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Tag: 1670

Eco: Characteristica and Calculus

Gottfried Wilhelm von Leibniz, Dissertatio de Arte Combinatoria

Gottfried Wilhelm von Leibniz (1646-1716), Dissertatio de Arte Combinatoria, an excerpt from his first doctoral dissertation, Dissertation on the Art of Combinations, Leipzig, 1666. This work is in the public domain in its country of origin and other countries and areas where the copyright term is the author’s life plus 100 years or less. 

“The theme of invention and discovery should remind us of Lull; and, in fact, Lull’s ars combinatoria was one of Leibniz’s first sources. In 1666, at the age of twenty, Leibniz composed his own Dissertatio de arte combinatoria (Gerhardt 1875: IV, 27-102). But the dream of the combinatoria was to obsess him for the rest of his life.

In his short Horizon de la doctrine humaine (in Fichant 1991), Leibniz dealt with a problem that had already troubled Father Mersenne: how many utterances, true, false or even nonsensical, was it possible to formulate using an alphabet of 24 letters?

The point was to determine the number of truths capable of expression and the number of expressions capable of being put into writing. Given that Leibniz had found words of 31 letters in Latin and Greek, an alphabet of 24 letters would produce 2432 words of 31 letters.

But what is the maximum length of an expression? Why should an expression not be as long as an entire book? Thus the sum of the expressions, true or false, that a man might read in the course of his life, imagining that he reads 100 pages a day and that each page contains 1,000 letters, is 3,650,000,000.

Even imagining that this man can live one thousand years, like the legendary alchemist Artephius, it would still be the case that “the greatest expressible period, or the largest possible book that a man can read, would have 3,650,000,000,000 [letters], and the number of truths, falsehoods, or sentences expressible–that is, readable, regardless of pronounceability or meaningfulness–will be 24365,000,000,001 – 24/23 [letters].”

We can imagine even larger numbers. Imagine our alphabet contained 100 letters; to write the number of letters expressible in this alphabet we would need to write a 1 followed by 7,300,0000,000,000 (sic) zeros. Even to write such a number it would take 1,000 scribes working for approximately 37 years.

Leibniz’s argument at this point is that whatever we take the number of propositions theoretically capable of expression to be–and we can plausibly stipulate more astronomical sums than these–it will be a number that vastly outstrips the number of true or false expressions that humanity is capable of producing or understanding.

From such a consideration Leibniz concluded paradoxically that the number of expressions capable of formulation must always be finite, and, what is more, that there must come a moment at which humanity would start to enunciate them anew.

With this thought, Leibniz approaches the theme of the apochatastasis or of universal reintegration–what we might call the theme of the eternal return.

This was a line of speculation more mystical than logical, and we cannot stop to trace the influences that led Leibniz to such fantastic conclusions.

It is plain, however, that Leibniz has been inspired by Lull and the kabbala, even if Lull’s own interest was limited to the generation of just those propositions that expressed true and certain knowledge and he thus would never have dared to enlarge his ars combinatoria to include so large a number of propositions.

For Leibniz, on the contrary, it was a fascination with the vertiginous possibilities of discovery, that is of the infinite number of expressions of which a simple mathematical calculation permitted him to conceive, that served as inspiration.

At the time he was writing his Dissertatio, Leibniz was acquainted with Kircher’s Polygraphia, as well as with the work of the anonymous Spaniard, of Becher, and of Schott (while saying that he was waiting for the long-promised Ars magna sciendi of the “immortal Kircher“).

He had yet to read Dalgarno, and Wilkins had still not published his Essay. Besides, there exists a letter from Kircher to Leibniz, written in 1670, in which the Jesuit confessed that he had not yet read Leibniz’s Dissertatio.

Leibniz also elaborated in the Dissertatio his so-called method of “complexions,” through which he might calculate, given n elements, how many groups of them, taken t at a time, irrespective of their ordering, can be ordered.

He applied this method to syllogisms before he passed to his discussion of Lull (para. 56). Before criticizing Lull for limiting the number of his elements, Leibniz made the obvious observation that Lull failed to exploit all the possibilities inherent in his combinatorial art, and wondered what could happen with variations of order, which could produce a greater number.

We already know the answer: Lull not only limited the number of elements, but he rejected those combinations that might produce propositions which, for theological and rhetorical reasons, he considered false.

Leibniz, however, was interested in a logica inventiva (para. 62) in which the play of combinations was free to produce expressions that were heretofore unknown.

In paragraph 64 Leibniz began to outline the theoretical core of his characteristica universalis. Above all, any given term needed to be resolved into its formal parts, the parts, that is, that were explicitly entailed by its definition.

These parts then had to be resolved into their own components, and so on until the process reached terms which could not, themselves, be defined–that is, the primitives. Leibniz included among them not only things, but also modes and relations.

Other terms were to be classified according to the number of prime terms they contained: if they were composed from 2 prime terms, they were to be called com2nations; if from 3 prime terms, com3nations, and so forth. Thereby a hierarchy of classes of increasing complexity could be created.

Leibniz returned to this argument a dozen years later, in the Elementa characteristicae universalis. Here he was more generous with his examples. If we accept the traditional definition of man as “rational animal,” we might consider man as a concept composed of “rational” and “animal.”

We may assign numbers to these prime terms: animal = 2, and rational = 3. The composite concept of man can be represented as the expression 2 * 3, or 6.

For a proposition to be true, if we express fractionally the subject-predicate (S/P) relationship, the number which corresponds to the subject must be exactly divisible by the number which corresponds to the predicate.

Given the preposition “all men are animals,” the number for the subject (men), is 6; the number for animals is 2; the resulting fraction is 6/2 = 3. Three being an integer, consequently, the preposition is true.

If the number for monkey were 10, we could demonstrate the falsity of either the proposition “all men are monkeys” or “all monkeys are men:” “the idea of monkey does not contain the idea of man, nor, vice versa, does the idea of the latter contain the former, because neither can 6 be exactly divided by 10, nor 10 by 6” (Elementa, in Couturat 1903: 42-92). These were principles that had all been prefigured in the Dissertatio.

Umberto Eco, The Search for the Perfect Language, translated by James Fentress, Blackwell. Oxford, 1995, pp. 271-5.

Eco: From Leibniz to the Encyclopédie

Gottfried_Wilhelm_Leibniz_c1700

Johann Friedrich Wentzel (1670-1729), Gottfried Wilhelm Leibniz (1646-1716), circa 1700. This work is in the public domain in its country of origin and other countries and areas where the copyright term is the author’s life plus 100 years or less. 

“In 1678 Leibniz composed a lingua generalis (in Couturat 1903). After decomposing all of human knowledge into simple ideas, and assigning a number to each, Leibniz proposed a system of transcription for these numbers in which consonants stood for integers and vowels for units, tens and powers of ten:

Umberto Eco, The Search for the Perfect Language, p. 270

Umberto Eco, The Search for the Perfect Language, p. 270. 

In this system, the figure 81,374, for example, would be transcribed as mubodilefa. In fact, since the relevant power of ten is shown by the following vowel rather than by the decimal place, the order of the letters in the name is irrelevant: 81,374 might just as easily be transcribed as bodifalemu.

This system might lead us to suspect that Leibniz too was thinking of a language in which the users might one day discourse on bodifalemu or gifeha (= 546) just as Dalgarno or Wilkins proposed to speak in terms of nekpot or deta.

Against this supposition, however, lies the fact that Leibniz applied himself to another, particular form of language, destined to be spoken–a language that resembled the latino sine flexione invented at the dawn of our own century by Peano.

This was a language whose grammar was drastically simplified and regularized: one declension for nouns, one conjunction for verbs, no genders, no plurals, adjectives and adverbs made identical, verbs reduced to the formula of copula + adjective.

Certainly, if my purpose were to try to delineate the entire extent of the linguistic projects undertaken by Leibniz throughout the course of his life, I would have to describe an immense philosophical and linguistically monument displaying four major aspects:

(1) the identification of a system of primitives, organized in an alphabet of thought or in a general encyclopedia;

(2) the elaboration of an ideal grammar, inspired probably by the simplifications proposed by Dalgarno, of which the simplified Latin is one example;

(3) the formulation of a series of rules governing the possible pronunciation of the characters;

(4) the elaboration of a lexicon of real characters upon which the speaker might perform calculations that would automatically lead to the formulation of true propositions.

The truth is, however, that by the end of his career, Leibniz had abandoned all research in the initial three parts of the project. His real contribution to linguistics lies in his attempts at realizing the fourth aspect.

Leibniz had little interest in the kinds of universal language proposed by Dalgarno and Wilkins, though he was certainly impressed by their efforts. In a letter to Oldenburg (Gerhardt 1875: VII, 11-5), he insisted that his notion of a real character was profoundly different from that of those who aspired to a universal writing modeled on Chinese, or tried to construct a philosophic language free from all ambiguity.

Leibniz had always been fascinated by the richness and plurality of natural languages, devoting his time to the study of their lineages and the connections between them. He had concluded that it was not possible to identify (much less to revive) an alleged Adamic language, and came to celebrate the very confusio linguarum that others were striving to eliminate (see Gensini 1990, 1991).

It was also a fundamental tenet of his monadology that each individual had a unique perspective on the world, as if a city would be represented from as many different viewpoints as the different positions of its inhabitants.

It would have been incongruous for the philosopher who held this doctrine to oblige everyone to share the same immutable grillwork of genera and species, without taking into account particularities, diversities and the particular “genius” of each natural language.

There was but one facet of Leibniz’s personality that might have induced him to seek after a universal form of communication; that was his passion for universal peace, which he shared with Lull, Cusanus and Postel.

In an epoch in which his english predecessors and correspondents were waxing enthusiastic over the prospect of universal languages destined to ease the way for future travel and trade, beyond an interest in the exchange of scientific information, Leibniz displayed a sensitivity towards religious issues totally absent even in high churchmen like Wilkins.

By profession a diplomat and court councillor, Leibniz was a political, rather than an academic, figure, who worked for the reunification of the church. This was an ecumenicism that reflected his political preoccupations; he envisioned an anti-French bloc of Spain, the papacy, the Holy Roman Emperor and the German princes.

Still, his desire for unity sprang from purely religious motives as well; church unity was the necessary foundation upon which a peaceful Europe could be built.

Leibniz, however, never thought that the main prerequisite for unity and peace was a universal tongue. Instead, he thought that the cause of peace might be better served by science, and by the creation of a scientific language which might serve as a common instrument in the discovery of truth.”

Umberto Eco, The Search for the Perfect Language, translated by James Fentress, Blackwell. Oxford, 1995, pp. 269-1.

Eco: The Dictionary–Synonyms, Periphrases, Metaphors

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Mary Beale (1633-1672), Portrait of John Wilkins, c. 1670-2, © The Royal Society. Wikipedia and the Royal Society website state that the original portrait was held by the Royal Society since it was acquired in the 1670’s. 

Wilkins‘ language provides names for 2,030 primitives, that is to say, species. These species include not only natural genera and artifacts, but also relations and actions. From these latter are derived the verbs.

As in Dalgarno, Wilkins used the copula + adjective formula for verbs, so “I love” is, again, “I am lover.” Besides this, the grammatical particles allow for the expression of tenses and modes for the verbs to be and to have as well as for pronouns, articles, exclamations, prepositions, conjunctions; the accidental differences express number, case, gender and comparatives.

But 2,030 primitive terms is still far too few to support discourse on a wide enough variety of topics. To increase the range of his language, Wilkins provided at the end of his Essay a list of 15,000 English terms not directly represented in his language, indicating the way that these might still be expressed.

The first way was by synonyms. For terms not included among the original 2,030, the list seeks to find the semantically closest primitive. To translate Result, the list suggests using primitive terms such as Event, Summe or Illation, without specifying in which context one should use the most appropriate synonym.

The list of possible synonyms can sometimes be very complex; for Corruption Wilkins suggests Evil, Destruction, Spoiling, Infection, Decay or Putrefaction. Some lists are even comic, as in the sequence of synonyms box-chest of drawers-ark-dresser-coffin-table.

The second way is periphrasis. The final dictionary records the term Abbie which has no corresponding primitive. There are primitives, however, for both Colledge and Monk. Thus, through periphrasis, Abbie can be rendered as Colledge of Monks.

The third way is that of the so-called transcendental particles. Faithful to his conception of a componential semantics based on primitive terms, Wilkins argued that there was no need to provide an additional character for Calf, since it is possible to express the same concept through Cow + Young, nor a primitive for Lioness when there was both a primitive for Lion and a marker for the feminine gender.

Thus in his grammar, Wilkins provided a system of transcendental particles (which then become a system of special markers for writing and pronunciation) that amplified or changed the meaning of the characters to which they were linked.

The 48 particles were articulated into eight classes, though there was little system in the classification. In fact, Wilkins drew from the Latin grammar the idea of different terminations such as “inceptives” (lucesco, aquosus, homunculus), “segregates” (gradatim or verbatim), endings indicating place (vestiarium) or agent (arator).

Sometimes these markers were essentially grammatical; as happens with those of gender, but for others Wilkins also took into account rhetorical devices such as metaphor, metonymy and synecdoche.

The particles in the class “metaphorical-like” indicate that the terms to which they are apposited are to be taken in a figurative sense. In this way, the primitive root can be modified so as to mean original, or light to mean evident.

Other particles seem to indicate relations such as cause and effect, container and thing contained, function and activity. Here are a few examples:

like + foot = pedestal

like + dark = mystical

place + metal = mine

officer + navy = admiral

artist + star = astronomer

voice + lion = roaring

Unfortunately, this incorporation of rhetorical solutions adds an element of imprecision to the entire system, and this weakens the project as a whole. Although Wilkins gave a list of examples showing the correct use of the particles, he was forced to acknowledge that they were just examples.

This list remains open, and its further elaboration is left to the inventiveness of the individual speaker (p. 318). Once set the speaker free to invent, and it is hard to avoid the risk of ambiguity.

Still, it is important to observe that–if the presence of a particle can produce ambiguity–its absence proves without any shade of doubt that a given term must be taken literally. This represents an advance of Dalgarno, in whose system there was nothing to indicate when terms should be understood literally or figuratively.

The fact is that Wilkins the author of a philosophical grammar seems to be working against Wilkins the inventor of a philosophic a priori language in real characters. Wilkins‘ attempt to take into account the figurative side of language also is certainly an interesting effort; however, it affects the precision of his language and its original claim to reduce the ambiguities present in ordinary language.

Note that, in order to render his language as univocal as possible, Wilkins had even decided to eliminate from the tables names of mythological (therefore non-existent) beings such as Sirens, Griffins, Harpies and Phoenixes, which could be at most written in natural language as proper names of individuals (for an analogy with Russell’s preoccupations, see Frank 1979: 160).

Wilkins also admitted that his language was unsuited to capturing the minutiae of food and drink, like different types of grape, jam, coffee, tea and chocolate. The problem could naturally be solved, he claimed, through periphrasis; yet it is easy to foresee that to do so the language would have been overloaded with a lot of new, awkward syntagms, as happens today with papal encyclicals, where video-cassettes become sonorarum visualiumque taeniarum cistellulae, and advertising men turn into laudativis nuntiis vulgatores.

Besides, in Latin it would have been possible to avoid such monstrosities by coining new words such as videocapsulae or publicitarii (see Bettini 1992), while Wilkins‘ language seems to have closed the door to neologisms. The only way to escape this difficulty would be to assume that the list of primitives was open.”

Umberto Eco, The Search for the Perfect Language, translated by James Fentress, Blackwell. Oxford, 1995, pp. 245-8.

Eco: Comenius

Labyrint

Jan Amos Comenius (1592-1670), Labyrinth of the World and Paradise of the Heart, the initial version was completed in 1623, while the first edition was published in 1631. The entire work is posted in an electronic edition. This work is in the public domain in its country of origin and other countries and areas where the copyright term is the author’s life plus 100 years or less. 

“The British quest was also influenced by the presence of Comenius (Jan Amos Komensky). In fact Comenius was a member of the Bohemian Brotherhood, a mystic branch of Hussite reformers, and he played a role–albeit a polemical one–in the Rosicrucian story (cf. his Labyrinth of the World, 1623, in Czech).

Thus he was inspired by religious ideals which were alien to the scientific purposes of the English milieu. On this complex cultural geography see Yates (1972, 1979): one is really facing a web of different projects, at once similar and antithetical, in which the search for a perfect language was but a single aspect (see Rossi 1960; Bonerba 1992; Pellerey 1992a: 41-9).

Comenius‘ aspirations must be seen in the framework of the tradition of pansophia, yet his pansophic aims were influenced by educational preoccupations. In his Didactica magna of 1657, he proposed a scheme for reforming teaching methods; for, as he observed, a reform in the education of the young formed the basis upon which any subsequent political, social and religious reform must be built.

It was essential that the teacher furnish the learners with a set of images that would stamp themselves indelibly on their imaginations. This meant placing what is visible before the eyes, what is audible before the ears, what is olfactory before the nose, gustatory before the tongue, and tactical before the touch.

In an earlier manual for the teaching of Latin, Janua linguarum, written in 1631, Comenius was first of all concerned that the learner should have an immediate visual apprehension of what was being spoken of.

Equally he was concerned that the images and notions that the learner was studying in the Latin lexicon be arranged in a certain logical order.

Thus lessons progressed from the creation of the world to the elements, to the mineral, vegetable and animal kingdoms, etc.

By the time of the Didactica magna Comenius had begun to rearrange his notions according to the suggestions of Bacon. In 1658 there appeared the Orbis sensualium pictus quadrilinguis, which represented his attempt to present a figured nomenclature which would include the fundamental things of the world together with human actions.

So important were the images that Comenius delayed publication until he was able to obtain satisfactory engravings that were not mere ornaments, but bore an iconic relation with the things represented, for which the verbal names appeared as nothing but titles, explanations and complements.

This manual was prefaced by an alphabet in which every letter was associated with the image of a particular animal whose voice recalled the sound of the letter–so that the result resembles Harsdörffer’s onomatopoetic fantasies concerning the sounds of German.

Therefore the image of a crow is commented by “Die Krähe krächzet, cornix cornicatur, la cornacchia gracchia, la corneille gazoüille,” or, for a snake, “Die Schlange zischtet, Serpens sibilat, il Serpe fsschia [sic], le Serpent siffle.”

Comenius was a severe critic of the defects of natural languages. In his Pansophiae Christianae liber III (1639-40), he advocated a reform that would eliminate the rhetorical and figurative use of words, which he regarded as a source of ambiguity.

The meaning of words should be fixed, he demanded, with one name for each thing, thus restoring words to their original meanings.

In 1668, in the Via lucis, Comenius offered prescriptions for the creation of an artificial universal language. By now, pansophy was more than an educational method; it was a utopian vision in which a world council was supposed to create the perfect state along with its perfect philosophical language, the Panglossia.

It is interesting to consider that Comenius had in fact written this work before 1641, when, after wandering through the whole of Europe in the course of the Thirty Years War, he had taken refuge in London.

Via lucis certainly circulated, in manuscript form, in the English milieu at that time (see, for example, Cram 1989).

Although Comenius was never to construct his new language in extenso, he had broached the idea of a universal tongue which had to overcome the political and structural limitations of Latin.

The lexicon of the new language would reflect the composition of reality and in it every word should have a definite and univocal meaning, every content should be represented by one and only one expression, and the contents were not supposed to be products of fancy, but should represent only every really existing thing, no more and no less (see Pellerey 1992a: 48).

Thus, on one side we have a utopian thinker, inspired by Rosicrucian ideals, whose goal was to discover a pansophy which aimed at picturing the unmoving and harmonical connection of every element of the creation, so as to lead the human mind to an unceasing quest for God; on the other side, rejecting the possibility of rediscovering the original perfect language, and looking, for educational purposes, for an easy artificial method, Comenius became the forerunner of that search for an a priori philosophical language that would later be implemented by English utopian thinkers whose inspiration was more scientific than theological or mystical.”

Umberto Eco, The Search for the Perfect Language, translated by James Fentress, Blackwell. Oxford, 1995, pp. 214-6.

Eco: Lullian Kabbalism, 2

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Jan Amos Komensky, or Johann (John) Amos Comenius (1592-1670), from Opera didactica omnia, Amsterdam, 1657. This work is in the public domain in its country of origin and other countries and areas where the copyright term is the author’s life plus 100 years or less.  

“Numerology, magic geometry, music, astrology and Lullism were all thrown together in a series of pseudo-Lullian alchemistic works that now began to intrude onto the scene. Besides, it was a simple matter to inscribe kabbalistic terms onto circular seals, which the magical and alchemical tradition had made popular.

It was Agrippa who first envisioned the possibility of taking from the kabbala and from Lull the technique of combination in order to go beyond the medieval image of a finite cosmos and construct the image of an open expanding cosmos, or of different possible worlds.

In his In artem brevis R. Lulli (appearing in the editio princeps of the writings of Lull published in Strasbourg in 1598), Agrippa assembled what seems, at first sight, a reasonably faithful and representative anthology from the Ars magna.

On closer inspection, however, one sees that the number of combinations deriving from Lull’s fourth figure has increased enormously because Agrippa has allowed repetitions.

Agrippa was more interested in the ability of the art to supply him with a large number of combinations than in its dialectic and demonstrative properties. Consequently, he proposed to allow the sequences permitted by his art to proliferate indiscriminately to include subjects, predicates, rules and relations.

Subjects were multiplied by distributing them, each according to its own species, properties and accidents, by allowing them free play with terms that are similar or opposite, and by referring each to its respective causes, actions, passions and relations.

All that is necessary is to place whatever idea one intends to consider in the center of the circle, as Lull did with the letter A, and calculate its possible concatenations with all other ideas.

Add to this that, for Agrippa, it was permissible to add many other figures containing terms extraneous to Lull’s original scheme, mixing them up with Lull’s original terms: the possibilities for combination become almost limitless (Carreras y Artau 1939: 220-1).

Valerio de Valeriis seems to want the same in his Aureum opus (1589), when he says that the Ars “teaches further and further how to multiply concepts, arguments, or any other complex unto infinity, tam pro parte vera quam falsa, mixing up roots with roots, roots with forms, trees with trees, the rules with all these other things, and very many other things as well” (“De totius operis divisione“).

Authors such as these still seem to oscillate, unable to decide whether the Ars constitutes a logic of discovery or a rhetoric which, albeit of ample range, still serves merely to organize a knowledge that it has not itself generated.

This is evident in the Clavis universalis artis lullianae by Alsted (1609). Alsted is an author, important in the story of the dream of a universal encyclopedia, who even inspired the work of Comenius, but who still–though he lingered to point out the kabbalist elements in Lull’s work–wished to bend the art of combination into a tightly articulated system of knowledge, a tangle of suggestions that are, at once, Aristotelian, Ramist and Lullian (cf. Carreras y Artau 1939: II, 239-49; Tega 1984: I, 1).

Before the wheels of Lull could begin to turn and grind out perfect languages, it was first necessary to feel the thrill of an infinity of worlds, and (as we shall see) of all of the languages, even those that had yet to be invented.”

Umberto Eco, The Search for the Perfect Language, translated by James Fentress, Blackwell. Oxford, 1995, pp. 130-2.

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