What is the Arrow chasing a soldier paradox and can anyone explain it?

[adjacent]

Do anyone know the "Arrow chasing a soldier" paradox?I don't know the actual name,so couldn't google it.
Someone throws an arrow at a soldier.The soldier runs away to flee.The arrow actually travels faster than the soldier,so it should hit him.The paradox states that as the distance halves and halves,the arrow actually does not touch him.I want an explanation on this.
Note:I can google this, only if I know the name.

 

[Enigman]

http://en.wikipedia.org/wiki/Zeno's_paradoxes
You are looking for an hybrid restatement of Achille's turtle in terms of Arrow paradox...
Achilles and turtle paradox:
In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead.

Arrow paradox
If everything when it occupies an equal space is at rest, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless

 

[phinds]

You might also be interested in Zeno's Paradox

 

[Enigman]

You might also be interested in Zeno's Paradox

The arrow, achilles and turtle and Dichotomy paradoxes are all Zeno's paradoxes.
Dichotomy paradox is more or less a restatement of Achilles and turtle and both deal with space, while Arrow paradox deals with time.

 

[adjacent]

Hmm...I was thinking if a PF member could give a non-Calculus answer to this.The web answers seems disturbing and difficult to understand.

 

[Borek]

a non-Calculus answer to this

Ancient Greeks could not understand the idea that sum of infinite number of elements can be finite. It can.

http://en.wikipedia.org/wiki/Geometric_series

 

[Enigman]

I don't like the statement of Achilles and torty paradox but math's sound.

 

[adjacent]

Ancient Greeks could not understand the idea that sum of infinite number of elements can be finite. It can.

http://en.wikipedia.org/wiki/Geometric_series

I don't see any non-Calculus here.It's full of calculus(I don' know Calculus yet.)

 

[Enigman]

I don't see any non-Calculus here.It's full of calculus(I don' know Calculus yet.)

See the vid. its pretty simplistic.
Random factoid: Zenosparadox is a cheat code in Age of Mythology: The Titans expansion. It spawns all heroes of both campaigns at your primary town center. Okay I'm off to learn Alice Chess...
E

 

[Dale]

Hmm...I was thinking if a PF member could give a non-Calculus answer to this.The web answers seems disturbing and difficult to understand.

If the arrow's position is $p_A=v_A t$ and the target's position is $p_T=v_T t + d_0$ then the arrow hits the target when $p_A=p_T$ or by simple algebra $t=d_0/(v_A-v_T)$.

Without calculus it is clear that the arrow reaches the target in a finite amount of time despite the fact that there are an infinite number of points along its trajectory. Actually calculating the infinite sum as an infinite sum requires calculus, but we know from algebra what the answer must be even if we cannot calculate it directly without calculus.

If you want something more than the algebraic answer then I think you need go ahead and learn calculus.

 

[Enigman]

http://en.wikipedia.org/wiki/Zeno's_paradoxes
You are looking for an hybrid restatement of [STRIKE]Achille's turtle[/STRIKE] Dichotomy paradox in terms of Arrow paradox...
Achilles and turtle paradox:
In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead.

Dichotomy paradox
That which is in locomotion must arrive at the half-way stage before it arrives at the goal

Arrow paradox
If everything when it occupies an equal space is at rest, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless

Correction.

 

[HallsofIvy]

"Zeno's paradox" is based on the assumption that it is not possible to pass through an infinite number of points in a finite time. That is incorrect.

 

[adjacent]

"Zeno's paradox" is based on the assumption that it is not possible to pass through an infinite number of points in a finite time. That is incorrect.

Hmm.For example,If someone travels a distance of 1m,the 1m can be divided into infinite pieces.Right?

 

[phinds]

Hmm.For example,If someone travels a distance of 1m,the 1m can be divided into infinite pieces.Right?

Mathematically, yes. Whether or not space is actually quantized is an open question, but in either case it doesn't matter in terms of how fast/far you can move

 

[Mandelbroth]

Hmm...I was thinking if a PF member could give a non-Calculus answer to this.The web answers seems disturbing and difficult to understand.

Well, you are asking us to not use calculus to answer a calculus question, or at least a question that is best answered in terms of the machinery of calculus. If you want a good answer, calculus is the way to go.

 

[adjacent]

See the vid. its pretty simplistic.
Random factoid: Zenosparadox is a cheat code in Age of Mythology: The Titans expansion. It spawns all heroes of both campaigns at your primary town center. Okay I'm off to learn Alice Chess...
E

Haahha.The most precious cheats to me are:
JUNK FOOD NIGHT
ATM OF EREBUS
TROJAN HORSE FOR SALE
PANDORAS BOX
L33T SUPA H4X0R

These cheats are enough to win even in titan mode.
oh:I forgot, O CANADA for emergencies. lol

 

[adjacent]

Well,Thank you all.I've now understood the Paradox well.

 

[Enigman]

Haahha.The most precious cheats to me are:
JUNK FOOD NIGHT
ATM OF EREBUS
TROJAN HORSE FOR SALE
PANDORAS BOX
L33T SUPA H4X0R

These cheats are enough to win even in titan mode.
oh:I forgot, O CANADA for emergencies. lol

[off topic hijack]I can handle titan mode campaigns... random map not so much...
The trick to campaign is never do what your objective says.The campaigns are built based on triggers, so don't complete the starting objectives just act towards the final objective. Eg.
*For the first scenario Kraken attacks the docks objective kill it. Don't, let it live while you beef up your army and resources. The next wave won't come till its dead.
*Osiris box scenario you just get out of prison. Objective look at the osiris box- don't as soon as you do timer starts to the next shuffle...just get triggers out of the way and you have thrown a wrench in the AI.
Oh and I made a mistake Zenosparadox is for random god powers, mixed up with atlantisreborn
[/off topic hijack]If you have mathematica you could check this wolfram demo out:
http://demonstrations.wolfram.com/ZenosParadoxAchillesAndTheTortoise/
In case you don't, you could download the free CDF player
http://demonstrations.wolfram.com/download-cdf-player.html
(wolfram demos are a pretty decent waste of time so its worth it)

 

[Tobias Funke]

Zeno's paradoxes seem like something that we only have second-hand accounts of, which are then repeated in distorted or incomplete versions in freshman calculus books.

My understanding is that the paradoxes, if we give the benefit of the doubt and call them that, are only paradoxes when taken together; Zeno was trying to argue that both infinitely divisible space and discrete space are impossible, which is why he came up with multiple scenarios. If Achilles can never catch the tortoise assuming infinitely divisible space, that's no paradox if space is actually discrete, is it?

I never liked or could take seriously the "solutions" in those calculus books. They're just showing off and kind of missing the point, since if you're going to solve it using techniques the Greeks would clearly reject outright, you may as well make it easier on yourself, walk up to the wall and touch it, and say QED. Claiming that you need calculus to show that a faster runner will eventually overtake a slower one is just going to reinforce the common belief that mathematicians are crazy people who prove (column proofs of course...) obvious things.

 

[phinds]

Zeno's paradoxes seem like something that we only have second-hand accounts of, which are then repeated in distorted or incomplete versions in freshman calculus books.

My understanding is that the paradoxes, if we give the benefit of the doubt and call them that, are only paradoxes when taken together; Zeno was trying to argue that both infinitely divisible space and discrete space are impossible, which is why he came up with multiple scenarios. If Achilles can never catch the tortoise assuming infinitely divisible space, that's no paradox if space is actually discrete, is it?

I never liked or could take seriously the "solutions" in those calculus books. They're just showing off and kind of missing the point, since if you're going to solve it using techniques the Greeks would clearly reject outright, you may as well make it easier on yourself, walk up to the wall and touch it, and say QED. Claiming that you need calculus to show that a faster runner will eventually overtake a slower one is just going to reinforce the common belief that mathematicians are crazy people who prove (column proofs of course...) obvious things.

Tobias, I mean no disrespect but I think you are the one that has it wrong. It doesn't matter whether space is continuous or discrete, the basic argument of the "paradox" has the same problem and yes the calculus solutions DO show the proper argument whether "common people" like it or not.

Positing that space is discrete would be unproven speculation.

 

[Tobias Funke]

Tobias, I mean no disrespect but I think you are the one that has it wrong.

Could be. I never claimed to have it right and I should have made that more clear in my first post. That doesn't mean that most other people have it right though--reading about one of his paradoxes which is quickly solved by an infinite sum doesn't make one an expert.

It doesn't matter whether space is continuous or discrete, the basic argument of the "paradox" has the same problem and yes the calculus solutions DO show the proper argument whether "common people" like it or not.

Positing that space is discrete would be unproven speculation.

You may be right about discrete space not coming into play. I thought his arrow paradox was an argument against discrete space, but I guess maybe it works with continuous space as well*. The calculus solution is an argument, and of course I believe it, but it seems obvious that Zeno didn't really believe that he couldn't walk to the wall. Presumably, he wanted an argument in "his own terms" and I don't know if those fit the bill (plus, summing the distance instead of time, which I see done a lot, is obviously not a proof; at any step you can name, he's not at the wall with or without calculus).

I'm not taking this too seriously and was partially joking with my physical proof, but then again, I don't think anyone can deny that touching a wall is a perfectly valid proof of the statement "I can touch a wall". I'd even say it's the proper proof :)

*edit: I may have been thinking of the stadium paradox, but he may not have had discrete space in mind for that either.

Also, I want to add that the calculus solutions seem to ultimately come down to accepting axioms about infininity that are at the very heart of the paradoxes. If Zeno was simply arguing that one couldn't reach a wall because it requires going halfway first, then we can just consider a wall twice as far away and go half that distance and we've reached the wall. He was arguing that motion itself was impossible, I guess because of some problems with "actual infinity" instead of "potential infinity". So even adding that first 1/2 of the infinite series is sort of begging the question, and if we somehow truly didn't know that motion was possible, I'd be torn between Zeno's arguments and the calculus ones.

 

[Enigman]

I'm not taking this too seriously and was partially joking with my physical proof, but then again, I don't think anyone can deny that touching a wall is a perfectly valid proof of the statement "I can touch a wall". I'd even say it's the proper proof :)

You are not the first guy with the same attitude or even the first one to come up with that pun:
According to Simplicius, Diogenes the Cynic said nothing upon hearing Zeno's arguments, but stood up and walked, in order to demonstrate the falsity of Zeno's conclusions. To fully solve any of the paradoxes, however, one needs to show what is wrong with the argument, not just the conclusions. This is usually done by using convergent series and calculus, a tool at Zeno's time 2000 years in making. Zeno wasn't trying to prove motion or space or anything else was impossible but rather the paradoxes were meant IMO to show the limitations of math as of then.
An interseting read:
http://physics.bgsu.edu/~gcd/Spacetime3.html

 

[Borek]

Zeno wasn't trying to prove motion or space or anything else was impossible but rather the paradoxes were meant IMO to show the limitations of math as of then.

That was always my understanding. Paradoxes, when they are in an obvious contradiction to the testable reality, are a great way of showing something is wrong with the theory.

Was it Feynman who said "in the end it is always nature that is right"?