Dear Farhad,

In your document says:

"If the traditional hypothetical 3-dimensionality of space, or the 4-dimensionality of space-time were really involved in physical effects, then all physical effects would be described by: (1L3=1L3), or (1L4=1L4) equations."

I think that here are the answers you are looking for, especially in the part of Einstein field equations:

ALL SPECIAL RELATIVITY EQUATIONS OBTAINED USING GALILEAN TRANSFORMATIONS & EINSTEIN FIELD EQUATIONS OBTAINED USING ONLY GAUSSIAN CURVATURE (ZOOM THEORY) VERSION II

Dear Farhad,

I read your article. It describes the difficulties faced by many theories. Keep in mind the theory is only imagination of scientist. For physics proof of reality is only experiment. According spatial dimensions experiment is extremely simple. In 1D space the force does not depend on distance, in 2D space the force decreases in proportion to the distance, in 3D space the force decreases in square to the distance, in 4D space the force decreases in cube to the distance etc. Observations confirm that there is 3D space. More in

Article Spatial Dimensions

Article New Concept of Space

Best regards Ilgaitis

Laud the question. It captures deep imagination, and imagination allows us to illuminate reality.

Because or material (physical) body is 3-dimensional, we percept surrounding space as 3-dimensional one. But people are not only material bodies, because they posses consciousness, which is related with non-material (energetic) world. This world is the Time which posses many dimensions. We percept the Time as one-dimensional flow which spreads from the past to the future. But we can visite in our mind our past and suppose different variants of the future. We will rigidly connected with the presence (for our body) until we think that we are only material bodies following from the past to the future. But astronomical observations of professor Kozyrev demonstrate that the past, the presence and the future exist simultaneously where the past and the future are the reflections of the presence (as in mirror). When our consciousness percepts this observable fact we''ll consider our bodies as projections of the multi-dimensional energetic body only as 3-projection of multi-dimensional subject. Material body is materialized state of the multi-dimensional energetic body.

I reply comments as I received:

1. Dear Sergio,

thank. As I mentioned all physical effects are considered due to the action of forces, field strength or curvature which are all kinds of curvature and of 1/L^2 dimension in geometric units (field theory). There is no physical quantity of 1/L^3 or 1/ L^4 dimension. Chern-Simons and Yang-Mills integrands are products of 1/L^2 with 1/L (connection component) and 1/L^2 with 1/L^2 curvature components. Please read again my paper.

2. Dear Ilgaitis,

thanks. Your remark "in 2D space the force decreases in proportion to the distance, in 3D space the force decreases in square to the distance, in 4D space the force decreases in cube to the distance etc. Observations confirm that there is 3D space." is not correct. These are confirmed just by assumption of cetrain formula for force which presuppose the result, i. e. circulus vitiosus. In 2D space there are two potential or connection components (x/r^2, y/r^2) and you can have Laplace equation. I you consider one of them with negative sign then you get a field strength or force.

No observation confirm 3 or four dimensionality of space or space-time. Otherwise the question of four dimensionality became no millennium question.

Dear Farhad,

Imagine that the universe is an orange (spherical) and you lived on the surface, the action of forces, field strength or curvature would be 1 / L, that means that the dimensions are not independent? I would not know what to tell you, but I think that if they are independent, only that you live on the surface

Dear Sergio,

First of all physics describes only the interaction between dynamical systems and not the shape of things. Secondly physics should be describes by curved (!) degrees of freedom (independent dimensions) and not dimensions including dependent ones. Then otherwise one has the unsolvable problem of applying classical or quantum constraints with two different results for physics. Thirdly since curvature is by definition in differential geometry always (on and manifold!) a differential two form with (1/L.L) components (not 1/L), therefore also on a curved two dimensional surface any kind of curvature forces and field strength as curvature act as 1/L.L. If you integrate a differential two form over a (L.L.L) volume you get no invariant result! Furthermore there is no proof for three dimensionality of space or four dimensionality of space-time, neither theoretical/mathematical or experimental. These are just faiths no more. Otherwise you had a proof at least after the question is announced as a millennium problem of physics!

Dear Farhad,

First, geometry influences our perception of forces, as explained by general relativity, and we can say then the interaction between dynamical systems depends on the shape of the universe (the shape of things).

Second, degrees of freedom are marked by the number of dimensions although some movements are not so obvious, an inhabitant on the surface of the spherical orange that I have set as an example has 2 degrees of freedom, but is embedded in a 3-dimensional space.

Third, the spherical curvature on a 3-dimensional surface is as you say 1 / R ^ 2 but note that in my example it is a 2-dimensional surface embedded in a 3-dimensional Euclidean space, so the circular curvature in that bi-dimensional surface is 1 / R

In short, if there are no equations of more than 3 dimensions (except the Einstein equations) does not mean that there are no more independent dimensions, it would be normal, if those equations refer to phenomena experienced by three-dimensional inhabitants in a hyper-surface of a hypersphere for example

Dear Sergio,

I am not sure wether you read carefully my arguments, but try to understand my position which is a topological one. The topological situation of a manfiold does not care its embeding. Moreover onem ust first prove the number of degrees of fredeom of any space and than use or speak about it.

As I argued before there is no experimental or mathematical possibility to prove the independency of more than two dimensions in order to speak about a manifold with more than two degrees of freedom! All such manifolds are just by tradition but falsely assumed in view of Banach-Tarski paradox.

The main ingredient in Einstein equations is the curvature which is the component of a diffeential two form and as such a two dimensional geometrical object.

Moreover if one is not able to integrate or solve analytically even the general three body problem, which if you could solve would mean the existence of three independent spatial dimensions dynamically. Then one is not able to explain three dimensionality of a space dynamically or analytically or globally.

All such assumptions are of more than two dimensional curved manifolds in physics and mathemtics are purely local assumptions without any global or integral topological warranty. As I mentioned in my announced paper there is no existence and uniqueness theorem for 4D manifolds including 4D space-time.

From geometrical point of view conditions on the number of components of physical quantities, e. g. vectors and tensors which are defined by the base manifold dimensions like gauge conditions or identities reduce the dimensional structure of the actual base manifold. This is what is not understood yet among colleagues. And again if there were a proof about four dimensional space-time or three dimensional space the it would be announced as an answer to the millennium problem of physics from the year 2000.

“…And again if there were a proof about four dimensional space-time or three dimensional space the it would be announced as an answer to the millennium problem of physics from the year 2000.…”

Thank heaven a bottle with a good cognac is 3-dimensional…

Cheers

@Farhad: “General three body problem is not integrable or solvable analytically which shows that the present assumption of three dimensional space is wrong.”

I’m guessing that you mean the Newtonian three-body problem, right?

In which case, since there is no general analytic solution in two dimensions either, would you argue the impossibility of two-dimensional space?

Just curious....

To Sergey: you mean vodca not cognac ?! Moreover it looks just after drinking 3D not before. So try to prove the independency of those 3D just before you drink!

Ypa

To John: the general three body problem has nothing to do with physical properties like Newtonian or else. Look in Abraham-Marsden's "Foundations of Mechanics" or Arnold's "mathematical foundations of classical mechanics". Then you understand what I mean with non integrability of general three body problem. General two body problem and even restricted three body problems with some two degrees of freedom structure are integrable. This hints that manifolds with two curved independent dimensions or two degrees of freedom are globally existing. But no dynamical system with more than two degrees of freedom are stable agianst small perturbations. You can see the problem even in Y = X^3 function which has no minimum but Y= X^2 function does have!

More than curious ...

@Farhad: “the general three body problem has nothing to do with physical properties like Newtonian or else. “

Your statement that I quoted clearly said otherwise. If your reference to the three-body problem was more general than the Newtonian case, then at least it has to be compatible with the Newtonian case as a special instance of the more general case. So my question stands.

“But no dynamical system with more than two degrees of freedom are stable agianst [sic] small perturbations”

You seem to be confusing mechanical degrees of freedom with spatial dimensions. Until that’s cleared up, I cannot justify investing time in reading referenced documents. The issue we have here is clear in its own context.

John you should look in those books in order to understand that the mechanical degrees of freedom has to do with independent dimensions of space-time. One should know what the meaning of general three body is before beginning a discussion about it. Moreover Mechanical problems are described either in phase space or polarized phase space like the position coordinate space. But even if people do other ways logic and experiments as the basis of physics prove that both are related categories to describe physics.

I explain ones more my position, it is your problem if you try to understand it or not! As before I mean always curved dimensions or degrees of freedom.

If the general three body problem which is a problem with three degrees of freedom could be integrated analytically then one could assume directions of those three body as three independent dimensions of space. It would be a dynamical proof of three independent dimensions in space. But such an integration is not possible (!) showing that any assumption of three independent dimensions for space in physics is not possible. This impossibility is proved also by the famous Bancah-Tarski paradox. Moreover one find the proof of this impossibility in those books according to the Morse-Smale theory and KAM theorem that: dynamical systems (including any type of mechanical systems) with more than two degrees of freedom = two space dimensions are not stable against the small perturbations = dynamics described by differetial equations of the system.

The traditionally assumed and still unproved three dimnsionality of space (see the 5. millennium problem of physics) did not care the problem of independency of these dimensions or degrees of freedom. Then degrees of freedom has to do with the dimensions of space time in view of the fact that any physical quantity which is assumed to cary some degrees of freedom , say electromagnetic vector potential is some function of space time dimensions and the number of its components depends on the assumed number of dimensions of space-time base manifold.

All basic unsolved problems in physics and even mathematics are related in some sense with the usually assumed local dimensions of the manifold whereon the problem is defined and the globally possible degrees of freedom of the problem. Insofar unsovablity of physical and mathematical problems like the general three body problem and Navier-Stokes problem (in three dimensions) shows that the assumed three dimensions for space in physics are dependent. See also my announced paper where I expalin that if the notion of three dimensional space with three independent dimensions = three spatial degrees of freedom is correct and it is assumed that physical bodies are in the same sense three dimensional, than these bodies should exploded in view of the independencies of those three dimensions. The Banch-Tarski paradox descibes the same fact in a dual manner.

“…The traditionally assumed and still unproved three dimnsionality of space (see the 5. millennium problem of physics) did not care the problem of independency of these dimensions or degrees of freedom.…..”

Yeah, the number of dimensions of the spacetime of some system is equal to the number of independent degrees of freedom for changing states of utmost fundamental basic elements of the system; and nothing else.

That Matter’s spacetime must have 3D space, if it exists and changes basing on some binary fundamental elements, was showed in the outstanding Carl Friedrich Freiherr von Weizsäcker’s “UR-hypothesis” in early 1950-th, see the papers “Eine Frage Über die Rolle der quadratischen Metrik in der Physik”. Zeitschrift für Naturforschung, 7 a: 141, (1952); and “Komplementarität und Logik”. Die Naturwissenschaften, 42: 521–529, 545–555, (1955.)

“Ur” in translation on contemporary English is “bit”, but von Weizsäcker, to underlain the fundamental role of these elements, used this Old German "Ur".

And indeed, from existent experimental data seems quite convincingly follows, that Matter’s spacetime is the [5]4D absolute Euclidian “empty container” with the metrics (cτ,X,Y,Z,ct), where cτ is the “coordinate time” coordinate/dimension; X,Y,Z, are the spatial coordinates/dimensions, ct is the “true time” coordinate/dimension.

Because of the energy conservation law every Matter’s object, and the system “Matter” as a whole, constantly and always are moving in the 4D sub-spacetime (cτ,X,Y,Z) [and simultaneously in the ct-dimension], only with 4D speeds of light; when the 4 coordinates/dimensions in this sub-spacetime indeed correspond to 4 independent degrees of freedom at changes of the fundamental logical elements (FLE) states, from which [FLEs] everything in Matter, i.e. particles, bodies, galaxies, fields, etc. are made.

With a very non-zero probability everything in Matter is/are some disturbances in the 4D dense lattice of the FLE, which [the lattice] fills the 4D empty, and so principally “flat”, container. More see https://www.researchgate.net/publication/273777630_The_Informational_Conception_and_Basic_Physics DOI 10.5281/zenodo.16494.

Cheers

To Sergey:

Yeah, yeah, I knew Weizsäcker personally and discussed with him several times. He did never assert to prove the 3-dimensionality of space any time. Moreover binary structure need not 3D space but only 2D space!

I repeat myself: it is always curved dimensions like those over ellipsoid.

It is a logical disease among scientists to embed 2D facts in 3D space or 4D space-time.

Binary structure of information theory (see 2 degrees of freedom of photon from physical side and Shannon-theory from mathematical side) and binary structure of standard logic deny the necessity of 3D space (see Turing theorem).

3D and more-D spaces is paradoxical according to Hausdorff-Banach-Tarski theorem.

3- and more-D are assumed only locally without any global, topological or integral authorization or definition.

All 3D or 4D arguments for space and space-time are circulus vitiosus.

4D space-time is mathematically unreasonable in view of the fact that there is no existence and uniqueness theorem for smooth 4-manifolds (see my paper and A. A. Kosinski, "Differential manifold").

All Gauge theories over 4D space-time are instable , but on 2D S^2 stable (see Atiyah-Bott 1981, 1982 and Jean-Pierre Bourguignon, H. Blaine Lawson Jr. 1981, etc.).

4 space-time dimensions in QFTs are dependent since one needs several gauge conditions, constraints, regularization and renormalization to extract physically meaningful results from those 4DQFTs. All these conditions and applied methods do only a reduction of number of space-time dimensions to 2 curved degrees of freedom.

If you are so certain about physical 3D and 4D manifolds write a paper and answer the millennium question of physics and got the Nobel prize and drink vodca or cognac so much as you like.

@Farhad,

I don’t have time to list all the non sequiturs in your reply, but I’ll mention one. You presumed to know the level of my knowledge of the general three-body problem with nothing more to go on than the fact that I asked you to clarify a statement you made about it, and you have still not answered my question with the simple “yes” or “no” that would have sufficed. I have to expect that your other work is similarly burdened by unjustified assumptions and confused associations.

The fact that the Newtonian general three-body problem has no closed-form analytic solution follows from the nature of the nonlinear math involved, not the number of dimensions in which it is formulated. And if you meant General Relativity instead of Newtonian gravity, you could have made a stronger statement by using the general two-body problem. But it’s irrelevant anyway.

It is clear that you did not ask an innocent question on ResearchGate, you had some agenda to advance. You are not the first to do so, and probably not the last.

Clearly we cannot communicate, so I’ll make no further attempts except for one suggestion: if you really wish to be taken seriously, you must do a better job of appearing to be someone who deserves to be taken seriously. Failing that, I can only recommend that you study the material at the website:

http://math.ucr.edu/home/baez/crackpot.html.

Dear Farhad,

"If you are so certain about physical 3D and 4D manifolds write a paper and answer the millennium question of physics and got the Nobel prize and drink vodca or cognac so much as you like. "

Here it is:

Preprint ALL SPECIAL RELATIVITY EQUATIONS OBTAINED USING GALILEAN TRA...

Are you able to find an error? So why is not it published? Ah, I do not pertain to any endogamous club of academics and since they do not like what I say better do not analyze it in detail.

All the possibilities of the current theoretical physics have been more than analyzed with great detail, to move forward a change of paradigm is necessary, a jump as it was in his time the relativity of Einstein

Mr. Fowler

I have no time to read your dreams. Read those books to understand general three body problem.

Dear Sergio,

as long as there is no proof for the 5. millennium problem of physics all the rest 4D physics are dubious. Thus 4D physics can not even know what energy is (Feynman lectures, 1), do not understnad the quantum theory and accept electron interference as a secrete (Feynman, "The character of physical law), needs a lot of dimensional conditions in order to come somewhat near the experimental results and can not quantize the gravity. The incompatibility of quantum theory and general relativity (gravity) as the two basic theories of physics shows the disastrous sitution of theoretical physics not only according to my very earlier arguments but also according to Hawking's later opinion (A brief history of time, updated and expanded edition) that there is a basic mistake in 4D assumption. For Hawking one - for me at least one - of the two basic 4D theories is wrong.

But I did my part. The hounorable community do not like to discuss difficult matters and like to dream the 4D physics. They do not any change of paradigm, so they do not publish anything beyond the standard dreams. I published 2003 in internet: "Dimensional structure of space-time" (https://arxiv.org › gr-qc), etc.

To Farhad:

“…I knew Weizsäcker personally and discussed with him several times….”

That can be so, but from this fact doesn’t follow that

“….. He did never assert to prove the 3-dimensionality of space any time…..”

is correct; simply at your discussions this problem wasn’t touched.

Again, von Weizsäcker put forward the “UR-hypothesis” in early 1950-th, where it was shown that for Matter be based on binary elements the Matter’s space must be 3D space – see, for example https://en.wikipedia.org/wiki/Digital_physics Sec. “Weizsäcker's ur-alternatives”

Though in this Wiki article in this section it is stated that the hypothesis was “first publicized in his book The Unity of Nature (1971)” – when that was in 1950-th, see SS post above; and that this theory was “theory of ur-alternatives (theory of archetypal objects)”, i.e. possibly a theory of something from, say, Carl Jung’s collective unconscious psyche – that is rather popular practice in some “scientific community” and the article’s author seems is a member of this community, for whom it was necessary to write that the “It from Bit” doctrine (1971) was the first; nonetheless the 3-dimensional space is mentioned.

And this hypothesis is rather actively, though without some results, is developed in physics, for example by [if I correctly remember, when read the papers long time ago] Lyre and a few his co-authors.

Cheers

Dear All,

Some of you confuse degrees of freedom of mechanical systems with dimensions of space. For example, the human body has several thousand degrees of freedom, but only 3D. More of shape and dimensionality in

Article Shape and Dimensions of Space

Dear Ilgaitis,

you confuse the notion f degree of freedom completely. It is defined according to the notion of free mofbilit in regurus books as "independent dimension". It is a geometrical/ topological notion, e. g. as the index of a differential operator on a manifold. So if one speaks of "several thousand degrees of freedom" one should be able to prove the independency of those and their definition them as such an index. Human beings has thousend dreams but only two degrees of freedom to move freely on the earth. not You should look on B. Russell's "An essay on the foundation of geometry" in order to end your confusion!

You confuse also the notion of dimensions. They are ASSUMED usually in physics and mathematics as purely local tools and if their assumed number is n>2 , then they becomes dependent dynmically if one use them in differential equations. Banach-Tarski paradox for n>2 is a reflection of this fact. But one has to prove their global existence for a manifold by topological tools. Otherwise Morse-Smale theory did not prohibit the assumption of more than two degrees of freedom.

Dear Ilgaitis,

I had a look on your quoted paper where you cited my paper "Dimensional structure of space-time". Your citation from me about 3D space is not correct. I proved and wrote there that there is no proof for more than 2D curved space (with 2 degrees of freedom).

Dear Sergey,

it seems that my reply on you is lost somewhat. I do it in a very compact form here. Specially the information theory is based on binary (2D) basis and Turing theorem proved it very fundamentally that any information (including physical knowledeg) can be given according to such 2D basis.

Dear Farhad,

By definition the Degrees of freedom is number of independent motions that are allowed to the body or number of possible independent relative motions between the pieces of the mechanism.

According to human body the proof is very simple: each finger I can move in the many independent motions.

According Your article. I found: "consequently there is no physical and mathematical evidence for a space-time with four independent dimensions" End of quotation. From it I conclude that 3 spatial dimensions according you is possible. Sorry if I am wrong.

Dear Ilgaitis,

the direction of finger motion should be independent. The notion of independent motion is not defined otherwise. Independency of direction is a geometrical/topological problem. Look on my previous argument about the three directions in the dynamics of general three bod problem whereby its non solvability shows the absence of three independent spatial directions. Dynamics is also a matter of topology if one apply the differential geometry to physical quantities like the potential as connection one form, etc.

Dear Farhad,

The motion is change of position. The definition of independent motion is extremely simple: The motion is independent if it is not affected by the other motions or position of other bodies. Where is problem to move each finger independently??

According 3 body problem. The non solvability shows only impotence of mathematics. It is not argument against 3D space.

The proof of space dimensionality is very simple: Mark the position of your phone on desktop. You get 2 dimensions x and y. Then lift the phone above desktop vertically. You get third dimension z. In the new position the coordinates x and y are not affected. Therefore there is 3D space. You cannot found next position where all 3 coordinates are not affected. So the fourth dimension does not exist.

Where is problem??

Dear Ilgaitis,

this is my last reply to your remarks.

Your notion of dimensions as that of a lot people including "impotence mathematicians" is a local one. This is pure assumption without any global and logical guarantee.

Dependency of directions in space are given usually if one of them can be mapped into other ones, i. e. it becomes by this map some components in other directions. It is a kind of prepanducularity barouwed from the cartesian conception of space but it works also on curved spaces such as two sphere or ellipsoid. Moreover independency of coordinate solutions or directions has a rigurus mathematical method, otherwise the index of differential operators which show the number of independent coordinate functions on a compact manifold on any sphere with assumed locally 2n dimensions would not be 2 and not the assumed 2n.

If you can show that such an index on such compact manifold is more than 2 than you become a "potent" mathematician and every body who like 3 dimensionality is happy. Although than you should prove the falsity of Morse-Samle and KAM theorems which show the unstability of dynamical systems with more than 2 degrees of freedom. bon viayage.

The computer denied my correction: bon voyage.

Is there a relationship between the number of dimensions of the universe and the number of basic physical quantities?

For the last question by Mohamed.

Depending how one distiguishes "basic physical quantities" my answer would change, but I argue that only two kind (categories) of basic physical quantities exists confirming the two curved degrees of freedom.

There are according to the wrong traditional cartesian dominant point of view in science a lot of undefined "basic" quantities in use, since in this "scientific" world view one attachs a "name" to any subjective new "thing" that one imagine and then believe that that imagined thing exists. Therefore since a lot of undefined quantities were used in mathematics and physics, Lebesgue noticed that only well defined (in my opinion logically defined without circulus vitiosus),quantities should be used in mathematics (see his letter to E. Borel quoted in A. Connes "non commutative geometry"). Also therefore Feynman complained tin his lectures hat physicits do not know what energy is! So energy is like the gauge potential and momentum (also a potential in symplectic geometry) a gauge transformable quantity. Thus there is no definition for connection, potential or potential energy in physics and mathematically it is a gauged quantity locally assumed. Globally its integral over a closed chain is equal to the surface integral of related curvature. On the other side electromagnetic field strength and spatial curvature are gauge invariant empirical quantities. Even so if one considers the dimensional order of physical quantities which range from (1/L) dimensional momentums, gauge potentials, currents and masses, temperature, frequences to the (1/L.L) dimensional field strenghs and curvatures (including not well defined forces (see B. Russell, "Principles of Mathematics")) in geometric units. And if one reasonably do not distinguish between any of the first category as components of differential one forms and also between any of the second category which are components of differential two forms, then one varifies that only two basic physical quantity categories exists.

One can argue the same number two for true basic physical quantities also in another similar way. I add this next.

Dear Farhad,

"There are according to the wrong traditional cartesian dominant point of view in science a lot of undefined "basic" quantities in use, since in this "scientific" world view one attachs a "name" to any subjective new "thing" that one imagine and then believe that that imagined thing exists."

I see that our ways of thinking are similar ...

To define a single point in space you have to cross all the dimensions to make that possible, however you have to define the "units of measure" such as meters, centimeters, millimeters, etc. It is even called "dimensionally correct" to the equations .

These "units of measure" are a dimension, exactly equal to x, y, z.

And if they are treated as a dimension, you can demonstrate many things that currently have no explanation, as I show in my work.

Dear Sergio,

there is no definition for point at all. If you look rigiorus mathematical encyclopadies you never find a definition for. Pioncare tried to do but gaved up. Hilbert did not even try to define point and begann with its assumption, etc.

Moreover what space you assumed for your point. One should define a space globally by integral invariants as Gauss did for two sphere in his Theorema egregium. A local assumption of a space with some dimensions or directions is a pure assumption with no global authoriaztion. Just the "measure theory" break down for spaces with more than two degrees of freedom. One of its reminiscences is the Hausdorff-Banach-Tarski paradox for n>2 manifolds.

So before any thing to do one has to prove the existence and uniqueness of some smooth 4D manifold. But there is no Existence and uniqueness for smooth$ manifolds (A. A. Kosinski, "Differential manifolds").

Dear Farhad,

"there is no definition for point at all. If you look rigiorus mathematical encyclopadies you never find a definition for. Pioncare tried to do but gaved up. Hilbert did not even try to define point and begann with its assumption, etc."

Exactly, is that if you think about it, you can not define a point in 3D no matter how small you define it, it could always be smaller, so it would not be a point anymore.

And as the clearest definition of dimensions is, all the information needed to define a point in space, it is clear that a 4th dimension is necessary to cut the other 3 and define the point as something unique and unequivocal.

"Moreover what space you assumed for your point. One should define a space globally by integral invariants as Gauss did for two sphere in his Theorema egregium. A local assumption of a space with some dimensions or directions is a pure assumption with no global authoriaztion. Just the "measure theory" break down for spaces with more than two degrees of freedom. One of its reminiscences is the Hausdorff-Banach-Tarski paradox for n>2 manifolds."

I have assumed a point in our 3 usual dimensions (x, y, z) and for this I have used a 4th dimension that cuts it (the Zoom, Zz) with which we are talking about the universe being a hyper-sphere, with our 3 dimensions in its hyper-surface. And why a hyper-sphere? because it is the only way to provide the universe with logical limits (replacing our conception of time with the new Zoom dimension as I show)

"So before any thing to do one has to prove the existence and uniqueness of some smooth 4D manifold. But there is no Existence and uniqueness for smooth$ manifolds (A. A. Kosinski, "Differential manifolds")."

I do not know all that mathematical treatment to which you refer, first the logic and then the mathematics to prop it up, never the other way around

Dear Sergio,

it seems that you repeat your ideas without proving them before.

You assumed a point in our 3 usual dimensions. You have to prove that 3 INDEPENDET dimensions exists globally before. (Note that this is not possible within our standard knowledge including mathematics and physics).

Then you have to define a point (This is not possible and not given).

This is the logic of matter.

The mathematical notion of "cut" used by you is a local one. Before one defines a point through cut of some curves, one has to define those curves globally and logically and also the cut at least globally.

Other wise you are right. I think it is enough.

I am explaining a hypothesis, that hypothesis is demonstrated by obtaining all the equations of special and general relativity using the galileo transformation and the Gaussian curvature, something that without this hypothesis is impossible as it shows the current way of obtaining the same equations with the most complex transformation of Lorentz and Bianchi identities.

Remember that to demonstrate the spherical shape of the earth it is not necessary to define the dimensions separately, in the same way that it is not necessary for to demonstrate the hyperspherical form of the universe

even so, I still think that with the definition of point as the set of crossed dimensions to achieve a single object with no possibility of error,

and with the definition of dimension, as the displacement perpendicular to any other displacement and that as a whole is able to define a point,

it is enough to define the dimensions as well as the point.

Perhaps it is that our concept of time of that 4th dimension is wrong and that is why what you say happens?

Apart, I think, and correct me if I'm wrong, that the problem of mathematicians and their manifolds, is that they always think in a Euclidean way, they add dimensions but do not use one that "closes" the others and that from the point of view of inhabitant who does not know the whole if they will be, but the whole set does not

1. Gaussian curvature in Theorema Egregium is before relativity and has nothing to do with it. General relativity may follow Gausian curvature but in the wrong generalization by Riemann (Riemann withdraws his related wrong paper from1864, but mathematician either do not even know this wrong paper or don't care about its withdraw).

2. "hypotheses non fingo" as Newton wrote.

3. Dimension has also no global definition in the case n>2. The case n = 2 is done by that theorem of Newton.

4. A striaght line (for its prependucular, etc.) is not given. Only curved geodesics over compact 2-manifolds like 2-sphere or ellipsiod can be defined by variation principle. no more.

5. The surface of earth is given empirically and need no further definition.

6. For definition of dimension "as the displacement perpendicular to any other displacement" is a local one. Try it globally then you need some well behaved integrals in three dimensions and this is impossible in view of difficulties of measure theory in n>2 related to Banach-Tarski paradox.

But I think we discussed enough. I give no more reply for yours.

Dear Farhad,

1- Gauss curvature is the way to express a curvature extrinsically, the Riemann tensor is the way to express it intrinsically, there is a relationship between the two, the Riemann tensor can be expressed with its bi-dimensional Gaussian curvatures as long as it has diagonal metric. What I do not know why you name it and has no relationship, is the Theorema Egregium

2- "Gravity is a force" also from Newton

3- Since there is no definition, do we live in a two-dimensional world then?

4- a hyper-sphere (3-sphere), for example, has 4 dimensions perpendicular to each other, with 3 geodesic lines

5- If a two-dimensional surface like the earth can be given in an empirical way and does not need more definition, a three-dimensional hyper-surface like the universe if it needs it?

6- That definition of dimension is local because it refers to a point, but obviously I am referring to all the points of the manifold and if all the points comply, that definition is global

Dear Christian,

tank you for your warm words. I had a rapid look on your essay.

Indeed you made rigourus thoughts about physics. Also I remarked in my erlear quoted work "Dimensional structure of space-time" that the space-time structure should be based on the dynmaical behaviour of observable physical systems and hence on their structural stability which determines the number of degrees of freedom to only two curved dimensions without any need to embed them in a imagined space with three dimensions. I agree also that basically there is no essential difference between classical and quantum theories and symplectic geometry give us a general framework for physics. But just therefore I do not understand why the time play a dominant role in your method. Then time can be completly absent from pthe hase space and symplectic mechanics. Of course there are very basic rempirical and logical easons to neglect the time from any physical theory in view of its dependency on spatial distances or coordinates. I described those arguments in the mentioned work. Thus one eliminates the time component of electromagnetic potential in QED or in any other quantization, e. g. Weyl gauge in Chern-Simons theory, etc. This elimination of time as dynamical variable is equivalent if we eliminate the time dimension in the space-time base manifold of QED or any QFT. It is of course related to the basic antisymmetry of field strength as two form component and the constant symplectic two form which an abstract field strength or curvature form.

Thus the symplectic topology is only given in two dimensions or two degrees of freedom which is astounishin for mechanics expert as Zehnder told me. For me this shows that CM and QM or QFTs are theories for two degrees of freedom and the rest is constructed and problematic as the constraints in four dimensional teories prove. Note that integral Maxwell equations which are somewaht unusual in lectures are two dimensional integrals as well as any field strength or curvature is a differential two form which necessiates only two curved dimensions or degrees of freedom. So Maxwell theory is globally (by integrals) two dimensional although it is custums locally in a 4D custume. In other words 4D space.time is just an local assumtion without any global authorization, i. e. not valid everywhere on the manifold.

At any rate if one believe on the 4D space-time, no matter where they come from, one has to prove the 4D space-time announced as the basic millennium qestion. I personally do not believe on that in view of the two dimensional structure of physics, differential topology and integral topology where the Euler characteristics of any comapct even dimensional manifold like 2n-spheres is only two irrespective of its locally assumed even dimensions. Whereas the same characteristic for odd dimensional spheres is zero! Moreover also that one should note that there is no existence and uniquesnes theorem for smooth 4D manifolds at all.

Let me at the end make a comment about algebras. Local aspects of algebras are imagined. So one has to look on their globally invariant aspects and this is outside of algebras in the integral or global topology. Similarly vector spaces are also purely locally imagined but their invaraint aspects as inner product are always some two dimensional or quadratic structure irrespective haw many local dimensions one attached to the vector space. So more topology than algebra.

Kindly

Dear Christian,

the fact is that empirical results manifest the same two diemnsionality (tdf) as the differential geometric or topological facts do. Therefore your aaffiliation to accelerator physics is welcome.

But you should explain explicitly how you extract two dimensionality from accelerator physics for me in short.

Moreover it was known theoretically that 4D Dirac spinor equation decomposes in concrete cases to two 2D spinor equations. Is your results the same?

In my view the 4D phase space (q1,p1,q2,p2) is de facto or equivalently in polrized case the 2D position space (q1, q2) in view of the p1 = R.q2, q2 = - R.q1, where R is some constant curvature like the magnetic H or B.

Dynamism in dynamical theories is manifested by the variation of spatial coordinates as it is known from usual (non extended) phase space mentioned before and from symplectic geometry of mechanics. There is no need for time as an independent coordinate in dynmaical theories. Also the Morse -Smale theory of structurally stable dynamical systems do not use time coordinates. There one may consider the two transversal V vector field components (as it is required by the two degrees of freedom for the structural stability of dynamical system) locally represented as above V1 ~ p1, V2 ~ p2, where R is again the constant component of the curvature differential two form (~ symplectic two form) manifesting the two dimensionality of the model. One may then introduce the time as a coordinate dependent on one of the curved spatial coordinates and rewrite one of them y ~ a.t where a is the constant acceleration.

As I mentioned before if one assumes time as an independent coordinate then one is confronted with constraints where the problem of application of quantum or classical constraints is not solved in QFT. At any rate the appearence of constraints in any theory manifests the dependency of assumed independent coordinates. And the necessary application of constraints in any theory reduces the number of degrees of freedom to two.

The issue is the right mathematical representation of the Universe. This is the starting step of Physics and the purpose of Geometry, before involving other physical phenomena such as forces or inertia. The word dimension comes from mathematics, and more precisely from differential geometry. To locate an event we need 3 spatial coordinates and one coordinate for the time. To do this we can use any conventional chart, with the simple condition that there is a way to go from one set of coordinate in a chart to a set of coordinates in another chart. This is precisely the definition of a manifold with 4 dimension. Then we can go further and deduce from the principle of causality that there is a non Euclidean metric on the universe, which represents its physical part. From there one can study the motion of a material body which is, by definition, an object which occupies a given location at each time. The Geometry of General Relativity can be built from few, precise assumptions which can be checked by everybody.

An issue which is commonly overseen is that of rotation. All material bodies have a specific arrangement with respect to an orthonormal spatial frame. This is still true for molecules and we can assume that this is a characteristic of all material objects, including elementary particles. As a consequence the location must be completed by the arrangement, as well as the motion. This gives "relief" to the structure of the physical universe, which is not just a collection of points. And the geometric representation of the motion of a material body requires 6 parameters. But they are of a geometric nature, and this does not imply that the manifold itself is 6 dimensional. Mathematically this is done in the fiber bundle formalism.

The last reply is logically and physically incorrect for example in view of the remark "To locate an event we need 3 spatial coordinates and one coordinate for the time."

This "need" is only correct in the case if the reference frame is in abolute rest. But since nothing can be consiodered n absolute rest nor logically neither empirically, therefore such a 4D reference frame is not the correct one. Thus any 4D dynamical system is constrained by definition as any CFT and QFT do and thereby the assumed 4 dimensions become dependent by definition (~ constraints). That one do so is the relict of ancient geocentric world view in the modern physics in view of the absence of scientific proofs of 4 dimensionality in the mean time.

In a dynamical (moving) reference frame as any reference frame in nature one needs only two curved independent dimensions (two degrees of freedom) to describe any stable motion with respect to the reference frame. About unstable one may not speculate in science in view of the fact that one can not do any experiment to confirm his statements.

This is only the mistake of one sentence. I may consider the rest of reply later.

For once forget about "reference frame". Think about locating a plane, a ship, or just to the postman. Have you ever met someone who uses a "frame" to locate a point ? Ask a pilot or the captain of a ship. Only in physics books you find this kind of fantasies.

Christian,

To be precise...

When we say "location" in Physics, it is to set up a procedure such that who ever follows the procedure can locate the point. The procedure itself does not matter, and several procedures can be used if the are compatible. The best example is the cosmic ladder used in astronomy. So there is no use of "phase space" or "momentum", which are abstract entities, well beyond the simple study of location.

When studying the procedures to locate a point we see that they depend on the observer. In the usual Galilean Geometry an event in time can be located in time similarly by all observers, the genuine innovation of Relativity is to say that this assumption is not necessary valid.

Most of the mistakes in Relativity and Physical Geometry come from the habit to use representations by orthonormal frames which have no experimental substance.

This is the starting point. How do you define a motion, a momentum ???

Why do we need "emergent" when basic geometry can work ?

The basic issue is that all physical quantities are located : we need some representation of the location to use coordinates. Moreover all other physical quantities (starting with rotation) must be defined in fiber bundles, so we need a principal bundle to start the procedure. A principal bundle, for the spin group Spin(3;1) can be defined from a tetrad. It can the be extended to a Clifford bundle, and then a complex Clifford bundle, which is necessary for the momentum and Dirac spinor.

The Quantum Mechanics axioms are just theorems, which can be proven. Observables are just statistical estimators of physical quantities, they do not represent physical quantities per se.

Pilot or ship captain or postman thinks and works with logically and mathematically undefined approximate concepts and tools. In science one should be able to define empirically or logically his concepts. Otherwise science became again the approxiamte, non logical science as in the time of Galilei, Newton and Huygens where one believed on absolute space and time. Nevertheless also if one accept this kind of approximate science as ingenieur do, then he must accept that the number of dimensions of universe is APPROXIMATELY 4. Although he could not define what this approximation is and would remain nebulus as before.

The 4 dimensional QM axioms (Heisenberg, Born, Jordan commutators, 1925) are as the authors remarked assumptions and can not be proved. Only one of them, the first relation by Heisenberg (1925) has a phenomenologic background!

Energy is a concept in physics : there is a quantity, which can be represented by a scalar, which is exchanged between objects in a process, continuous or not. Energy is larger than "kinetic energy", which is specific to Mechanics, and actually well defined only in Newton's Mechanics (but even the rotational kinetic energy is a bit convoluted). So it depends on the process, and how the state of the object is measured. Kinetic energy is related only to the geometric part (a change of the geometric state of the body). As any physical quantity it follows the principle of locality : what happens at a point is ruled by quantities which are defined at this point. This principle notably forbids action at a distance.

I do an extensive use of Clifford algebras, which appear as a "demultiplication" of the 4 dimensional space. I have written a purely mathematic note on the subject, with some interesting results (such as exponential and Cartan algebra). But it appears that in Physics we need a complex structure (this is actually the genuine innovation of Dirac's equation).

The energy is located at the point where the process occurs, so for any material body at its location.

The measure of the location and the motion of a material body are very physical and practical endeavor. And it seems that it is abstract enough to confuse many physicists, without the need to call for more complicated concepts.

It seems strange to give more physical meaning to kinetic energy or momentum than to location or motion. One does not measure the kinetic energy or the momentum, only their variation, in a definite process, which occurs at some location. This holds for energy as well as any other physical quantity : it is located.

So there is no "relative" or "absolute" kinetic energy : energy (whatever its nature) is a flow, and not a stock, it manifests in a process, that is in the interaction between different objects in a system.

And it is necessary to include the rotational momentum and the rotational kinetic energy. They are of the same nature than the corresponding translational quantities, so they should be measured in the same units. There is a natural universal constant relating the measure of length and time (which do not use the same procedures), which is c (which happens to be speed of propagation of force fields, but this assumption is not necessary to introduce c). The measure of translational kinetic energy involves naturally c. But there is no "natural" measure of the rotational kinetic energy : a rotational motion can be measured in rad / s, degree /s or turn / s. So actually one needs an additional constant, and it is not too difficult to see that it is some Planck constant.

In another forum on RG there has been a discussion about Science and Philosophy, lasting for years. I appreciate both, but their purpose are no the same. Science aims at finding how the world work. The first step is to define the objects of the science: what are the phenomenon that we will study. This is done in words, this is a bit general and vague but can be clearly understood by anybody who is interested. Theories and formalisms come after that : if we want to check a theory it is necessary to agree on the meaning of an experiment. This is one of the reason why the usual quantum physics is wrong : its supporters tells that it always provide the right answer, but nobody agree about they mean.

Philosophy has other purpose, to provide the payj to a good life for the Greeks, as well as Nietsche, or to tell why the world is it so. So philosophy calls for skepticism : am I sure of what see ? This is the path to metaphysics, or art, which are both useful and of interest.

Aber das Leben ist schwer genugen, um die Dinge einfach zubehalten !

At the end is the concept of location.

~ Location ideas:

irrespective what one means with location literally the scientific concept of locality as in mathematics is problematic. The locality concept and those of location are coordinate dependent if one try to describe these concepts regorusly . One the other hand empirical results and physical measurments give finite values and are global concepts given by some inner product or invariant integrals of usual theoretical ocal physical quantities like field strength. Only these global values are scientific results not local ones.

The fundamentality of two degrees of freedom arises exactly according to this very empirical stand point.

The location of an object has a clear meaning for all humans, except perhaps for some physicists. OK, let us say that we try to explain how the world works to everybody except the latter.

Humans and a lot of scientists have a "clear meaning of location" but they can not describe it in a well defined logical manner without stumbling in contradictions and they can not apply it to an empirical description of nature. Otherwise the "clear meaning of 4D location" which is understood also by Einstein did not result in the basic contradiction between the theory of general relativity and quantum theory that are both based on the same 4D "clear meaning of location". We discuss here about a scientific problem and not "humans clear meaning of location". This is not the place to discuss what humans "clear meaning of location" is.

1) Location of a material body is a fact which is well understood by - almost - every body. The issue if it is "absolute" or "relative" has no meaning. The measure of a location is done with respect to an observer, as any other physical phenomenon. It uses charts, which depend on the observer.

2) It is strange to assert that the measure of a location should involve Quantum Theory, a set of axioms of which nobody has been able to give a physical interpretation. Next time you take a plane check with the pilot if he has revised his quantum theory.

Different interpretations can be given.. The number of actual dimensions attributed to space itself is just 3.

But if you have to describe the configuration of a two body problem, then you need perhaps 5 or 6 independent variables.

To describe the state or wave function of a quantum problem

really involves infinitely many.

In mechanics the independent variables are denoted by the

q(i)

Phase space of a gas can have 6N dimensions where N is the number of simple atoms in it. 3N are spatial and 3N are for momentum.

Some of the parameters may be angles which do not have

visibility. In this sense some models are said to have for example to have 10 dimensions.

It can be fashionable for some scientists to use complicated words to identify simple facts or objects. Denying common sense is just pedantry.

OK, next time you purchase a ticket for a travel you find your words. Good luck !

In my book, at the precise page 28, I deal with Quantum Mechanics, and I start from how the "state of a system" is usually represented in Statistical Mechanics. This is not my representation, just an example of what is commonly used in Mechanics. This part on Quantum Mechanics, and notably the theorems, does not use any physical assumption, in particular on the Geometry of the Universe.

The 2d chapter is about quantum mechanics and does not introduce any physical assumptions because QM is not a physical theory, just a collection of mathematical theorems.

The physical part starts at chap 3 and the assumption about the representation of the universe is clearly stated.

You have good books, but you do not read them.

A major problem in mathematics and physics is that beyond a few not well known exceptions mathematicians and physicists are not depply enough informed about the others disciplins. In order to discuss the concepts of dimension, degrees of freedoms and number of dimensions in the universe or any globally stable manifold it is not enough to repeat the usual inaccorate notions which are just sophisticated and purely local version of the undefined traditional notions of dimensions, etc. Then it result in the unsolved crisis of matematics and physics, e. g. "The foundational crisis of mathematics" and "The crisis of incompatibility of quantum theory and general relativity theory".

As an example the concept of infinite dimensional space, like the Hilbert space in mathematics and physics is fundamentally wrong and leads to unsolvable problems. Hilbert's defending of infinity in mathematics is based on laughable "synthetic a priori judgments" of Kant's (5+7 = 12) example which is logically absolutely wrong if one considers the Russell-Whitehead's "Principia Mathematica" where in order to define the operation of addition they need more than 300 pages of definitions from which at least some are results of empirical concepts a posteriori. Nevertheless mathematician and physicists use the infinite dimensional tools without thinking. This is only one exmaple of inaccurateness in science. So as Hermann Weyl wrote: “We must learn a new modesty. We have stormed the heavens, but succeeded only in building fog upon fog, a mist which will not support anybody who earnestly desires to stand upon it. What is valid seems so insignificant that it may be seriously doubted whether (mathemtical branch of) anlaysis is at all possible.”

But if analysis is not possible at all what mathematicians and physicists will do at all?

1. In Mathematics the concepts of Geometry, in Algebra or Differential Geometry are well established and known of the professionals.

2. There are different "infinite", the most usual is the countable (corresponding to the cardinality of integers). Vector spaces with unaccountable dimensions are difficult to use,

3. The axioms of QM are theorems which can be precisely proven, they hold only for countable dimensions.

vector spaces with

Proposition 35 : the universe can be represented by a four dimensional real manifold

In my world material bodies inhabit the universe, which is the container. Which can be represented by a manifold.

Phases spaces are a representation common in Mechanics. Whenever there are constraints a phase space is no longer a vector space but a manifold. Moreover the big problem of this common representation is that, whenever we consider a force field, defined everywhere, the phase space has an infinite dimension. A well known problem in Mechanics, that I address precisely in the chapter 2 : actually the state of a system is defined by a map.

Perhaps in your world material bodies inhabit phase spaces, then I understand that you have some difficulties to find your way.

A material body has first a location, second a motion. Moreover it has an arrangement and a rotational motion. But it you definitively want to keep the Newton's Mechanics...which acknowledges 3 spatial dimension. Have your way !

I am not responsible for writing mistakes in my reply. Sometimes shows the computer mistakes which are not and sometimes it do not accepts rapid corrections.

I would not use the word evolution at all, maybe change.

Motion is change in position.

In ordinary space one single point is identified by 3 parameters, in phase space by 6.

Hausdorf dimension in topology, math, may be fractional.

It is very difficult to comment on the number of dimensions in the Universe.

We know, however, that the number of dimensions required to construct theories describing certain features of the Universe varies with the most commonly accepted 4 dimensions proposed by the special relativity theory. The more abstract constructions require many extra dimensions but they remain very speculative.

It is worth mentioning that there are still some debates about the most basic problems like the n-body problem. In the original question there is a statement: “General three body problem is not integrable or solvable analytically which shows that the present assumption of three dimensional space is wrong.” According to some authors this problem can be solved, albeit, in the terms of series expansion. I have attached a paper on the global solution of the n-body problem.

Reading the interesting paper linked to the question, I noticed the statement about the Poincare conjecture, which should be updated considering the proof provided by Perelman.

https://www.claymath.org/millennium-problems-poincaré-conjecture/perelmans-solution

The question about dimensionality sets the scene for more general questions related to the topology of the Universe. According to some theories the Universe is a multi-connected topological structure making interpretations of dimensionality even more difficult.

The series expansion is not an analytical method but just an approximated method. I mentioned already that in view of some early replies: they may assume that the number of spatial dimensions in the universe is approximately three, but physical and mathematical models do not work with approximate number of dimensions but with exact number of dimensions. Therefore this approximate notions of number of dimensions in the universe are irrelevant in scinece. This statement apply also to those who integrate the general three body problem by series expansion. By the way the series expansion is more or less a trick and logically irrelevant.

The Poincare conjecture is from the stand ponit of rigorous topology a wrong generalization of the topological properties of 2D two sphere to the fantasy object of 3D three spheres. Because the dimension is a topological invariant and one may not generalize 2D properties of two sphere to some 3D object. Poincare was in this respect fundamentally wrong! Thus also his "biological" argument with human ear shows that he was not good in logic and topology. There are further nebulous and fundamentally wrong relations related to him which are however accepted in the scientific community.

Also the so called proof of this conjecture which recals more a comedian theatre

than a rigorous scientific presentation is by no means a proof of a topological question. Then the whole work of R. S. Hamiton , ..., Perelman with respect to this conjecture, e. g. Ricci flow is of purely local conception, i. e. of purely differential topological nature. Differential topology unfortunately do not care about dimensional or global restrictions. Only integral invariants do care about dimensional circumstances. Thus no integral relation is given for this so called proof. But the conjecture is a global conception which needs a global or integral topological proof!

There is and there can be no mathematical or experimental proof for the three dimensionality of space and four dimensionality of the universe. Nor mathemaics, specially differential topology, neither physics possess any tool to do it. Specially differential topology is based on a two dimensional closedness in view of d^2 = 0 and *d* ^2 = 0. It is amyzing that the dual of this relations is called the Poincare lemma according to which the boundary of a boundary is empty.

The three dimensionality of space and also the assumption of an idependent forth dimension, the time, are relicts of the geocentric world view in science.

Note further that there are a lot of restricted three body problems which are in fact genaral two body problem which are of course integrable. One has to consider the notion of small perturbations exactly. Insofar any condition which transform the general three body problem into a restricted three body problem has nothing to do with the general problem but with restricted problem.

“…..There is and there can be no mathematical or experimental proof for the three dimensionality of space and four dimensionality of the universe. Nor mathemaics, specially differential topology, neither physics possess any tool to do it….”

- yeah, there cannot be principally any mathematics’s proof of existence of any physical, i.e. material, event/effect/process, in mathematics simply there are no mathematical objects, say, “Energy”, “Momentum”, “Star”, “Gravity”, “particle”, etc.

However, as that is rigorously proven in the Shevchenko-Tokarevsky’s indeed philosophical “The Information as Absolute” conception https://www.researchgate.net/publication/260930711_the_Information_as_Absolute DOI 10.5281/zenodo.268904, there absolutely fundamentally exist nothing else than some informational patterns/systems of the patterns, that are elements of the absolutely fundamental and absolutely infinite “Information” Set, including, say, “Matter” is nothing else than some informational system/the Set’s element; and human’s consciousness, which studies Matter in the science “physics” is only some informational system also.

And just because of Matter is the system, which exists and constantly changes – as a whole and as every material object – particles and systems of the particles, basing on rather small set of fundamental and fundamentally universal laws/links/constants, when

- in spite of the other system, a consciousness exists and constantly changes basing a set of fundamental and fundamentally universal laws/links/constants that fundamentally differ from the Matter’s set above,

- because of both system systems are made from the same stuff “Information” is able sometimes adequately decode some laws/links/constants in material objects/systems.

At that Matter is rather simple, however very smartly designed/created, informational system, which practically for sure exists and operates basing:

- on simplest binary – in accordance with von Weizsäcker’s outstanding 1954-55 years “Ur hypothesis”, logics, and so, as von Weizsäcker had showed that for to be a binary system Matter should have 3D space, the Matter’s space is 3D space,

- that, of course, by no means follows from any mathematics, and

- as well as Matter is based on reversible – in accordance with the outstanding Fredkin-Toffoly finding, logics that reversible objects/systems don’t lose energy at transformations and interactions, just so Matter doesn’t lose the energy outside in the Set, and in Matter so the energy conservation law acts, and to actualize this point in Matter’s spacetime besides “true time”, t, special dimesion “coordinate time”, τ, exists,

- that, of course, also by no means follows from any mathematics.

And, because of that objects/effects/processes in Matter exist/happen in accordance with universal laws/links/constants ; including that can be described by a few fundamentally universal measurable parameters, mathematics is extremely effective at analysis of physical tasks, remaining at that as nothing else/more than an extremely effective tool.

In the SS&VT conception above, besides the phenomena/notions “Matter” and “Consciousness”, a number of other fundamental and so Meta-physical phenomena/notions are properly defined also, including “Energy”, “Space”, “Time”, and so “Dimension”,

and basing on the conception and existent experimental data in the Shevchenko-Tokarevsky’s informational physical model https://www.researchgate.net/publication/273777630_The_Informational_Conception_and_Basic_Physics DOI 10.5281/zenodo.16494 it is shown, that Matter’s spacetime is absolute [5]4D Euclidian “empty container” in the reality, and/or [5]4D Euclidian spacetime in physical theories, has metrics (cτ,X,Y,Z, ct).

More see the links above.

The to read last SS posts in the threads https://www.researchgate.net/post/What_is_the_most_precise_definition_of_time_And_what_is_the_easiest_way_to_describe_time

and https://www.researchgate.net/post/Why_has_the_view_of_WW_Engelhardt_about_SRT_clock_synchronisation_problems#view=5dd0036aa4714b3e8f2afe32

would be useful also.

Happy New Year coming!

Cheers

Sometimes degrees of freedom can be confused with dimension.

Degrees of freedom is roughly the number of independent paramenters to describe some theory.

When they say that some M theory is in 10 or more dimensions, they really mean degrees of freedom, which are angles, parameters or anything.

For me this is an improper usage of the term. Spatial dimensions are only 3 and one more for space-time.

It is very astonishing that people who comments scientific works believe religiously on four dimensional space-time: although its proof is announced since about 20 years as the Millennium problem of physics (string 2000, Ann arbor) and no one dares to prove it.

Science is not the place for stories and dreams like those recalling Weizsäcker that I new very well than those write stories about him.

One who dares to say anything scientifically about 3D space or 4D space-time he or she should integrate first of all the general three body problem that is not possible at all!

Then he should disprove the Morse-Smale theorems and KAM theorems, Peixoto theorem about the structurally stable dynamical systems which proved for all time that only dynamical system with two degrees of freedom are stable against small perturbations.

I will not disturb those dreams asking about the impossibility of quantization of the major 4D model of all time, the gravity in view of its 4 dimensionality! Those people can dream further.

As the name says it "dimension" refers to a mathematically defined property of a manifold. That is the number of independent parameters that we need in a chart to locate a point. So far, as anybody can see, we need 4 coordinates to locate a point, so our universe has 4 dimensions. Now as usual some academic can call for his own pervert definition of dimension and says that the number is 3.741 or anything else, who care ?

I can not comment childish statements about dimensions of universe. Nor the concept of dimension neither the number 4 of dimensions are defined in a logical/ global manner in a mathematical encyclopedia. Both physicists and mathematicians assume the number of dimensions locally. This has no global or integral value on the whole underlying space. This is not definition but assumption, only local assumption without any logical value.

Dear Jean Claude,

The number of dimensions are the parameters used to define a point in space, totally agree.

But let me test you, could you define a point in space and tell me what are the dimensions you have used?

Ask the pilot of a ship...

Dear all,

And what would a pilot of a ship tell me? What would be the dimensions you would have used to point a point?

So vivid discussion after seems quite clear SS post above…

“…Sometimes degrees of freedom can be confused with dimension…..”

- the degrees of freedom aren’t of course, some dimensions, as well as there are no some abstract “degrees of freedom”. That are in every case just concrete degrees of freedom for some concrete object/system [we remember that there exist nothing else than some informational patterns/systems of the patterns, see the SS post above], i.e. some informational pattern/system, to change its/her/his parameters/ informational senses.

So that

“…Degrees of freedom is roughly the number of independent paramenters to describe some theory.……”

- contains seems as some misprint; it should be as “number of independent degrees of freedom is equal to the number of independent parameters”. And, besides, correspondingly - to number of corresponding dimensions, i.e. of the possibilities for the parameters to change their values. Parameters change their values in their dimensions, when the degrees of freedom remain be at that the same, though at changes some new degrees of freedom can appear, etc.

The concrete systems, which have concrete independent degrees of freedom and corresponding parameters so exists and changes in some systems of dimensions, and changes of concrete parameters values are motions of elements of the system in corresponding concrete dimensions; in physics that is formulated sometimes that the possibilities/dimensions are concrete dimensions of concrete spaces.

Thus the spaces in physics, i.e. systems of dimensions where physical parameters can – and are - be changed,

because of, again, Matter, which physics studies, is a simple and rigorously organized logical system, which exists and changes in accordance with a small number of fundamental and universal laws/links/constants, and so mathematics is extremely effective tool at describing/analysis of objects/events/processes in Matter, the physical spaces in certain sense are very similar to spaces in mathematics.

With, nonetheless, essential difference: mathematics is essentially “static” informational system, where in every concrete theory “all had happened” when the set of axioms of the theory was formulated. So, e.g., the real numbers exist always, already – and always be fixed - on their possibility/dimension to change a number value the “infinite real number line/axis”.

However Matter is fundamentally dynamical system, which always constantly changes, and so besides the concrete, purely Matter’s parameters, it has absolutely fundamental and universal possibility to change -“true time”- ct-dimension; and purely Matter’s “coordinate time”- cτ-dimension.

So in physics the system “Matter” exists and changes in corresponding absolute Euclidian “spacetime”, that is composed by Matter’s possibilities to actualize its independent degreases of freedom.

At that, besides the true time dimension, Matter has 4 utmost fundamental and universal, i.e. that work on the utmost fundamental Planck scale, independent degrees of freedom to change their state

– 3D in space in accordance with von Weizsäcker’s “Ur - hypothesis”, and in accordance with Fredkin-Toffoli finding [see the SS post above] binary reversible fundamental logical elements (FLE), and so utmost fundamental and universal Matter’s spacetime is the absolute [5]4D Euclidian spacetime with metrics (cτ,X,Y,Z, ct),

Where every material object moves with 4D speed of light in the 4D sub-spacetime with metrics (cτ,X,Y,Z), and, simultaneously, with the 1D speed of light in the true time ct-dimension.

Of course in Matter there are a number of other parameters of concrete material systems, and so it is possible to add to the 5 dimensions above a lot of other dimensions. For example that can be a “charge dimension” and practically all/every material objects so would be in some points with coordinates in the spacetime say, (cτ,X,Y,Z, ct, q); and, say, if somebody would comb dry hair with a dry comb, this somebody and the comb would move in the q-dimension in opposite directions, etc.

However such cases aren’t essential, at least now, and so at least in mechanics the Matter’s spacetime above , and some configuration spaces in this spacetime, etc. should be used.

Happy New Year coming!

Cheers

“…An artist perceives the truth.….”

- it seems interesting – what “truth” what artist perceives, that other humans don’t perceive? There are so many truths in the World…

“…A philosopher understands the truth …”

Yet more interesting – what philosopher what “truth” has understood/understands…..

“…the scientist quantitatively describes the truth. ….”

- when scientists indeed mostly quantitatively analyze the truths, understanding, at that, what they do; and paying no any attention to about what truth what philosopher claimed that he understood …

Though the above relate to the mainstream philosophy, of course; the indeed philosophy is indeed science, which indeed studies utmost fundamental truths, and next examples how the indeed philosophy indeed helps science see last SS posts in the threads

https://www.researchgate.net/post/Number_of_universe_dimensions#view=5e0769d1d7141baf555b8b3e

https://www.researchgate.net/post/Is_Chalmers_so-called_hard_problem_in_consciousness_real#view=5cefe4224921eeaeb72973d0

https://www.researchgate.net/post/Why_has_the_view_of_WW_Engelhardt_about_SRT_clock_synchronisation_problems#view=5e0780702ba3a15ce62c170b

Happy New Year coming

Cheers

If I new what kind of comments would be written about such a serious scientific question, I wouldn't introduce it.