[namaan]

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]

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]

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]

the first step, which is half the distance,can also be considered a distance in and of itself?

[damocles]

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.

[JP]

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]

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]

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]

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]

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.

[bizerl]

Lovely discussion people. Just wanted to try and boil it down to words I understand.

so...

This Zeno paradox (as a pure mathematical construct and not in any way an observable phenomenon) is resolved because although it plays the "infinity" card (infinite amount of steps), it also implies another "infinity" (namely the smallness of those steps) which cancels it out. I guess if you look at it another way and say that the first step is an infintely small step, you can't then turn around and say it is a finite time, because then you could use the opposite argument and say that if a journey is taken in infinitely small time steps, but in each time step there is a finite distance travelled, nothing takes any time.
So based on the Zeno paradox the universe could be granular or infinite and it wouldn't change the results, as long as your were consistant with how you applied it.
I lost some of the argument but that's kind of where I ended up with it. Too simplistic?

Also, in terms of real stuff, I was led to believe that when you get down to fundamental particles, they exist in a kind of probability field. Wouldn't this resolve the granular universe by saying that each particle could exist anywhere, in infinite detail?

[damocles]

Sorry, Bizerl, I was not meaning to get over-technical and leave you behind. There are some very deep philosophical issues involved, and I guess my own interest was in putting some of them before the theoretical physicists who participate in this forum to see how they would react to some of the ideas in my own mind.

You are quite right. The Greek philosophers and mathematicians had a significant problem with the concept of infinity, and lacked the mathematical tools to deal with it. That is another aspect of Zeno's paradoxes. With the mathematical approaches we have been taught, especially the inclusion of differential and integral calculus, we have effective and consistent ways of dealing with the infinite and infinitesimal. That does not address philosophical questions of whether or not we should consider the infinite and the infinitesimal as existing in the real world, but as far as science is concerned it does provide us with reliable terms for observation and description of the real world in a physical model.

From here on, I am getting to a personal, and perhaps slightly tongue-in-cheek viewpoint. There are huge differences between the ways that chemists and physicists look at things. There are also huge differences between the work that theoretical physicists actually do, and the way that they present their findings to a lay audience. When they are actually working, researching, and discovering things, they are manipulating high-powered and completely inaccessible mathematical expressions as they develop and explore mathematical models in an attempt to describe the physical phenomena that they are trying to address. When they try to report their finding to a lay audience, they often use analogies and make philosophical pronouncements that their work really does not warrant, but they find themselves unable to communicate in lay-accessible terms in any other way.

Now physicists like aesthetic simplicity in their models, and often this search for aesthetic simplicity is very productive of new results. For example, chemists recognize and work with three types of sub-atomic particle -- protons, neutrons, and electrons. They are very happy with that picture because it represents a simplification of their previous conception of 80 or 90 individual elements with puzzling patterns and differences in their properties -- a chaotic picture indeed! But theoretical physicists were intrigued by the fact that protons and neutrons had nearly the same mass, but different electrical charge, and that isolated protons were stable whereas isolated neutrons underwent radioactive decay. So they worked hard to produce a mathematical model where protons and neutrons were symmetrically equivalent -- where a proton could emerge as a neutron under some sort of symmetry operation. When this model was developed, it suggested that there should be a number of other sub-atomic particles, some of which were known from previous experiments, and some of which were discovered when they were specifically looked for. These other particles had approximately the right masses and lifetimes that the mathematics had suggested. The theory was very fruitful.

From bizerl:

Also, in terms of real stuff, I was led to believe that when you get down to fundamental particles, they exist in a kind of probability field. Wouldn't this resolve the granular universe by saying that each particle could exist anywhere, in infinite detail?

You certainly have reproduced pretty much what a physicist would probably tell you in the first sentence. The problem is in the last three words of your post. The Copenhagen interpretation of quantum mechanics, which seems to be the only viable one although many scientists are trying to come up with alternatives, says that not only is there an unavoidable minimum uncertainty in any measurement you make on a particle (well, most measurements), but that this uncertainty is a property of the particle, and not just a failure of the measurement method. The problem is that if you try to look at the detail of a particle -- for example to examine its shape or its parts -- then the uncertainty in the measurement of each part is greater than the uncertainty in the measurement of the whole: that is that the parts of a particle are blurred or defocussed over a larger amount of space than the particle itself occupies. That is certainly the sort of thing that the mathematics says; I am not at all sure of my translation of it back into the real world! But those last three words are the essence of the problem.

[yor_on]

I fully enjoyed your eh, wiev look back? tillbakablick as we say in Swedish. Furthermore, to ignore what people thought before us isn't that clever, it speaks more of hubris than understanding to me. Although we have come quite a bit on the way to getting those ideas into some proper reference frame, possibly

[bizerl]

Sorry, Bizerl, I was not meaning to get over-technical and leave you behind.

It doesn't take much to leave me behind! And thank you for the consideration and the great reply.

This may be the right thread for another mathematical anomaly from the depths of my highschool classes. I seem to remember a technique to convert a repeating decimal to a fraction (see here). When applied to 0.99999..., this result = 1. Is there a difference between 0.99999... and 1? I've always thought of 1-0.99999... as infinitesimal. Is that right?