"Depth is the first and most primordial dimension" (Maurice Merlau-Ponty).
"The laws of nature become simpler and more elegant when they are expressed in higher dimensions" (Peter Freund).
"I am trying to describe a unified world. But since we are in a world of duality, I can only express it through dimensions" (Harold Klemp).
The concept of dimension
In general, the dimension of a system is the number of degrees of freedom it possesses, or the number of basic entities that are necessary to be able to express any entity of the system by means of these basic entities. The basic entities are independent −or orthogonal− to each other.
For example, the classical physical space has 3 dimensions. Every point in space can be expressed by projections onto 3 axes orthogonal to each other. In this case, each dimension has two opposite directions (north-south, east-west, up-down).
In mathematics, the concept of dimension is initially linked to Euclidean space In, a generalization of physical space to n dimensions. Any
point in space can be expressed by a sequence of n real numbers (positive or negative), each corresponding to a dimension. The concept of dimension in mathematics also applies to other spaces: vector, topological, etc.
In linguistics, dimensions are the hypothetical basic, essential or primary concepts from which it would be possible to express any other concept. We speak in this case of "semantic dimensions".
In philosophy, the dimension is a characteristic possessed by any entity that makes it exist, that is to say, it is the being, the essence of the entity.
Absolute and relative dimensions
A circumference is a two-dimensional object contemplated at the absolute or extrinsic level, from 2D space. But at the relative or intrinsic level, contemplated from inside the circumference, it is dimension 1 because only one parameter is needed to define a point on the circumference. Similarly a sphere at the extrinsic level is dimension 3 and at the intrinsic level is dimension 2, because only 2 coordinates (latitude and longitude) are needed.
A geometric object immersed in a higher dimension is also considered a space, which is called a "variety". A variety of (intrinsic) dimension n is called an n-variety. So a circumference is a 1-variety and a sphere a 2-variety.
The concept of "field"
Since the time of the ancient Greeks, mathematics and physics have been inseparable. Newton made no distinction between mathematics and physics, and both were viewed from a philosophical point of view. Newton was a "natural philosopher".
A key concept in the evolution of physics was that of "field," created by Faraday. With the field concept, all forces of nature could be expressed as a field. The field associated with a force source is the space surrounding that source, such that at each point in space a force of a certain intensity is associated with it. Before Faraday, the force between distant bodies was believed to be an instantaneous interaction, a mysterious "action at a distance" of a nonlocal type.
Hyperspace
A hyperspace is a space that has 4 or more dimensions. This concept is not only mathematical, but also applies to the physical world.
Gauss was the first to consider the surfaces of 3D space as spaces in themselves, using local coordinates (intrinsic or relative) and establishing curvature as a local property.
Riemann was the first to establish a mathematical foundation for the geometry of multidimensional space. And he was also the first to state that nature finds its natural domain in the geometry of multidimensional space.
Riemann's three major contributions to multidimensional geometry were:
The generalization of the Pythagorean theorem to n dimensions.
The generalization of the concept of surface as a space: the n-dimensional geometric spaces (which today we call "varieties").
The introduction of a metric tensor describing the curvature at each point of a geometric space. It is a generalization of the Pythagorean theorem for curved spaces (with curvature). The metric tensor consists of a collection of numbers associated to each point of the space that describes the curvature of a geometric space of n dimensions. For a 2D space 3 numbers are needed. For a 4D space 10 numbers are needed. The Riemann tensor is a generalization of the Faraday field concept.
For Riemann, force is a consequence of geometry. Electricity, magnetism and gravity were caused by a deformation of our 3D universe in 4D space.
Einstein, in his theory of special relativity (1905), considered time as the fourth dimension. Einstein used Riemannian geometry for his theory of general relativity, integrating space and time into a concept he called "spacetime" in an intrinsic space (4-variety). The gravitational field is explained as a curvature of spacetime.
The fifth dimension (the fourth spatial dimension) was introduced by the Kaluza-Klein theory (created by Kaluza and later improved by Klein). This theory united Einstein's theory of general relativity (gravity) and Maxwell's electromagnetic theory. Light emerged from the distortion of the geometry of the fifth dimension. This fifth dimension was topologically equivalent to a circle.
Einstein, during the last 30 years of his life, tried to develop a "unified field" theory, a term he coined for a theory that would explain all physical laws. But in his time the nuclear forces (weak and strong) had not yet been discovered.
The geometry of n-dimensional space, the geometry of hyperspace, is the current path followed by physics to try to achieve a "theory of everything", the Holy Grail of physics. According to Michio Kaku [1996], hyperspace theory may be able to unify all known laws of physics into a single theory. The problem that such a theory is not falsifiable, because the higher dimensions are not detectable.
In the hyperspace paradigm, matter and forces may simply be different vibrations of hyperspace.
The Kaluza-Klein theory inspired modern string theory. According to this theory, subatomic particles are tiny vibrating strings in a 10-dimensional space. The 6 extra dimensions we cannot detect because their size is smaller than the Planck length.
According to the M theory (proposed by Edward Witten), there are 11 dimensions. It uses supersymmetry (a hypothetical symmetry connecting fermions and bosons). That is why it is also called "superstring theory". It contemplates objects called "p-branes" of different dimensions, being p the dimension of the brane (a 1-brane is a string, a 2-brane is a membrane, etc.). This theory integrates quantum physics and generalized relativistic physics, but is not yet complete.
Superstring theory is so abstract that it is indistinguishable from mathematics. Some mathematicians such as Isadore A. Singer of MIT have claimed that perhaps superstring theory should be considered a branch of mathematics, regardless of its possible application in physics. At a deep level, physics is so abstract that it does not seem to make any reference to the physical world.
Witten has suggested that a general formula for the theory M would require a new mathematical language, since the present mathematical language is not sufficiently abstract.
MENTAL, a Multidimensional Language
MENTAL is a multidimensional language. It is based on 12 semantic dimensions. Each dimension corresponds to an archetype of consciousness, philosophical category or universal semantic primitive. They are degrees of freedom operating in an abstract space of 12 dimensions. In turn, each dimension is dual. The set of the 12 dimensions, together with their duals, constitutes a universal language.
The advantages of the multidimensionality (or multidimensional paradigm) of MENTAL are:
Everything is simplified, clarified or resolved by looking at things from a higher (or deeper) point of view.
It allows to act from the source of power, that is, from the conscience.
It favors creativity.
It unites reductionism and holism. In general, it unites opposites. Dimensions connect the intuitive to the rational, the generic to the specific, etc.
Analogies between the multidimensional of MENTAL and the theory of hyperspace in physics
Space.
In physics, space is hyperspace, a generalization of 3D space.
In MENTAL, space is abstract. Space is formed by relationships (local and non-local) that are dynamically created.
Space-time.
In physics, since Einstein's theory of relativity, space and time are united in space-time.
In MENTAL, space and time are united and both are abstract.
Power.
In physics, from hyperspace, one has the ultimate power. "Any civilization that masters hyperspace theory will become lord of the universe" (Michio Kaku).
In MENTAL, we have the maximum possible power, but beyond the physical level.
"Theory of everything".
Physics is looking for a unifying theory based on hyperspace that contemplates the various fields (electromagnetism, quantum physics, relativity, etc.
MENTAL is a theory (and a practice) of possible worlds. It allows to contemplate and integrate formal sciences (mathematics, computer science, artificial intelligence, cybernetics, linguistics, etc.), depth psychology and philosophy. All this is achieved in a simple way by the paradigm of primary archetypes.
Field.
In physics, the concept of field allows to unify all forces, where each type of force is linked to points in space.
In MENTAL there is the concept of generic parameterized expression, which allows to describe a "field" of expressions, each one associated to the values of the parameters.
Multidimensionality.
The outer nature finds its natural scope in the geometry of multidimensional space, in the theory of hyperspace.
The inner (mental) nature finds its natural scope in multidimensionality.
Relationship between mathematics and physics.
According to A.N. Whitehead, "Mathematics at its deep level is inseparable from physics." It is more appropriate to say that "Physics at its deepest level is inseparable from mathematics". According to Galileo, nature is written in the language of mathematics. The mathematical world transcends the physical, it is more general than the physical. The physical world is a particular manifestation of the mathematical world.
Einstein said that "In a certain sense, therefore, I maintain that it is true that pure thought can grasp reality, as the ancients dreamed."
According to Michio Kaku, "Perhaps one of the by-products of the pursuit of unification by physicists will also be the unification of mathematics." Indeed, at a deep level, physics must encounter the archetypes of consciousness, of which physics is a manifestation.
Conclusions
MENTAL, as a multidimensional language, can help formalize the higher dimensions of the physical world and thus assist in its unification. It is the "new mathematical language" that Witten needs. This is justified because the physical is a restricted manifestation of the primary archetypes. Discovering the dimensions of the internal reality must logically help to discover the dimensions of the external reality, of the physical world. There really is a descending hierarchy:
MENTAL → Mathematics → Physics
The semantic dimensions of MENTAL are the true dimensions of internal and external reality. They constitute the essence of reality.
Addenda
The fourth dimension
The third dimension appeared in Renaissance art. Until then all representations were two-dimensional. The most representative example of 3D painting is Leonardo's "The Last Supper".
Since the ancient Greeks there has always been speculation about the possibility of our universe having more than the 3 spatial dimensions. In general, there was talk of the "fourth dimension", of a transcendent dimension of reality.
Interest in the possible existence of a fourth dimension of space reached its peak between 1870 and 1920. This interest was reflected in literature, art and in some scientific theories. In art it influenced the development of cubism. In mathematics he promoted the generalization of Euclidean geometry.
Edwin Abbot Abbot, in his book "Flatland" describes a "square being" living in a 2D world. This being is confronted with inexplicable phenomena caused by a 3D being.
Charles Howard Hinton coined several neologisms to describe elements of the fourth dimension. He used the word "tesseract" (tesseract) in his work "A New Age of Thought". He also invented the words "ana" and "kana" ("up" and "down" in Greek) to describe the two opposite directions of the fourth dimension.
Dimensional analysis
Dimensional analysis is a branch of physics that expresses physical quantities by the elementary quantities of mass (M), length (L) and time (T). For example, the dimensional expression for force is MLT−2 In the SI system (International System of Units) 7 basic quantities are used.
Bibliography
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Bragdon, Claude Fayette. A Primer of Higher Space (The Fourth Dimension). Nabu Press, 2010.
Ibáñez, Raúl. La cuarta dimensión. ¿Es nuestro universo la sombra de otro? RBA, colección El Mundo es Matemático, 2010.
Green, Brian. El universo elegante. Crítica, colección Drakontos, 2005.
Hinton, Charles Howard. Relatos científicos. Siruela, 1988.
Hinton, Charles Howard. A New Era of Thought. Cornell University Library, 2009.
