Movement Paradox

[Zezon Vice]

So I don't know exactly where this concept came from and I know it has been thought of before me, perhaps even on this very forum in a thread I was incapable of finding, but I present this idea anyway.

If Space-Time can be divided infinitely into smaller halves and eventually into infinitely small components of a whole, then doesn't that mean that for motion to occur that the object doing so is moving over an infinite distance? That would mean an infinite amount of energy would be required to move something and even then if it moved an infinite amount of space then really it only moved an infinitely small amount of space leaving it infinitely close to where it moved from.

So I have heard arguments saying that say there is only an infinite amount of divisions possible in theory, but this doesn't seem to make sense in the slightest to me for multiple reasons.

My own answer originally was that we aren't in a constant flow universe. Rather our universe is the result of a massive series of equations all controlling a different element and creating what we know to be Laws. One most importantly for the rate at which the others can derive an answer. Ill call this the time Law. This is what we get, the answers as though they were the frames per second flashing with their answers compiling as our universe. Then there are those that regulate the variables in the equations. Which is an infinite cycle I realize. Then I thought there must be a Law that dictates how much variation is allowed in one flash of the time Law.

So this time Law is a key, or rather the one controlling it is, because with limits we can force things to answer DNE = does not exist. These DNE entries in the time Law would force things to "flash" from being to not back to being with different answers which then gives change and the illusion of motion.

What is your take?

 

[nexion]

This is called Zeno's paradox, and if I remember correctly, it was put to rest by the creation of calculus. I don't exactly know the math though.

EDIT: The idea is that a finite answer exists for an infinite sum of fractions.

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

 

[themirror]

Somewhat related:
http://en.wikipedia.org/wiki/Cantor_set

 

[Black Rose]

My own answer originally was that we aren't in a constant flow universe. Rather our universe is the result of a massive series of equations all controlling a different element and creating what we know to be Laws. One most importantly for the rate at which the others can derive an answer. Ill call this the time Law. This is what we get, the answers as though they were the frames per second flashing with their answers compiling as our universe. Then there are those that regulate the variables in the equations. Which is an infinite cycle I realize. Then I thought there must be a Law that dictates how much variation is allowed in one flash of the time Law.

So this time Law is a key, or rather the one controlling it is, because with limits we can force things to answer DNE = does not exist. These DNE entries in the time Law would force things to "flash" from being to not back to being with different answers which then gives change and the illusion of motion.

What is your take?

Cellular automaton

skip to minute 14:00
https://www.youtube.com/watch?v=60P7717-XOQ#

 

[Vrecknidj]

If Space-Time can be divided infinitely into smaller halves and eventually into infinitely small components of a whole, then doesn't that mean that for motion to occur that the object doing so is moving over an infinite distance?

Space-time cannot be divided infinitely. That's the solution. There is no paradox.

 

[Etheri]

Space-time cannot be divided infinitely. That's the solution. There is no paradox.

I'd like some reasoning and / or explenation as to why...

If Space-Time can be divided infinitely into smaller halves and eventually into infinitely small components of a whole, then doesn't that mean that for motion to occur that the object doing so is moving over an infinite distance?

A sum of an infinite ammount of positive terms can still be finite. So no, there's no infinite distance, which is the flaw in your reasoning.

 

[Duxwing]

Movement, or motion, is a change in distance (dD) during a period of time (t). All variables must have units, and in SI, these would be the meter and the second. Since the meter is a finite distance, it can theoretically chop the universe into tiny but finite pieces. No infinite division implies no paradox because distance can now be defined rather than measured. Problem solved.

-Duxwing

 

[nexion]

Movement, or motion, is a change in distance (dD) during a period of time (t). All variables must have units, and in SI, these would be the meter and the second. Since the meter is a finite distance, it can theoretically chop the universe into tiny but finite pieces. No infinite division implies no paradox because distance can now be defined rather than measured. Problem solved.

-Duxwing

1. Measurements are systems imposed on reality by man (ie. they have no objective existence)
2. Have you never heard of centimeters, millimeters, nanometers?

Values in the universe tend to not be discrete numbers. No, the universe operates in infinitesimals. Nothing is ever discrete, man again imposes these discrete values onto reality, and thereby gives reality bounds within which the weak human minds have understanding.

 

[Etheri]

Nothing is ever discrete, man again imposes these discrete values onto reality, and thereby gives reality bounds within which the weak human minds have understanding.

-Everything periodic can be represented in a discrete set (fourier). (thus waves can be represented using discrete values, even in reality.)
-In quantum, bound states are generally discrete, and thus so are the corresponding eigenvectors (energy levels, etc). This implies the energy lost / gained to go between these levels is also quantified (discrete levels). Also, what happened to the idea of 'quanta' (photons, electrons and other particles with quantified energy, despite not being precisely defined.)

@Duxwing : I don't see your point. While your approach is the typical numeric approach, this is not the one used in maths and theoretical physics. Also, while I agree that every distance CAN be put into meters and any period of time can be put expressed in seconds (given respective POV), I do not see why SI changes anything over other measurement systems. The results should, after conversion, be the same regardless.

 

[Architect]

Spacetime isn't infinitely dividable, there is a hard limit at the Planck length. Fundamentally space-time is digital or discrete.

 

[Etheri]

Spacetime isn't infinitely dividable, there is a hard limit at the Planck length. Fundamentally space-time is digital or discrete.

Spacetime (4D) or phase space (6D)?
As far as i'm aware, the limitation is on px and x, and there are no direct correlations between py and x, for example? How do you condense phase space into spacetime? (I simply do not know or understand, these are not rethorical questions)

 

[Duxwing]

-Everything periodic can be represented in a discrete set (fourier). (thus waves can be represented using discrete values, even in reality.)
-In quantum, bound states are generally discrete, and thus so are the corresponding eigenvectors (energy levels, etc). This implies the energy lost / gained to go between these levels is also quantified (discrete levels). Also, what happened to the idea of 'quanta' (photons, electrons and other particles with quantified energy, despite not being precisely defined.)

@Duxwing : I don't see your point. While your approach is the typical numeric approach, this is not the one used in maths and theoretical physics. Also, while I agree that every distance CAN be put into meters and any period of time can be put expressed in seconds (given respective POV), I do not see why SI changes anything over other measurement systems. The results should, after conversion, be the same regardless.

I used SI as an example for the sake of the argument:

All variables must have units, and in SI, these would be the meter and the second.

Nevertheless, perhaps I should have explicitly defined why I chose the meter and second. Regarding your proof from quantum physics, it is readily apparent that we're solving different problems: you proved the quantification of matter and energy (I'm not sure about Fourier, though) while I proved that space isn't quantifiable without an axiomatically true system of measurement, and that therefore one can move about without having to cross infinitely many "checkpoints".

Moreover, even if man (and woman. child, and trans/nonvestite, too-- curse English!) were constrained by having to cross these checkpoints before reaching his destination, the very first checkpoint would be 1/infinity units of distance away, which, if my fuzzy math is correct, should be traversable in 1/infinity units of time. Moreover, the model in which he cannot move must be wrong because otherwise, no-one would know about it because the lips of the smart-aleck philosopher who thought of this riddle would have been mercifully zipped shut.

-Duxwing

PS I mean, I'm all for theory, but this is ridiculous!

 

[Etheri]

We have diffrent approaches, and I think while our outcome (this theory is wrong) is the same, I do not think your argument holds true.

I used SI as an example for the sake of the argument:

All good. I didn't mean to attack your choice, I meant to raise the question 'why must we make a choice, isn't it entirely arbitrairy?' (as long as our systems are, in relation to eachother and on themselves, well defined.)

I proved that space isn't quantifiable without an axiomatically true system of measurement, and that therefore one can move about without having to cross infinitely many "checkpoints".

I don't see why this is true. What is wrong with an infinite ammount infinitesimals making up an finite length? This isn't even far fetched, it's one of the main principles of differential calculus. I'm not arguing in favour or against OP, i'm arguing that there's a major flaw in your proof / argument.

the very first checkpoint would be 1/infinity units of distance away, which, if my fuzzy math is correct, should be traversable in 1/infinity units of time.

Exactly, there's a correlation between dx and dt (iff there's a correlation between x and t, atleast), and thus you can work with an infinite ammount of infinitely small parts. I see no paradox, only a lack of maths.

 

[BigApplePi]

If Space-Time can be divided infinitely into smaller halves and eventually into infinitely small components of a whole, then doesn't that mean that for motion to occur that the object doing so is moving over an infinite distance?

1/2 + 1/2 =
1/4 + 1/4 + 1/4 + 1/4 =
1/8 + 1/8 + 1/8 + 1/8 + 1/8 + 1/8 + 1/8 + 1/8 =
1/16 + ... + 1/16 =
1/32 + ... + 1/32, etc.

 

[Architect]

Spacetime (4D) or phase space (6D)?
As far as i'm aware, the limitation is on px and x, and there are no direct correlations between py and x, for example? How do you condense phase space into spacetime? (I simply do not know or understand, these are not rethorical questions)

What is "Phase space"? I'll answer by saying probably something from Deepak Chopras belly button.

Space time can be described by any coordinate system you choose, typically Euclidean but that isn't necessary. It can be non isotropic, for instance in a bowl of jelly. There are substances that have non isotropic properties, meaning that (say) when you press it the material translates the compression in a horizontal direction (jelly is kind of like this). That kind of space is non Euclidean and needs description by tensors instead of vectors.

The only time phase comes up that I've ever seen is if you are in a polar coordinate system. This is still Euclidean - the 4 dimensions are still orthogonal, but are more conveniently described as circles. Electromagnetic radiation falls into this cateogry.

Regardless Planck Length still holds because it is a fundamental unit.

 

[BigApplePi]

Regardless Planck Length still holds because it is a fundamental unit.

Why can't we break this "Planck Length"? I am not a physicist. If it has to do with Heisenberg, I last saw that not everyone goes with him.

 

[Architect]

Why can't we break this "Planck Length"? I am not a physicist. If it has to do with Heisenberg, I last saw that not everyone goes with him.

All science is subject to disbelief and alternate theories so it's natural there are people who question the Uncertainty principle - however, that theory is quite verified, and so the haters would need to come up with something better.

Planck Length, approximately 1.616199 × 10-35 meters, is a beautiful little expression using only three of the most fundamental physical constants; the speed of light, Planck's constant and the gravitational constant, thus tying general relativity (gravity) and quantum mechanics together nicely.

It's so small that we'll surely never probe this level, but according to uncertainty equations this is the smallest length at which we can even theoretically measure anything. It seems to be built into the uncertainty of the universe, below this we cannot probe. This has led some people to hypothesize that the universe is actually a digital program running on a computer, and there are experiments to verify or disprove this hypothesis.

So ultimately, to the best of our knowledge, the universe/space/time (there is an equivalent Planck Time) is not continuous, but very much discrete. We move through space by hopping across Planck Length units, and time ticks forward in Planck time units, if you like.

 

[Jah]

Well... Considering Relativity, you never actually move.

Everything else moves in relation to you, but you'll always remain the center of your own perspective of the universe, (the size of the universe being exactly (age of universe) * c . Starting from you in the center, and all the way out, still expanding.)

seems legit.


You'll only move relatively to someone else, and the whole paradox kinda breaks down as soon as you introduce calculus.

(actually it's kinda the crux of Calculus, as Calculus is about what the relation between space and time is on the infinitesimal level. otherwise stated as the infinitely small space crossed in the infinitely small time or dx/dt. )

 

[Architect]

Well... Considering Relativity, you never actually move.

Everything else moves in relation to you, but you'll always remain the center of your own perspective of the universe, (the size of the universe being exactly (age of universe) * c . Starting from you in the center, and all the way out, still expanding.)

No, you can verify your motion by the acceleration required to bring you up to speed (measured as an apparent gravitational field in your ship). Once you are at a set velocity it does appear that the rest of the universe is moving instead of you, which is why they are described at two inertial frames with the speed of light as constant to both. This constancy of the speed of light also is proof of your motion (relative to another inertial frame) because your clocks and rulers will measure differently. The quick and dirty measurement is to check in with your twin; whoever is younger is the one that moved faster.

and the whole paradox kinda breaks down as soon as you introduce calculus.

What?

 

[Jah]

Zeno's paradox... breaks down...
doesn't it ?

I mean, it's all about slicing movement into finer and finer parts of space and time, which basically boils down to infinitesimals, that is to say calculus.


Even if you're accelerated, the edge of the universe, that is to say, the furthest away a photon can be from you, will still be (age of universe) * c.
It might red-shift, but it will still move away from you at the speed of light... thus you'll always be in the center of the expanding universe. right ?

 

[BigApplePi]

You just said something and that's what I recall my objection was:

It's so small that we'll surely never probe this level, but according to uncertainty equations this is the smallest length at which we can even theoretically measure anything.

What does the ability to measure something have to do with dividing it? Just because my knife can't cut it doesn't mean it can't split spontaneously, wear out or evaporate. That would make whatever is at that level smaller.

I don't recall what I read about String Theory and its vibrating strings and how they relate to Planck's constant, but if the strings were of that length, the vibrations themselves would be smaller.

 

[Etheri]

Zeno's paradox... breaks down...
doesn't it ?

I mean, it's all about slicing movement into finer and finer parts of space and time, which basically boils down to infinitesimals, that is to say calculus.

Even if you're accelerated, the edge of the universe, that is to say, the furthest away a photon can be from you, will still be (age of universe) * c.
It might red-shift, but it will still move away from you at the speed of light... thus you'll always be in the center of the expanding universe. right ?

That's exactly what i've been trying to say. /agree

I don't recall what I read about String Theory and its vibrating strings and how they relate to Planck's constant, but if the strings were of that length, the vibrations themselves would be smaller.

The 'strings' aren't actual strings as we see them in our 'reality'. They're more a conceptual. However, i'm not read up on string theory enough to really go in on it

What is "Phase space"? I'll answer by saying probably something from Deepak Chopras belly button.

Space time can be described by any coordinate system you choose, typically Euclidean but that isn't necessary. It can be non isotropic, for instance in a bowl of jelly. There are substances that have non isotropic properties, meaning that (say) when you press it the material translates the compression in a horizontal direction (jelly is kind of like this). That kind of space is non Euclidean and needs description by tensors instead of vectors.

The only time phase comes up that I've ever seen is if you are in a polar coordinate system. This is still Euclidean - the 4 dimensions are still orthogonal, but are more conveniently described as circles. Electromagnetic radiation falls into this cateogry.

Regardless Planck Length still holds because it is a fundamental unit.

I may have used the term 'phase space' a tad too freely. http://en.wikipedia.org/wiki/Phase_space.
What I meant was : Heisenberg states :

http://upload.wikimedia.org/math/6/d/b/6db69c40c8c74596f81ab37f5bf091de.png

This only limits x and px, y and py, z and pz. (p being momentum, x y z being orthogonal directions). Phase space is a 6D representation of a particle, 3D for coordinates (x,y,z) and 3D for momentum (px, py, pz). Rather than describing a particle through (x,y,z,t), you describe a particle through (x,y,z,px,py,pz).
Classical physics : Every particle is represented by a point in this space. (a single vector)
Quantum : every particle is not a point in this 6D space, but a volume of h³. (h = planck's constant).
In both cases, this space is still euclidean! (Tho the subjects are no longer points in the case of quantum.)

(In classical terms, thus back when gibbs introduced this idea, they would integrate over the phasespace to find statistical properties. In quantum mechanical terms, these integrals changed into sums. These sums / integrals were used to find theoretical expressions for thermodynamical expressions (or rather, since they were experimentally known, help understand them.)

Where lies my problem : I know heisenberg from planck's constant, and the limitation on the product ([dv]]x *[dv]px). This is a product. Reducing the spread (thus incertainty) on x by increasing the incertainty on px is theoretically possible! As far as planck's length and the limitation sole on 'x' (and perhaps t?), I was told* that there is no consesus, tho there are hypothesises in regards to plancks length etc.
Either way, feel free to enlighten me.

(Tl dr, i'm familiar with the limitation on the product, not on the individual terms. Thus I was questioning if you can chop up spacetime (4D) from what I know.)

*Yes, I know, I should do my own research. In my defence, a professor in quantum should be a fairly reliable source of info, since it should be his expertise?

 

[Psychic Child]

Plank's Length is like pixels of the reality.

 

[Architect]

You just said something and that's what I recall my objection was:
What does the ability to measure something have to do with dividing it? Just because my knife can't cut it doesn't mean it can't split spontaneously, wear out or evaporate. That would make whatever is at that level smaller.

I don't recall what I read about String Theory and its vibrating strings and how they relate to Planck's constant, but if the strings were of that length, the vibrations themselves would be smaller.

I think the problem is that you're mixing up physics and your imagination. You can imagine dividing up something further in a mathematical sense, but our present knowledge of physics just says you can't. You get to a point where further division is impossible, because you can't have certainty of what you're dividing, and where it is. If you could imagine being that small (which doesn't make sense because now you're greatly smaller than the wavelength of light and width of a proton so therefore can't see or measure anything because it would be like trying to see a football by throwing the stadium at it, but bear with me). All you would 'see' is some kind of 'quantum foam' where everything is uncertain.

I won't get into String Theory because I think that theory is bullshit, and at any rate there is zero experimental proof.

 

[Psychic Child]

I think the problem is that you're mixing up physics and your imagination. You can imagine dividing up something further in a mathematical sense, but our present knowledge of physics just says you can't. You get to a point where further division is impossible, because you can't have certainty of what you're dividing, and where it is. If you could imagine being that small (which doesn't make sense because now you're greatly smaller than the wavelength of light and width of a proton so therefore can't see or measure anything because it would be like trying to see a football by throwing the stadium at it, but bear with me). All you would 'see' is some kind of 'quantum foam' where everything is uncertain.

I won't get into String Theory because I think that theory is bullshit, and at any rate there is zero experimental proof.

you are just like one of my old friends

 

[BigApplePi]

@Architect

I think the problem is that you're mixing up physics and your imagination. You can imagine dividing up something further in a mathematical sense, but our present knowledge of physics just says you can't. You get to a point where further division is impossible, because you can't have certainty of what you're dividing, and where it is. If you could imagine being that small (which doesn't make sense because now you're greatly smaller than the wavelength of light and width of a proton so therefore can't see or measure anything because it would be like trying to see a football by throwing the stadium at it, but bear with me). All you would 'see' is some kind of 'quantum foam' where everything is uncertain.

I have the impression you are looking for hard physics ... that is things we know. The question I'm asking about division or smaller than that length should have nothing to do with human beings and what we ourselves can or can't do. Imagination, active dividing, and relative sizes have to do with human beings. You did say, "our present knowledge of physics just says you can't." That I will buy.

I won't get into String Theory because I think that theory is bullshit, and at any rate there is zero experimental proof.

Is that theory bullshit because there is no proof or simply because it is a theory? I don't think it makes sense to call all theories "bullshit" if they are possibilities. Theories describe possibilities and they are at present neither true nor false.

 

[Psychic Child]

I dont understand this planck lenght myself too actually.
It is still possible to mathematically divide it.
And BTW, there is no reason to believe anything.
all the 5 senses can be illusion.
And all I had done is read about planck's lenght.
Show me the proof, show me the experiement, show me the result, show me that what am i seeing are not illusions, and i am not hypnotized.
I dont even know if I am hypnotized at this moment or not.'
Do you people actually exists or not?
There is nothing to believe.
In a way all we have are possibilities.
Neither true nor false.
All is BS.

 

[Psychic Child]

whatever who cares

 

[Architect]

I have the impression you are looking for hard physics ... that is things we know.

I don't know what hard (or presumably soft) physics is, but yes I'm talking about things we know. For instance, we know that the speed of light is constant in differing inertial frames. We've theoretically derived this and have verified it in day to day life (muons from the upper atmosphere wouldn't make it to the surface without this being true, and those are easy to see in a cloud chamber).

Planck length has is a theoretical result and we are nowhere near verifying it experimentally. So we see here three levels of ... 'truth' ...

1. At the most abstract level is math. If you can't describe it mathematically then you can't do any physics with it.
2. Next you need some theoretical physics basis for the idea. You don't have to have this to proceed to the next step, but ultimately if you have the final step you have to have this one
3. Finally you need experimental verification. As stated this can and does occur before theoretical explanation.

The idea of infinite division stops at 1. Planck length stops at 2, and there is no 3. because we are no where near measuring distance at those scales.

So to say we can imagine distance less than Planck means nothing, I can also imagine angels flying out of my ass too. For it to have as much validity as Planck then we need a competing physical theory for why Planck is not the smallest distance - and we don't have a hint of that.

Is that theory bullshit because there is no proof or simply because it is a theory? I don't think it makes sense to call all theories "bullshit" if they are possibilities. Theories describe possibilities and they are at present neither true nor false.

String theory is at 2, but is a poor one. It's not particularly falsifiable, really its more descriptive than prescriptive. A good theory is one that tells us something we didn't already know, then we go out and find out yes, it is true. For example, GR predicted anolomies in the orbit of Mercury because of its proximity to the Sun. Einstein had the best week of his life when an eclipse in South America verified that the aberration in Mercuries orbit agreed with what his theory predicted.

I dont understand this planck lenght myself too actually. It is still possible to mathematically divide it.

Yes, see above.

 

[Duxwing]

We have diffrent approaches, and I think while our outcome (this theory is wrong) is the same, I do not think your argument holds true.


All good. I didn't mean to attack your choice, I meant to raise the question 'why must we make a choice, isn't it entirely arbitrairy?' (as long as our systems are, in relation to eachother and on themselves, well defined.)

I don't see why this is true. What is wrong with an infinite ammount infinitesimals making up an finite length? This isn't even far fetched, it's one of the main principles of differential calculus. I'm not arguing in favour or against OP, i'm arguing that there's a major flaw in your proof / argument.


Exactly, there's a correlation between dx and dt (iff there's a correlation between x and t, atleast), and thus you can work with an infinite ammount of infinitely small parts. I see no paradox, only a lack of maths.

Now, looking back on my point, I think discrete, rather than quantifiable, would have been a better word; I was trying to avoid a malapropism with a root "quantum," yet still communicate the idea of discrete chunky-ness. Nevertheless, I entirely agree with you that an infinite number of infinitesimals can add up to a finite length. *high-five for excellent proof*

-Duxwing

 

[scorpiomover]

What you are talking about, is one of Zeno's Paradoxes. It's based on Archimedes' Axiom of Continuity, that given any 2 points, there is a point between them. Either Archimedes' Axiom is true, i.e. that the universe is continuous, or Archimedes' Axiom is false, i.e. that the universe is discrete.

If the universe is continuous and the axiom is true, then by Zeno's argument, motion should be impossible. Yet we see that motion occurs all the time.

If the universe is discrete, then Archimedes' Axiom is false. But Archimedes' Axiom is an axiom required for Calculus. So if Archimedes' Axiom is false, then we cannot say that Calculus is true. But we can see from our calculations, that Calculus works, and is the basis for most of Physics, especially quantum physics. So Calculus works, and that in turn shows that the universe is continuous.

Either way, we have a problem. Hence why it is called a paradox.

 

[Jah]

Either Classical or Quantum models of the universe, in other words.


For us, it seems that quantum models yield the most consistent and precise predictions, thus we can say that the quantum model seems to be the most beneficial to use. (being of a purely pragmatic mindset now, I refuse to call it "correct" or "False" )


The universe may well be continuous, but to us it seems to stop at the planck-length and appear discrete.
(that is, we seem unable to look deeper than the planck-length. so to us, that's the edge of the infinitesimal, the Chunks of the chunky universe. )


Who cares about whether or not the model "actually" is True. it works.

We just have to sort out when it works and when it doesn't and apply the most appropriate model(s) when we try to deduce/induce another theory to aid us in further studies.

 

[Etheri]

What you are talking about, is one of Zeno's Paradoxes. It's based on Archimedes' Axiom of Continuity, that given any 2 points, there is a point between them. Either Archimedes' Axiom is true, i.e. that the universe is continuous, or Archimedes' Axiom is false, i.e. that the universe is discrete.

This is correct

If the universe is continuous and the axiom is true, then by Zeno's argument, motion should be impossible. Yet we see that motion occurs all the time.

This is false

If the universe is discrete, then Archimedes' Axiom is false. But Archimedes' Axiom is an axiom required for Calculus. So if Archimedes' Axiom is false, then we cannot say that Calculus is true. But we can see from our calculations, that Calculus works, and is the basis for most of Physics, especially quantum physics. So Calculus works, and that in turn shows that the universe is continuous.

Either way, we have a problem. Hence why it is called a paradox.

Requirement for calculus : correct, yet calculus proves how motion is possible while archimedes' axiom is true. There is no paradox, only bad understanding of maths.

Zeno's paradox is faulty, and calculus rips through it. Regardless of whether or not the universe is discrete, regardless of archimedes' axiom being true or false, zeno's paradox is wrong.

 

[SpaceYeti]

It's a silly paradox, because infinitely small points in space require zero time to cross.

 

[pernoctator]

You mean infinitely small time.

 

[SpaceYeti]

I don't see what the difference is. An infinitely small number is point infinite zeroes followed by a one... but due to infinite zeroes, you never get to the one. Infinitely small = 0. Just like point infinite nines = 1.

 

[pernoctator]

Well I don't have much of a math background, but here's how I see it: We're talking about infinitely small fractions -- in other words, 1 / infinity, the reverse of which is [infinitely small number] * infinity = 1. But, 0 * infinity = 0. That's a pretty significant difference, because it would mean all distances take 0 time to reach and we are currently everywhere.

 

[Etheri]

I don't see what the difference is. An infinitely small number is point infinite zeroes followed by a one... but due to infinite zeroes, you never get to the one. Infinitely small = 0. Just like point infinite nines = 1.

Well I don't have much of a math background, but here's how I see it: We're talking about infinitely small fractions -- in other words, 1 / infinity, the reverse of which is [infinitely small number] * infinity = 1. But, 0 * infinity = 0. That's a pretty significant difference, because it would mean all distances take 0 time to reach and we are currently everywhere.

It's as pernoctator states : There's quite the diffrence between something which 'approaches 0' and something which is 'identically zero'. Limits as "0 * inf" are undefined, but can be found. The answer greatly depends how 'zero' this 'zero' is (and how inf the inf is in comparison!).

Spoiler

The easiest way to see this is to look at a simple rational function. x^a / x^b, then looking at the limit for x->0 and for x->inf.
if a == b, then this limit is a defined number.
if a > b then limit x -> 0 is 0 and limit x -> inf is inf.
if a < b, then limit x -> 0 is inf and limit x -> inf is 0

you can always write the resulting function as x^(a-b), but nothing stops you from seeing it as a product of x^a * x^(-b). Obviously, if a = b, then this function is simply x ^a * x^(-a) = 1, which clearly has 1 as limit everywhere!

the limit of exp(-x)*x^n for x -> inf, for any given n, is always 0, simply because exp(x) is dominant over any power of x. The simplest thing is usually to look at the limit of the product directly.
exp(-10) = 4.5e-5
exp(-100) = 3.7e-44
To counter this effect you'd need a function which gets to big numbers equally fast, which is impossible with a simple power function. (Note that it is possible to counter these two numbers I gave as example.)

 

[BigApplePi]

@Architect I abstracted just the part where you talked about theory since I wanted to comment on that.

Originally Posted by BigApplePi Is that theory bullshit because there is no proof or simply because it is a theory? I don't think it makes sense to call all theories "bullshit" if they are possibilities. Theories describe possibilities and they are at present neither true nor false.

I can also imagine angels flying out of my #%& too. For it to have as much validity as Planck then we need a competing physical theory for why Planck is not the smallest distance - and we don't have a hint of that.

Any theory is made up by human beings, not nature. Human beings love to find possible explanations, proof or no proof. What you said about angels is one of the possibles, but not adequately explanatory for most since it's hard to make the connection.

... A good theory is one that tells us something we didn't already know, then we go out and find out yes, it is true.

Spoken like a good experimental physicist. However String Theory does offer a possible explanation and although we may never prove it, it has a certain appeal. String Theory may not yet be physics, but perhaps we can call it, "metaphysics" ... and many find that legit.

 

[Duxwing]

@Architect I abstracted just the part where you talked about theory since I wanted to comment on that.


Any theory is made up by human beings, not nature. Human beings love to find possible explanations, proof or no proof. What you said about angels is one of the possibles, but not adequately explanatory for most since it's hard to make the connection.

Spoken like a good experimental physicist. However String Theory does offer a possible explanation and although we may never prove it, it has a certain appeal. String Theory may not yet be physics, but perhaps we can call it, "metaphysics" ... and many find that legit.

I love theory. It's so nebulous, so profound, and so much fun! Where else can you take a system to the limit of absurdity and then push it even further?

-Duxwing

 

[Paintzee]

Imagine a length, any length approaching zero to approaching infinity. This is the x axis. Now imagine a second length, its only criteria be that it be equal in length to the first length. This is the y axis. These two lengths represent any length in space. If the length is defined as one unit, using Pythagoras theorem the hypotenuse is an irrational number, the square root of two. This length in space must also exist if the other two lengths are absolute, therefore none of the lengths can be absolute. An absolute point cannot exist, only a distribution around a point. For a black hole no singularity just a distribution within the event horizon.

If you have an adjacent area of space it’s distribution must overlap its neighbour therefore Zeno’s paradox does not exist.

 

[scorpiomover]

This is correct

This is false

Judgement without any basis of explanation.

Requirement for calculus : correct, yet calculus proves how motion is possible while archimedes' axiom is true. There is no paradox, only bad understanding of maths.

Zeno's paradox is faulty, and calculus rips through it. Regardless of whether or not the universe is discrete, regardless of archimedes' axiom being true or false, zeno's paradox is wrong.

Fraid not. Calculus is based on the limit function. The limit function is really a logical statement, which does not say what happens when you reach the point being examined, only that if you take any non-zero error margin, then there exists an open neighbourhood in the domain around the point, whose results all fit within the error margin. This in turn means that a limit doesn't establish absolute truth, only that whenever you are testing for a value with a certain margin, the value should always fit within your error margin, provided that everything involved is continuous, because if it is all continuous, then the value couldn't go that far away from everything around it, and so also has to fit within the error margin.

In terms of continous motion, what calculus says, is that if you were to take a video of the motion, no matter how many frames per second your video takes, the individual sequenced frames appear as if the object is in motion, and from that, we infer that the object should be in motion, without actually saying so, or having any proof of such.

That is why, when it comes to limits, like a differential, we say that [ f(x+h) - f(x) ] / h tends to df/dx, as h tends to 0, but never actually say it becomes or equals or is df/dx.

Calculus is deliberately phrased in this way, to avoid even touching the end limit of the breakdown of motion, where Zeno's Paradox can occur. In calculus, we simply gloss over this, and let people assume that the convergence results in a reality, like this:

It's a silly paradox, because infinitely small points in space require zero time to cross.

 

[BigApplePi]

Infinitely small = 0. Just like point infinite nines = 1.

Infinite is undefined so it doesn't equal anything. You can trudge out there as far as you want though ... until you get tired.

 

[SpaceYeti]

Infinite is undefined so it doesn't equal anything. You can trudge out there as far as you want though ... until you get tired.

If infinite were undefined, then it could be used as a variable in algebra. It actually means that the number we're discussing "has no limit". Something so small that there's no limit to it's smallness has to be zero, because you can't get smaller than that.

Another way of thinking about it is as point infinite zeroes with a one on the end. There's no limit to the number of zeroes, which means the one doesn't actually exist. If the one was at a finite position, then we'd have an actual fraction, but the 1 is effectively nonexistent because you never get to it, because there's no limit to the number of zeroes which precede it.

 

[scorpiomover]

If infinite were undefined, then it could be used as a variable in algebra. It actually means that the number we're discussing "has no limit".

Define "has no limit". For instance, how many digits does it have? We covered it in 1st-year of my maths degree. There is no such number as infinity. You can only write it as a symbol about which you can make the logical statement, that whatever number exists, it is bigger. But it doesn't have a proper numerical definition, like 2.

Something so small that there's no limit to it's smallness has to be zero, because you can't get smaller than that.

That's another thing covered in 1st-year real analysis. You can't prove that the smallest number is zero. But you can look at the null sequence, and show that "1/x tends to 0 as x tends to infinity", which is the standard terminology), meaning, that 1/x gets closer and closer to 0 as x gets bigger and bigger. It doesn't make much of a difference in high-school, because high-school maths is planned out to be rather simplistic and not get into too many philosophical issues. But when you get to study it in university, at least in British ones, you have to learn it properly, and then you discover that it's actually extremely tricky.

 

[SpaceYeti]

Define "has no limit". For instance, how many digits does it have? We covered it in 1st-year of my maths degree. There is no such number as infinity. You can only write it as a symbol about which you can make the logical statement, that whatever number exists, it is bigger. But it doesn't have a proper numerical definition, like 2.

Has no limit as in "not limited". As in "there is no end". If it had a quantifiable number of digits, then we just figured out it's limit, and that makes it necessarily not infinite. And while I agree that reality contains no such thing as infinite in any meaningful way, the concept can still be considered.

That's another thing covered in 1st-year real analysis. You can't prove that the smallest number is zero. But you can look at the null sequence, and show that "1/x tends to 0 as x tends to infinity", which is the standard terminology), meaning, that 1/x gets closer and closer to 0 as x gets bigger and bigger. It doesn't make much of a difference in high-school, because high-school maths is planned out to be rather simplistic and not get into too many philosophical issues. But when you get to study it in university, at least in British ones, you have to learn it properly, and then you discover that it's actually extremely tricky.

Yes, the closer a number gets to being divided by zero, the closer it gets to infinity... because something divided by infinity is zero. I already shared my proof. Agree or disagree, but basically everything you're saying is irrelevant. A number that is zero point infinite zeroes, meaning those zeroes never end, is zero, because anything which would follow those infinite zeroes never happens, as the zeroes never end such that you ever get to those final numbers.

 

[scorpiomover]

@SpaceYeti

Has no limit as in "not limited". As in "there is no end". If it had a quantifiable number of digits, then we just figured out it's limit, and that makes it necessarily not infinite. And while I agree that reality contains no such thing as infinite in any meaningful way, the concept can still be considered.

What about the super-number of "has no limits", the number beyond that number? And the number beyond that number? And the number beyond that? And, more importantly, what about the number, that is beyond all of those numbers, a number that is beyond even the description of "there is no end"?

Yes, the closer a number gets to being divided by zero, the closer it gets to infinity... because something divided by infinity is zero. I already shared my proof. Agree or disagree, but basically everything you're saying is irrelevant. A number that is zero point infinite zeroes, meaning those zeroes never end, is zero, because anything which would follow those infinite zeroes never happens, as the zeroes never end such that you ever get to those final numbers.

That's what people are taught in high school. But the only way to prove it, is to prove it, literally, that is, to calculate 1 divided by infinity. You can't infinity in a calculator, or write it down on paper. So we can't actually be sure of what 1/infinity is. We can only know it, intuitively.

 

[SpaceYeti]

Yes, we humans are far too stupid to ever understand an idea that isn't written down in it's longest possible format and in front of us, ignore that it's us who writes it down such that it gets written in the first place! Ignore that the reason infinity cannot be written in literal numbers, but instead we use language or that goofy sideways 8, is due to lack of space to write it, not lack of understanding the concept!

By the way, in school you're taught that you can't divide by zero, not that it equals infinite. Same difference, though, as infinite is useless and renders a mathematical formula un-solvable. Infinity's not a real thing, but that doesn't mean we can't consider or understand the concept.

You don't actually argue because you think you have pertinent information, do you? You just like to nay-say. I would ask what your problem is, but you've already spilled your guts about how smart kids picked on you, so now you hate smartness. Good on you, but wouldn't you rather just jerk-off, or something?

 

[scorpiomover]

Yes, we humans are far too stupid to ever understand an idea that isn't written down in it's longest possible format and in front of us, ignore that it's us who writes it down such that it gets written in the first place! Ignore that the reason infinity cannot be written in literal numbers, but instead we use language or that goofy sideways 8, is due to lack of space to write it, not lack of understanding the concept!

Easy, buddy. You have the capacity to understand. But that ability doesn't reach potential, unless you make the effort to use it.

If you mean that you can't write down infinity in any symbol, you just pointed out that you can. If you mean that you can't write down infinity in a finite number of decimal digits, you can't write down a seventh in a finite number of decimal digits either. Or a third. Or a thirteenth. Or plenty of fractions.

That, you could have figured out by yourself.

By the way, in school you're taught that you can't divide by zero, not that it equals infinite.

Actually, I recall being taught BOTH. You're taught to not divide by zero, when trying to solve algebraic equations, because then one can end up with an indeterminate answer. You're also taught that 1/0 is infinity.

Then when you get to study maths at university, you discover that much of what you learned was a general rule of thumb, that is a huge simplification of complex concepts, and you realise that your teachers were dumbing things down for everyone.

Same difference, though, as infinite is useless and renders a mathematical formula un-solvable. Infinity's not a real thing, but that doesn't mean we can't consider or understand the concept.

Actually, it depends on the situation. In many situations, it makes the nature of the mathematical formula resolve to an infinite number of potential solutions, and removes anything we could have learned from it. In many situations, it's fundamentally necessary to solve the problem. It all depends on the problem.

You don't actually argue because you think you have pertinent information, do you? You just like to nay-say.

Yeah. Sure. I'm an typical nay-saying INTP, and you're just a super-capable, never-wrong INTJ?

I would ask what your problem is, but you've already spilled your guts about how smart kids picked on you, so now you hate smartness.

I didn't get picked on by the kids who were really good at maths and science. They liked me. I got picked on by the popular kids, the kids who liked to claim they knew everything, to impress the girls. You know the type. They're the ones who get annoyed when other people have different views, that are so reasonable, that they can't pretend they are stupid, and then try to discredit their viewpoints, by claimng ad hominem attacks. Yeah, I don't like THAT. But I wouldn't call it "being smart". I believe they're called "jerk-offs".

Good on you, but wouldn't you rather just jerk-off, or something?

See what I meant about jerk-offs?