A question of paradox

[gray]

Well, I thought I might as well see if anyone here would like to discuss this idea with me. Basically, I've been thinking of paradox's today, and one in particular struck me as interesting. If you know what I'm talking about, that would save time, basically, it's the theory that since there are an infinite number of point's between any two locations, travel between the two should be impossible. Example; You are running toward a wall, but in order to get to the wall, you must first travel half the distance, and then half of the that remaining distance, and then once again, eventually, you are very close to the wall, but you must still half the distance, and therefore ever reaching the wall would be impossible under this thinking. As far as I know this is one of the proofs that science as we know it is likely no where near as developed as it could be, but I would like to talk about this if anyone feels like it.

Oh, and if I misused any terms in the above paragraph, I apologize, no need to point and laugh, I will fix it if you call it to my attention.

 

[Ronyo]

So.. You're basically saying that we never reach the wall because we always have half of the current distance to reach it? I agree to a point, since I know that we actually do not touch the wall, electromagnetism causing repulsion has us hovering a few atom's width away at all times. ^__^

 

[gray]

Thats true, which helped explain it to me for a while, but what about when there is no wall? Reaching a certain point on a track would have the same affect right?

 

[Ronyo]

Well, you could say that you needed to get 'here' but since there is little chance of you getting in the exact spot, there would always be a difference.

Random paradox! The rule: All rules have at least one exception, the exception of this rule being that there are no exceptions.

I love that rule X3

 

[Kawamura]

The answer's in calculus, namely, limits. In maths, you can take certain infinite sums and they'll converge onto a finite answer.

 

[vampire seduction]

Mmmm calculus. I miss it ;_;

 

[gray]

Thats true, I've been reading up on it, and math does seem to play a large part of this, still, I may need to start studying the math now, in order to understand how that works. I'm not doubting it or anything, I'm just very curious, I large part of what I know comes from learning things this way, finding a question and chasing it back to it's roots.

 

[Ronyo]

What about other paradoxes? Or was this thread about this one particular paradox?

 

[gray]

Not at all, lets see, what another good one. Well, I suppose you could use this same one it relation to time, but I have my own theories there. Let see, anybody have any other ones they would like to bring up?

 

[Raziel99]

Yes, but you have to remember that the argument you are posing is that whatever the first object is moves in half distances, i.e., they move half the distance at one time. Humans move on semi-constant distances, i.e., they change their velocities, but they generally move at a constant velocity that is not affected by the distance from the end point. Now, it would be impossible to get against a wall by a atoms width precision, you already said why. But for fixed open points on a plane, you always can get within a atom or two's distance from it.

 

[Kawamura]

Yes, but you have to remember that the argument you are posing is that whatever the first object is moves in half distances, i.e., they move half the distance at one time. Humans move on semi-constant distances, i.e., they change their velocities, but they generally move at a constant velocity that is not affected by the distance from the end point. Now, it would be impossible to get against a wall by a atoms width precision, you already said why. But for fixed open points on a plane, you always can get within a atom or two's distance from it.

Speed nor velocity have nothing to do with the paradox.

 

[Raziel99]

Yes, but you have to remember that the argument you are posing is that whatever the first object is moves in half distances, i.e., they move half the distance at one time. Humans move on semi-constant distances, i.e., they change their velocities, but they generally move at a constant velocity that is not affected by the distance from the end point. Now, it would be impossible to get against a wall by a atoms width precision, you already said why. But for fixed open points on a plane, you always can get within a atom or two's distance from it.

Speed nor velocity have nothing to do with the paradox.

I know that, but I put that to have a difference between half distances and constant/semi-constant distances.

 

[Kawamura]

Yes, but you have to remember that the argument you are posing is that whatever the first object is moves in half distances, i.e., they move half the distance at one time. Humans move on semi-constant distances, i.e., they change their velocities, but they generally move at a constant velocity that is not affected by the distance from the end point. Now, it would be impossible to get against a wall by a atoms width precision, you already said why. But for fixed open points on a plane, you always can get within a atom or two's distance from it.

Speed nor velocity have nothing to do with the paradox.

I know that, but I put that to have a difference between half distances and constant/semi-constant distances.

And why would that matter for Zeno's paradox?

 

[gray]

I asked a few friends about it today and got into a rather interesting discussion, a lot of it can largely be rationalized by math, but I still think there much to say about it. It seems math is almost always related to these types of things, I was also having fun talking about the "birthday problem" does anyone know about that one?

Oh, and thank you Kawamura, I had forgotten the name, thank you for reminding me.

Edit: The arrow paradox or fletcher's paradox would also be an interesting idea to bring up here, another of Zeno's wonderful ideas.

 

[Ronyo]

I've never heard of the arrow or fletchers paradox before

 

[gray]

It's a rather odd idea, and not as fleshed out in my opinion, but it goes something like this.

"In the arrow paradox, Zeno states that for motion to be occurring, an object must change the position which it occupies. He gives an example of an arrow in flight. He states that in any one instant of time, for the arrow to be moving it must either move to where it is, or it must move to where it is not. It cannot move to where it is not, because this is a single instant, and it cannot move to where it is because it is already there. In other words, in any instant of time there is no motion occurring, because an instant is a snapshot. Therefore, if it cannot move in a single instant it cannot move in any instant, making any motion impossible. This paradox is also known as the fletcher's paradox—a fletcher being a maker of arrows."

 

[Mr Master]

The issue I had with the initial paradox was the statement about the "infinite number of points" which really only works if you can divide the space with infinite levels of granularity, i.e. if the "points" in question are merely mathematical figures, "imaginary" points, that have no actual dimension. That's like, mathematically, you have planes which stretch off infinitely with absolute flatness. Which, as we know, doesn't happen in real life.

So the objection I had to the initial paradox is the idea that it applies to real life, the "can't get anywhere" observation. Because in real life, we don't deal in infinite mathematical points, the actual points we deal with have measurement in space-time, and therefore are finite between objects. An Zeno's paradox can't apply to real-world situations very well, because eventually, half-distances are going to be too small to measure, and the effectiveness of the one object on the other will be dealt with by other forces (generally magnetic repulsion, in essence, if I understand the theory on how atoms of one object actually "touch" or not-touch atoms of another object).

And in the arrow paradox, that's because Zeno could not recognize the attribute of velocity or vector as a component of reality, and the "snapshot" mentioned above is purely visual. If a single instant of reality were to be entirely captured, it would have to include all attributes of space-time, not just the visual, and therefore it would include vector and velocity. So the only point of the paradox is that the comprehension of the measures was incomplete (which is also a comment on the constantly-evolving state of science; imagine what attributes are present in reality that we can't even conceive of now, but which will be common knowledge in a hundred years).

And here, knock yourself out (that's where gray got the quote above). The problem with paradoxes is that they are excellent thought experiments, but the physicality renders them kind of pointless in any practical sense.

 

[gray]

The issue I had with the initial paradox was the statement about the "infinite number of points" which really only works if you can divide the space with infinite levels of granularity, i.e. if the "points" in question are merely mathematical figures, "imaginary" points, that have no actual dimension. That's like, mathematically, you have planes which stretch off infinitely with absolute flatness. Which, as we know, doesn't happen in real life.

So the objection I had to the initial paradox is the idea that it applies to real life, the "can't get anywhere" observation. Because in real life, we don't deal in infinite mathematical points, the actual points we deal with have measurement in space-time, and therefore are finite between objects. An Zeno's paradox can't apply to real-world situations very well, because eventually, half-distances are going to be too small to measure, and the effectiveness of the one object on the other will be dealt with by other forces (generally magnetic repulsion, in essence, if I understand the theory on how atoms of one object actually "touch" or not-touch atoms of another object).

And in the arrow paradox, that's because Zeno could not recognize the attribute of velocity or vector as a component of reality, and the "snapshot" mentioned above is purely visual. If a single instant of reality were to be entirely captured, it would have to include all attributes of space-time, not just the visual, and therefore it would include vector and velocity. So the only point of the paradox is that the comprehension of the measures was incomplete (which is also a comment on the constantly-evolving state of science; imagine what attributes are present in reality that we can't even conceive of now, but which will be common knowledge in a hundred years).

And here, knock yourself out (that's where gray got the quote above). The problem with paradoxes is that they are excellent thought experiments, but the physicality renders them kind of pointless in any practical sense.

Hmm, I see what your saying, and thank you for giving a link to the wiki.
I guess thats the point about all this that a like to think about however, like you said, it mostly only work in mathematical theory, and if thats the case, it brings up the question of how well mathematics portrays reality, and through that, physics as well. I know it's very easy to rebuke that statement, but I still believe it raises a few good point at the very least.
Oh, and for the explanation of the arrow paradox, the reason velocity is not factored in is because in a single instant, velocity does not exist, there is no psychical or chemical force we can measure without multiple 'snapshots' of the object being measured that is called velocity, it's merely an idea that helps us calculate other things, much like time itself.

Oh by the way, thank you everyone for contributing to this, I love these conversations oh so much ^^

 

[Kawamura]

The issue I had with the initial paradox was the statement about the "infinite number of points" which really only works if you can divide the space with infinite levels of granularity, i.e. if the "points" in question are merely mathematical figures, "imaginary" points, that have no actual dimension. That's like, mathematically, you have planes which stretch off infinitely with absolute flatness. Which, as we know, doesn't happen in real life.

So the objection I had to the initial paradox is the idea that it applies to real life, the "can't get anywhere" observation. Because in real life, we don't deal in infinite mathematical points, the actual points we deal with have measurement in space-time, and therefore are finite between objects. An Zeno's paradox can't apply to real-world situations very well, because eventually, half-distances are going to be too small to measure, and the effectiveness of the one object on the other will be dealt with by other forces (generally magnetic repulsion, in essence, if I understand the theory on how atoms of one object actually "touch" or not-touch atoms of another object).

The thing with the paradox is you are dealing with an infinite number of distances because we're talking about fractional parts of length. Doesn't matter the length you use. You're simply going by halves, which you can take infinite amounts of (half of a half of a half of a half of a... etc). That you can't measure is also not a problem.

 

[Kawamura]

it brings up the question of how well mathematics portrays reality, and through that, physics as well.

What, you've found a problem?

Oh, and for the explanation of the arrow paradox, the reason velocity is not factored in is because in a single instant, velocity does not exist, there is no psychical or chemical force we can measure without multiple 'snapshots' of the object being measured that is called velocity, it's merely an idea that helps us calculate other things, much like time itself.

Actually, velocity does exist at any one time. Here we get back to calculus (and why physics these days is calculus based). Maybe it doesn't count if you have the whole equation describing the position of the object, though.

Also: time is no more an idea that helps us calculate other things that space is.

 

[Mr Master]

Oh, and for the explanation of the arrow paradox, the reason velocity is not factored in is because in a single instant, velocity does not exist, there is no psychical or chemical force we can measure without multiple 'snapshots' of the object being measured that is called velocity, it's merely an idea that helps us calculate other things, much like time itself.

But you see the arrow paradox was created at a time before we even had the conception of space-time, that a thing is not ever unconnected from it's position and velocity and context in gravity. They thought the world was entirely stable, stationary, and didn't conceive of it spinning on an axis while simultaneously spinning in an orbit while simultaneously rotating in an arm of a galaxy that was itself hurtling away from the initial point of the big bang.

The only information the paradox talks about is visual. And it's true, in an unconnected visual-only snapshot, you only see the moving arrow as stationary. But that's because it's a single instant's information of only one single attribute of that arrow, what it looks like in mid-air, and the paradox assumes that that's all there is and all that can be measured, so that's why it's an impossible situation, self-contradictory. But it's only impossible from that one very specific and limited frame of reference; if you view it with modern information, it's not a paradox at all.

The thing about paradoxes (and thought experiments in general) is that there might be a frame of reference where things make sense, where the information that seems contradictory actually is shown to be consistent when viewed from another angle, with additional information. I think that's what makes some theories, like General Relativity, fascinating, because it can explain phenomena using a new context and information.

 

[Mr Master]

And now we're getting the actual physicist involved! Yay!

The thing about halves is, I may be reaching for my water bottle, and the halves may continue to fraction down beyond the realm of perception, and that's fine, but it's not going to stop me from picking my bottle up and taking a sip. I may never actually "touch" the bottle, but the other elements of physics, the atomic valence fields or what have you, are going to kick in while the distance is still halving and I'm going to get a good grip. That's what I'm saying. I get that the halves will mean you'll never actually touch, but I'm still drinking my refreshing beverage right now, so...

 

[Kawamura]

Well, yeah. But there are still an infinite number of halves. Just.. that series adds up to the distance between you and your water bottle and you get a cool, refreshing beverage. The halves only mean you'll never touch if you're an ancient Greek, who has no idea about calculus and infinite sums. Greeks didn't like the infinite and it didn't like them.

 

[Mr Master]

That must be why their culture crumbled, and ours will LIVE FOREVER!

FOREVAR!

 

[Ronyo]

This is turning into a very interesting thread o.o