Zeno of Elea

[Victor Eremita]

Back before I went to university, I knew about Zeno's paradoxes. But how I knew was weird. There was a really strange young teen book in the school library which combined Zeno's example of the tortoise and Achilles with Aesop's fable of the tortoise and the hare. If I remember correctly, the tortoise challenged the hare to a race because of the mocking he was suffering from the hare; but instead of racing; the tortoise outlined Zeno's paradox and the hare conceded the race without racing! **LAME**

Anyway, Zeno has some interesting paradoxes which have logical merit (maybe), but obviously (aka practically) make no sense at all. The famous Tortoise and Achilles example is reproduced here:

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 metres,” 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-metre head start. Would you say that you could cover that 10 metres between us very quickly?” “Very quickly,” Achilles affirmed. “And in that time, how far should I have gone, do you think?” “Perhaps a metre – 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?” “Fine,” 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.

WTF?!? I can definitely outrace a damned turtle. It's called velocity Zeno, look it up. My velocity is greater than the turtle’s, so even if the turtle has 10 metres advantage, if he’s moving at 1 metre per hour and I’m moving at 20 metres per hour, I will be 9 metres ahead of the freaking turtle by a hour.

Can someone tell me the philosophical significance of Zeno’s paradox, because it sounds so dumb to me.

[Deftil]

We all know Achilles will overtake the tortoise. But at the same time the tortoise's argument seems to be logically airtight. If there isn't anything wrong with any specific part of the tortoise's logic, then it's a demonstration of a logically solid argument leading to a false conclusion.

From a mathematical standpoint there are ways to handle the problems with calculus, but this may not completely solve the philosophical issues raised.

these may help you
Zeno's Paradoxes (Stanford Encyclopedia of Philosophy)

Zeno's paradoxes - Wikipedia, the free encyclopedia

[Victor Eremita]

What do you mean it's logically airtight? There's something missing in the tortoise's story, he ain't giving the whole kitten kaboodle.

Let's break it down

P1. Tortoise must have p metres head start.
P2. Tortoise can go x metres, in the the time Achilles can go p metres
P3. Tortoise can go another x/2 metres in the time Achillies can go another x metres
P4. Tortoise can go another x/4 metres in the time Achillies can go another x/2 metres
P5. Tortoise can go another x/8 metres in the time Achillies can go another x/4 metres
And so ad infinitum.

What's the problem? Zeno's gots to give me a good reason why Achilles must keep slowing down his velocity, from p to x to x/2 to x/4, etc.

Isn't it logically conceivable that:

P1. Tortoise must have p metres head start.
P2. Tortoise can go x metres, in the the time Achilles can go p metres
P3. Tortoise can go another x/2 metres in the time Achillies can go another p/2 metres
P4. Tortoise can go another x/4 metres in the time Achillies can go another p/4 metres
P5. Tortoise can go another x/8 metres in the time Achillies can go another p/8 metres

So for example if p = 10, x = 1 as in this story,

P1. Tortoise must have 10 metres head start.
P2. Tortoise can go 1 metres, in the the time Achilles can go 10 metres
P3. Tortoise can go another 1/2 metres in the time Achillies can go another 5 metres
P4. Tortoise can go another 1/4 metres in the time Achillies can go another 2.5 metres
P5. Tortoise can go another 1/8 metres in the time Achillies can go another 1.25 metres

So? No problem right? At P3, Achillies has already kicked the tortoise' ass.

[paulhanke]

Originally Posted by Victor Eremita

So for example if p = 10, x = 1 as in this story,

P1. Tortoise must have 10 metres head start.
P2. Tortoise can go 1 metres, in the the time Achilles can go 10 metres
P3. Tortoise can go another 1/2 metres in the time Achillies can go another 5 metres
P4. Tortoise can go another 1/4 metres in the time Achillies can go another 2.5 metres
P5. Tortoise can go another 1/8 metres in the time Achillies can go another 1.25 metres

So? No problem right? At P3, Achillies has already kicked the tortoise' ass.

... actually, this particular paradox is more like:

P1. Tortoise must have 10 metres head start.
P2. Tortoise can go 1 metres, in the the time Achilles can go 10 metres
P3. Tortoise can go another 1/10 metres in the time Achillies can go another 1 metres
P4. Tortoise can go another 1/100 metres in the time Achillies can go another 1/10 metres
P5. Tortoise can go another 1/1000 metres in the time Achillies can go another 1/100 metres
... ad infinitum ... and Achilles never catches the Tortoise :-)

This paradox is implicitly based upon a mathematical model of reality in which space-time is infinitely divisible. That is, given any finite distance no matter how small, you can always divide that distance into even smaller units. So how valid is this model of reality? (Hint: a quantum physicist might dispute that space-time is infinitely divisible.)

[VideCorSpoon]

This was part of one of the first posts I did on the forum on eternal recurrence where I uses Xeno's paradox to support my position.


In Xeno’s paradox… say I shoot an arrow at a target 20 feet away. That arrow must travel that 20 feet in order to hit the target. But here is the rub. In order for that arrow to travel that twenty feet, it has to travel ten feet (half the distance) to get to its destination FIRST. BUT!!!! Here’s where it starts getting complicated. In order for it to travel that half distance (10 feet), it has to travel five feet FIRST. In order for it to travel the five feet, it has to travel the half distance of 2.5 feet FIRST.

Every time you cut the distance in half, look at the problem from a fresh perspective, as though the twenty feet did not exist, only the ten.

The way we measure things will quickly reduce and we eventually come to zero. BUT!!! Here’s where it gets abstract… We recognize that the arrow was shot, and that it traveled twenty feet. However, we also realize that in order for it to have traveled that twenty feet, it had to make it half way, and half of that way, and half of that way, etc. THE PROBLEM IS THAT THAT HALF WAY WILL CONTINUE TO REDUCE LOGICALLY BUT IT STILL TRAVELS THE DISTANCE.

THERE WAS A REDUCTION IN DISTANCE IN THE SAME TIME THERE WAS EXTENSION OF DISTANCE!!!!!!! HENCE THE PARADOX.


This a link to the thread. The full post (with Xeno's paradox in it) is at post #6.

http://www.philosophyforum.com/forum...ecurrence.html

[paulhanke]

Originally Posted by VideCorSpoon

THERE WAS A REDUCTION IN DISTANCE IN THE SAME TIME THERE WAS EXTENSION OF DISTANCE!!!!!!! HENCE THE PARADOX.

... or was Zeno just a sleight of hand artist? ... in the arrow paradox Zeno asks us to consider a model of the world where time is made up of an infinite number of points (that is, time is not continuous) ... intuitively, this seems reasonable as we're all familiar with the notion of an "instant"; but mathematically, this is impossible ... a point in time has zero temporal extent (the duration of an instant is zero) ... but no matter how many points you sum up, the sum total will always be zero - stated another way, in Zeno's arrow-paradox model of the world there is no such thing as time (he stole it away without you noticing!) ... and how realistic is such a model?

[VideCorSpoon]

But how could Xeno be a slight of hand artist??? It’s a paradox underlining a logical fallacy. It is not a trick of oratorical prowess. Xeno does indeed posit that time is a so called “multitude of points” (if a way.) But that implies that time is continuous because the basis for the paradox is that the arrow will never reach half its distance because of the sheer limitless amount of points it must intersect. Xeno’s cutting half of a half of a half of a half of a half, etc, etc, etc. Again, limitless points.

The notion of an “instant” does not seem to be a factor in the paradox though. The paradox underlines the notion that an “instant” to a point cannot happen because the arrow is in constant motion and is never able to reach that “instant” to begin with.

Also, I do not agree with your supposition that though we are familiar with a broad notion of an “instant,” it is a mathematical impossibility because “a point in time has zero temporal extent.” This statement is extremely problematic. It is extremely axiomatic and tautological. It can very well be argued that zero is an “extended” numerical value.

It’s not that Xeno so subtly “stole time,” it’s just that that particular notion does not fit in a normative scientific framework the way we have come to understand time,space, distance, Egg McMuffins, Quantum physics, etc. How realistic is such a model? It’s as realistic as Burridan’s A$$ paradox. It’s not meant in a literal context, only in an abstract context.

[Deftil]

Originally Posted by Victor Eremita

What do you mean it's logically airtight? There's something missing in the tortoise's story, he ain't giving the whole kitten kaboodle.

I'm afraid I didn't make my point well.

OBVIOUSLY something is missing because we KNOW Achilles will catch up with and overtake the tortoise.

The argument at first seems logical while yielding an obviously false conclusion. That's the point. On further investigation the argument is found to be flawed. But it's not necessarily apparent at first.

[Arjen]

Originally Posted by paulhanke

... or was Zeno just a sleight of hand artist? ... in the arrow paradox Zeno asks us to consider a model of the world where time is made up of an infinite number of points (that is, time is not continuous) ... intuitively, this seems reasonable as we're all familiar with the notion of an "instant"; but mathematically, this is impossible ... a point in time has zero temporal extent (the duration of an instant is zero) ... but no matter how many points you sum up, the sum total will always be zero - stated another way, in Zeno's arrow-paradox model of the world there is no such thing as time (he stole it away without you noticing!) ... and how realistic is such a model?

Boy, does that remind me of Euclides' Geometry...

[paulhanke]

Originally Posted by Arjen

Boy, does that remind me of Euclides' Geometry...

... I don't know whether to take that as a compliment or an insult!