Infinity Paradox and contineous variables in physical quantities

[black_hole]

The famous infinity paradox is there are infinite fractional numbers between lets say 55 and 56. Then how it happens that a car's speed reaches from 55 to 56..?? Just how it happens? because it would have been impossible for the speed of car to "cross" infinite numbers.

 

In the same way I see the cursor on my monitor screen moving from left corner to the right corner. Why it happens so..?? because between the two corners, there have to be infinite fractional (numbers) distence.

 

It is noticeable, however, that cursor do not move contineously. It only move pixel wise. There are a few number of pixels on a monitor screen and the cursor only moves from one pixel to the next and in this way reaches the other corner, giving an illusion to us that it contineously move from one corner to other.

 

I think this situation accounts for this infinity paradox. The "empty space" everywhere is consist of those kind of "pixels" (i.e. quantum distences) and every moving physical object dose not move contineously but "pixel wise".

 

So in this way, contineous variables of physical quantities are not possible. Similarly any contineous graph of any physical quantity will also be impossible. Every physical movement in this world is a step wise movement and not a contineous movement.

 

Same conclusion applies to time also. Time, then, is also not a contineous variable. Time also proceeds step wise. By definition, time is the duration between events. So if events occur step-wise then "duration between events" also occur step wise. Therefore, time is also not a contineous variable, in this way.

[swansont]

1. This sounds a bit like Zeno's paradox. (aka Xeno, and there are actually several paradoxes)

 

2. Space and time are thought to be quantized, or at least current theories can't deal with them at some small scale. These are the Planck length (~1.6 x 10^-35 m) and the Planck time (~10^-43 s).

[Pinch Paxton]

Well, it depends on how the tiniest movement in the universe works. I think that it would be based on the release of energy. The energy passes from one tiny object to the next. The distance that it has to travel is important, and its speed. The slower it can travel, the smaller the tiniest distance becomes, but there must be a minimum distance that anything can move. The speed is dependant on how often the energy is released. A pixel on a computer can wait before it lights up. The longer it waits to light up, the slower the movement can be. Maybe you are right, maybe life is similar to pixel movement.

 

Pincho.

[Tom Mattson]

Swansont is right, this is equivalent to Zeno's paradox of motion which states that no destination can ever be reached, because, at any point in the motion, you always have at least half the remaining distance in front of you.

 

The reason Zeno's paradox fails is that, in his argument, he assumes that the infinite series of powers of 1/2 diverges to infinity, when in fact it converges to 1.

 

It goes like this:

 

Assume you are traveling at speed v and must cross a distance L in time T.

 

Before you can get to L, you must get to L/2. Once you have gotten to L/2, you must then get to 3L/4, etc. This continuous halving of distances can be represented by the series:

 

Distance=L/2+L/4+L/8+...

 

Note that it takes an infinite number of steps to travel the distance. Since it is not possible to perform an infinite number of steps in a finite time, it is not possible to cross any distance L in time T at speed V.

 

Zeno's mistake is in assuming that an infinite number of steps cannot be done in a finite amount of time. In other words, he tacitly assumes that it would require an infinite amount of time, and so could never be done.

 

Is that true? Let's see.

 

T=L/v=(1/v)Σ_n=1^oo(1/2)ⁿ

 

The series on the right is a geometric series. In order for Zeno's argument to hold up, it would have to blow up to infinity--but it actually converges to 1.

 

edit: typo

[Sayonara]

I prefer the solution "but I can get places" (mainly because I don't have to revise my maths for that one).

[AntiMagicMan]

It has been long accepted amongst scientists that space and time are quantised. There is no infinity paradox.

[Tom Mattson]

It has been long accepted amongst scientists that space and time are quantised.

 

No, it hasn't. The only scientific journals in which you can find discussion on quantized spacetime are the ones which discuss quantum gravity. All such theories are far from being finished, let alone tested.

[black_hole]

It has been long accepted amongst scientists that space and time are quantised. There is no infinity paradox.

 

You are not right. At least Bertrand Russell could not find the solution to this "infinity paradox". In fact I got familiar to this "infinity paradox" by reading Russell. He himself accepted in his writings that he had been unable to find the solution.

 

I also have seen similar situation in the writings of Will Durant. Blaming "logic" for its clever negative use, Will Durant quoted the example of Zeno's paradox which he considered to be such a "true" logic which resulted in (apparently - my own words in bracket) wrong conclusion. In this way Durant (wrongly) concluded that even true logic may not lead to true conclusion.

 

And Mr. Tom,

 

Thankyou for writing down the series that result in final answer of 1.

 

But real "paradox" starts after this 1...

 

Because to cover the distance of 1, you have to cover the distance of 0.5, and so on.

 

I think real answer to this situation lies in quantum distance. Limited space is divided into these limited number of "quantum distances". So we only have to "cross" these limited number of "quantum distances" in order to cover that "limited space".

 

We directly "jump" from one "quantum distance" to the next, in this way we easily cross those limited number "quantum distances".

[AntiMagicMan]

No, it hasn't. The only scientific journals in which you can find discussion on quantized spacetime are the ones which discuss quantum gravity. All such theories are far from being finished, let alone tested.

 

I am of course reffering to the planck length, it is meaningless to talk about something that is smaller than a planck length, so to all effects and purposes space is quantized.

[Tom Mattson]

Thankyou for writing down the series that result in final answer of 1.

 

But real "paradox" starts after this 1...

 

There is no "after this 1".

 

Actually' date=' I made a typo, the series should have read:

 

T=L/v=(1/v)Σ[sub']n[/sub]L(1/2)ⁿ

 

(I left out the L in the summand last time).

 

Anyway, the sum of the series is the total time that it would take to cover the entire distance. Zeno assumed that the sum of an infinite number of terms is necessarily infinite, and it is not.

[Tom Mattson]

I am of course reffering to the planck length, it is meaningless to talk about something that is smaller than a planck length, so to all effects and purposes space is quantized.

 

But quantized space does not appear in any working model that been tested by experiment.

[black_hole]

There is no "after this 1".

 

Actually' date=' I made a typo, the series should have read:

 

T=L/v=(1/v)Σ[sub']n[/sub]L(1/2)ⁿ

 

(I left out the L in the summand last time).

 

Anyway, the sum of the series is the total time that it would take to cover the entire distance. Zeno assumed that the sum of an infinite number of terms is necessarily infinite, and it is not.

 

Thank you for reply.

 

If it is so that sum of that series is finite number, then what is your conclusion about "contineous variables".

 

Are contineous variables of physical quantities exist....???

 

If every movement is contineous, then what is the concept of "quantized"...???

[Tom Mattson]

If it is so that sum of that series is finite number' date=' then what is your conclusion about "contineous variables"?

[/quote']

 

I am a bit confused by your "if....then...?" question. I don't see the connection between being able to sum an infinite series and continuous variables.

 

Are contineous variables of physical quantities exist....???

 

Presently, all working physical models contain continuous variables to represent spacetime coordinates, and there is no problem with it. But it iis thought that, to be consistent, and to carry the quantum revolution "all the way", that spactime itself must be quantized. Only time will tell whether or not this is experimentally justified.

 

If every movement is contineous, then what is the concept of "quantized"...???

 

Currently in quantum theory, dynamical observables of bound states are quantized. These include energy and angular momentum. Also, in the standard model of particle physics, charges (I mean that in the general sense to include EM, Weak and Strong charge) are quantized. All of this can take place in a continuous spacetime.

[black_hole]

I am a bit confused by your "if....then...?" question. I don't see the connection between being able to sum an infinite series and continuous variables.

 

Refer to my very first post, I tried to reject the infinity paradox (i.e. impossibility in covering limited distence) using the anology of the motion of cursor on the monitor screen. Cursor motion is not contineous, i.e. it is pixel wise. So this pixel movement is not a contineous movement. In order to overcome the infinity paradox situations, in every day motions, I applied the conclusion of that anology to all the physical motions i.e. by saying that all motion is just like a pixel motion which is a discrete motion and is not contineous. Now IF really it is so, i.e. if in fact all motion is pixel like motion (the concept of pixel wise motion is not absurd altogether, there are already voices of quantized space time, present in science circles), then it means that in fact no contineous variable can exist.

 

But if you have some other solution to this infinity paradox, then pixel wise motion is not applicable to all the motion. Then contineous variables are possible to occur.

 

But if the only solution to infinity paradox lies in that pixel wise (i.e. quantized) motion, then existence of any contineous variable is out of possibility. I know according to your formula series (if it is really valid, because still I have not understood why your final 1 is not still dividable...?), you have another solution to infinity paradox. So pixel wise motion and thus the denial of the existence of contineous veriables is un-necessary.

 

Presently, all working physical models contain continuous variables to represent spacetime coordinates, and there is no problem with it. But it iis thought that, to be consistent, and to carry the quantum revolution "all the way", that spactime itself must be quantized. Only time will tell whether or not this is experimentally justified.

 

Just suppose for a moment that your formula series is not accurate, i.e. it is not the true solution to that infinity paradox, then all the contineous variables in physics models are really have the problem. For contineous movement cannot even cover the distance of 1.

 

Currently in quantum theory, dynamical observables of bound states are quantized. These include energy and angular momentum. Also, in the standard model of particle physics, charges (I mean that in the general sense to include EM, Weak and Strong charge) are quantized. All of this can take place in a continuous spacetime.

 

If angular momentum is quantized i.e. there is no possibility of contineous spin motion, then why linear momentum is not quantized...??? Why there is a possibility of contineous linear movement....????

[Tom Mattson]

Refer to my very first post' date=' I tried to reject the infinity paradox (i.e. impossibility in covering limited distence) using the anology of the motion of cursor on the monitor screen. Cursor motion is not contineous, i.e. it is pixel wise. So this pixel movement is not a contineous movement. In order to overcome the infinity paradox situations, in every day motions, I applied the conclusion of that anology to all the physical motions i.e. by saying that all motion is just like a pixel motion which is a discrete motion and is not contineous. Now IF really it is so, i.e. if in fact all motion is pixel like motion (the concept of pixel wise motion is not absurd altogether, there are already voices of quantized space time, present in science circles), then it means that in fact no contineous variable can exist.

[/quote']

 

I agree, that if Zeno were correct, and that motion in continuous space were not possible, then it would have to be as you say.

 

But if you have some other solution to this infinity paradox, then pixel wise motion is not applicable to all the motion. Then contineous variables are possible to occur.

 

Right.

 

But if the only solution to infinity paradox lies in that pixel wise (i.e. quantized) motion, then existence of any contineous variable is out of possibility. I know according to your formula series (if it is really valid, because still I have not understood why your final 1 is not still dividable...?), you have another solution to infinity paradox. So pixel wise motion and thus the denial of the existence of contineous veriables is un-necessary.

 

The infinite series that I presented is the one that is used in the argument for the infinity paradox. The "1" is dividable--that's why I can write it in terms of a series!

 

The point is that all the terms in the series add up to 1, and that series represents the total time it would take to traverse the distance.

 

When Zeno formulated his argument, he assumed that the series of terms would have to add up to infinity, simply because there were an infinite number of them. But what he didn't know was that an infinite series can add up to a finite number, and this one does.

 

Just suppose for a moment that your formula series is not accurate, i.e. it is not the true solution to that infinity paradox, then all the contineous variables in physics models are really have the problem. For contineous movement cannot even cover the distance of 1.

 

I agree: If the series actually diverges, then we cannot have continuous motion.

 

If angular momentum is quantized i.e. there is no possibility of contineous spin motion, then why linear momentum is not quantized...??? Why there is a possibility of contineous linear movement....????

 

I don't understand your puzzlement. Logically, it is possible to start with position and momentum operators with continuous spectra, construct a Hamiltonian, and get bound states with quantized energies and angular momenta. The program that accomplishes precisely that is quantum mechanics.

 

But your question seems to be from a point of view that quantized angular momentum implies quantized linear momentum, which is not true.

[AntiMagicMan]

Erm wait a second, someones maths has gone wrong somewhere here. It is indeed true that the series [math] \[\sum\limits_n {\frac{1}{{n^2 }}} \] [/math] converges to 1, but zeno was talking about the series [math] \[

\sum\limits_n {\frac{1}{{2n}}}

\][/math] which is divergent (it is half of the harmonic series, which tends to infinity).

 

But still there is no paradox because we all know it is possible to cover a finite distance in a finite time. Regardless of how much you subdivide 1 unit, it is still only 1 unit, and if you are travelling at 1 unit per second, after 1 second you will suprisingly enough have covered 1 unit.

[Dave]

(Just a minor point: [math]\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}[/math] for anyone that might be a bit confused )

[Tom Mattson]

Erm wait a second' date=' someones maths has gone wrong somewhere here. It is indeed true that the series [math'] \[\sum\limits_n {\frac{1}{{n^2 }}} \] [/math] converges to 1, but zeno was talking about the series [math] \[

\sum\limits_n {\frac{1}{{2n}}}

\][/math] which is divergent (it is half of the harmonic series, which tends to infinity).

 

No, Zeno's series was the geometric series with (1/2)ⁿ as the summand. Since subsequent terms share a common ratio of 1/2 (which is less than 1), the series converges.

[black_hole]

The infinite series that I presented is the one that is used in the argument for the infinity paradox. The "1" is dividable--that's why I can write it in terms of a series!

 

The point is that all the terms in the series add up to 1, and that series represents the total time it would take to traverse the distance.

 

Ok the series according to formula may add up to 1 but why is that if I contineously divide 1 by 2 on my home scientific calculator, and carry on this process, I always find smaller and smaller answer tending to infinite small number...???

 

I mean why calculator answer dose not confirm your formula series..???

 

But your question seems to be from a point of view that quantized angular momentum implies quantized linear momentum, which is not true.

 

Consider the linear motion of a car. That linear motion is directly linked with the rotating motion of its wheels. Rotation motion is quantized, so why linear motion (which is directly linked to that ratation motion) is not quantized...???

[Radical Edward]

Ok the series according to formula may add up to 1 but why is that if I contineously divide 1 by 2 on my home scientific calculator' date=' and carry on this process, I always find smaller and smaller answer tending to infinite small number...???

[/quote']

 

 

did you remember to add them up?

 

1/2 + 1/4 +1/8 + ........ =1

[black_hole]

Mr. Tom, I think now I get your point. You mean (1/2+1/4+1/8.....) = 1 or in other words you mean (0.5+0.25+0.125......) = 1

 

This is an interesting fact. Thanx for sharing with us.

 

But let me doubt again. The answer 1, we can get by applying the formula that uses the infinity as a symbol. But actually when we contineously divide 1 by 2 and contineously carry on the process, I do not think that process would ever end.

 

In this way the answer 1 seems only to be a tendency and not the actual answer. What you say about it....???

[AntiMagicMan]

No, Zeno's series was the geometric series with (1/2)^{n[/sup'] as the summand. Since subsequent terms share a common ratio of 1/2 (which is less than 1), the series converges.}

 

^{Oh bloody hell, my mistake, only reading the first two terms of a series is never a good thing.}

[Tom Mattson]

But let me doubt again. The answer 1' date=' we can get by applying the formula that uses the infinity as a symbol. But actually when we contineously divide 1 by 2 and contineously carry on the process, I do not think that process would ever end.

 

In this way the answer 1 seems only to be a tendency and not the actual answer. What you say about it....???[/quote']

 

The sum of the series is 1.

 

But even if it were only a "tendency", that wouldn't matter. The point is that it is finite. You seem to be thinking that crossing the distance would take an infinite amount of time because it would take an infinite amount of time to add the numbers. But that is not right, because the series is the time it would take to cross the distance. It doesn't matter that it has an infinite number of terms, and it doesn't matter what real number it converges to. As long as it is finite (which it is: even if you don't believe the series sums to 1, it' s certainly less than, say, 1.1), then the time to cross the distance is finite.

[Guest pete de breeze]

this may seem pedestrian but the voyager space probe was set to accelerate using orbit sling shot effects from the suns satellites.with the minimal friction in space concievably acceleration should proceed anabated.given enough time and barring contact in space shouldent it eventualy reach warp speed and beyond?

[Tom Mattson]

this may seem pedestrian but the voyager space probe was set to accelerate using orbit sling shot effects from the suns satellites.with the minimal friction in space concievably acceleration should proceed anabated.given enough time and barring contact in space shouldent it eventualy reach warp speed and beyond?

 

No, it shouldn't.

 

First, there is the fact that the gravitational force of the planets is position-dependent, and becomes weaker with increasing distance.

 

Second, there is the fact that once the probe has passed the planet, the planet attracts the probe, thus working against the acceleration away from the solar system!

 

Third, what you have got to understand is that the space probe is not even close to approaching relativistic speeds, and so there is no apparent difference between the Galilean and Lorentz velocity addition law. But when the speed of the probe approaches the speed of light, it is found that a constant force does not produce a constant acceleration. There is a "diminishing returns" effect by which it costs more energy to attain that extra bit of speed, such that it takes an infinite amount of energy to attain light speed.