm
@farmerman,
This paradox puzzled mathematician for centuries because the logic thinking about it is true and Achilles would never catch up or pass the tortoise, given a billion years to do so.

It was the invention of the zero or lending the zero from another culture that mathematicians could work it out easily.
Alan McDougall
 
  0  
Reply Mon 10 Apr, 2017 03:18 pm
@layman,
layman wrote:

Infinite division has nothing to do with it. If I take one step, I have already traversed an "infinite" number of points. And I can do that again. And again....

The problem is with thinking that "infinity," as either a mathematical or intuitive concept, has any "real" meaning.

Take a yardstick, for example. In theory, I could subdivide that into an "infinite" number of sub-pieces. If I do that, and then add them all up, what do I get? An infinite number? No, I get one yard.

Mental conceptualizations have no physical effect on the yardstick whatsoever.


You are absolutely correct!
0 Replies
 
farmerman
 
  1  
Reply Mon 10 Apr, 2017 06:04 pm
@Alan McDougall,
was it a lesson in history??
Fil Albuquerque
 
  1  
Reply Tue 11 Apr, 2017 03:49 am
@farmerman,
One is clear....those who appeal to metrics the most are the same that **** on it the easiest.
farmerman
 
  1  
Reply Tue 11 Apr, 2017 05:43 am
@Fil Albuquerque,
Ya want to explain with perhaps an xample or two?

Dimensional analyses allows one to explain a phenom in terms of a totally different phenom, so long as the field equations " rhyme".
 

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