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 MENTAL vs. Lingua Characteristica Universalis, by Leibniz

MENTAL vs. Leibniz's Lingua Characteristica Universalis
 MENTAL vs.
LEIBNIZ'S LINGUA
CHARACTERISTICA
UNIVERSALIS

"To generate the whole from nothing, the one is enough" (Leibniz).

"I do not conceive of any reality without genuine unity" (Leibniz).

"A metaphysical reality sustains and engenders the material universe" (Leibniz).


The Philosophy of Leibniz

The principles

Gottfied Leibniz −philosopher, mathematician, scientist, historian, jurist, librarian and politician− was a universal genius. He made important contributions in very diverse fields: metaphysics, epistemology, mathematics, logic, physics, chemistry, economics, geology, philosophy of language, philosophy of mind, philosophy of religion, jurisprudence and history. King George I of Great Britain described him as "a walking encyclopedia".

Leibniz's philosophy is based on a number of interrelated principles:
Truths of reason and truths of fact

According to Leibniz, truths of reason are necessary propositions, that is, they cannot be denied. Their negation implies contradiction. Truths of reason are based on the principle of non-contradiction, they are self-evident or reducible to others that are necessarily true. Mathematical truths are an example of truths of reason. They are a priori truths.

Within the truths of reason are the "identical" truths, which are known by intuition and are self-evident.

The truths of fact are contingent. They refer to the sphere of possibility. Everything that exists in reality could be otherwise. Its negation is possible. The opposite of a factual truth is conceivable. For example, "Columbus discovered America" is a factual truth because it could have been the opposite way (Columbus did not discover America). They are based on the principle of sufficient reason. They are a posteriori truths.


Monads

Monadology was Leibniz's contribution to metaphysics. It is described in his work "Monadology" (1714), a short text written toward the end of his life. Leibniz saw the universe as consisting of fundamental entities which he called "monads". This term comes from the Greek "monas", meaning "unity" or "logos". Monadology would come to be the treatise on monads or the science of unity.

Leibniz was against the Cartesian dualism mind-body and thought-extension. He harmonized the monistic (the inextensive and substantial) and the pluralistic (the manifestations of substances in the world). In this way, Leibniz harmonized theology and philosophy.

According to Leibniz, the universe is full, it is full of "monads", the characteristics of which are as follows:
The Lingua Characteristica Universalis, of Leibniz

Leibniz devoted his whole life to the great project of creating a universal language that would be useful for reasoning, for the discovery of truth and as a tool to enhance human thought (in the same way that a microscope or telescope enlarges our vision). I knew Llull's Ars Magna. He also knew Descartes' work on the Mathesis Universalis (it is known from a letter he wrote to Mersenne in 1629).

Leibniz did not intend to create a language for human communication (spoken and written), although he devoted time and effort to study this subject, for he was fascinated by the richness and plurality of natural languages. He considered the idea of simplifying the grammar of Latin, something similar to what Peano tried at the end of the 19th century with the latino sine flexione (Latin without declensions). He finally abandoned the universal language project because he saw that it was impossible to discover the mother tongue or Adamic language from which all natural languages must have arisen. From then on he focused on a more viable project: to try to develop a universal language for science, a useful instrument for the discovery of truth.

Leibniz's concern to formulate a universal method for reasoning was set forth in his doctoral thesis De Arte Combinatoria, in 1666 (when he was only 20 years old), where he sets forth his project of constructing an "alphabet of thought" and where he analyzes the power and limits of the art of combination. An alphabet based on simple concepts and in which all other concepts would be combinations of the simple concepts. This project was clearly influenced by Llull's Ars Magna, a work that he considered insufficiently generic and that his Art tried to generalize.

In Elementa Characteristicae Universalis Leibniz developed the principles presented in his doctoral thesis, suggesting to use prime numbers to encode concepts. He attempted to apply a model of arithmetical analysis to concepts in general. The arithmetic model is that every integer is uniquely decomposed into prime factors. Leibniz assigns a "characteristic number" to each attribute. To each entity he assigns the product of the characteristic numbers of its attributes. To find out if an entity has a certain attribute, the characteristic number of the entity is divided by the characteristic number of the attribute. If it is divisible, then the entity has that attribute. For example, the concept "space" is represented by the number 2, the concept "between" by the number 3, and the concept "all" by the number 10. A composite concept, such as "interval" would be expressed as the product of these three numbers: 2*3*10, i.e., "space between all".

Diagram of
"Of Combinatorial Art"

In Lingua Generalis he suggested replacing the Arabic numerals 1 to 9 by the letters of the Latin alphabet (b c d f g g k l m n), respectively, and using the vowels (a e i o u) for the units and powers of 10 (1, 10, 100, 1,000, 10,000), respectively. For example, the number 81,374 would be coded as "mubodilefa". This system had the advantage that it allowed the syllables of a word to be permuted.


Characteristics of the Lingua Characteristica Universalis

The characteristics of this universal language were as follows: Leibniz confesses his great admiration for the Aristotelian syllogistic theory, the essence of which is demonstrative. However, he believed that this type of logic was insufficient to deal with complex problems. He wanted to extend and generalize Aristotelian logic and go beyond the closed, complete and perfect system conceived by Kant, to make it a universal method for the discovery of new truths. In this sense, he divided logic into two parts:
  1. The logic of demonstration. It is the traditional logic, which allows to formalize reasoning and deduction. These logical or symbolic calculations would reflect the processes of human thought. "All our reasoning is nothing but connections and substitutions of characters."

  2. The logic of invention. It would be based on the combinatorics of symbols, according to operational logical rules (applied to the basic symbols) to lead from the known to the unknown, to new truths. The real strength of the calculus would reside in the combinatorial rules. This new logic would be an "algebra of thought" or a "general algebra", the art of combining concepts.
Leibniz was one of the first modern philosophers to realize the power of formalism in general, based on representational linguistics, and of reasoning in particular. He wanted to formalize all areas of knowledge, including the demonstrative aspects. His formalization would be based on syntax, semantics and pragmatics. He wanted to systematize all knowledge by means of a universal scientific method (or a science of logic) that would allow him to discover the intelligible structure of the universe and its deep and essential unity. This science of logic he calls "Scientia Generalis" (SG). Once developed, it would be the solution to all the problems of science. It was the task of the SG to discover which of these basic concepts of human thought.

The Calculus Rationator, as a purely formal "blind calculus", anticipates the algebra of logic (Boolean algebra) and the computer language of computers and even artificial intelligence. Leibniz is considered the first scientist of computation. He was the first to think of logic as calculus and as the foundation of mathematics itself. Leibniz is also the inventor of binary arithmetic, an arithmetic with metaphysical implications, since it is a calculation with 0 (associated with nothing) and 1 (associated with everything or God). "To generate the all from nothing, the one is enough". On this subject, Leibniz understood the enormous power of simplicity.

Leibniz built (between 1674 and 1694) a calculating machine, a prototype machine for stepwise mathematical calculations that improved on Pascal's (which only added and subtracted). It was the first machine capable of performing the 4 arithmetic operations (addition, subtraction, multiplication and division). Only two were built. The one in the picture is in the National Library of Lower Saxony.

However, Leibniz wanted to go beyond arithmetic. He dreamed of a logical calculating device. In this sense, the Calculus Rationator could be automated to generalize his arithmetic machine.

The universal philosophical-mathematical-psychological-scientific-symbolic-logical language intuited by Leibniz was not carried to completion because of its overly ambitious character. Leibniz's merit was not only to formulate this great project, but also to try to carry it out throughout his life, an attempt sustained by his universalist philosophy and by the faith he had in the realization of this project.

Nor did he present his new logic, a generalization of Aristotelian logic. And neither, consequently, the universal science nor the complete encyclopedia of human knowledge. The same thing happened to Descartes with his Mathesis Universalis.

However, Leibniz's universalist ideas exerted a remarkable influence on later authors. The historical effects of the LCU are visible today:
MENTAL and the Lingua Characteristica Universalis

As in the case of the Mathesis Universalis Cartesian, there are numerous parallels between the LCU and MENTAL: In addition to these aspects, MENTAL adds more features: Leibniz's LCU was not carried to completion because of its overly ambitious character. Descartes, however, was less ambitious and also more fruitful, as was the union of algebra and geometry to give rise to what we know today as "analytic geometry". With MENTAL, with profoundly simple primitives, they produce maximum creativity and maximum fruit.

In conclusion, MENTAL is a proposal for a simple (and at the same time powerful) and abstract universal language, totally in line with the language conceived and sought by Leibniz.


Addenda

Leibniz's infinitesimal calculus

Leibniz came up with the idea of this calculus in 1676 and published his results in 1684. Newton had conceived the same idea some years before Leibniz, but did not publish his results until 1687. The notation proposed by Leibniz was adopted as the simplest, and is still used today.


Gödel and the Lingua Characteristica Universalis

Various authors have believed that Leibniz's universal language project was an unfounded and overly optimistic fantasy. However, Gödel believed that Leibniz's universal language was feasible and that its eventual development would revolutionize mathematics, both theoretically and practically.

Gödel was a lifelong student of Leibniz's work, especially during his four decades at Princeton, when he studied all the works in the university library. He also ordered a copy of Leibniz's voluminous manuscripts.

Gödel discovered that Leibniz's works had been partially censored. Several books published around Leibniz's time referred to paragraphs, passages, pages and chapters of Leibniz's works that did not exist. And that there were practically no concrete and detailed descriptions of his universal language in Leibniz's publications, despite the fact that Leibniz speaks in his writings of "my invention". Gödel interpreted this to mean that there was a systematic conspiracy that had suppressed essential parts of this subject. Gödel was convinced that Leibniz had successfully completed his universal language project. When Gödel was asked about those who had prevented his invention from coming to light, he replied, "Those who wish to prevent people from becoming more intelligent."

Gödel believed that the same group that had censured Leibniz centuries earlier was after him, to prevent people from understanding the true meaning of his own discovery (the incompleteness theorem): that mathematics is not a self-contained (or self-sufficient) system, that it cannot have superficial or formal foundations, and that one had to investigate along the line or philosophy traced by Leibniz, that is, the search for the primary concepts (of the intuitive and deep type) and their combinatorics.

Gödel considered himself the intellectual heir of Leibniz and worked on the search for the philosophical language. He drew up a list of 18 fundamental intuitive categories or concepts: reason, cause, substance, accident, necessity, value, God, cognition, force, time, form, content, matter, life, truth, idea, reality, and possibility.

When his wife (Adele) became ill, and had to be admitted to a hospital, Gödel, without Adele's protection, refused to eat because he was sure they were trying to poison him, and he died of starvation. Gödel's writings on Leibniz are archived in the Firestone Library at Princeton. They are written in archaic 19th century German, in shorthand form, with unconventional mathematical symbolism and Latin phrases (probably original quotations from Leibniz).

On the other hand, Leibniz's formalism is believed to have influenced Gödel in working out his incompleteness theorem. Indeed, there is a certain analogy between Leibniz's system of coding concepts (simple concepts as prime numbers and composite concepts as product of prime numbers) and Gödel's system of coding sentences (gödelization).


Gödel and monadology

Gödel described his general philosophy as a monadology with a structure similar to that of Leibniz [Wang, 1990]. He drew an analogy between monadology and the reflection principle of set theory. This principle refers to the universe or set U of all sets. This set U is said to "reflect" the universe of all sets. According to this principle, if C is a set, U contains C, all its subsets, the set containing all its subsets, and so on. The analogy between monads and U is that in both cases we have a universe of objects, and that each object is a universe that resembles the whole.


Monadology vs. MENTAL

From the time of Leibniz to the present day, with the exception of Gödel, a great follower of Leibniz, monadology has been considered an arbitrary, even eccentric philosophy. It was also too ambitious, claiming to cover everything from God to the inanimate.

However, an analogy can be drawn between Leibniz's monads and MENTAL, in its two aspects: the deep (the primary archetypes) and their manifestations (the expressions), a vision that generalizes Gódel's vision, which is limited only to sets.

This view of monadalogy has its main antecedents in:
  1. Plato, who once called the Ideas or Forms, the antecedents of the present primary archetypes, "monads".

  2. Nicholas of Cusa, who developed a monadology based on the principle that "everything is in everything", a principle he attributed to Anaxagoras. According to this philosopher, the unity of all things (the universe) exists in plurality (the di-verse), and plurality exists in unity. Each thing exists in act "reducing" (reflecting) the whole universe. The universe reduced in each thing makes of each thing a unity that can be called "monad".

  3. Modern fractal theory, where an object with fractal structure contains the same pattern at all scales. And holograms, where the part contains the whole (the entire image) and the whole is also contained in each part. Both structures (fractals and holograms) were conceived by Leibniz.
The characteristics of MENTAL as monadology are:
Monads in ancient Greek philosophy

The ancient Greek philosophers referred to the monad as the first thing, the seed, the essence, the builder, the foundation, the immutable truth, etc. Numbers only express different qualities of the monad. Unity was not considered a number, but the father of all numbers. Unity exists in all things, but goes unnoticed.

The circle is the source of all subsequent forms, the matrix in which all geometric patterns develop. The Greek term for the principles represented by the circle was "monad," from the root "meneim" (stable being) and "monas" (unity).

Unity is implicitly present in every number, for n×1 = n and n/1 = n. Unity always preserves the identity of everything it encounters.

The monad is the common denominator of the universe. Every circle we see or create is a profound sentence about the transcendental nature of the universe. The center is dimensionless. The periphery has infinite points. The circle represents unity and relates zero (the center) to infinity (the circumference). Relates nothingness to everything. It relates the limited (the surface) and the unlimited of the periphery. Symbolizes unity, eternity and enlightenment. Each circle represents the monad, the complete universe.

Just as 1 is the father of all numbers, the circle is the father of all forms. The point is the essence of a circle. A point has no extension, so it is impossible to draw. Nothing exists without a center that founds it: the nucleus of an atom, the heart of a body, the capital of a nation, the sun of the solar system, the black hole of a galaxy, and so on. The point is the source of everything. A point by expanding creates a circle.


Bibliography