I think the continuum of scale can be accounted for within the 4 dimensions of spacetime. I think the 4 orthonormal dimensions basis vectors each exit in Complex Space. I was pleased to find out that Quantum Effects can manifest in our macroscopic world rather blatantly, about 3/4th of the way through this film:

Horizon How Long is a Piece of String? full version

http://www.youtube.com/watch?v=VbodZKrAOGg

I recently read the paper:

Minimal length, maximal momentum and Hilbert space representation of quantum mechanics

http://arxiv.org/abs/1205.0158v1

and was able to recognise classical concepts expressed in Hilbert Space.

I could be wrong about what I think I see, but the question remains: What can be done with a technology that fully exploits the nature of space and time?

That the length of a piece of string might be different at difference scales reinforces the idea that the position - a measurement along the continuum - of scale is a necessary parameter for locating an object in space.

Consider our body at each level of scale. There is always a 3-D volume associated with our body. However, as we move down in scale, this 3-D volume holds different objects - organs, cells, proteins, etc. How can this same 3-D volume hold different objects in the same 'space'?

It cannot. Different 3-D volumes hold these objects, separated by scale.

The continuum of scale is mutually perpendicular to all three of our 'standard' dimensions.

Combining these 3-D volumes constitutes a 4-D hypervolume - along the continuum of scale.

This is not a question of accuracy, since we observe different objects at different scales. Accuracy provides more details about the same objects. If we encounter different objects, we have moved beyond the concept of accuracy.

I agree with your line of reasoning:

"How can this same 3-D volume hold different objects in the same 'space'?

It cannot. Different 3-D volumes hold these objects, separated by scale.

... This is not a question of accuracy, since we observe different objects at different scales.

Accuracy provides more details about the same objects."

Space allows the encoding od information at the different sizes of volume. The volume of a molecule hold the electrical potential of its bonds, which allows energy trasnport for a muscle cell to contract, which allows mw ro pick up a cup of coffee.

Correct - If we include scale as a fourth dimension, then we can expect to devise equations which traverse scale, since scale becomes an aspect of location of objects.

The question of measuring the distance between a book and a molecule of a pen, both sitting on the same table, becomes manageable. Considering only 3 dimensions, this distance (which should be the same as any other 3-D distance if we live in only a 3-D world) and the position of the two objects will be extremely difficult to define as different levels of accuracy.

I like this:

"since scale becomes an aspect of location of objects"

Consider an isolated volume of objects where some are moving and others are at rest relative to each other. (or, all are moving or all are at rest relative to each other.) Within the scale of this volume the length of time ans space are consistant and things like Conservation of Momentum and Energy hold.

When the observed volume is increased and more objects are added the initial volume of consideration is perturbed. I will assume the larger observed volume retains Conservation of Momentum and Energy. One could look at the initial volume and find it to be empty or greatly depleted of objects.

Is it possible that Momentum and Energy are factors in a scaling dimension?

JW:

I think you might find that 'observed volume' requires a perspective - and scale - from which to observe.

If we consider ourselves at rest and some volume containing objects is observed to increase, then there are two situations to consider: 1) the contained objects will also be observed to increase in volume (expand relative to ourselves) or 2) the contained objects will be observed to stay the same relative to ourselves - and hence an increase in 'space' will be observed between the objects and the volume.

In 1), Conservation of Momentum and Energy should be retained - relative to the objects and the volume - since nothing will appear to change within the volume. However this may not be true from our observations, as we will observe them all to increase - and potentially the momentum and energy will appear (be observed from our scale) to increase. This is, however, a relative situation, since objects in the volume will continue to be the same - relative to the scale of the volume itself.

In 2) the contained objects remain the same relative to ourselves, yet the volume is observed to increase (still relative to our scale). Relative to the volume, these objects will appear to shrink. So long as we use our scale, the conservation laws should be correct.

So Conservation of Energy and Momentum would be relative to the scale observed from. If there was movement in scale, then conservation would depend upon the scale of observation and against what measuring sticks - those 'in motion' or those 'at rest'.

Some interesting things can be considered for the second situation - along with a spherical volume and objects sitting on the outside of the sphere. The sphere will appear to expand, while the objects will attempt to stay 'at rest' on its surface. If constant movement at the same scale is velocity - what would constant expansion appear like of a volume moving in scale - constant acceleration? Is there another term used for constant acceleration?

And what would a 'constant expansion' appear like to those resistant objects on the surface - with the surface pushing up on them? Consider Einstein's equivalence principle.

If we looked out at the universe and it appeared to be expanding - might it be movement in scale (relative to ourselves)? Would not this appear like constant acceleration being a part of that universe (relative to ourselves) - with the potential for appearances of conservation laws to be violated when only considered against our single scale?

But this is way outside the box for current physics... even if it might make sense.

This line of readoning "In 2) the contained objects remain the same relative to ourselves, yet the volume is observed to increase (still relative to our scale). Relative to the volume, these objects will appear to shrink. So long as we use our scale, the conservation laws should be correct." looks promising. I will re-read what you wrote a few times and may have something like: "How do quantum eddects manifest as classical phenomina?". It's late, will get back to you in the morning.

Consider an imaging program that provides a virtual 3-D presentation of a patient's body to a surgeon. This surgeon could virtually perform an operation on the patient. But what if the surgeon also wanted (or needed) information about the patient at the cellular level or the protein level. The virtual 3-D information at these levels would require additional 3-D volumes of this same patient - each 3-D level associated with objects at different levels of scale. So, again, we need 4 measures to locate objects in physical space.

If we only live in a 3-D world, then these different 3-D levels have to the same patient's 3-D volume - different objects occupying the same 3-D volume. Our experience of reality does not allow different objects to occupy the exact same space.

The first who reflect on the question of the universe being different at different scales are the ancient atomists. That interest was renewed after the invention of the telescope and the microscope.

If you look at the sky and that you gradually reduce the aperture angle then you will gradually see biggest structures (see hubble deep space images). If you look at close object and that you gradually reduce the aperture angle then you will gradually smaller structures.

There is also a relation between spatial scale, time scale involves in phenomenon observed at a given spatial scale, time when structure at a particular spatial scale appear in the evolution of the universe.

LB:

Would you say that scale is needed to locate an object in space?

I would say yes. In image analysis, when we look for structures, we have to analyse the images at different spatial scales. This framework of analysis is called scale-space. If you are looking at mountains on the surface of the earth, you first have to specify the scale because there are mountains at all scales. A grain of sand is a mountain at a fine scale. The notion of surface is scale dependent. At a certain scale, the skin of your body is a surface, at much finer scale it is not a surface. At a certain scale it make sense to think of the surface made by a tree, at another one it does not make any sense.

So, if we need to specify the scale of an object to provide its location, this should mean we actually have a 4-dimensional physical space (length, width, height, scale) that we see and experience around us.

This would mean the several thousand year-old paradigm of a 3-D physical world is too limited and needs to be expanded to a 4-D physical model of the world which incorporates scale (an admitted physical continuum of the world around us) as that fourth dimension.

I note that the 'direction' of scale is mutually perpendicular to our traditional 3 dimensions (a requirement of a geometric 4th dimension). We can 'slide' a 3-D set of axes along the 'direction', or continuum, of scale. We can construct a 4-D image of our body by taking the 3-D volume of our body at our level - with organs and bones - and 'sliding' that volume along the continuum of scale and observing it changing to cells and proteins. This is 'moving in the direction' of scale.

A chemical or nuclear explosion is a fast-moving occurrence that includes movement in all four physical dimensions. As we grow from an egg to a person, we 'move' in scale. So, just as we cannot move equally in all 3 traditional physical dimensions ('up' is much more difficult that left or right), neither can we move equally in the scale dimension.

We need different tools to see in the two directions of scale - a telescope to see 'upwards' and a microscope to see 'downwards'.

So an aspect of reality - the continuum of scale - fulfills the geometric characteristics of a 4th physical dimension (required to locate an object in space and provides a direction mutually perpendicular to our other dimensions).

We should be including this physical continuum in our models of reality - expanding our concept of space to four dimensions.

In the scale-space image analysis framework, an image L(x,y) (2D scalar field) is transformed into its scale-space 3D scalar field L(x,y,scale). L(x,y,scale) is useful only when we have no apriori knowledge of the size of the pattern we are looking for. If you want to locate a point, then L(x,y) is sufficient.

The notion of coordinate is an objective position of a point. When we deal with structure occupying space, we have to deal with a measurement problem involving scale. Even for locating a point in the Universe, you cannot use real numbers because in general they are made of an infinite number of decimals. You will have to decide about an accuracy. If you decide that an accuracy of 100 light years is sufficient, then everything in this galaxy would end up into a single location and that might be sufficient for the purpose of simulating the expansion of the universe with computers.

for the evolution of the universe, the smallest scale merge with the biggest scale the further we go in the past and the evolution universe is now mostly the biological and societal evolution at the mid range of scales. So the scale dimention is not linear but circular.

It is important not to confuse the symbol systems we use and the underlying concepts those symbols represent. Decimals are not the same as the Real Numbers. There are constraints and limitations on decimals, which are not true of Real Numbers. Infinite decimal expansion is such a limitation on the decimal representation, which is not true of the underlying Real number. Note that Pi can be written as 10 base Pi, so it need not be an infinite decimal (however our usual integers might then become infinite decimals).

There are several issues with measurements and decimals - among them is the problem that decimals can only refer to measurements near the same scale. Using 6 significant digits, we cannot connect measurements (say, using equations) between our scale and the molecular scale. Then there is the problem you refer to, which is the need for infinite decimal expansions - even for such simple values as 1/3.

Two situations come out of this difficulty:

1) A science built upon the decimal numeric system will be limited in what it can define across scale. This means that the mathematical tools we use for measurements affect what we can measure and what we can define in terms of equations. In essence, the use of decimals constrains results in science to similar scales, which, in turn, will break science into levels of scale which can be managed by similar scale. Note this is what we find in considering disciplines of science today.

2) With decimals affecting measurements, through limited accuracy, it is possible we are confusing accuracy with scale. If scale is required to locate an object in space and our tools are unable to connect objects at significantly different levels of scale, then we might conclude that accuracy is the problem, and scale is not somehow real. However, the continuum of scale is a true part of our experience of reality and should not be confused with issues of decimal limitations.

In physical theories, the variables and constants are all real numbers (complex numbers are made of real numbers) which cannot be represented with decimal numbers of finite length. The actual measurements of physical quantities are of finite accuracy and precision. There are always measurement errors (This would not be the case if the universe is fundamentally an information system as some physicists speculate). In practice what is important is to have enough significant numbers in the measurement which means a ratio error/measurement that is small at a given scale of interest

This is the crux of what is needed for major advances in physics - a new, more powerful numeric system which provides manipulable values for complex numbers (as opposed to the current undefined "i" variable).

Just as decimals provided a much more powerful numeric system than ratios/fractions and could represent real numbers, so we now need a more powerful numeric system which can represent complex numbers.

A few properties of this new numeric system:

It will define what a negative base is for logarithms and exponentials, providing values for negative base numbers.

Having defined a negative base, -2 to the 1/2 power can be represented as an actual numeric value - as can all complex numbers. This means we could provide a measurement using a complex value - and then manipulate it using arithmetic operations.

Just as decimals added an exponential operation to the definition of a numeric value (eg. 1 * 10^1 + 2 * 10^0 = 12), a complex numeric system will include some form of DiffIntegral operation to it. I will suggest that it could be based on the natural log 'e', so that it can integrate and differentiate indefinitely 'up and down'. It will also allow for fractional/real valued differentiation.

Such a new mathematical tool will open up large expanses of measurements and science - in part by re-defining what 'accuracy' refers to and how an 'error term' is defined. This will be due to a change in the definition of what a positional digit is - such that the position of a 'decimal digit' will not be used for these new values.

One outcome of this will be that accuracy will be 'measured' differently and so the confusion we currently have over scale and accuracy will become clear - or at least clearer. Note that positional digits do not represent accuracy for fractions. The concept of accuracy doesn't really change, however how we 'measure' an error term will change, as it did when we moved from fractions to decimals.

5/18 +/- 1/72 is a different error measurement than .27 +/- .01

I think the concept of 'scale', as presented here, is unnecessary complication. It seems nearly the same as 'resolving power', which, roughly speaking, only limits the precision available when locating objects in space. But 3D coordinates are still satisfactory and remain valid. Or, another way: if you only want to use numbers (say coordinates) from bounded range only (not exceeding 100, for example) then, indeed, you need something extra to precisely say what is the real size of the object under observation. Does this trick deserve the name of fourth dimension? I don't think so.

Marek:

There is an interesting distinction between 'resolving power' and scale. Resolving power is about how well you can see the objects in view. Using a microscope, we are no longer resolving the objects in view - we start to see different objects. And the objects we see at one level of scale (say organs and tissue) are not the same as at another level of scale (say molecules and proteins).

Resolution should not produce this change in the objects seen. Movement in a direction, to a different 3-D space would.

Note that the combined tissue and organs in our body take up the 3-D volume of our body - as do the combined set of all cells, as do the combined set of all proteins and molecules. How can these different objects take up the same 3-D volume?

Donald,

description 3D is indeed too small to accommodate all the information about the object. Giving the location of a pen (x, y, z) we do not give its size. Therefore, we use a trick. Saying "pen" we average description of the contractual term, "collapsing" dimensions of the pen so we can hold the other three dimensions describing its location. Here the pen size is zero (point). The word "pen" does not have the physical size. Simply put, from experience, we know what it is.

I would say that the scale determines the level on which objects are treated conventionally as a points. We reduce dimensions scaling contractual structure to zero, in order to describe its position in the 3D system.

Does this mean that the scale can be considered as a dimension? Well, it shows some kind of location, some kind of distance. So, why not? But between abstractions. From physical point of view there is no pen, no desk, etc. Scale doesn’t bring some additional direction in geometry.

@Krzysztof Wypych

I believe that, geometrically speaking, scale acts the same as a 4th dimension. If we consider our body at the different (scale) levels of objects within our body - at the skeletal level, at the capillary level, at the cellular level, at the protein level, at the molecular-atomic level. Taking each of these levels of our body - each level consists of the 3-D volume of our body filled with objects of that scale level.

First - how can the same 3-D volume be filled by different objects? Only if those objects do not actually take up the same space.

Second - combining the 3-D volumes at each (scale) level we will obtain a 4-D hypervolume. Consider how to visualize these different scale levels on a computer screen - we have organ level NMR 3-D pictures. We cannot fill up these images with the other (scale) levels unless we include these other levels as different 3-D volumes.

Third - If we can combine all these levels of the same 3-D volume, and then pick an organ-level object in that 3-D body, we will have to 'travel' between levels to observe what exists at the same location as that organ at different levels. This travel will be in a direction mutually perpendicular to the 3 axes of our 3-D space.

Take the videos 'Cosmic Zoom' ( http://www.youtube.com/watch?v=VgfwCrKe_Fk ), 'Powers of Ten' ( http://www.youtube.com/watch?v=0fKBhvDjuy0 ), or Cary and Michael Huang's great 'Scale of the Universe' ( http://htwins.net/scale2/ )

These are travels in scale and, most obvious when travelling smaller, one does not have to 'move' in our normal 3 dimensions to travel in this scale dimension.

Donald,

I think that there is something important asscociate with this scale idea. The idea of space as defined by a set of dimensionless points is wrong. Space is made of locations at certain scales. This way, the numbers specifying the location do not need to have infinite accuracy.

Well, if we are talking about locations only, then 3D is necessary and sufficient. What is the dimensionality? It is the number of independent components to describe the elements of space, or, if you will, the number of their degrees of freedom. Surely, there also is a time (some locations are changing) - this is why a 4D space-time concept. By the way, the "effect of scale" is present in time domain too - isn't it? But this is, more or less, the area of our human perception. I see no special objections to talk about man more dimensions: just add 3 components of gravity field and 6 more related to electric and magnetic field. Suddenly you have 13 unquestionable dimensions. But why stop here? Quarks have (discrete!) colors ...

I'm unable to say whether or not this is our "experience".

@William Jackson: Imagine a single photon with energy equal to all the energy of our Universe. Divide this energy by a Planck constant and you have an answer, albeit grossly overestimated. Nobody knows this number but the limit itself obviously exists.

@Marek Gutowski

I think it is important to separate the abstract dimensions, introduced via mathematical theories, and the experiential aspects of space.

Mathematical theories of reality have introduced 4, 10, 11, 13 (and other) mathematical dimensions - which we do not experience in reality. Experimental results supporting these dimensions presume the mathematical underpinnings in a tautological way.

This is not the same as the experiential aspect of space we designate as 'scale'. Scale is a mutually experienced aspect of space, regardless of any underlying theories of space. The physical characteristics of scale we experience of space fit the definition of an additional dimension - strongly suggesting our 3-D model is inadequate to the task of representing scale.

This is not a new dimension introduced via some mathematical theory of reality. It is, instead, a result of actual experiential characteristics of space, which do not fit our traditional 3-D mathematical model introduced some 2500 years ago.

@Louis Brassard

[Firstly - thank you for even suggesting there might be something to this idea]

To address the issue of 'accuracy' and scale: Accuracy presumes the object which is being measured to be somehow static such that we can be more and more precise in our measurement of the object. However this is not what we find - since as we increase our accuracy by orders of magnitude, the object itself changes, becoming composed of different objects with different characteristics at smaller scales. The surface of a wooden table ceases, at some level of scale, to be a smooth surface that we can continually add measurement accuracy to. It becomes comprised of wood 'cells' and, at smaller scales, molecules and atoms (in motion). A measurement of the surface of the table to, say 12 or 14 significant digits (presume base 10 values), becomes a completely different measurement at the molecular or atomic scales - we can no longer measure to the 'surface of the table'.

Note that this discussion has assumed a 3-D model of reality - since in a 4-D model, we would need to account for scale as part of the location we are measuring. In this case the 'more accurate' measurements would actually be attempting to measure to a line in the direction of scale, rather than a supposed point in a 3-D model.

Something like a 3D object we can "draw" in space using a fast moving point. And the interior of the atom is actually a collection of such "points". Measure the distance between such fast changing position "objects" is practically impossible. This what looks like a 3D object is then just an approximation of it. In fact, it may be a point change, changing location, which looks like it was three-dimensional thing. Thus, even trying to measure the distance on the micro level we need to make an average. This means that we must take, for example an atom as fixed point. Navigating through the scale lies in the fact that we average the subsequent levels of complexity of motion and treat them as a 3D points (homogeneous stretched in 3D solids), giving them at the same time the names like: particle, atom, proteins, tissue, organ, person, planet, galaxy ...

But these names doesn’t represent real 3D objects, but our impressions. These "objects" exist only in our imagination. In reality, these complexes are some part of one big body. We just divide it for our purposes. From my point of view they are not 3D. They just use 3D to make their job. Object-oriented approach is very convenient and allows for quick calculations, where the response time to a stimulus plays a key role in the survival of the species, so a brain does not do anything else but calculates and displays the results: averaged 3D objects. However, navigating through the objects of this type is a virtual journey. In terms of additional >physical< direction the scale does not bring any one. To get to the atom from the galaxy we have to overcome the distance from the contractual galactic center (point) to the contractual center of the atom (point). Likewise, if we only act between atoms. These are not real points, although in relation to the size of the galaxy subatomic particle seems to be the point.

In my opinion, problem lies in other matter. Mankind has great difficulty with understanding the essence of time. And this is the main obstacle to make the next step forward. Although the watch shows the time, common, classic looking at watch is not enough to understand the time. Therefore, I propose to consider time as a break in the action (of space = energy) and ponder on the > physical< relationship of time with mass (time = mass). Let's end with a tautology. Treating time as a break in the action and in consequence as inertia we are dealing with a concept we are able to compare to the other concept and are able to distinguish them each other. We obtain natural, believable and conceivable frame of reference.

As to the math instead. Mathematics is only language in which we can describe the universe (not necessarily we have to). We create this language. Therefore, we first need to understand what we want to describe to then construct a tool to describe, and not vice versa. The interpretation of reality is in my opinion logic, not math.

Dear Donald,

I find your intuition fascinating. Relating to other streams on RG this question can be extended or enhanced by several other "dimensions". The questions about: why math is such a good of a model, or how the model (i.e. math) was conceived and fits the reality, or whether the same model can be applied to all scales of our observable reality, beg the answers or at least a mention. Maybe our observable reality is so deceitful that we need to express it in something simpler that we call space. Krzysztof complements the question of space with a question of time. Traditional math considers any additional vectorizable variable in conjunction with other variables as a new dimension.

There is no assurance or certainty that the physical space fulfills the assumptions of math, which are, that the same is valid in every spatial point of physical space and every point in time. Existence of special theory of relativity and quantum physics muddles this simple picture significantly. We know from experience that things do not scale down and start behaving differently, so our question about scaling is highly nontrivial. There is a new math in it as well as new physics.

Krzysztof mentions that math is only a language. That is true but there is always a mutual relationship between the language and the object, especially when natural language evolves to be formalized by restricting the meanings. Logic can be equally recognized as a field of math (a formal language) or as a premath intuition that comes from a desire for formalization. Human brain always restricts the notions only to be expanded later for generalizations. So the contraction and reduction of meaning is prerequisite to the formulation of a formal language. However expansion comes when a formal language is insufficient and breaks down requiring the introduction of a new layer or metalanguage to describe more complex a phenomenon. Human brain works in rather nonlinear manner.

@Krzysztof Wypych

Thank you for your comment - a interesting one, since it touches on several very important items.

First, I am suggesting we make too much of our geometric and mathematical models. We tend to use them as if they were reality. Thinking our experiential space must be 3-D is making too much of our model. Points are not something we experience - especially geometric points. These only exist in a geometric model, not in reality. We only postulate such a point object when we use a geometric model as part of a theory of reality - allowing it to be comprised of geometric points.

On the other hand scale is something we experience - and so, if we are have a holistic view of reality (one big body), it needs to be accounted for in our models of space. We should not presume a 3-D model, but should justify our use of it.

I note your example of "To get to the atom from the galaxy we have to overcome the distance from the contractual galactic center (point) to the contractual center of the atom (point)." Simply in stating the problem you reduce the galaxy and the atom to a point. Please attempt to locate this point of the center of the galaxy to the accuracy of the center of an atom and note the difficulty of determining the location of this point with the galaxy being comprised of all the other atoms in the galaxy. You have assumed a 3-D model with geometric points in laying out the problem and have presumed you can 'simply' calculate the position of any such point.. How would you do this practically? I think this would not be the simple task you seem to think and I think the difficulties will be related to the differences in scale of the galaxy and the atoms.

What if, instead of geometric points, the particles which make up an atom are actually 4-D objects that have some component of scale to them? Then, in attempting to locate these particles in a 3-D model, it will appear to us as if we projected the 4-D object onto a 3-D space. There is a high likely hood that this projection will make the object appear to be in more than one 3-D location (just as projecting a 3-D object onto a 2-D surface can make the 3-D object 'spread-out' over the 2-D surface).

Lastly, I agree with you that science needs a proper definition of 'time'. From my perspective, if the 'time' of 4-D space-time gets pushed out by a 4-D scale model, what happens to the 4th D of 'time'?

There is a relatively simple solution and definition - time is the measure of change. Any change will involve time. So no matter the number of dimensions we ascribe to space, time can always act like (be modelled as) another dimension in which change can occur. If we say reality is 4-D, with the 4th D being time, and there is any change in this 4-D space-time, then we have mis-construed what time is. How can there be change (time) in 4-D space-time? If we we use 4-D space-time as our model today, how can the Big-Bang be an expanding 3-D space - or a changing 4-D spacetime? We have broken our model of space-time in either case.

@Boguslaw Stec

Your suggestion of reality as 'deceitful' is great!

And I agree there is an interplay between mathematics and reality.

In terms of formal languages and mathematics, there were attempts to place mathematics on a logical foundation in the early 1900's. Between Godel's Incompleteness proof and the problem of self-referential statements (eg. 'this statement is a lie', or 'the set of all sets'), it would appear that mathematics and logic may not 'bottom out' anywhere and continually need meta-layers and possibly work in non-linear fashion also. (So I think we are mostly in agreement here.)

I agree that we are at a time of non-linearity, where we need new mathematical tools and scientific concepts - including, as Krzysztof states, a different concept of 'time'.

I also believe this will be a meta-level that is more expansive and inclusive (eg. including scale) rather than a shift to more primitive geometric concepts such as 1-D points or 2-D strings.

What if scale is a physical dimension - what concepts of reality would change, simply due to this change in perspective? I suggested one in my previous post of 4-D particles projected onto a 3-D model. How would this dimension appear to us as and how would movement in this direction appear to us as?

Heat might be understood as movement in scale (how do we 'feel' heat if it only occurs at the atomic level?).

To your point of reality being deceitful: I will suggest that this scale dimension would appear to us earth-bound 3-D sight humans as an exponential dimension, such that constant movement (velocity) in it would appear to us as if it were constant accelerated movement. The accelerating universe might just be constant movement in this dimension. (I will suggest we consider gravity, but that would be a hugely non-linear concept.) If accelerated motion includes movement in this direction, how would the length of a yardstick appear as it moved in scale - would it appear to change length? Could this (non-linearly) change the interpretation of Einstein's equations - without changing the equations themselves?

Our perspective of reality is heavily influenced by who we are, and the perspective we start from. This perspective should be understood as a starting position, not as the end point and certainly not as the presumed model that we must use forever (eg. our 3-D model is over 2500 years old and comes from a 3-D sight-based perspective.) Changing our perspective can have radical effects on our understanding, without making any change to the tested equations and experiments made to date.

Donald,

Most models that expressed as a set of equations are usually valid for a given spatial and temporal scale range and the problem of scales do not arise in such models. We use different models for different phenomenal domains, each one being at a given spatial and temporal scale. Planetary models, stelar system models, galactic models, cosmos models, atomic models, elementary particle models, fluid models, cell models, animal models, etc all of them are conceived in isolation at a given scale.

We usually speak of the different scales as the hiearchical organisation of reality. It is not certain that the best way to conceive such hiearchies is to used a continuous scale variable given that these hierarchies are clearly discrete and occuring abruptly around scale factor of 100 or 1000. There is a clear link between structural hierarchy and history or time. The structural hieararchy or taxonomy are now interpreted as phylogeny.

@Louis Brassard

So, if we are to treat the whole of reality as a whole - then shouldn't all these levels of scale, which we have separate models for, be considered as parts or levels of that whole reality?

It occurs to me that all these levels do comprise a larger whole. I also wondered why we should break up our models into these levels, if reality is such a whole. The conclusion I came to was that it is our mathematical tools which cause us to break up reality into these levels of scale. We can only model levels of scale because our tools can only model levels of scale.

A particular culprit could be our means of taking a measurement - our way of representing quantities. If this aspect limits us to levels of scale, then all our models will also be so limited.

In looking into this, our decimal numeric system (and base-positional variations, eg. base 2, base 12) came into focus. It is a 1500 year old system (used by science since about Galileo's time) that has under-pinned all scientific measurements over the past ~500 years. We don't even think of altering or improving it, since it is such a presumed aspect of measurements and mathematics. However, it is also our system of measurements which require us to designate a level of scale to these measurements.

We cannot add a measurement of 3.256 +/- .003 with a quantity of .0000005182 +/- .0000000003 due to the difference in scale and accuracy. We must limit our measurements (and models) to close levels of scale because we cannot calculate with widely different scale measurements.

Might it be possible to overcome this limitation? How might this be accomplished?

I suggest it can be by inventing or defining or discovering (depending upon how you believe such things come about) a new, more powerful numeric system. One which can allow measurements across scale.

We have Integers and whole numbers. We have Rationals and fractions. We have Real numbers and decimals. How about Complex numbers and a new numeric system? Integers can be defined using just addition and subtraction. Rationals need the additional reversing operations of multiplication and division in defining fractions. Reals add exponentiation (and logarithms) in defining decimals. What if complex numbers add integration and differentiation for the next numeric system?

Integration and differentiation (esp. using 'e' as a base) could cross scale - with, say, positional numbers being first, second, etc. integral or differential operations.

Using 'e' as the base, we could define a value for sqrt(-1) and resolve Euler's identity. The system would require negative bases - an undefined area for mathematics. Then complex numbers could be represented by singular values - something our current representation of complex numbers cannot do - with (x + iy).

Enough for now.

Donald,

I know that the positioning of such a purely theoretical formation like point is physically impossible. I gave an example, that in some way visualizes the method we use. Simply put, for practical purposes in place of the object we enter dot and its specific location coordinates.

I feel, however, that the essence of this discussion is the same issue that I raised in the topic: "The Fifth Dimension", although seemingly topics seem unrelated. Apart from the nature (physical?, not physical?) scale is continually used to the virtual "rolling" dimensions to the point, which makes the description, which we use in everyday life is simplified. Thus, the scale hides part of dimensions. Makes that they are perceived and presented differently.

I'm glad you agree with me that there is a need to change the concept of time, because the axis of problem is the time.

If time is assigned to each spatial dimension, then time from the nature of things would not be one-dimensional. Of course, time is a measure of change. But I think the critical point is the information about what is changing and how? So, what exactly is going on? In my view, any direction can be relatively reduced, that means interrupted the operation of it, and thus distinguished by the relative change of its value, that means by the relative change in its size. So, I think time has a geometric origin and the popular sentence 3D + time is a simplified approach. To understand the origin of this simplification is required dynamic frame of reference. This could be a "swelling sphere" that already I had described in some way as the 5D system, though, as to the number of dimensions we could of course argue. As a rule, however, it is important that the system volume changes. And it may do it asymmetrically. But any calculation of time 3D (5D?) for a given object would be rather complicated. We would have to count the value of each direction, and in big object directions are entangled (in the colloquial sense), so from a mathematical point of view, the calculation would be the massacre. Meantime, the scale allows us to bring virtually 3D area, which is occupied by physical system to 0D. This causes that remain to us free three directions in which we can attach a simplified image of the object. In this approach, we draw some abstraction, such as e.g. a cube, but then we lack the space to draw what causes the cube to move in a certain direction, because in order to show it geometrically, we need to draw a specific physical changes governing cube ( loss and gain of the value in a particular direction, that means the time) such as e.g. the subsequent pictures. But then there would be no longer a cube (and indeed there are no physical cubes). Therefore, we write by words that cube is attracted by the planet and draw the arrow next to showing the turn (->). The point is that such a simplification causes that the time becomes one-dimensional. We treat the things like the points, so we average the interruption on three axes to interruption on one axis. We get as a result 3D virtual object and the virtual time (age of object). By introducing this simplification to study we consider the virtual elements. And that's what we have to remember, otherwise we lose the true nature of time, as we examine our impressions, which presents to us our brains, not reality.

Manuel:

I do not think things are as simple as you present in your post above.  Observing collective actions on a galactic scale is not something we observe directly - movement of galaxy clusters for instance.  Observing movement of atomic structures or quark interactions are really inferred from other observations (which you discuss).

So "unambiguous empirical evidence" is not so unambiguous - which you actually put forth in the beginning of your Article.  The exact same 'facts' have been used to support different theories over the millennia - from phlogiston to ether to quantum mechanics.

Further, as I have attempted to discuss in previous posts, the underlying tools of physics - mathematics - are not static and have changed significantly over time. These changes have caused tremendous alterations of physical theories, since the tools available for taking measurements change.  Without complex numbers and non-Euclidean geometry, our current physics would not be possible.  In fact there are many 'facts' which could not be stated without these underlying tools.

These developments are only a couple hundred years old (it seems a change in underlying tools takes awhile to impact the sciences using these tools). A more important tool underlying science is the decimal numeric system - a tool that has not really changed in about 1500 years - 800 for scientific use.  A change in such fundamental scientific tools will significantly impact what science is able to measure, to calculate, and to theorize about.  Whatever theories we have today will likely not survive the change in a new set of mathematical tools.

"If something exists we can observe its scale" has some real problems with it, since we could 'observe' galaxies several hundred years ago - we just didn't know that those specks behind the telescope were much much larger than a star.  It took many tens of years of observation and comparative analysis before we realized that 1) Our planet existed in a 'galaxy' 2) that other galaxies exist and that 3) galaxies move collectively.

We really need to consider our position as part of an historical context and not the end of a train of inquiry.  This also means we are much less important that we would hope ourselves to be - since many things will change long after we have passed away.