# The Naked Scientists Forum

### Author Topic: What is the Xeno motion paradox?  (Read 11576 times)

#### David Cooper

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##### Re: What is the Xeno motion paradox?
« Reply #25 on: 09/08/2012 23:16:26 »
It's really quite simple: at every level of chopping, you have a granular solution, each one with a finer grain than the one before it, but because there are an infinite number of these granular solutions, the solution isn't granular. In other words, it's turtles all the way down.

The infinity is the elephant in the room - it's there, but people don't want to see it and have trained themselves not to do so in order to maintain their belief that calculus solves a problem which it actually fails to address. We all know that the time is finite and that things can actually move - Zeno understood that fully well. His paradox is focused squarely on the infinity issue, and calculus doesn't engage with that. Calculus gives you exact answers to questions like "what would the answer be if you could add everything up infinitely", but it simply does it through a trick.

#### William McCormick

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##### Re: What is the Xeno motion paradox?
« Reply #26 on: 09/08/2012 23:24:51 »

An arrow does not just move ahead. Any object in this universe made of matter, is expanding and contracting, constantly, at some frequency. When you get down to the increments, that are less then the movements of the arrows expansion and contraction, you will find that your timing of the arrows contraction and expansion, will be more important then the increment it moved.

That is why we use infinity, to describe just such a proposed experiment. The variables become to great. To silly to try to guess at, or confirm.

Sincerely,

William McCormick

#### Geezer

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##### Re: What is the Xeno motion paradox?
« Reply #27 on: 10/08/2012 00:13:13 »
David,

I think you better set down precisely the version of the paradox that you are referring to so that we can test the logic to see if it even is a paradox, and, unless you have certain mystical powers, it might be best if you stop claiming that you know what Zeno was thinking.

#### JP

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##### Re: What is the Xeno motion paradox?
« Reply #28 on: 10/08/2012 02:04:54 »
JP knows his math David :) I'm pretty sure on that one, doesn't mean that one can't have a different opinion though, just as I'm trying to see where that limit is thought as. To me the reason could be that a second is a limit of sorts whereas a infinity? On the other hand "It's a concept, which in Zeno's paradox means that you divide the interval into halves without limit.  But at the same time, you chop the times in half without limit.  Treating this as a limiting process you can prove that the time is finite."

So i split a second in 2, then 2 again, then 2 again. then 2 ... ad infinitum, but how and where do I reach that limit? What in calculus demand that I can't just keep doing so for ever JP? There has to be some hidden logic to this reasoning that I'm missing.

In calculus, you can.  That's why I was careful to say above that I'm not talking about quantum effects.  In classical physics and calculus, there is no issue in chopping up time and distance into as small a chunk that you want.  There is a problem with saying you have an infinitely small segment unless you're precise about what infinity means.  Infinity is not a  number like 1, sqrt(-1) or Pi.  Infinity has to be used with care as a concept for very large things.  It does have meaning, but since it's not a number, it doesn't make sense if you try to add, divide or multiply by it.  You can (and physicsts often do) use it similarly to a number to mean things can increase forever or decrease forever without stopping, but that's a concept, not a single number.

Again, the Zeno-like paradox is to say that if it takes you 1 second to walk a distance, you can chop that distance up into as many segments as you want.  If you do that, each segment is of finite time.  But if you keep chopping, eventually you're adding up an infinite number of finite times which must be infinite.  The problem is that if you really have an infinite (meaning increasing without limit) number of steps, then each step takes an infinitely short (decreasing without limit) time to cross.  For example, if you chop it in half, each segment takes 1/2th of a second to cross.  2*1/2=1 second.  If you chop it into 3rds, each takes 1/3rd of a segment, and 3*1/3rd=1 second.  If you shop it into n segments, each one takes 1/nth of a second, and n*1/n=1.  Calculus basically says that the proper way to treat this is to take n*1/n seconds and then to let n get bigger without limit.  Since n*1/n=1 for all possible real values of n, then the limit has to be 1 no matter how big n gets, even if it's allowed to increase forever without limit.

Zeno's paradox is a bit more sophisticated, since it's basically chopping the segment in half, then that half in half, then the next half in half, etc., so it's not as obvious that it has to sum to 1, but it's a similar proof in calculus.  You can write the sum of a finite series and calculate the sum as you chop it up without limit.  (The answer was actually found before calculus--thanks to Matthew for pointing that out to me--but calculus really made it rigorous and showed how to treat it within a firm mathematical framework.)
« Last Edit: 10/08/2012 02:07:14 by JP »

#### William McCormick

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##### Re: What is the Xeno motion paradox?
« Reply #29 on: 10/08/2012 02:35:12 »
JP knows his math David :) I'm pretty sure on that one, doesn't mean that one can't have a different opinion though, just as I'm trying to see where that limit is thought as. To me the reason could be that a second is a limit of sorts whereas a infinity? On the other hand "It's a concept, which in Zeno's paradox means that you divide the interval into halves without limit.  But at the same time, you chop the times in half without limit.  Treating this as a limiting process you can prove that the time is finite."

So i split a second in 2, then 2 again, then 2 again. then 2 ... ad infinitum, but how and where do I reach that limit? What in calculus demand that I can't just keep doing so for ever JP? There has to be some hidden logic to this reasoning that I'm missing.

In calculus, you can.  That's why I was careful to say above that I'm not talking about quantum effects.  In classical physics and calculus, there is no issue in chopping up time and distance into as small a chunk that you want.  There is a problem with saying you have an infinitely small segment unless you're precise about what infinity means.  Infinity is not a  number like 1, sqrt(-1) or Pi.  Infinity has to be used with care as a concept for very large things.  It does have meaning, but since it's not a number, it doesn't make sense if you try to add, divide or multiply by it.  You can (and physicsts often do) use it similarly to a number to mean things can increase forever or decrease forever without stopping, but that's a concept, not a single number.

Again, the Zeno-like paradox is to say that if it takes you 1 second to walk a distance, you can chop that distance up into as many segments as you want.  If you do that, each segment is of finite time.  But if you keep chopping, eventually you're adding up an infinite number of finite times which must be infinite.  The problem is that if you really have an infinite (meaning increasing without limit) number of steps, then each step takes an infinitely short (decreasing without limit) time to cross.  For example, if you chop it in half, each segment takes 1/2th of a second to cross.  2*1/2=1 second.  If you chop it into 3rds, each takes 1/3rd of a segment, and 3*1/3rd=1 second.  If you shop it into n segments, each one takes 1/nth of a second, and n*1/n=1.  Calculus basically says that the proper way to treat this is to take n*1/n seconds and then to let n get bigger without limit.  Since n*1/n=1 for all possible real values of n, then the limit has to be 1 no matter how big n gets, even if it's allowed to increase forever without limit.

Zeno's paradox is a bit more sophisticated, since it's basically chopping the segment in half, then that half in half, then the next half in half, etc., so it's not as obvious that it has to sum to 1, but it's a similar proof in calculus.  You can write the sum of a finite series and calculate the sum as you chop it up without limit.  (The answer was actually found before calculus--thanks to Matthew for pointing that out to me--but calculus really made it rigorous and showed how to treat it within a firm mathematical framework.)

Due to the reality that exists, isn't the paradox then as stated, just a mathemtical exercise, admittedly or obviously not taking into consideration any actuality or reality?

Sincerely,

William McCormick

#### yor_on

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##### Re: What is the Xeno motion paradox?
« Reply #30 on: 10/08/2012 02:53:15 »
Ahh, cool JP :) Now I see how you meant. And it's a basic principle of calculus as I remember, very vaguely though :) n*1/n=1 I mean.

#### yor_on

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##### Re: What is the Xeno motion paradox?
« Reply #31 on: 10/08/2012 02:59:14 »
William, to me it's logic and it is correct, at least for calculus. And as we use that kind of math for all sort of daily needs and constructions, and find it working? But ask JP of what he didn't discuss instead :)

#### Geezer

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##### Re: What is the Xeno motion paradox?
« Reply #32 on: 10/08/2012 03:08:58 »

Due to the reality that exists, isn't the paradox then as stated, just a mathemtical exercise, admittedly or obviously not taking into consideration any actuality or reality?

There are various versions of the paradox and there is no definitive Zeno version. That was lost to antiquity. As David is using it to claim that Calculus is somehow defective and that space is granular, I think it's important that he specifies his particular version of it.

However, the version I'm aware says something like "at an instant in time, an arrow.........." The counterargument is, because time flows, there are no "instants in time", and if you use instants of time you have just assumed that time is granular which will lead to the inevitable conclusion that space is also granular.

#### namaan

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##### Re: What is the Xeno motion paradox?
« Reply #33 on: 10/08/2012 03:13:46 »
David, you might be interested in Georg Cantor (http://en.wikipedia.org/wiki/Infinity) where it's mentioned:

Georg Cantor formalized many ideas related to infinity and infinite sets during the late 19th and early 20th centuries. In the theory he developed, there are infinite sets of different sizes (called cardinalities).[2] For example, the set of integers is countably infinite, while the set of real numbers is uncountably infinite.

There is a particular limit problem I'm working on at the moment that goes like this:

limit as x->1 of 1/lnx - 1/(x-1)

As x approaches one, both 1/lnx and 1/(x-1) approach infinity. So you could say in this case "infinity" minus "infinity" equals 1/2, which is what the value this expression takes on at exactly one. Clearly, these are different "sized" infinities, or I believe the technical term is these are infinities of different cardinality?

Anyway, I'm pretty sure the correct way of treating infinity is as a tendency, rather than strictly a number. It is the tendency for a measure to grow without bound. It isn't any particular number. This implies that even if you were to define a symbol for some sort of discreet infinity, something that represented more a hard boundary than a tendency, the infinity we're used to would still grow without bound past that boundary, because...that's what it's defined to do!

And of course if it grows without bound, it naturally represents an infinite family of numbers, which is precisely why you can't treat the symbol for infinity, itself, as a number: the symbol takes on meaning relative to expression in which it is used. Do correct me if I'm wrong, that's as far as I understand at the moment.

#### William McCormick

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##### Re: What is the Xeno motion paradox?
« Reply #34 on: 10/08/2012 03:37:04 »
William, to me it's logic and it is correct, at least for calculus. And as we use that kind of math for all sort of daily needs and constructions, and find it working? But ask JP of what he didn't discuss instead :)

But the tip of the arrow is going to be found going backwards, or at least at different velocities, at certain cycles of the arrows, expansion and contraction in the real world, once the increments of the arrows movement, get very small. All depending on the arrows veocity.

So the whole theory is void once you pass a certain point of decreasing increments. The arrow at the midway point in time, may not be at the midway point. Because the arrow itself is expanding and contracting at its own rate based on the material it is made from. By increasing the level of accuracy you uncover variables not considered before.

So if you were to call the first half of the arrows journey n, and the second half of the arrows journey n, then n*1/n may not equal 1     Ha-ha.
By claiming you split the arrows journey in half you declared both halves equal. There should be no problem with that equation but in real life there is.

That would be the case in the real world, using very exacting measurements. For most non exacting purposes it would be true.

That was the purpose of infinity, if you don't know, and you cannot measure it in this lifetime, or you do not wish to waste a lifetime, call it infinity. If someone wants to go there let them, you already turned down the job.

Sincerely,

William McCormick

#### David Cooper

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##### Re: What is the Xeno motion paradox?
« Reply #35 on: 10/08/2012 22:24:58 »
Again, the Zeno-like paradox is to say that if it takes you 1 second to walk a distance, you can chop that distance up into as many segments as you want.  If you do that, each segment is of finite time.  But if you keep chopping, eventually you're adding up an infinite number of finite times which must be infinite.  The problem is that if you really have an infinite (meaning increasing without limit) number of steps, then each step takes an infinitely short (decreasing without limit) time to cross.  For example, if you chop it in half, each segment takes 1/2th of a second to cross.  2*1/2=1 second.  If you chop it into 3rds, each takes 1/3rd of a segment, and 3*1/3rd=1 second.  If you shop it into n segments, each one takes 1/nth of a second, and n*1/n=1.  Calculus basically says that the proper way to treat this is to take n*1/n seconds and then to let n get bigger without limit.  Since n*1/n=1 for all possible real values of n, then the limit has to be 1 no matter how big n gets, even if it's allowed to increase forever without limit.

All you're really saying is that calculus includes as one of its principles a rejection of a particular idea of infinity, so it isn't calculus that's solving the paradox at all, but a flat rejection of a particular idea of infinity is by itself being used to reject the arrow paradox on the basis that it involves a faulty kind of infinity. Calculus simply doesn't engage with the kind of infinity which Zeno is using.

The issue then is about the validity of the kind of infinity in the paradox. Now we're getting somewhere at last - I can actually see now that you're rejecting Zeno's kind of infinity on rational grounds and that the success of calculus appears to give you a justification for doing so, so you attribute everything to calculus. Your argument is essentially that there is no such thing as infinity in the sense in which it is used in the paradox, so the paradox is not valid on that basis: there can be no such thing as the kind of infinity which Zeno had in mind because it can never be reached by anything - there is no forever that's long enough to make one, and if you're cutting somthing in half, then that half in half, then the resulting quarter in half, and so on forever, there is again no possible way of ever getting to an infinitieth, so the concept of infinity which Zeno was using is a kind of infinity which can never be realised. That is good thinking. I like it.

But there's a problem with rejecting Zeno's kind of infinity - his kind of infinity still has to apply to the inherent nature of a universe where you can chop things into smaller and smaller chunks forever. If it doesn't, you're going to have a universe where a finite amount of chopping must be capable of hitting the grain. If there is no grain, you have necessarily introduced Zeno's kind of infinity and made his paradox valid. Either way, the universe must be granular to solve the paradox.

#### David Cooper

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##### Re: What is the Xeno motion paradox?
« Reply #36 on: 10/08/2012 23:11:34 »
David, you might be interested in Georg Cantor (http://en.wikipedia.org/wiki/Infinity) where it's mentioned:

Georg Cantor formalized many ideas related to infinity and infinite sets during the late 19th and early 20th centuries. In the theory he developed, there are infinite sets of different sizes (called cardinalities).[2] For example, the set of integers is countably infinite, while the set of real numbers is uncountably infinite.

That's the kind of thing I was referring to, though I've only heard about it through radio programmes which I can't link to, but from what I've heard there are people who are now doing useful new kinds of maths with different kinds of infinities. At the bottom of this page - [link removed as an experiment to find out why this post is being rejected for using a blacklisted term] - it says:-

Quote
There is a whole theory about the sizes of infinite sets, much of which was developed by the mathematician Georg Cantor. It is a very deep theory, much of which depends on several things that are far beyond the scope of hub pages. If you are interested and have a little bit of mathematics training, I would recommend visiting here http://people.umass.edu/gmhwww/382/pdf/09-infinite%20sizes.pdf. You will learn many things such as the nice theorem "there are an infinite amount of sizes of infinity".

[http://nauticaricky.h u b p a g e s.com/hub/Different-Sizes-of-Infinity - the only way to get there (for anyone who feels the desire to) is to copy that, remove the spaces and paste into address bar.]

The reason I brought up the idea of different kinds of infinity here was simply to discourage anyone from wasting time by going down that route as an objection to my argument on the basis that cutting things in half, then the half in half, then the quarter in half, etc. is the wrong kind of infinity - I wrongly thought it probably was a different kind of infinity, and had that been the case there was a an easy way to show that it wasn't relevant. Having looked back at JP's comment about it, I realise now that I misinterpreted his reply and thought he was rejecting the idea that there were different kinds of infinity, but he was actually just saying that they were the same kind.

Quote
...
And of course if it grows without bound, it naturally represents an infinite family of numbers, which is precisely why you can't treat the symbol for infinity, itself, as a number: the symbol takes on meaning relative to expression in which it is used. Do correct me if I'm wrong, that's as far as I understand at the moment.

I think you're right. I think the kind of infinity Zeno's paradox involves cannot exist at all, and the consequence of that is that our view of the nature of the universe as something that could be chopped up forever without hitting the grain is wrong. There is no finite number in an infinite series that can't be reached by going through that series, and all numbers in an infinite series are finite. The only way the universe can have the property that you can't hit the grain is if it's built on infinities of the kind Zeno was using, completely beyond the reach of finite values.
« Last Edit: 10/08/2012 23:16:38 by David Cooper »

#### JP

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##### Re: What is the Xeno motion paradox?
« Reply #37 on: 11/08/2012 03:08:50 »
Zeno's paradox requires you make some assumptions about the mathematical model you're using.  It's pointless to say you're using the "real model of the universe," since there is no such thing that we've discovered yet.

I was perfectly clear that I was talking about the classical Zeno's paradox in which it is assumed that the universe can be chopped up into arbitrarily small pieces.  Calculus resolves that, as it (and limits) were invented to deal with things being chopped up into arbitrarily small (or large) pieces.

I also noted that there are good reasons to believe that you do hit some fundamental limits when chopping things up due to quantum effects, and that version of Zeno's paradox is not resolved by calculus and limits.
« Last Edit: 11/08/2012 03:12:42 by JP »

#### damocles

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##### Re: What is the Xeno motion paradox?
« Reply #38 on: 12/08/2012 06:42:30 »
A little off the main line of this discussion, but I would submit that we have very good evidence that space cannot be anything but infinitely divisible -- that is, that it cannot be pixellated. Why not? Because there is a relatively small and finite number of ways in which uniform elements may be stacked in an infinite array. They correspond to the 400-odd crystallographic space groups, each with its own particular symmetry.

The point is that none of them can possibly be isotropic, (that is have perfect spherical symmetry), and so with any stacking of finite elements in three-dimensional space there must be preferred directions, that would readily show up in a macroscopic observation. The preferred directions would even have to be a convenient several degrees apart.

The fact that we have no observations that indicate any absolute or preferred directions in space seems to me to be a very solid indication that there is no pixellation, at any degree of fineness, of real space.

#### yor_on

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##### Re: What is the Xeno motion paradox?
« Reply #39 on: 12/08/2012 13:08:48 »
I'm a slow learner Damocles.
Are you suggesting that if uniform elements only can be stacked certain ways, but not other, that this stacking also should be expressed by matters grouping in space? The 'preferred directions' I can see but for that to define the grouping of galaxies for example? Surely interesting, so you better go deeper there :)

#### namaan

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##### Re: What is the Xeno motion paradox?
« Reply #40 on: 12/08/2012 14:45:54 »
Assuming I correctly understand your post Damocles, rather than zero pixellation at all degrees of fineness, couldn't another possible reason be that space wasn't "stacked" piece by piece to begin with? But that all the 'pieces' of space were laid in place instantaneously?

I think the only reason I somewhat understood your post was that this "preferred direction" you mentioned showed up while programming the area in a 2d-neural network I worked on as a hobby sometime back. There was some sort of bias in the direction that the algorithms were working because I unwittingly laid out the neurons and the connections between them in some predefined way.

#### damocles

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##### Re: What is the Xeno motion paradox?
« Reply #41 on: 13/08/2012 04:28:53 »
Suppose that space is 'pixellated' in some way, and that there are finite sized chunks of distance as a result.
Premises:
(1) That space is 'full'. It has to be for the concept of space to make any sense -- else the question of what is between the finite atoms of space becomes a meaningful one.
(2) That the geometry of space is locally Euclidean. I suspect that this stipulation is one of convenience rather than necessity.

It follows that:
(1) We cannot inquire into the shapes of the pixels or atoms of space -- spatoms -- because the idea of measurement of distance on a scale smaller than these supposed entities has no meaning.
(2) Any unit of distance must be an integral number of spatoms
(3) Spatoms must be distributed to fill the whole of an Euclidean 3-dimensional space.

The most isotropic tiling of three dimensional space that is possible has the symmetry of a cubic close-packed crystal lattice. There are 2-dimensional sections of this lattice that are triangular close-packed.

Imagine a measurement made of a large scale object along an edge of a triangle in this lattice. It will involve, say, N spatoms. Now rotate the object through 30° without changing the plane. It now involves 2N/√3 spatoms -- roughly a 20% increase.

We do not observe a 20% variation between a minimum and maximum distance measurement when we rotate a real object in any direction.

#### damocles

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##### Re: What is the Xeno motion paradox?
« Reply #42 on: 13/08/2012 14:09:53 »
Make the 20% of the last posting a 15% -- more accurate, but it is immaterial to the thrust of the argument anyway. (Moderator, my communication seems to freeze when I try the 'modify' link on my original posting).

#### CZARCAR

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##### Re: What is the Xeno motion paradox?
« Reply #43 on: 14/08/2012 10:03:52 »
the first step, which is half the distance,can also be considered a distance in and of itself?
« Last Edit: 14/08/2012 10:06:26 by CZARCAR »

#### damocles

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##### Re: What is the Xeno motion paradox?
« Reply #44 on: 14/08/2012 10:16:16 »
the first step, which is half the distance,can also be considered a distance in and of itself?
That is true of a crystal lattice, but not of an atomically pixellated space. The first step, like any of the other steps, must take the integer value 1.

It is a bit like the original naive concept of an indivisible atom -- you can conceive of half an atom in such a world, but it cannot exist.
« Last Edit: 14/08/2012 10:18:28 by damocles »

#### JP

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##### Re: What is the Xeno motion paradox?
« Reply #45 on: 14/08/2012 14:12:21 »
Damocles, I agree with you that there can't be a lattice of space pixels.  But it could also be the case that there is a smallest meaningful or measurable distance (or time), but that these distances aren't located at fixed positions in space.  In other words, our ideas of space and time might break down at a small enough distance, which doesn't require that these "smallest" distances arrange themselves in to a lattice.

From what I understand, this is what some quantum theories of gravity propose, but I'm out of my depth in trying to understand the details.

#### yor_on

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##### Re: What is the Xeno motion paradox?
« Reply #46 on: 14/08/2012 19:41:13 »
Quite interesting, and I will have to reread you Damocles. As for JP:s suggestion it also make me think of the observer dependencies noted in relativity, relative 'a absolute globally true distance' there. As in the often cited example of the infalling muon.

#### yor_on

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##### Re: What is the Xeno motion paradox?
« Reply #47 on: 02/09/2012 08:52:37 »
Still interesting Damocles, I'm revisiting it, and a use of latices I haven't seen before. I adhere to the idea of  a 'flow' myself, or 'no pixelation' if one like. If we assume this as a fact then what does it make of particles of rest-mass, can we assume a pixelation there? And space, what does it make of a vacuum. Something without structure? Distance exist, as well as 'motion', both related to the observer. And with matter a property containing a lot of space.

#### damocles

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##### Re: What is the Xeno motion paradox?
« Reply #48 on: 02/09/2012 10:17:55 »
I have difficulty understanding why we cannot have a physical space that is continuous and infinitely divisible. It is surely the case with the (one dimensional) space of real numbers. I am not used to dealing with spaces that take into account general relativity with high gravitational fields, large distances, or relative speeds close to c0.

It seems to me that there is nothing in quantum mechanics or quantum electrodynamics -- a relativistic theory with virtual pair creation/destruction etc., that has been proved empirically to be extremely accurate on a local scale -- that says that space is not infinitely divisible or that space without some matter in it is impossible (not counting the "vacuum field" as "matter", of course).

It is a mathematical fact, and not too difficult to prove, I should imagine, that any locally Euclidean space of dimensionality greater than 1 cannot be both isotropic (uniform in every direction) and pixellated (completely filled with a stacked set of subspaces with similar geometry to each other).

Aristotle and Plato believed that space had to be filled with some matter, in order to hold things apart. Aristotle considered that the presence of nothingness was an oxymoron, and that if it were not for the presence of the "lumeniferous aether", material things as we know them would all collapse together. Zeno devised his several paradoxes to prompt closer attention to the supposed impossibility of motion that he considered a concomitant of current views of the world. Leucippus and Democritus (pre Aristotle) considered that the world was full of just two types of stuff -- substance, and void. The matter was in indivisible lumps (a tomos = indivisible) of various shapes and sizes, which were kept apart by their continuing motion, and by continually bouncing off each other, and that the different properties of different material arose from different shapes of the "atoms" of substance. The void was a continuous (and presumably infinitely divisible) backdrop to these "atoms".

I envisage the world as lumpy matter in continuous motion in an infinitely divisible void. This is much the way that Leucippus and Democritus saw it 2500 years ago. There is nothing in such a view that is problematic for any of the science that I do. But then I am only a chemist, not inquiring closely into substructures of protons, neutrons, and electrons. Things may or may not be different for a particle physicist or a theoretical cosmologist.

#### JP

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##### Re: What is the Xeno motion paradox?
« Reply #49 on: 03/09/2012 01:20:45 »
Damocles, I don't think there is any reason why space can't be continuous and infinitely divisible.  The only major issue is that we know our theories break down at small enough scales (the Planck time and length).  The breakdown of our theories doesn't mean that suddenly space and time become discretized.  It just means that we need a new theory.

There is a good reason to think that this new theory might involve space-time doing some funny things, however.  General relativity describes gravity in terms of the geometry of space-time.  Quantum mechanics describes all other forces in terms of quantized particle interactions.  For consistency, it's natural to expect that on a small enough scale, gravity should behave like particles.  Since the fabric of space-time is so intimately connected to gravity, it's natural to expect it might take on quantized, particle-like properties.

And of course this quantization happens on scales so small that we haven't even been able to detect it in particle accelerators, which are capable of breaking apart even protons and neutrons.

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##### Re: What is the Xeno motion paradox?
« Reply #49 on: 03/09/2012 01:20:45 »