"Language is an isomorphic representation or model of the world" (Wittgenstein).
"A logical representation of facts is a thought" (Wittgenstein).
"Vision or imagination is a representation of the eternal, true, and unchangeable" (Willia Blake).
Representation
A "representation" is a reality that replaces, imitates, or reflects another reality. For example, a map can represent a territory, a painting a landscape, a portrait a person, a musical score a musical composition, etc.
The main characteristics of the representation are:
There can be many forms of representation, while what is to be represented (the represented) is unique. Representation implies alternatives, possibilities. There can be textual, graphic, sonorous, etc. representations. Here we are interested in linguistic representation, which is textual, linear, one-dimensional.
There can be many aspects of representation, depending on the aspect we want to capture of reality: attributes, causal relationships, structure, processes, etc. These aspects can be independent or interdependent.
There can be many levels of representation, depending on the level of detail to be considered.
The only complete representation of an object is the object itself. The best representation of something is the one that represents itself. Any other representation is incomplete and imperfect, that is, it is only an approximation of reality.
A representation is isomorphic when the elements (including their relationships) of reality have their corresponding elements in the representation. In this case, the elements of the representation must be simpler or more abstract than the elements of reality.
The representation always looks for the optimal form, the most adequate and of the maximum possible simplicity, which makes explicit the characteristics of the reality to be represented.
A representation is a way to reduce complexity, since the representation is simpler than what is represented.
A representation is all the better the more compact it is, without losing its expressiveness.
A representation is another level of reality, so it, in turn, could be represented. Therefore, there can be higher order representations.
We can say that a representation is a complete map of the reality to be considered, according to a certain aspect and according to a certain level of detail, which tries to reflect in a more understandable way that reality, and which has a content of significance.
Language and Representation
When a representation is applied to a language, we have a syntax of that language. There can be many syntactic forms of the same language. Syntax is a set of symbols or signs and their relationships. Symbols are preferable over signs because symbols refer to themselves, whereas signs require external interpretation. Signs are conventional; that is why there are multiple languages.
It is clear that −as in Magritte's famous pipe painting− that formal language is a representation or manifestation of the true language, which is semantics (what is represented), which is neither expressible nor representable.
The specular conception of language
Reality is very complex, and through language we make that reality understandable by considering only the essential, creating a "map", a reflection or a representation of reality. This is what is called "specular conception of language".
This conception or relationship between language and reality is the one that has lasted the longest in time. It dates back to Aristotle: the grammatical structures or categories of language are a reflection of reality. He established the famous 10 philosophical categories from the analysis of the Greek language.
This idea was expanded by Locke (language is an aspect of nature), by Descartes (language is a representation of mental images), by Nietzche (language is a manifestation of reality, as a symptom is a manifestation of a disease) and by Wittgenstein (language is an isomorphic representation or model of the world).
Representation Theory
Representation theory (RT) is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations in a vector space. The vector space is called "representation space" and can be defined by means of real numbers, complex numbers, p-adic numbers, etc.
Abstract algebraic structures are groups, associative algebras, rings, bodies, vector spaces, Lie algebras, etc. The most basic abstract algebraic structure is the group structure, on which the RT is based, since almost all abstract algebra structures can be considered as groups endowed with additional operations and axioms. For example, a ring (the generalization of the integers) can be considered an abelian group together with an additional operation.
The RT simplifies. It reduces abstract algebra problems to linear algebra problems, which are simpler and more understandable. RT can be said to study the symmetry of linear spaces and representations of associative algebras.
The RT was born in 1897 with the articles published by the German mathematician F.G. Frobenius on finite groups. Other pioneers were Burnside, Schur and Brauer.
Frobenius systematized abstract algebra by means of mathematical logic and axiomatic procedures. He formalized the concept of representation of finite groups (abelian and non-abelian), symmetry groups and alternating groups.
A symmetry group is the set of permutations of a finite set under the operation of composition of permutations.
An alternating group is a subgroup of a symmetry group formed by the even permutations.
The so-called "Langlands program" attempts with its conjectures and investigations to directly relate RT to number theory. The correspondence between these two fields of mathematics would make it possible to transfer a problem from number theory to RT and vice versa.
Main concepts of RT
The main concepts of the RT are:
The elements of a group G are considered permutations, which are static elements.
Permutations are represented by arrays of binary digits, which are dynamic elements (actions). Each row of each matrix contains only one 1 (the rest are zeros).
The matrices form a group GM under the matrix multiplication operation. This operation is associative, but not commutative.
All matrices have determinant 1.
The neutral element is the unit matrix.
The matrices are invertible, which are the inverse elements of GM.
Matrices operate on vectors of a vector space V (of dimension equal to the order of G) to obtain concrete permutations.
In short:
The elements of a group are represented as matrices of binary numbers, thus establishing a correspondence between group theory and number theory.
RT transforms a group into another group of symmetries transformations.
Example
A 2-element permutation can be represented by a 2×2 matrix. In general, a permutation of n elements can be represented by a matrix of n×n. And m permutations in a row can be represented as the product of the m matrices corresponding to each of the permutations.
Finding a representation of a symmetry group is equivalent to the problem of finding matrices whose square is the identity matrix. In the 2-element case, {a1, a2}, the matrices are
(
1
0
)
0
1
and
(
0
1
)
1
0
The first matrix is the identity matrix and corresponds to the permutation P0 = (a1, a22), and the second corresponds to the permutation P1 = (a2, a1)
The dimension of the vector space is equal to the order of the group. For groups of infinite elements, the vector space in which the group is represented is of infinite dimension (e.g., a Hilbert space).
Applications of the RT
RT is a fundamental theory in its own right, with important applications in other branches of mathematics (number theory, combinatorics, geometry, probability theory, etc.) and in other fields of science, physics especially, since it describes how the symmetry group of a physical system affects the solutions of the equations describing that system.
The RT has given rise to numerous generalizations. The most important is that associated with categories: algebraic objects can be considered categories, and representations as functors from the category of objects to the category of vector spaces.
The use of categorization as a tool in RT has given rise to higher RT (Higher Representation Theory).
MENTAL, a Universal Language of Representation
Representation and primary archetypes
Representation is a psychological phenomenon in which the mind or consciousness organizes or "collapses" into certain essential, simple and direct forms, which are the primary archetypes. The primary archetypes are the invariants of consciousness. Representation, like perception, is structured around these simple concepts that are the primary archetypes.
There are primary representations, which are invariant, and there are many secondary representations formed by combinations of these primitives to represent entities from the different fields of mathematics: number theory, algebraic geometry, model theory, differential geometry, operator theory, algebraic combinatorics, topology, etc.
There are many ways to represent a language. The natural, simplest and most effective representation is the one based on the primary archetypes, the archetypes of consciousness. These archetypes are inexpressible, that is, they are unrepresentable. Only their concrete manifestations are expressible or representable.
MENTAL is a universal language of representation. It is a language of natural representation because it is based on the archetypes of the consciousness, the concepts with which we structure internal and external reality: their properties and their static and dynamic relationships. Moreover, these primary archetypes constitute the limits of the expressible.
It is applicable to any domain, not only to group theory.
It unifies theory and practice.
Allows to represent all types of expressions: extensive and intensive, static and dynamic, active and passive, etc.
Connects representation (the external) with semantics (the internal). Primary semantics is never lost.
Allows to represent data, information and knowledge, as well as all kinds of mathematical, logical and computational entities.
Allows to represent aspects of expressions.
The primitive "representation"
Representation is one of the dimensions of consciousness. MENTAL has a primitive that is representation, which is a variant of substitution, potential substitution:
(x =: y) // x represents y
There may be strings of representations (higher order representations):
(x1 =: x2 =: ... xn =: y)
It means: x1 represents x2, x2 represents x3, ... , xn represents y. In short, all expressions x1, ... , xn represent y.
Normally, in the expression (x =: y), x is a name. But, as with immediate substitution, x can be any expression, that is, any expression can represent any other expression. In this case we can speak of "imaginary representations", for example, (a+b =: 33). It is a situation analogous to imaginary substitutions, such as (i*i = −1), where i is the imaginary unit.
In general, x is simpler than y. But this is not necessarily so for imaginary expressions.
Every expression, in principle, stands for itself: (x =: x), just as every expression stands for itself: (x = x). Both expressions imply each other:
⟨( (x =: x) ↔ (x = x) )⟩
MENTAL vs. RT
RT is an attempt at generalization and simplification of abstract algebraic structures, but it is not a general or universal theory. It is a restricted theory based on group structure. MENTAL is applicable to any domain, not just group theory.
Poincaré said that group theory is the essence of mathematics and that mathematics is just the history of groups. Indeed, groups are fundamental in mathematics but more fundamental are the primary archetypes.
The primary archetypes of MENTAL are categories (philosophical and mental). MENTAL unifies RT and category theory.
Finite groups can be constructed as a combination of simple groups, in a process analogous to numbers as a product of prime numbers. But simple groups are not simple. MENTAL is founded on primary archetypes, which are the simplest conceptual elements. The "conceptual atoms" are not the simple groups but the universal semantic primitives.
RT is a theory. MENTAL is a theoretical and practical language.
RT only applies to static structures. MENTAL allows to represent all kinds of expressions: static and dynamic, active and passive, generic and specific, descriptive and operational, etc.
RT is an artificial theory. MENTAL is a language based on natural archetypes.
MENTAL is the most appropriate language for representing symmetries because its primitives are opposites or duals.
A group can be defined in MENTAL by sequences and substitutions.
MENTAL unites syntax and semantics. And it unites lexical semantics and structural semantics.
Groups are important in mathematics because they unite algebra and geometry through the concept of symmetry. MENTAL allows expressing algebraic geometry, which is a more generic field.
In RT there is loss of semantics. With MENTAL primitives there is never a loss of semantics.
MENTAL allows higher order representations. With RT this is not possible.
Ejemplo
In MENTAL, matrices are represented by sequences of sequences (as many sequences as the length of each sequence).
In the case of 2-element permutations, the corresponding matrices are represented by the names P/0 and P/1:
( P/0 =: ((1 0) (0 1)) ) // matrix identity
( P/1 =: ((0 1) (1 0))) )
We can also represent unitary sequences:
( s/1 = (0 1) ) ( s/2 = (1 0) )
And represent the above permutations by these sequences:
( P/0 =: (s/2 s/1) // matrix identity
( P/1 =: (s/1 s/2)
By virtue of the inverse evaluation associated with the representation, we have the following properties (*/m indicates matrix product):
These properties are equivalent to the composition of permutations Pi-Pj.
A matrix transforms a vector into another vector within the vector space. For example,
( P/0 */m (a b) ) // ev. (a b)
( P/1 */m (a b) ) // ev. (b a)
Bibliography
Alperin, J.L. Local Representation Theory. Cambridge University Press, 1983.
Bierbach, Ludwig. Introducción a la teoría de la representación conforme. Editorial Labor, 1966.
Conrad, Keith. The Origin of Representation Theory. Internet.
Curtis, Charles W. Pioneers of Representation Theory: Frobenius, Burnside, Schur, and Brauer. American Mathematical Society. History of Mathematics, 1999.
Curtis, Charles W. Representation Theory of Finite Groups: From Frobenius to Brauer. The Mathematical Intelligencer, vol. 14, no. 4, 1992.
Eingof, Pavel y otros. Introduction to representation theory. Internet, 2011.
