What is the logical flaw in Xeno’s Paradox? It must have one, or I’d never have reached the keyboard to type this question.
Does anyone know?
The paradox assumes that there are an infinite amount of points between any two points. This is wrong.
It turns out that the the universe is quantized, which means there are a finite amount of discrete points between any two points in the universe. I’m not 100 percent sure, but think the smallest distance in the universe is a Planck Length.
The flaw is that it assumes that Achilles and the turtle slow down.
Let’s say the turtle has a 50 yard head start and travels one yard a second. Let’s say Achilles travels at 10 yards per second.
In one second, Achilles has gone 10 yards, the turtle 51.
Two seconds: A: 20 yards, turtle 52
Three seconds: A: 30 yards, turtle 53
Four: A: 40 T: 54
Five: A: 50 T: 55
Six: A: 60 T: 56
Achilles takes the lead and never looks back. In order for the turtle to remain in the lead, he and Achilles have to move slower and slower as time passes.
BTW, it’s Zeno.
Here’s a website that discusses Zeno’s paradoxes. (Apparently, there are four of them.)
http://forum.swarthmore.edu/~isaac/problems/zeno1.html
IANAM, but I’m pretty sure that the part of the fallacy is related to infinite series. I believe that the ancient Greeks thought that infinite series could not equal a number less than infinity. Actually, this is not true. The series: 1/2 + 1/4 + 1/8 + 1/16 + … actually equals 1.
So anyway, I’m pretty sure the real mathematicians will show up soon, if they haven’t already.
There are no flaws in Xeno, Warrior Prince! How dare you even make such an assertion!
(insert Xeno’s distinctive war cry here)
Actually, scanning the website I cited above:
“Originally, there were about forty [paradoxes] but only eight have survived.”
The website discusses the four most famous of his paradoxes.
"The Planck length is the scale at which classical ideas about gravity and space-time cease to be valid, and quantum effects dominate. This is the ‘quantum of length’, the smallest measurement of length with any meaning."
This is how the website I searched and found described Planck’s Length.
The key words to see here are “length with any meaning”, which I take to mean “length with any useful value”. That doesn’t mean any smaller lengths don’t exist. Zeno’s Paradox still holds true.
robby, please explain how “1/2 + 1/4 + 1/8 + 1/16 + … actually equals 1” is true.
The limit of the series 1/2[sup]n[/sup] + 1/2[sup]n + 1[/sup] + 1/2[sup]n + 2[/sup] … is 1. As n approaches infinity, the sum of the series converges on 1.
Whether or not this is an answer to Zeno’s Paradox is doubtful, since the series never actually reaches 1. It just gets closer and closer.
The correct answer to Zeno’s Paradox is to face the person mentioning it squarely, and punch the fucker right in the mouth.
I concur.
You did notice the three dots, right? I’m talking about an infinite series. If you want to get nitpicky, the limit of this infinite series. (I’m an engineer, not a mathematician! :p)
From the website I cited above:
[sub]Yeah, I know, it’s an ellipsis. [/sub]
that the infinite series 1/2 + 1/4 + 1/8 + … converges to 1.
Definition: A sequence a[sub]1[/sub], a[sub]2[/sub], a[sub]3[/sub] … of real numbers is said to converge to a real number a, if, for any real number ê (I couldn’t find an eta), there is an integer N such that |a - a[sub]n[/sub]| < ê, for all n > N.
Definition: A series a[sub]1[/sub] + a[sub]2[/sub] + a[sub]3[/sub] + … of real numbers is said to converge to a real number a if the sequence of partial sums a[sub]1[/sub], a[sub]1[/sub] + a[sub]2[/sub], a[sub]1[/sub] + a[sub]2[/sub] + a[sub]3[/sub], … converges to a.
Pick an arbitrary number ê. Because there is no largest integer, there exists an integer larger than 1/ê. Call it k. 2[sup]k[/sup] > k > 1/ê, so 1/2[sup]k[/sup] < 1/k < ê. For any integer m, |1 - (1/2 + 1/4 + 1/8 + … 1/2[sup]m[/sup])| = 1/2[sup]m[/sup]. For all n > k + 1, |1 - (1/2 + 1/4 + 1/8 + … 1/2[sup]n[/sup])| = 1/2[sup]n[/sup] < 1/2[sup]k[/sup] < ê.
So 1/2 + 1/4 + 1/8 + … converges to 1.
Aren’t you glad you asked?
Personally, I like the Planck’s length explanation. Has anyone actually looked into the possibility that, when distances get very small, “position” takes on a different meaning?
NB: Math and Philosophy Departments at most universities tend to frown on the use of Hansel’s final argument.
My factoid condensation of the answer: The fallacy is assuming that it takes a finite amount of time to travel an infinitesimally small amount of distance. It doesn’t: it takes an infinitesimally small amount of time.
It turns out you can add up an infinite number of infinitesimally small numbers and still get a normal, non-infinite number as a result. So even though there are an infinite number of points an infinitesimal distance apart between the start and the finish, it’s still possible to cover the course in a finite amount of time.
The ancient Greeks didn’t know this, though, and so they were rightly perplexed by the paradoxes.
Warning: IANAM. (I can’t spell “infinitesimally,” either.)
Plank’s length doesn’t enter in. Like all good paradoxes, this is a question about math/philosophy/logic, not reality! 
Wait, so Zeno’s paradox is FALSE? Ah ha! I’ll get you now, you turtle!
The fallacy is the assumption that our modern tortoises are the same as the tortoises Xeno was talking about. Archaeologists have recently uncovered Grecian urns that depict a tortoise from about the same era as Xeno. It had long spindly legs and could run like crazy! So Achilles really couldn’t outrun a tortoise!
The other fallacy is the assumption that Achilles WANTED to pass the tortoise. You know how some people are afraid to step on a crack? or afraid to have a black cat cross their path? Well, Achilles was afraid to pass a tortoise. This was one sick hero.
(Thanks to Severin Dardin.)
I think what Zeno’s Paradox well illustrates is that there are limits to any given mathematical approach in describing the natural world. In this case, Maurice Greene proves that an infinite set of fractions isn’t an efective way of measuring velocity.
A paradox is not “true” or “false.” Paradoxes just point out problems or contradictions in our our way of thinking.
I think you (and the website) are probably correct that the Planck length is not smallest length, but I still believe the universe is quantized. It just may be quantized at a smaller length.
The universe may or may not be quantized. I think it is not. However, Zeno’s paradox assumes that it is not quantized and you don’t need to invoke quantization to resolve it. It is a paradox only if you make the (erroneous) assumption that the sum of an infinite number of positive terms must be infinite. That is the crux of it.
What’s sick about that - that shell’s not going to break up much in the digestive tract, passing that’s going to hurt like hell.
True or not, the quantization of the universe doesn’t have anything to do with Zeno’s paradox. The Wumpus has provided the best explanation so far.
While it is true that you have to traverse an infinite number of small distances to reach your keyboard, the fallacy is in assuming that it will take a finite amount of time to cross each of them. Zeno just didn’t understand infinitessimals, and can we blame him so many centuries before calculus was invented?