THE MATHEMATICAL LANGUAGE

"Mathematical language is the only universal language" (Alain Connes).

"It is common to speak of 'the language of mathematics'. But is mathematics really a language?" (F. David Peat)

"To interpret mathematical notation, it is necessary to know what kind of grammar it uses" (Stephen Wolfram).

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The Problem of Mathematical Language
Mathematics is considered to have a language of its own, a language that is abstract and distinct from natural language:

"Nature is written in the language of mathematics and its characters are triangles, circles and other geometrical figures, without which it is humanly impossible to understand a single word of it; without this, one finds oneself lost in a dark labyrinth" (Galileo, in "Il Saggiatore", The Assayer, 1616).

This approach of Galileo was truly revolutionary in his time, replacing the ancient concept of nature as a superficial and concrete organization of material substances by a deep and abstract structure, that provided by mathematics.

But does a mathematical language really exist? We can affirm that it does not exist because mathematics has no formal language. It only has some basic abstract-type symbols that have been created throughout history, and on which there is general consensus. The most important ones are:

Numerical symbols:
0 ... 9 + − × · ÷ / √ ∞ π e Φ i ∑ Π ∫

Comparison symbols:
= ≠ < > ≥ ≤ ≡

Set theory symbols:
∅ ∩ ∪ ⊃ ⊇ ⊈ ⊂ ⊆ ∈ ∉ { }

Logical symbols:
∀ ∃ ¬ ∧ ∧ ∨ → ↔

Of all these symbols, the digits 0 to 9 are the foundation of numerical language, a truly universal language that is part of our daily lives. "Digits constitute today the only true universal language" (Georges Ifrah).

The power of modern mathematics lies in its symbols, which encapsulate or compact precise concepts. However, in mathematics, anyone is free to use any notation, as long as he explains it. The real fact is that mathematical notation is not universal, as it is interpreted differently depending on the subject, the author, and even the context.

Among the shortcomings of mathematics in its linguistic aspect are the following:


Lack of grammar

Mathematics lacks formal language because it lacks grammar. It is often claimed that mathematical notation is not a mathematical problem, that it is a linguistic problem, but this is not true, because the fundamental or primitive concepts of mathematics should be reflected in the semantics of mathematical language, along with its corresponding syntax.


Mixing with natural language

There is no pure mathematical language, since it is mixed with natural language, i.e., it relies on natural language. An example is the definition of mathematical group structure, which is defined as follows:

1. There is a set G
   
   
2. There is a binary inner operation (symbolized by "*") defined on G, i.e., for any pair of elements, x and y of G, x*y is an element of G: x*y∈G
   
   
3. There exists an element e (called neutral) such that x*e = e*x = x for every element x of G
   
   
4. For every element x of G, there exists another element x' (called "inverse of x") such that x*x' = x'*x = e
   
   
5. The elements of G satisfy the associative property: x*(y*z) = (x*y)*z

The problem is that natural language is ambiguous, that is, it admits many interpretations.

An advance toward pure mathematical language was the emergence of algebra, where letters (representing variables) are used to free oneself from the bondage of the word. The use of symbols in algebra represented a great qualitative leap, opening a new era in mathematics.


Polymorphism

In mathematical language there is polymorphism, that is, the semantics of some operators vary depending on the arguments. Operators are said to have semantic "overloading". For example:

• Vector and matrix addition and subtraction operations use the same symbols as addition and subtraction of numbers.
  
  
• The operation of subtracting arrays is represented by the same symbol as subtracting numbers.
  
  
• Exponential notation. In this case, the polymorphism is compounded by the fact that there is no explicit operator. Indeed, the notation xn can mean (depending on the nature of the base): numerical power, Cartesian power, matrix power, and even n-fold application of a function. In all these cases, the exponent indicates repetition n times over the argument (base) of an operation to be inferred from the context.
  
  

  Base	Interpretation	Expression.
  Number	Numeric power	rn = r*...*r...
  Set	Cartesian power	Cn = C·...·C
  Matrix	Matrix power	An = A·...·A
  Function	Successive application	fn(a) = f(f(...(f(a))))

  
  In the case where the exponent is −1, it can mean inverse number, inverse matrix or inverse function.

Bidimensionality

The syntax of mathematical language is two-dimensional, which has the virtue of readability because it is intended for human consumption, but has the disadvantage that it is not suitable for computer processing (or is very difficult), since linearity is required. Two-dimensionality appears in exponential notation, in superscripts and subscripts, in matrices, in operations such as summation and productorio, in integrals defined between two numerical limits, etc.

The most frequent two-dimensionality is the exponential notation. This notation is considered a very important advance in mathematics. Intuited by Napier, it was Descartes who generalized the practice of writing "powers" as we do today. Thus, x³ replaces x·x·x Descartes made this notation public along with other refinements of algebra in the third appendix (Geometry) of "The Discourse on Method" in 1637. The notation includes two levels: base and exponent. The latter is expressed with a smaller symbol. When there are more than two levels there are serious difficulties of representation.


The numerical coding

The decimal system of numerical representation is firmly established in our culture and is considered one of the great conceptual advances in human history. But from a linguistic and combinatorial point of view, it is not generic enough:

1. There is no possibility that a number can be expressed by other numbers and not just digits. For example, the encoding 2(−15)7 consists of three numbers, the second being a negative two-digit number. A positional type interpretation could be 2×10² − 15×10 + 7 = 57. In Vedic Mathematics a restricted version of this system is used, as the "positional sign" on a digit is admitted. For example, 329 is interpreted as 3×10² − 2×10 + 9 = 289.
   
   
2. Numeric encoding is based on so-called "place values", which are powers of base 10. There is no reason why these place values can also be any number, as long as it follows a generic pattern. For example, from a computational point of view, it is more appropriate to use Cantor's system, which allows any rational number p/q to be represented exactly by a finite sequence and thus without loss of precision. For example, a number as simple as the rational 1/3 is 0.33333? has infinite decimal places, so, without being an irrational number, it cannot be "captured" by a finite representation. The most logical would be to use Cantor's system. [see Applications - Mathematics - The Paradox 1 = 0.999...]

Another way of exact representation of rational numbers is by continued fractions [see Applications - Mathematics - Continued Fractions].


Semantic gap. Dehumanization and complexity

There is a gap between mathematics and the human way of thinking. Mathematics, in its present form, is not humanistic. It is a cryptic, unnatural and intuitive language, oriented only for specialists or initiates.

• At the syntactic level, mathematics has developed a cryptic language, difficult to understand, dehumanized. "Mathematical notation is full of signs and symbols that people don't normally use and that most people don't understand" (Stephen Wolfram).
  
  
• At the semantic level, mathematics does not clearly reflect meaning, it either forgets or eliminates it, for it is sometimes blind, mechanical, of mere calculation, when it should be a form of description and manipulation of meaning. Calculus should not be blind, but should express the semantics of both the problem and the steps associated with the solution.

This linguistic gap can lead to some mental deformation or impairment.

• "Mathematics produces damage to consciousness by separation, by boundaries, by not perceiving and working with unified consciousness" (Ken Wilber).
  
  
• "There is no doubt that prolonged engagement in mathematical exercises in economics leads to atrophy of judgment and intuition" (John Kenneth Galbraith).

All this has caused mathematics to be perceived as an abstruse discipline distant from everyday reality. And the learning of mathematics as a heavy, difficult and artificial activity, unnatural and dehumanized.


Expressive limitations

Mathematical notation suffers from serious expressive limitations, even the most elementary ones, which, on the other hand, computer languages do have.

If there are expressive limitations in the simple, there are even more problems with complex phenomena that cannot be modeled with traditional mathematics such as distributed systems, client-server systems, shared systems, etc., as well as to express programming paradigms such as: object-oriented, event-oriented, agent-oriented, aspect-oriented, generic, evolutionary, multidimensional, etc. programming. There is also no formalism for parallelism, concurrency or synchronization mechanisms, as in computer languages.

It is precisely because of these expressive limitations of mathematics that computer languages had to be invented. If the mathematical language had been sufficiently powerful in this sense, there would have been no need to invent so many computer languages, a veritable tower of Babel.

There are also expressive limitations that prevent describing the phenomena of quantum physics, including:

• Superposition of states.
  It consists in a property of a quantum particle possessing several states at the same time, although by performing a measurement the particle is forced to move to one of the superposition states (projection postulate).
  
  
• Quantum entanglement.
  It consists in the fact that two particles (electrons, neutrons, photons) that have once shared the same property (e.g., spin) remain interconnected, no matter how far apart they are, even millions of kilometers, such that a change in one of them instantaneously produces a change in the other.
  
  
• Multiple or diffuse location of a particle.
  A particle (e.g., an electron) is simultaneously in several places at once.
  
  
• Instantaneous jumps of a quantum entity in space. For example, an electron jumps from one orbit to another without passing through intermediate space. Or the instantaneous jump of a quantum entity across a potential barrier.

Lack of algorithmic language

Traditional mathematical notation allows one to represent mathematical objects, but not mathematical processes. Many mathematicians claim that mathematics is only about formal demonstrations. From this point of view, computation has no place in mathematics.

Therefore, there is no canonical (or standard) language for representing algorithms. A kind of (informal) pseudocode, a mixture of formal and natural language, is often used. There is talk of a new discipline: "computational mathematics", a mathematics that deals with algorithms and their experimentation.

Is computer science different from mathematics or part of it? According to Donald Knuth, "Mathematics deals with theorems, infinite processes, and static relations, while computer science deals with algorithms, finite constructions, and dynamic relations." However, the concept of algorithm should be one of the core concepts of mathematics.

Since there is no dynamicity, there is no time in mathematics. This is logical, because time is a physical concept. But space is also a physical concept, and yet mathematics deals with abstract spaces. Therefore, there should also be "abstract time".


Confusion between equality, substitution and equivalence

To say that a thing is equal to itself is a truism (a tautology), which does not add knowledge. And to say that two things are equal is a contradiction. In this second case, it is better to say that there is equivalence between the two, as when we express that the sum is commutative: x+y ≡ y+x, which refer to the same thing. In this sense, Frege distinguished between sense (the form of expression) and reference (the object referenced by the expression). There can be many equivalent senses associated with the same reference.

Examples:

1. The expression x+0 = x interpreted as an equation gives us (subtracting x from both sides) 0 = 0. Interpreted as a calculation, it indicates that the sum of x and 0 is 0.
   
   
2. The expression i² = −1 should be interpreted, not as an equation, but as a substitution. Therefore:
   
   
   • It is not true that i is the square root of −1.
   • i must be interpreted as a mathematical entity such that its square is −1.
   • When i² appears in an expression, it is replaced by −1.
   
   
3. The expression ε² = 0, which defines an infinitesimal, must also be interpreted as a substitution, because if it were an equation, it would follow that ε = 0.
   
   
4. The definition of the golden ratio is Φ = 1 + 1/Φ.
   
   As an equality, it is an equation, whose solution is Φ = (√5 + 1)/2.
   
   As a substitution, it is a recursive expression:
   
   

Substitution is the key operation that allows us to define imaginary expressions in general, including imaginary numbers, infinity and infinitesimal. In this sense, substitution is the "mathematical bridge" that allows us to connect the real and the imaginary.

There is also confusion between equality and equivalence. For example, the physics formulas f = m·a and E = m·c² are not equalities (even if one says e.g. "force equals mass times acceleration") but equivalences.

"To say of two things that they are equal is foolish, and to say of a thing that it is equal to itself is useless" (Wittgenstein). In this sense, equality is meaningless; only substitution and equivalence make sense.


Confusion between parameters and arguments of a function

There is little clarity between the definition of a function (with its parameters) and the application of the function (with its arguments). In short, that there is no clear difference between arguments and parameters. Given an isolated expression, it is not known whether the variables are arguments or parameters. This problem was addressed by Alonzo Church in his lambda calculus [see Comparisons - MENTAL vs. Lambda Calculus].


Magnitudes

Mathematics does not contemplate the concept of magnitude, something so common in everyday life and in physics. A magnitude is composed of a number (quantity) and a unit. For example, in colloquial language we say, for example, "3 oranges". However:

• There should be a symbol, a formalism that connects quantity with unit.
• It should be possible to operate with magnitudes and between magnitudes.
• Functions should be able to use magnitudes as arguments or as a result.
• The unit should be able to be any mathematical entity and not only physical entities.

Higher order arithmetic operators

In arithmetic, the addition, product and power operators, as well as their inverses, are defined. But there is a need for a standard notation that allows higher order powers and their inverses to be expressed as well.


Division by zero

The result of the expression 0/0 is said to be "indeterminate", that is, it can be any number. But this is not admissible. We need an expression that is the result, otherwise the mathematics is operationally incomplete. The same is true of the expression 1/0, which is said to be "infinity" as its result.


Parallelism

This concept, so common in computer science, does not exist in mathematics. It refers to several processes running (or evaluating) in parallel, but it can also refer (or apply) to the existence of several mathematical entities occurring simultaneously, such as the concurrence of several states in quantum physics.


MENTAL, a Mathematical Language
With MENTAL these problems associated with mathematical language are solved and clarified:

• Grammar.
  Mathematical language cannot be a concrete language, with a particular syntactic grammar, because that would be limiting mathematics. Mathematics is essentially freedom and creativity. "The essence of mathematics lies in its complete freedom" (Cantor). A mathematical language must have a semantic grammar, that is, a grammar based on general concepts and the combinatorics of these concepts, which represent degrees of freedom, in order to be able to express all kinds of creative ideas. This semantic grammar is that of MENTAL. The development of mathematics is linked to mathematical language. A language based on abstract and generic concepts facilitates creativity.
  
  It is the same with the issue of the model of the mind. There can be no specific model of the mind because the mind has degrees of freedom, which are the universal semantic primitives and the same resources as the semantic grammar of MENTAL.
  
  The lack of a standard mathematical language is a reflection of its lack of grounding. In MENTAL, language and grounding go hand in hand. The universal semantic primitives constitute the language and are the foundation of mathematics.
  
  MENTAL is a mathematical language that opens consciousness, that provides freedom and provides deep insight into the underlying unity of all things.
  
  A particular language reflects a culture and its history, an identity and a specific way of looking at the world. Mathematical language has to be cross-cultural and universal by its very nature. And that universality is provided by semantic universals, the primitives of semantic grammar.
  
  According to Chomsky, there is an innate universal grammar common to all human beings. Here we argue that this universal grammar is based on the universal semantic primitives (or primal archetypes) of MENTAL, which are common to mind and nature.
  
  
• Mixed with natural language.
  MENTAL is a "pure" formal language, complete and self-sufficient. It does not need natural language. It is a language with perfectly defined semantics and syntax. It is like a computer programming language.
  
  
• Polymorphism.
  In MENTAL there is no polymorphism with overloading. Every operation has a precise and fixed semantics, where the arguments can be of any type. Only if the operation cannot be performed, the corresponding expression is self-evaluating.
  
  
• Bidimensionality.
  This problem disappears because MENTAL is one-dimensional. It can therefore be treated computationally.
  
  
• Numerical coding.
  Numerical encoding is generic in the sense that any component of a sequence can be any number or expression.
  
  Semantic gap Dehumanization and complexity.
  In MENTAL there is no semantic gap. The semantic primitives are common sense concepts, simple, clear and understandable. Therefore, it is a humanistic language. "Most of the fundamental ideas of science are essentially simple, and can, as a rule, be expressed in language understandable to all" (Einstein).
  
  
• Expressive limitations.
  With MENTAL, all kinds of relationships can be expressed, even the strangest ones in quantum physics.
  
  
• Lack of algorithmic language.
  MENTAL is an operational, procedural or algorithmic language. Every expression is evaluated and produces a result (which may be the expression itself). MENTAL is a language valid for both mathematics and computer science. The boundary between these disciplines is blurred.
  
  
• Confusion between equality, substitution and equivalence.
  In MENTAL these three concepts are clarified.
  
  
• Confusion between parameters and arguments of a function. In MENTAL these two types of elements are clearly differentiated.
  
  
• Magnitudes.
  Two types of magnitudes can be defined with MENTAL: quantitative and qualitative. In the former, the quantity can be any number. In the latter, there is a factor between 0 and 1 applicable to a quality.
  
  
• Higher order arithmetic operators.
  Higher order arithmetic operators can be expressed with MENTAL.
  
  
• Division by zero.
  In MENTAL, there are specific expressions to represent 1/0 and 0/0 expressions. Therefore, no exception occurs and the process can continue without interruption.
  
  
• Parallelism.
  Parallelism is simply contemplated with the conjunction operation, which is applicable to objects or processes.

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Addenda
The importance of language in mathematics

For André Weil, the development of new mathematical ideas parallels the development of new linguistic forms. Weil created a new mathematical language that allowed him to express mathematical concepts that were inexpressible with conventional mathematical language, This language emerged within the Nicolas Bourbaki group (of which Weil was one of its founders) of unification of contemporary mathematics. Weil's ideas became the foundations of what we now call "geometric algebra".

Alexander Grothendieck, a great innovative mathematician of the 20th century brought to life the outline of the new language created by Weil, devising a revolutionary new abstract mathematical language for algebra and geometry, which allowed mathematicians to express ideas that were previously impossible to express. Grothendieck wrote several books showing the great expressive possibilities of his language. In 1966 he received the Fields Medal for his contributions to geometric algebra. But the problem was that this new language was complex and difficult to learn. [see Comparisons - MENTAL vs. Grothendieck's Generalist Mathematics].

Alain Coones has created an entirely new language for understanding geometry. His non-commutative geometry is a modern take on the Riemannian view of geometry that goes beyond the geometry of Weil and Grothendieck, revealing an entirely new mathematical world. This geometry has proven to be a very powerful tool in its application to quantum physics, especially for string theory.


The J language

J is a powerful programming language that had its heyday in the 1960s. It was designed by the same author of APL, its predecessor, Kenneth E. Iverson [1995].

J is an executable mathematical notation. In fact, Iverson has written several mathematical texts using J as a notation.

J has a semantic orientation close to natural language. J consists of 70 verbs. The verbs are functions and are expressed with two characters.

Iverson intended to integrate mathematics and computer science, but it has the drawback that it takes a long time to learn (because of the excessive number of verbs) and because the notation becomes rather cryptic.

As opposed to J, MENTAL only requires learning 12 primitives, and its notation always clearly reflects semantics.


FMathL (Formal Mathematical Language)

FMathL is a project of the University of Vienna to create a system (and framework) aimed at the formalization of mathematical problems and their computer processing, including the automatic proof of theorems. Its main features are:

• Complete separation between descriptive and operational aspects.
  
  
• Use of the formal specification language Z as a means of automatically generating source code for a given programming language. Z is based on set theory, lambda calculus and first-order logic.
  
  
• Utilization of Donald Knuth's "literate programming paradigm" (literate programming). This paradigm is an alternative to structured programming. Programs are written in a manner similar to human language. Z code is included in the background.
  
  
• Use of the LaTeX language to represent mathematical expressions.
  
  
• Automatic generation of documentation.

Bibliography

• Cajori, Florian. A History of Mathematical Notations. Dover Books on Mathematics, 2011.
  
  
• Changeux, Jean Pierre; Connes, Alain. Materia de reflexión. Tusquets, 1993.
  
  
• Crosland, Maurice. The Language of Science. From the Vernacular to the Technical. The Lutterworth Press, 2006.
  
  
• De Cruz, Helen; De Smedt, Johan. Mathematical symbols as epistemic actions. Internet.
  
  
• De Cruz, Helen. Mathematical symbols as epistemic actions – an extended mind perspective. Internet.
  
  
• Guénon, René. Observaciones acerca de la notación matemática. La Gnose, 1910. Disponible online.
  
  
• Iverson, Kenneth E. J. Introduction and Dictionary.Iverson Software, 1995.
  
  
• Mazar, Joseph. Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers. Princeton University Press, 2014.
  
  
• O’Halloran, Kay. Mathematical Discourse: Language, Symbolism and Visual Images. Bloomsbury Academic, 2008.
  
  
• Scheinerman, Edward R. Mathematical Notation: A Guide for Engineers and Scientists. CreateSpace Independent Publishing Platform, 2011.