Zeno's paradox

[Henrik]

How is it a paradox?

Is it just seems to be parsing the model of reality incorrectly? Or what?

[antimatter]

It's a paradox because what he says makes sense logically, but it's wrong where physics defines the laws.

[Kyrisch]

Zeno's paradoxes, in short, represent the problem with the mathematical logic of the time which were later solved when Newton invented the Calculus.

[lakmilis]

Kyrisch.. I think the problem solving Zeno's paradox came even later than Newton actually... it was more like 18th century wasnt it or so.. when the real line was redefined?

 

(zeno's paradox is the one with the half distances all the time right?)

[Mr Skeptic]

In Zeno's time, people did not believe in infinities -- Zeno showed that movement requires you to move through infinite points. There may also be a quantum limit to movement that would invalidate Zeno's assumption that to get somewhere, you must first get halfway there.

[coke]

i assume you mean beating the turtle or the halfway one... Lets use the halfway one its simpler...

your speed stays constant, but supposedly it wouldnt work because to get to some goal, you have to get halfway there first... in reality he's simply taking a linear function, y = x where y is distance and x is time, and making an infinite amount of points of both y and x

 

Say you are trying to get somewhere 2 miles away. You walk at 1mph. This can be modeled by y = x. At x = 2, y = 2 so it will take you 2 hours. Now Xeno's way:

 

In each halfway step its x miles:

1 mile, 0.5 mile, 0.25 mile, 0.125 mile, 0.0625 mile,... and going on forever... but adds up to 2 miles

 

the time for each of these goals, at 1mph, is y hours:

1 hour, 0.5 hour, 0.25 hour, 0.125 hour, 0.0625 hour,... and going on forever... but adds up to 2 hours

 

You should have gotten over the 2 miles after 2 hours.

Xeno simply made an infinite amount of steps out of both x and y...

 

But his real trick is that he only pointed out the infinite amount of steps in x, thus making it seem impossible.

In logic, it's impossible. In math, the paradox can be simplified into a linear equation, i.e. y = x when relating to time.

 

The turtle paradox is the same concept, but a little more complicated.

Edited March 31, 2009 by coke

[stereologist]

The old math joke is:

 

A mathematician and and engineer are told that they are to stand on one side of a room and that on the other side of the room is a beautiful person of the opposite gender that they can have if they only follow the rule that in any given time they are only allowed to move half of the remaining distance.

 

They ask the mathematician what will happen. Sadly the person says, "I'll never reach the person of my dreams."

 

They ask the engineer what will happen. With a smile they say, "I'll soon get there for all practical purposes."

[coke]

engineer will get close enough

[J.C.MacSwell]

Reminds me of the kid with the slide rule:

 

"2 X 3 is 5.9...looking closely...5.99...what the heck, let's call it 6!"

[Sisyphus]

There are actually many of Zeno's paradoxes, not just one, and they were intended to demonstrate a particular philosophical view of the world. There's not really anything to "solve," as such, but there have been various explanatory accounts of them. Most notably, Aristotle (about a hundred years later) gives a pretty good explanations of the most famous of them in his Physics. More or less, just because something is divisible doesn't mean it is in fact divided, and so there is no problem. Yes, you can infinitely divide the distance into an unlimited number of steps, but so too can you divide the time - the finite time - needed to get there into as small moments as necessary. He basically makes the distinction between "potential infinities" and "actual infinities."

 

And, naturally, nowadays you'd most likely talk about them in terms of calculus, in which we deal with those pesky infinitesimals all the time...

[coke]

yeah sisyphus, i've got kind of stuck on trying to create the equation, i think it's something of this nature

[math]x\sum^{\infty}_{1}.5^n = my\sum^{\infty}_{1}.5^n[/math]

x= dist. covered

y= time passed

m= speed

n= goal number

but basically it simplifies to x = my... plus mathemetically you know .5 + .25 + .125 + .0625 ..... = 1

 

it would make more sense as y = mx, since usually x should be the independent variable, time. but since i already put x as distance in my last post...

Edited April 1, 2009 by coke

[the tree]

[imath]\sum^{\infty}_{n=1}(\frac{1}{2})^n =1[/imath] only with the condition that you can halve every number infinitely many times - it's the continuity of distance that Zeno's paradox demonstrates. Precisely that either time and distance are both continuos or they are both discrete - this was an important problem for thinkers thousands of years ago since a discrete world would be nessersary for the pythagorean view that the world was constructed entirely with the natural numbers.