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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:

[quote]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. [/quote]

Quote:

"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.

@Victor Eremita,

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

@Deftil,

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.

@Victor Eremita,

Victor Eremita wrote:

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.)

@paulhanke,

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.

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/creative-writing/951-do-you-belive-eternal-recurrence.html

@VideCorSpoon,

VideCorSpoon wrote:

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? Wink

... 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?

@paulhanke,

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.

@Victor Eremita,

Victor Eremita wrote:

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.

@paulhanke,

paulhanke wrote:

... 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...

@Arjen,

Arjen wrote:

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

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

@VideCorSpoon,

VideCorSpoon wrote:

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.

... but if we argue that zero is an "extended" numerical value, doesn't that render the description of the arrow paradox nonsensical?

Quote:

[INDENT] 1. When the arrow is in a place just its own size, it's at rest. [/INDENT][INDENT] 2. At every moment of its flight, the arrow is in a place just its own size. [/INDENT][INDENT] 3. Therefore, at every moment of its flight, the arrow is at rest.
[/INDENT]

That is, if we define zero as an "extended" numerical value then it is impossible that "2. At every moment of its flight, the arrow is in a place just its own size." ... a "moment" endures for zero time - but because we have defined zero as an extended numerical value, some amount of time passes within a given moment, which implies that the arrow moves within a given moment, which in-turn implies that at every moment of its flight, the arrow is in a place that is larger than its own size, thus rendering the description of the arrow paradox nonsensical.

@paulhanke,

Arguing that "zero can be an extended numerical value" was a juxtaposition to the supposition you posed which said, "a point in time has zero temporal extent." First, I am unaware of any paradox that is sensical to any degree. Paradoxes illuminate "holes" in normative frameworks, but mostly never patch those frameworks up.

If we did argue that zero is an extended numerical value, we only uphold the abstract values of Xeno's paradox. But more to the point, that very fact helps us provide some level of solution to the problem. The short summary of Xeno's paradox is that the arrow never gets half way because it has to go through half of that half, etc. The arrow is moving but not moving. The paradox lies in the fact that the arrow (theoretically) does not move because it is essentially traveling half of a half of a half, etc. It is in a sense going backwards while going forwards. It is the quintessential reductio ad absurdum example.

The quote that you mention is an loose account of Xeno's paradox from Aristotle's Physics. But look very carefully at the quote you posted. If we were to define zero as an extended numerical value, it would not negate the second premise because they are both (#1 and #2) bi-conditionally concurrent (i.e. A=B & B=A). So any axiomatic argument that necessarily

@VideCorSpoon,

VideCorSpoon wrote:

First, I am unaware of any paradox that is sensical to any degree.

... agreed - that's why I said it renders the description of the arrow paradox nonsensical ...

VideCorSpoon wrote:

The short summary of Xeno's paradox is that the arrow never gets half way because it has to go through half of that half, etc.

... is it? ... here's (Aristotle on) Zeno on the arrow: "The arrow in flight is at rest. For, if everything is at rest when it occupies a space equal to itself, and what is in flight at any given moment always occupies a space equal to itself, it cannot move." ... maybe you have the arrow paradox confused with another of Zeno's paradoxes?: "You cannot cross a race course. You cannot traverse an infinite number of points in a finite time. You must traverse the half of any given distance before you traverse the whole, and the half of that again before you can traverse it. ..." (translations by J. Burnett)

VideCorSpoon wrote:

But look very carefully at the quote you posted. If we were to define zero as an extended numerical value, it would not negate the second premise because they are both (#1 and #2) bi-conditionally concurrent (i.e. A=B & B=A). So any axiomatic argument that necessarily follows is illogical because the two premises cancel each other out.

I'm not sure I follow - how is a nonsensical antecedent #2 (from the standpoint of zero being an extended value) necessarily made sensical by virtue of antecedent #1? ... as far as I can see, the only relationship between the two is the (paradoxical) consequent (#3).

VideCorSpoon wrote:

Also of note is the fact that the ancient Greeks had no conception of zero the way we did.

... which could be one big reason why this was such a paradox for them - Zeno was messing around with infinities, but was not mathematically equipped to deal with the implications of those inifinities (e.g., an infinite sum of zeros is still zero) Wink

@paulhanke,

To posit anything nonsensical implies that it may have been sensical to begin with. I'm glad we both agree on a fundamental notion of paradoxes and semantic composition.

On the short summary of Xeno's paradox. It is first and foremost a paradox. Perhaps that is why the notion of a half's half becomes relevant because it is contradictory (hence the biconditional). Your translation of a basic Aristotelian text is very much different than my own, from the deeper contexts of Aristotle's terminology to the way his name is spelled. I think this is the greatest point of our discontention which is the translation aspect. But even in the Burnett example you provide, it underlines the same abstract notion I (and I would suspect we) are trying to convey, albeit through different standards.

The example you provided had 3 lines. These lines display a logical deductive proof formation. I am quite sure of this because the website where this was quoted from displays Xeno's proof in predicate proof form. The "nonsensical" antecedent #2 is made "sensical" in virtue of antecedent #1 because both #1 and #2 are inextricably connected because they form a bi-conditional. In a biconditional, an apple is red and red is the color of the apple. The consequent #3 extrapolates on the biconditional premises (#1 and #2) illuminating that it is in fact a biconditional (#3). But as a side note, the word "sensical" and "nonsensical" are problematic because they are relative terms and insufficient for the terms of the discussion.

But was Xeno messing around with infinites??? That seems a bit premature to say at this point. Also, the concept of zero for the Greeks is rudimentarily established in cosmogony, not mathematics.

:eek: (I would have done the winky emoticon too, but I thought this was far more irrelevant. LOL!)

@VideCorSpoon,

VideCorSpoon wrote:

Your translation of a basic Aristotelian text is very much different than my own, from the deeper contexts of Aristotle's terminology to the way his name is spelled. I think this is the greatest point of our discontention which is the translation aspect.

... which is one reason I pointed to the source of the translation - in going from one translation to another, I'm finding large discrepancies ... in fact, if I remember correctly Burnett states in his preface to "Early Greek Philosophy" that he did his own English translations because he couldn't agree with any of the existing English translations!

VideCorSpoon wrote:

... an apple is red and red is the color of the apple.

... given this example, wouldn't the bi-conditional of antecedent #1 be "When an arrow is at rest, it's in a place just its own size." ... and for what it's worth, it doesn't sound like bi-conditionals make very good antecedents in a syllogism:

1. Apples are colored red.
2. Red is the color of apples.
3. Therefore, ???

@paulhanke,

paulhanke;20278 wrote:

... 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?

And doesn't this continuity-discreteness tension remain with us still? How does Zeno connect to calculus? To transcendental numbers like pi?

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