Zeno's paradox

[Kyriakos]

Zeno, the ancient pre-Socratic greek philosopher concieved of a few paradoxes. This thread is about one of them, which is ussually reffered to as "Achilles and the turtle".

According to Zeno, if a turtle and Achilles were in a race, with the turtle given a headway into it, then Achilles, nomatter that he was so much faster, could never erase the gap between them.
The reasoning was that at each given time Achilles would be second to the turtle, for an infinite amount of moments. For example at moment 1, the turtle would be in the position X, whereas Achilles would be in the position X-Z. At Moment 2 the turtle would be in the position X2, and Achilles in X2-Z2 and so on.

This is a paradox, for the simple reason that it focuses on the amount of moments there are for all of which the turtle would be ahead of Achilles. Obviously at some moment in time Achilles would go past the turtle, but Zeno's point was that there were infinite numbers of moments for which Achilles would still be behind it, since any time can be divided into infinite parts. Thus since there is an infinity of moments for which Achilles is behind the turtle, and there is nothing bigger than an infinity, it follows, paradoxically, that Achilles would never go past the turtle.

I posted this in this forum, although, while it is philosophy, it has been linked to art too. I recall an article by Borges in which he claimed that Zeno's paradoxes were one of the precursors of Kafka's literature.
To me Zeno's paradox still seems interesting, in that - to my knowledge- there hasn't been any mathematical theory dealing with this problem, namely that whereas we can sense that at some point Achilles would go past the turtle, we cannot explain how this is possible, since it is our senses which bypass the infinity of points of time which go by until Achilles effectively is in front.

However there might exist in the realm of Mathematics something dealing with such issues. I recall learning of barriers for infinite groups (don't know the term in english, and am just translating word for word here), but im not sure if that could be applied in this issue.

Anyway, maybe there can be more hybrid philosophical threads here, since they are artistic too in a way, at least in my view :)

[cyberbob]

But there isn't an infinite number of moments between Achilles and the turtle. A moment may be infinitely divisible into smaller units but the moment itself is not infinite.

A second can be infinitely divided into smaller units, but it still only lasts one second. The only way for Achilles to not pass the turtle would be if time froze or if he was traveling at a slower velocity than the turtle.

Since velocity = distance/time and time is necessarily passing, Achilles will eventually pass the turtle if he travels at a higher velocity.

[Kyriakos]

Yes he will, but if he would have been a being that had an experience of the infinite by its senses, then wouldnt he be trapped in the infinite number of instances it woudl take for him to go past the turtle? :)

[MystyrMystyry]

There is a phiosophy begun by Aristotle called Ekphresis), but it deals with Art - if two people were in a totally dark cave and one descibes a work of Art to another, does it need to exist? Can it exist as it is being described in the moment in the cave?

They weren't really big on art those ancient philosophers, but they loved dark caves

[cyberbob]

Yes he will, but if he would have been a being that had an experience of the infinite by its senses, then wouldnt he be trapped in the infinite number of instances it woudl take for him to go past the turtle? :)

There's only an infinite number of instances or moments if you don't define them within some numerical unit like a second. In that case it becomes an abstract concept and thus isn't divisible at all just like vanity or love aren't divisible. If something can't be assigned some sort of numerical value then it cannot be infinite.

It's the same as saying that a missile can never hit a fighter jet because there's an infinite number of moments at any given unit of time. However the fighter jet is in motion at any given moment, the only way for there to practically be an infinite moment or an infinite number of moments would be if time stopped, which it cannot.

[Ecurb]

I was always befuddled by Xeno’s paradox until I recently opened a book filled with photos of ancient Greek artifacts. One artifact was the sculpture of a tortoise. This was not a flat, stubby-legged tortoise, but a sleek, bullet-shaped tortoise, with long spindly legs. It looked as if it could run like lightening!

This would seem to explain Xeno. But it doesn’t. Because Homer, who never lied about anything, once said that Achilles could beat any man or beast in a footrace if he wanted to.

What does this mean, “If he wanted to.”? Do you know how some people can’t step on the cracks in the sidewalk? Achilles couldn’t beat the tortoise. He was a very sick hero.

So much for Xeno……

[Cunninglinguist]

The paradox comes about when you assume that infinite divisibility is equated with infinite length/magnitude. Though this is not necessarily true, infinite length does not necessarily follow from infinite divisibility, and there have been many arguments contending this. The most intuitive one being that something like the number 1 has infinite divisibility, it can be cut infinite times. Yet, when analyzed in the context of a number line, it does not have infinite magnitude (note: when you analyze it outside of the context of a number line the number 1, and any number, for that matter, becomes a totally infinite entity).

[Heteronym]

To me Zeno's paradox still seems interesting, in that - to my knowledge- there hasn't been any mathematical theory dealing with this problem, namely that whereas we can sense that at some point Achilles would go past the turtle, we cannot explain how this is possible, since it is our senses which bypass the infinity of points of time which go by until Achilles effectively is in front.

It only shows that mathematics, for all its hype as an awesome science that reveals great truths, doesn't hold a candle to simple observation of the physical world.

Zeno's paradox is just word play and sophistry. Language is like mathematics, a self-contained system; you can prove anything with them so long as you make them coherent and don't compare them to the real world. That's why in some mathematics 2 + 2 can be 5, although a PhD in Maths wouldn't get you out of trouble if you tried to short-change a greengrocer.

[weltanschauung]

there is too a mathematical description of this paradox, its called the asymptote
http://en.wikipedia.org/wiki/Asymptote

am i wrong?

"you cannot reach zero through division"

[Cunninglinguist]

It only shows that mathematics, for all its hype as an awesome science that reveals great truths, doesn't hold a candle to simple observation of the physical world.

I don't think it shows that the content of math is flawed or fallible or so esoteric as to not correspond to factual things; moreover I believe it shows that the language through which the content of mathematics is expressed is imperfect, or, rather, misapplied at times. Zeno's paradox can ultimately be boiled down to a fallacy of equivocation in which Zeno treats the word infinite with a double meaning, namely, in one instance, as infinite divisibility, and in another as having infinite magnitude. Such equivocations (esp. when dealing with infinitude) are also possible in the mathematical language (though mathematicians have prescribed rules in attempts to prevent them). A simple example is: 1 times infinity equals 2 times infinity; cancel the infinities and you get 1 = 2. How this false is obvious on an intuitive level. By multiplying each number by infinity you render them without any properties of magnitude, yet they still retain their properties of divisibility, and in the sense that they are both infinitely divisible 1 and 2 (and any real, positive number) are equal. Yet, to express this equality in the conventional symbols is misleading and any mathematician would call the aforementioned example an abuse/misuse of the language.

I agree that Zeno's paradox is sophistry (I just call it a fallacy), but I don't believe that languages are entirely self contained (if we share the same definition of 'self contained'). If they were it would be impossible to winnow out the erroneous inferences without referring to intuition, which, for Zeno, is not the case. It is clear that he is equivocating, and one does not have to refer to the real world to see that his argument is unsound.

[aboad]

Borges has two absolutely fascinating texts on the paradox! One attempts to try to solve it (I assume it's the one Kyriakos mentions, but it's been a while and I can't be sure), with recourse to the various solutions that had been proposed throughout history. The other, however & predictably enough, ends concluding it's better it's left unresolved: a kind of mathematical work of art, for contemplation & elusion.

[YesNo]

I don't know much about Zeno's paradoxes except that I've heard they are overcome with convergent infinite series (infinite sums of numbers less than 1 which sometimes converge, that is, are equal to some finite number).

Maybe Zeno was secretly placing money on Achilles winning the race. Since everyone else, with any common sense, was doing the same thing he had to try to convince someone to take the other side of the bet or he wouldn't win anything. So he argued that the tortoise had to win. Or at worse couldn't lose.

I'm sure that's not the history of this, but who knows?

[Cunninglinguist]

I don't know much about Zeno's paradoxes except that I've heard they are overcome with convergent infinite series (infinite sums of numbers less than 1 which sometimes converge, that is, are equal to some finite number).

Maybe Zeno was secretly placing money on Achilles winning the race. Since everyone else, with any common sense, was doing the same thing he had to try to convince someone to take the other side of the bet or he wouldn't win anything. So he argued that the tortoise had to win. Or at worse couldn't lose.

I'm sure that's not the history of this, but who knows?

Convergent infinite series mathematically expresses that infinite magnitude does not necessarily follow from infinite divisibility. For example, take the infinite sum 1/2+1/4+1/8..., it converges on the finite number 1. Zeno assumes that all infinite sums are infinitely large, and in as far as we known him he never presents a proof nor even a reason to think that infinite sums wouldn't be finite.

[jajdude]

I figure that the paradox wherein one cannot reach a point that is "x" distance away since one always must go halfway --x/2 (or whatever fraction of the way) there, can be easily resolved like this:

If one is going toward point "x" one must also be going toward point "y" which is farther away than "x". At some stage one will pass point x on the way to point y.

I don't know how simplistic this idea is, and no doubt others have proposed it before, but for me this "solution" seems satisfying enough. Perhaps it's not a great answer but I kind of like it.

[Cunninglinguist]

I figure that the paradox wherein one cannot reach a point that is "x" distance away since one always must go halfway --x/2 (or whatever fraction of the way) there, can be easily resolved like this:

If one is going toward point "x" one must also be going toward point "y" which is farther away than "x". At some stage one will pass point x on the way to point y.

I don't know how simplistic this idea is, and no doubt others have proposed it before, but for me this "solution" seems satisfying enough. Perhaps it's not a great answer but I kind of like it.

Another conclusion is that you can never reach point x or point y or any point between the starting point and point x or y, and you can never even get off the starting line because to travel any distance would be to travel an infinite(ly large) distance. Ergo nothing changes...at least this is what Zeno concluded.

[HitlerProf]

Doesn't something like this do more to illustrate the limitations of language and the paradoxes created by said limitations rather than demonstrating an actual principle?

[jajdude]

I guess it is useful for mathematics. Of course it is absurd to say I cannot walk a mile when I can walk twice that easily, but for math at least the paradox provides interesting developments or possibilities. A mile is not infinite, nor is a minute, but math includes infinite elements. As for the limitations of language, you may have a good point there too.

[cyberbob]

Well a mile is a mathematically defined unit of measurement, as is a minute, whilst a moment is not.

There is no concrete definition of what constitutes 1 moment and how long that moment lasts.

So it pretty much does just demonstrate a flaw in language.