Infinity

[Yasa]

Quote by: saltinespike The only infinite thing I have totally thought through and have agreed with is an infinite series. Let's say you shoot an arrow at a bullseye. It makes it half way there then an quarter of the way there, then an eigth, and a sixteenth. Will the arrow ever hit? Yes. Let's suppose you shot the arrow from 3 feet away and hit right on bullseye. It's an infinite series. You could go to 1/209674326098 and it still would not have hit yet, but it has to hit, what's going to stop it? Infinite series = 1.

It's like one of Zeno's paradoxes. Imagine a light that is on untill the arrow is 1/2 the distance from the target. As it hits 1/2, it turns off (lets assume instantly). When it hits 1/4 from the target, it turns back on, and then off again when 1/8 remains and continues doing this untill it reaches the target. Will the light be on or off when the arrow reaches the target?

[saltinespike]

Quote by: Yasa It's like one of Zeno's paradoxes. Imagine a light that is on untill the arrow is 1/2 the distance from the target. As it hits 1/2, it turns off (lets assume instantly). When it hits 1/4 from the target, it turns back on, and then off again when 1/8 remains and continues doing this untill it reaches the target. Will the light be on or off when the arrow reaches the target?

You got me stumped. Haha. Well, not entirely, for I know that the answer is that you don't know. As it was mentioned before, infinity is not exact. Like (x+6)(x-7).

[Yasa]

Quote by: saltinespike You got me stumped. Haha. Well, not entirely, for I know that the answer is that you don't know. As it was mentioned before, infinity is not exact. Like (x+6)(x-7).

Yeah, infinity causes a lot of mind pain, haha. There are many, many paradoxes that are all fundamentally realated. You should check out zeno's paradoxes. Despite modern calculus (which suposedly "solved" them), they still are logical mind teasers.

[iclaudius]

What saltine is describing is fully explained by something called potential infinity. You can chop stuff up into infinite degrees, but they're not an actual infinity -- only a hypothetical one.

As for that particular paradox, the answer is, whatever the light was when the arrow was at rest. Unless you prescribe a condition for the light to be in at the point where the arrow hits the target, the light has no reason to be either on or off. The condition for the light changing is distance cut in half every time, and it will continue switching on and off as the increments become infinitesimally small as the arrow approaches the target. Because you can divide the distance between the arrow and the target infinitely, it would continue switching on and off as far as you cared to check it. Strictly speaking, however, it wouldn't be a physically demonstrable paradox, because there is no mechanism (and in fact, can be no mechanism) capable of differentiating between infinitesimal increments. Suffice it to say it would be switching on and off infinitely quick the instant just before the arrow hits the target, and would not "end" on the on or off position. For all intents and purposes, we might even say that at that instant, it would be both on and off. When the arrow hits the target, this condition for "off" or "on" is no longer applicable, so the light reverts to whatever condition it was in during the "rest" position.

[Yasa]

Quote by: iclaudius What saltine is describing is fully explained by something called potential infinity. You can chop stuff up into infinite degrees, but they're not an actual infinity -- only a hypothetical one. As for that particular paradox, the answer is, whatever the light was when the arrow was at rest. Unless you prescribe a condition for the light to be in at the point where the arrow hits the target, the light has no reason to be either on or off. The condition for the light changing is distance cut in half every time, and it will continue switching on and off as the increments become infinitesimally small as the arrow approaches the target. Because you can divide the distance between the arrow and the target infinitely, it would continue switching on and off as far as you cared to check it. Strictly speaking, however, it wouldn't be a physically demonstrable paradox, because there is no mechanism (and in fact, can be no mechanism) capable of differentiating between infinitesimal increments. Suffice it to say it would be switching on and off infinitely quick the instant just before the arrow hits the target, and would not "end" on the on or off position. For all intents and purposes, we might even say that at that instant, it would be both on and off. When the arrow hits the target, this condition for "off" or "on" is no longer applicable, so the light reverts to whatever condition it was in during the "rest" position.

Well it's pretty obvious it can't be a demonstratable paradox. All this stuff is moot. I could state it in a different way and say we start "even" and when we reach half way, we change to "odd", then back to "even" when reached 3/4, and so on. I don't see how you've actually answered this paradox at all though--why does it revert to whatever condition it was in during the "rest" position? You said it yourself, we might even say it's both on and off, which is clearly still a paradox.

[iclaudius]

Quote by: Yasa Well it's pretty obvious it can't be a demonstratable paradox. All this stuff is moot. I could state it in a different way and say we start "even" and when we reach half way, we change to "odd", then back to "even" when reached 3/4, and so on. I don't see how you've actually answered this paradox at all though--why does it revert to whatever condition it was in during the "rest" position? You said it yourself, we might even say it's both on and off, which is clearly still a paradox.

Assuming that there are no extraordinary constraints on the event, the light reverts to whatever condition it was in while at rest because it's resting when it hits the target and stops. The point I was trying to make was that the on/off of the light as the arrow approaches the target means nothing. If you want the light to remain in whatever condition it was in the infinitely small instant before it hit the target, you're out of luck. It's impossible to quantify the infinite, and in the end, all you will have demonstrated is that you can't place the constraints of finite value on the infinite. Thus, the condition of the arrow reaching the target must be defined independently of the stupid on/off behavior. I hypothesize that it simply reverts to whatever status it was at during resting, since we have no reason to believe it would be anything but.

[nm420]

Quote by: Yasa It's like one of Zeno's paradoxes. Imagine a light that is on untill the arrow is 1/2 the distance from the target. As it hits 1/2, it turns off (lets assume instantly). When it hits 1/4 from the target, it turns back on, and then off again when 1/8 remains and continues doing this untill it reaches the target. Will the light be on or off when the arrow reaches the target?

You'd get one wicked looking strobe light!

[5010]

The light's on-off state is not determined by the problem as defined. Why? Because the state of the arrow at the target is assumed to match the last state of the arrow before reaching the target, but by definition of the problem there are infinite states before reaching the target. Therefore the last state does not exist within the problem, and since the target state matches a state that does not exist, it is undetermined.

[Yasa]

Quote by: 5010 The light's on-off state is not determined by the problem as defined. Why? Because the state of the arrow at the target is assumed to match the last state of the arrow before reaching the target, but by definition of the problem there are infinite states before reaching the target. Therefore the last state does not exist within the problem, and since the target state matches a state that does not exist, it is undetermined.

I think you just restated the paradox. It's undetermined, but we know for a fact it will reach it's destination. If there is no last state, then how can it ever reach a target (a last state)? When is it that there is a cross over from being infinitely close to actually being at it's destination? It's the same as asking why do equations actually reach their limits?

[5010]

Quote by: Yasa I think you just restated the paradox. It's undetermined, but we know for a fact it will reach it's destination. If there is no last state, then how can it ever reach a target (a last state)? When is it that there is a cross over from being infinitely close to actually being at it's destination? It's the same as asking why do equations actually reach their limits?

'Infinitely close' is not a position, but instead an offset closer than any position. You can cross it but you can't stop on it because any position you stop on is defined as not it.

Asking whether the infinitely close is on or off is like asking whether the last decimal of an infinitely smallest number is even or odd. There isn't one.

[Yasa]

Quote by: 5010 'Infinitely close' is not a position, but instead an offset closer than any position. You can cross it but you can't stop on it because any position you stop on is defined as not it. Asking whether the infinitely close is on or off is like asking whether the last decimal of an infinitely smallest number is even or odd. There isn't one.

I understand what you're saying. But I'm wondering why we can reach the target if there are infinte steps between the initial position and the destination. Classic Zeno's paradox and still seems to break my logic. I think I need an explanation of the logic behind limits... something they avoid in high school mathematics/calculus.

[Lullaby Chainer]

Quote by: 5010 'Infinitely close' is not a position, but instead an offset closer than any position. You can cross it but you can't stop on it because any position you stop on is defined as not it. Asking whether the infinitely close is on or off is like asking whether the last decimal of an infinitely smallest number is even or odd. There isn't one.

So basically.. it's not so much that the problem is a paradox.. it just can't be done.

What does this tell us about infinity, the universe, and infinity in the physical realm?

[Yasa]

Quote by: Lullaby Chainer So basically.. it's not so much that the problem is a paradox.. it just can't be done.

Yet.. the arrow reaches the target, so it can be done.

[Lullaby Chainer]

Quote by: Yasa Yet.. the arrow reaches the target, so it can be done.

But it can't.. wait..

huh..

perhaps this problem is discovering the very smallest point in space?

What can't be done is that t-

f*CK.. now my head hurts. >:[

I'll come back when I have a complete thought.

[iclaudius]

Guys, guys, guys... The arrow does hit the target. It moves from rest through the air to the target, and then it stops. The light switching on and off is just a demonstration that you can go on dividing something forever and still be dividing because you can do that infinitely. The fact that you could divide infinitely does not affect the arrow, because it is still flying over a finite distance. That distance, no matter how many times you divide, is still finite. Even when infinitely divided, that space does not become bigger -- it's only divided into smaller pieces. The light turning off and on demonstrates what happens when you try to divide something infinitely: nothing. It continues switching on and off until it is finally switching so fast that it appears to be both on and off. When it hits the target, the light has switched between on and off infinitely fast, and because of this, it might as well be both on and off at once. There is no ending position for the light. Once the arrow stops, the light is either on or off, depending on the parameters of the event. It might be on because that's what condition it held at rest, or it might be off, because the exercise is finished, and it is not given a definition to be when the exercise has completed. Whatever the case, it doesn't matter -- the whole thing only shows what happens when you try to divide something infinitely. I hope this clears everythign up. If you are still confused, don't hesitate to ask questions. I suggest reading up on POTENTIAL INFINITY, as well.

[Yasa]

Can't say I read up on potential infinity. But, just to throw some more controversy out there... Despite it's finite distance, do you deny that there are indeed infinite steps required to get there (under the assumption space is infinitely divisible). How do we even start to move if we must first move half the distance of our target, finite distance, and half of that half before, and so on?

[iclaudius]

Quote by: Yasa Can't say I read up on potential infinity. But, just to throw some more controversy out there... Despite it's finite distance, do you deny that there are indeed infinite steps required to get there (under the assumption space is infinitely divisible). How do we even start to move if we must first move half the distance of our target, finite distance, and half of that half before, and so on?

Yes, I do. Those steps are only potential. You're only walking over an infinite distance if you divide it infinitely in the first place, which as we've clearly demonstrated, is just plain impossible. That's why it's called potential infinity.