Xeno's Paradox

I've always loved this paradox.

Mathamatically an arrow can never reach its target.

Say the arrow leaves the archer (Point A) and is pointed at the target (Point B). The arrow must travel half of the distance to point B before travelling all of the distance. Now from that point you must again travel half of the remaining distance. If you continue to do so (travel half of the distance) the arrow will never reach point B.

But of course in reality we know that it does.

So whats up with that, does maths fail to explain something? I don't know I'm not much of a maths person.

What are some thoughts about this paradox.

whats half of nothing?

Thats interesting, much pondering must be done.

Pfft ur jsut saying that if osmething travels half of something, then it's reached half way, now if it travels half of wahts left, its reached half of what's letft, now take whats left and when the arrow reaches half way it's gone half way, and then it can go antoehr half of that too.

Of couse if u keep halving somethign it wont make the full distance.

Ur jsut talking nonsense.

Of course an arrow can reach the desired target as it has enough velecity/momentum to do so. Half has nothing to do with any of it.

You would in time make the distance from a practical point of view but technically never make it.

you knew(willed) it would because of the laws of physics you are accostomed to in this universe...

but you really did not see that the points were not touching...

Its a logical paradox darage, not a literal one, we all know the arrow will hit its target.

But explainable as anything divided an infinite amount of times = 0.

(The whole concept of infinty is always a good way to bend your brain isnt it )

I found this on some maths site, apparently its the solution:

Zeno's Paradox Resolved

The faulty logic in Zeno's argument is the assumption that the sum of an infinite number of numbers is always infinite. While this seems intuitively logical, it is in fact wrong. For example, the infinite sum 1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ... is equal to 2. This type of series is known as a geometric series. A geometric series is a series that begins with one and then each successive term is found by multiplying the previous term by some fixed amount, say x. For the above series, x is equal to 1/2. Infinite geometric series' are known to converge (sum to a finite number) when the multiplicative factor x is less than one. Both the distance that Achilles travels and the time that elapses before he reaches the tortoise can be expressed as an infinite geometric series with x less than one. So, Achilles traverses an infinite number of "distance intervals" before catching the tortoise, but because the "distance intervals" are decreasing geometrically, the total distance that he traverses before catching the tortoise is not infinite. Similarly, it takes an infinite number of time intervals for Achilles to catch the tortoise, but the sum of these time intervals is a finite amount of time.

It doesn't equal 2.
it will equal to 1.99999999999999999999999 sort of number.

it can never reach the number 2.

Actually, 1.99 repeating is 2. It is 2 because of special property of the infinitely repeating number:

n = 1.99 rep
10n = 19.99 rep
10n - n = 19.99 rep - 1.99 rep
9n = 18
n = 2

Simple.

Although Zeno's arrow may not be traveling an infinite distance, it actually is spanning an infinite width. The problem stems to the formation of geometric dimensions. The distance between 2 points is infinite if no relative scale is taken into account. This is because a point has no height, width, or depth. Points are now thought of as locations, not a physical point. Still, an infinite series of distances ends up as a finite number. Basically, math plays a joke on us, where infinity=finite. Go figure.

I hate it when I'm writing a post, and during that time someone already answers the question I'm about to post.

Alwell, as is life.

I deleted the original content to this post.

[Edited on 10-11-2003 by StationsCreation]

a reply to: StationsCreation

I came across a video that disproves Zeno's paradox.
In a nutshell they conclude the infinite sum 1/2 + 1/4 + 1/8 + 1/16 .... = 1.

My issue with this is it approaches 1 doesn't = 1. By it's nature it's infinite so it never ends.

In the real world Zeno's Paradox will hold. If a person walking to a point walks half way, then half way again he will in fact never reach the destination. He'd have to walk forever.

Also, if he somehow did reach his destination then would he not be able to walk further?! Mathematically he can't because the sum ends at 1 but it's an infinite sum! It never ends.

This is my major issue using infinities. They are useful for mathematics to make real world predictions but in the real world they don't exist.

In conclusion I believe Zeno was correct. That is he is correct if space is continuous and not discrete but that is a totally different discussion.

It is not math it is philosophy .
Once you know that then you no longer need the question .

Except on the event horizon of a black hole .
But black holes just iggy laws of any kind lol .

It is not math it is philosophy .
Once you know that then you no longer need the question .

Except on the event horizon of a black hole .
But black holes just iggy laws of any kind lol .

It is not math it is philosophy .
Once you know that then you no longer need the question .

Except on the event horizon of a black hole .
But black holes just iggy laws of any kind lol .

a reply to: midnightstar

I disagree. It's mathematics based on the rational numbers and whether in reality you can divide by half indefinitely.
Taking Zeno (the philosopher) out of it I can rephrase the question.

Will 1/2 + 1/4 +1/8 + 1/16.... ever reach one. The answer is no. It will always be approaching 1 forever.

a reply to: Deluxe


No, that is not quiet it.

The problem is that in math, you can always divide something smaller and smaller. That is the infinite number line.

You can't do that to an arrow! An arrow is set number of the infinite distance (number line). You are comparing a discrete entity (16" arrow) to infinity (which passes through the 16" barrier on the way to smaller and smaller).

The concept gets really wild when you start branching off into the complex and real number systems (and equations) and run across discrete numbers! Like Riemann Zeta equation spitting out non-negative prime numbers when the non-trivial zeroes of the function are calculated (even the trivial zeroes are just negative multiples of 2). How can an infinite product with the square root of -1 being one of its factors along with infinite factorials the other and yet you end up on 3, 5, 7, 11, 13, 17, 19, 23, ..., etc??!!

That is more mind blowing than Zeno's Paradox!

a reply to: TEOTWAWKIAIFF

I was trying to keep it simple by just talking about the rational numbers. Numbers represented by p/q where p and q are integers without bringing in the other sets of numbers to complicate things.

My focus is really on the use of infinite sums in mathematics and if they really represent reality or if they are just a mathematical tool. I'm guessing most of mathematics is a tool. Obviously complex numbers don't exist in the real world but they are extremely useful to simplify some mathematics.

I'm thinking about this because earlier I was reading about quantized space and time in Physics.
If space and time are quantized then I wonder if this would rid modern mathematics of the infinities and singularities that pop up in Physics.

originally posted by: StationsCreation
So whats up with that, does maths fail to explain something? I don't know I'm not much of a maths person.

Mathematics is inherently limited by built-in paradoxes and is limited when it comes to representing things that happen in the real world -- particularly how poorly it handles nested infinities, and how it doesn't incorporate objective points of view. One of my favorite books is this one, which will make you think differently about math after you read it:
www.goodreads.com...

a reply to: Deluxe


It would be a nice, clean universe if we were stuck with just integers.

And it is because Zeno brought up infinities (half the distance, then the next half, etc) that he got his drinking buddies to get up and walk into walls (the other version of the paradox)! There is no paradox if you were to word the problem like this: There are 30 arrow lengths between the archer and the target; the arrow has to fly half the distance to hit the target (15 arrow lengths); then half again (7.5 a.l.), then again (3.75); The very next "half way there" is less than 2; If the arrow cannot shrink in distance, how many does it take to reach the target?

You see, you leave out infinities this way and never reach a "paradox" state!

Infinite sums are just integrals from some start point to infinity. A quantized version is exactly how integral calculus is derived! That is how they add the infinites together to reach an answer; they "chop up" the area under the curve into equal pieces then add them together; the individual "pieces" get smaller and smaller until they are lines under the curve filling in the area. The integral is shorthand for doing this process. If the curve becomes too complicated, there are substitution equations that give you an approximation. There are a few conditions and other "tools" to help the process but it pretty well worked out.

I don't think we can escape infinities!