Zeno's Paradox

Jun 27, 2015 2:55 AM

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Jan 2009

This thread was made in commemoration of @ClumzyPanda's membership. I tried to introduce something related to Pandas..something Yin and Yang, perhaps 1's and 0's. Something vs. Nothing. But I guess Zeno's Paradox will have to do.

So, Zeno's Paradox is a thought experiment where if you kept dividing the amount of space between you and your destination by 2 and walk that much distance, you will mathematically never get to where you're going regardless of how much you keep walking. If you apply this to money, if you kept dividing your money by 2 and only used that amount of money, you will never run out of money. Of course, we don't really have the .00000001 coins for that. And in physics, there's constants like the planck length to avoid this confusion from non-discrete numbers. What are your thoughts on this?

Great Explanations for this:

https://www.youtube.com/watch?v=EfqVnj-sgcc

https://www.youtube.com/watch?v=u7Z9UnWOJNY

BiddingGortonioJun 27, 2015 3:01 AM

Jul 20, 2015 4:03 PM

FrenchVegeta

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Jul 2013

The key point to remember here is to understand that if an infinite series is conditionaly convergent she could converge to a finite value . In fact if a series in conditionaly convergent , there exist a permutation of its terms in order to make her to converge towards any value we want . it has a name : "the Riemann rearrangement theorem"

For more infos :
https://en.wikipedia.org/wiki/Riemann_series_theorem

or simply google : 1+2+3+.....=-1/12 ;)

Jul 21, 2015 11:10 PM

BiddingGortonio

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Jan 2009

Thank you for this solution. Is this a solution? How does one identify Zeno's Paradox as a conditionally convergent series as opposed to an absolute one?

BiddingGortonioJul 21, 2015 11:29 PM

Oct 22, 2015 1:28 PM

FrenchVegeta

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Jul 2013

Sorry for not being very active in this club but i have started university this september but i'm currently trying to change of school (long story short ) and i dont have a lot of free time and i wanted to add something that counter-say what i said previously so , this wasn't a solution at all .

Here is what my analysis and logic math teacher said about the demonstration shown in the "numberphile" video :

(You need to know that in math , false can imply true , as an example 2=1 (false) , multiplied on both side by zero we have 2*0=1*0 <=> 0=0 (true) )

In the video they start by defining the series :

A = 1-1+1-1...+1-1=1/2 And that is false because here you assume this series as an convergent one or , a series is convergent if the sequence of its partial sums converges .

Here we can define this sequence by Sn = (-1)^n and looking at the definition of a sequence who have an finite limit : https://en.wikipedia.org/wiki/Limit_of_a_sequence if we chose epsilon equal to
0.3 (for example ) we have the proof that this sequence is a divergent one , so therefore for A .

But again contrary to what i thought at first this subject is not as easy as he seem to be ...

Mar 5, 2019 4:54 PM

FrankAiello

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Oct 2017

Zeno actually came up with several paradoxes. For example, the arrow paradox: take the flight of an arrow. It follows its path from some point A to some other point B. However, at each point of its flight, it must go through a sequence of intervening instants (a_0, a_1, ..., a_n). At each of these instants, the arrow is stationary. But how could a path made up of a sequence of "instants" (stationary points) ever make up the motion of the arrow from point A to point B? In other words, how would the arrow even move, given that if you sum up all of the instants that make up the arrow's path, then logically you should have no motion at all? Today, in calculus, this paradox is resolved with the introduction of a rigorous notion of instantaneous velocity:

https://www.themathpage.com/aCalc/instantaneous-velocity.htm

As the name suggests, instantaneous velocity is the velocity of an object at a certain time t. It is the limit of the average velocity. You find the instantaneous velocity by taking the derivative of the position function at some time t. With instantaneous velocity, an object actually does have a definite velocity at each instant (time t). Thus, it no longer makes sense to say that the object is stationary at each instant of its trajectory from point A to point B. Hence, paradox resolved.

The paradox you are referring to specifically is the Achilles and the Tortoise paradox. You have Achilles and the tortoise. Let's say the tortoise starts out any arbitrary distance ahead of Achilles (e.g. 5 meters). Then Achilles can never reach the Tortoise because he must first travel half the distance, then half of that distance (a fourth), then half of that distance (an eighth), and so on. Like the arrow paradox, this paradox is resolved with calculus. What we have is the infinite geometric series: 1/2 + 1/4 + 1/8 + 1/16 + ... which converges absolutely to 1. Thus, it is not the case that Achilles will never catch up with the tortoise simply because catching up with the tortoise requires traversing an infinite number of points. He will in fact catch up with the tortoise eventually. Hence, paradox resolved. Likewise, all of the other paradoxes Zeno gives can in one way or another be addressed with modern methods in mathematics such as the calculus.

Whether there is some ultimate scale of physical length (e.g. the Planck scale) beyond which spacetime can no longer be subdivided is a debated question. While mathematically, spacetime is typically modeled as a continuum, our current understanding of physics does suggest there should be a definite length scale beyond which spacetime would breakdown and no longer be continuous.

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Zeno's Paradox

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