We assume that any N-dimensional space can have an (N-1)-dimensional "boundary". However, if the boundary is limited to points, lines, and surfaces, and not to bodies, then three-dimensional space will be the maximum spatial dimension that satisfies this condition. What is the mathematical concept involved here?
Dear Chian Fan
The mathematical concept involved here is "dimensionality". In mathematics, the dimensionality of a space refers to the minimum number of coordinates or parameters required to specify any point in that space. For example, a point in two-dimensional space can be specified using two coordinates (x,y), while a point in three-dimensional space requires three coordinates (x,y,z).
In general, any N-dimensional space can have an (N-1)-dimensional "boundary", which is the set of points that lie on the edge or boundary of the space. However, if the boundary is limited to points, lines, and surfaces, then three-dimensional space is the maximum spatial dimension that satisfies this condition. This is because in higher dimensions, there are additional geometric objects such as hypersurfaces and higher-dimensional polytopes that can form the boundary of the space.
Therefore, the concept of dimensionality is central to understanding the relationship between the number of coordinates required to specify a point and the geometric objects that can form the boundary of a space. The mathematical concept of "manifold" is also closely related to the idea of boundaries in higher-dimensional spaces, as manifolds are spaces that locally resemble Euclidean space and can have boundaries of various dimensions.
Very good and professional answer.
So, let's go back to the physical world, if a polyhedron can be considered as a boundary, how can we distinguish between inside and outside? Assuming an infinitely small distance from the inside to the outside through such a "boundary", how can the substantial difference between the inside and the outside be reflected?
“…We assume that any N-dimensional space can have an (N-1)-dimensional "boundary". However, if the boundary is limited to points, lines, and surfaces, and not to bodies, then three-dimensional space will be the maximum spatial dimension that satisfies this condition. What is the mathematical concept involved here?….”
- that is indeed purely mathematical question, while so that
“…So, let's go back to the physical world,….”
- really well differs from the first quote. Physics and mathematics are fundamentally different sciences. Including, say, if in mathematics the spaces have dimensions that as a rule are “the same type” dimensions, Matter’s utmost universal “kinematical” spacetime is fundamentally absolute, fundamentally flat, and fundamentally “Cartesian”, [5]4D spacetime with metrics (cτ,X,Y,Z,ct), which is formally mathematical 5D space, however really it is composed from fundamentally different 3 types of dimensions:
- 3 “ordinary mathematical space ” dimensions X,Y,Z, where a certain “physical point”, say a particle, has 3 coordinates, when is at rest and if moves;
- “coordinate time”, cτ-dimension, which exists aimed at to realize the possibility to exist for two types of particles – particles and antiparticles, what isn’t a principally mathematical problem, though that can be described/analyzed in a mathematical construction; though a given particle/antiparticle moves in the 4D space with metrics (cτ,X,Y,Z) in all 4 dimensions mathematically by the same way;
- and the ct-dimension, which exists completely independently on the 4 space dimensions above, simply at any change of a material object’s state, independently on – that is motion in 4D space or inside the object, when it is fixed in the 4D space, this object fundamentally obligatorily moves in parallel also on some Δct in this dimension.
And, in this case finally: in physics there exist fundamental parameter of practically everything – “energy”, which at least till now doesn’t exist in mathematics as, say, a mathematical object or an mathematical operation, though, of course, mathematically can be described and analyzed, including as “energy dimension” [momentum dimensions in 3N dimensional configuration spaces for N-bodies systems, etc.]
Moreover, real Matter’s spacetime is [4+4+1]4D spacetime with metrics (cτ,X,Y,Z,g,w,e,s,ct), where the g,w,e,s dimensions relate to fundamental Nature Gravity, Weak, Electric, and Strong Forces, where every particle can have only unit length “coordinate” [seems Planck length], and “is at rest”, though has some degreases of freedom – and so “rotational dimensions”.
Etc., however, again, all the above really exists and happens first of all in the [5]4D spacetime above, which is fundamentally necessary by definitions of the absolutely fundamental phenomena/notions “Space” and “Time” and correspondingly relates to the ultimate Matter’s base – the (at least ) [5]4D lattice of (at least)[5]4D binary reversible fundamental logical elements [FLE] where:
- Matter has 3 XYZ space dimensions in accordance with the outstanding von Weizsäcker hypothesis that Matter is based on binary logics and so the ultimate base of Matter are some binary constructions “Urs” in old German – i.e. “bits” in English,
- cτ space dimension in accordance with the outstanding Fredkin-Toffoli finding provides action in Matter of the energy conservation law;
- while the topology of material objects in the 4D space [though till now we cannot exclude in the 5D “space”] is determined by the additional condition – every particle [with all its predecessors] from the Beginning moved in the 4D space distance D=ʃ0t(dx2+dy2+dz2+d(cτ)2)1/2=ct; though here can be well essential other physical effects – first of all the seems real expansion of the FLE-lattice, etc.
More see the SS&VT informational physical model, main papers are
https://www.researchgate.net/publication/354418793_The_Informational_Conception_and_the_Base_of_Physics ,
https://www.researchgate.net/publication/355361749_The_informational_physical_model_and_fundamental_problems_in_physics , and
https://www.researchgate.net/publication/369357747_The_informational_model_-Nuclear_Force
Cheers
spaces differ from one another in mereologic AND topologic properties ............
Dear Sergey Shevchenko
Your response carries more important questions.
I: “spacetime is fundamentally absolute, fundamentally flat, and fundamentally “Cartesian””
Several words describing properties here, absolute, flat, fundamental, Cartesian, need to be defined precisely.
Personally, I also consider the fundamental spacetime as absolute, flat, Cartesian, but it does not conflict with the relative spacetime, if we consider "relative" as a "modulation" of "absolute". If we consider "relative" as a "modulation" of "absolute".
II：“And, in this case finally: in physics there exist fundamental parameter of practically everything – “energy”, which at least till now doesn’t exist in mathematics as, say, a mathematical object or an mathematical operation, though, of course, mathematically can be described and analyzed, including as “energy dimension” [momentum dimensions in 3N dimensional configuration spaces for N-bodies systems, etc.]”
The mathematically describable energy is essentially the unification of the concept of energy. It is necessary to initiate a new discussion on this issue: to make it in the obvious place.
https://www.researchgate.net/post/Can_different_forms_of_energy_be_unified
Best Regards，Chian Fan
Dear Chian Fan,
“…Dear Sergey Shevchenko
Your response carries more important questions.
I: “spacetime is fundamentally absolute, fundamentally flat, and fundamentally “Cartesian””
Several words describing properties here, absolute, flat, fundamental, Cartesian, need to be defined precisely…”
- well, that is indeed fundamentally important– and just so in the philosophical 2007 Shevchenko-Tokarevsky’s “The Information as Absolute” conception, recent version of the basic paper see in
https://www.researchgate.net/publication/363645560_The_Information_as_Absolute_-_2022_ed
- all really fundamental phenomena/notions, first of all in this case “Matter”, “Consciousness”, “Energy”, “Information” - and just “Space” and “Time”, which are the fundamentally completely transcendent/uncertain/irrational in mainstream philosophy and sciences, including physics, are really scientifically precisely defined;
- and further in the SS&VT informational physical model [the links see SS post above], which is based on the conception and on all existent really reliable experimental data, , it is rigorously precisely shown that Matter’ spacetime is as that is written in the last SS post above.
The scientific definitions of the absolutely fundamental , and universal, “Logos” set elements, phenomena “Space” and “Time” [and other fundamental phenomena/notions above, though] are given on first dozen of pages in any of two main physical model papers that are linked in the SS post above; why the actualizations of “Space” and “Time” as the concrete space/time/spacetime of some informational system “Matter” are as they are [spacetime metrics is (cτ,X,Y,Z,g,w,e,s,ct),] see the rest in the paper.
More concretely in this case – that the Matter’s spacetime is “fundamentally absolute” – means that the spacetime is absolutely necessarily and obligatorily “appeared automatically” as absolutely inevitable actualization of these “Logos” set elements – any/every informational pattern/system, including “Matter” can exist absolutely only in some space [that can have arbitrary number of arbitrary dimensions], and can universally change absolutely only in [fundamentally one] time dimension.
The number and nomenclature of space dimensions is completely determined by degreases of freedom a pattern/system can change its state at which;
- since the ultimate base of Matter are binary reversible fundamental logical elements [FLE], which have 4 utmost universal degreases of freedom at its binary flip, the Matter’s utmost universal space is (3+1)D, with metrics (cτ,X,Y,Z);
- and this “space appeared” as the infinite by definition “empty container” for whole Matter just at the first FLE was created; with the time ct-dimension that compose the utmost universal Matter’s spacetime with [5]4D metrics (cτ,X,Y,Z, ct), though FLE has some other degreases of freedom, and so the space has more than 4 dimensions, see the SS post above.
Since FLE compose the ultimate (at least)[5]4D dense FLE-lattice, while everything in Matter is/are some disturbances in the lattice, and at any disturbance nothing can change FLE logical structure, while fundamentally nothing at all in whole “Information” Set can impact on time dimension, Matter’s spacetime is absolute in the sense that all impacts of material objects on the spacetime that are postulated in the SR/GR fundamentally are impossible; and so these SR/GR postulates are fundamentally wrong.
The spacetime is “fundamentally flat”, since the FLE-lattice is fundamentally uniform, any emptiness “is flat”, though.
Matter’s spacetime is “fundamentally Cartesian”, since, though any mathematical Euclidian space can have any sets of basic unit non-complementary vectors, the angles between which can be arbitrary, in Matter’s space that is possible only for the 3DXYZ space dimensions, but the cτ-dimension fundamentally – materially/physically – is orthogonal to any 3D space line.
This response is long already, so a comment to
“…II：“And, in this case finally: in physics there exist fundamental parameter of practically everything – “energy”, which at least till now doesn’t exist in mathematics…”
- will be, let hope, in other SS post.
Cheers
The comment to
“…II：“And, in this case finally: in physics there exist fundamental parameter of practically everything – “energy”, which at least till now doesn’t exist in mathematics…”
- see in SS post in https://www.researchgate.net/post/Can_different_forms_of_energy_be_unified
Cheers
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