Question about Zeno's paradox

[WoodenPupa]

Most explanations of this paradox that I've seen solve the roblem by demonstrating that since motion is physically possible, the paradox goes out the window. I am wondering, can the paradox be solved without resorting to this? In other words, what is the pure mathematical proof and the reasoning behind it?

 

[itachi]

which paradox are you referring to? there're 4 paradoxes of motion.

 

[WoodenPupa]

Sorry, I meant the stadium paradox (dichotomy).

But in any even, which paradox isn't the issue, because all of them can be defeated using demonstrations of actual motion.

 

[icejunkie]

if you know the answer, why ask the question?

 

[f95toli]

It is easy to show that there is no paradox. If you write down the equations you will see that you will get an infinte sum, BUT the sum converges wich means that the result is still a real number. So the "paradox" can be solved with pure math.

The fact that an infinte sum can converge to a real number was unknown until just a few hundred years ago so it is not surprising that this was considered to be a paradox 2000 years ago.

 

[WoodenPupa]

Originally posted by: icejunkie
if you know the answer, why ask the question?

What gives you the idea that I know the answer? The fact that motion exists doesn't solve the paradoxes mathematically. It demonstrates a physical refutation only, which was not the thrust of my question.

 

[WoodenPupa]

Originally posted by: f95toli
It is easy to show that there is no paradox. If you write down the equations you will see that you will get an infinte sum, BUT the sum converges wich means that the result is still a real number. So the "paradox" can be solved with pure math.

The fact that an infinte sum can converge to a real number was unknown until just a few hundred years ago so it is not surprising that this was considered to be a paradox 2000 years ago.

Very interesting. I guess my question now is, what is a convergence? How does an infinite series approach a limit such as a whole number?

 

[itachi]

err... convergence (divergence) is a property of an infinite series.. the dichotomy paradox can be modeled using a geometric series- half of the previous at each interval. the resultant series has a ratio of 1/2, which is less than 1.. satisfying the condition for the ratio test, therefore is convergent. you can also prove it using induction.

 

[ThatWasFat]

Well hang on a minute. An infinite series has a limit, but a limit still does not ever reach it's target, it' just gets arbitrarily close. If you are constantly getting 10^39 times closer every nanosecond, guess what. You still aren't going to mathematically get there. But you are arbitrarily closer at all times. I don't really see how an infinite series shows us anything. Mathematically, using limits it works, but I just don't see how it -really- works out.

 

[Velk]

Originally posted by: ThatWasFat
Well hang on a minute. An infinite series has a limit, but a limit still does not ever reach it's target,

It does actually if it's a converging series, which is what sum(x/2,x/4,x/8) etc is - it reaches that point when the terms reach infinity. It's a subject lesson in what the reality of having an infinite number of points in a line of fixed length is.

 

[unipidity]

And with Xeno, it DOES reach infinity. Obviously