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Tag: Dissertatio de arte combinatoria

Eco: The Problem of the Primitives

Gottfried Wilhelm von Leibniz, Dissertatio de Arte Combinatoria, frontispiece

Gottfried Wilhelm von Leibniz (1646-1716), Dissertatio de Arte Combinatoria, frontispiece, Dissertation on the Art of Combinations or On the Combinatorial Art, 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.

“What did Leibniz’s ars combinatoria have in common with the projects for universal languages? The answer is that Leibniz had long wondered what would be the best way of providing a list of primitives and, consequently, of an alphabet of thoughts or of an encyclopedia.

In his Initia et specimina scientiae generalis (Gerhardt 1875: VII, 57-60) Leibniz described an encyclopedia as an inventory of human knowledge which might provide the material for the art of combination.

In the De organo sive arte magna cogitandi (Couturat 1903: 429-31) he even argued that “the greatest remedy for the mind consists in the possibility of discovering a small set of thoughts from which an infinity of other thoughts might issue in order, in the same way as from a small set of numbers [the integers from 1 to 10] all the other numbers may be derived.”

It was in this same work that Leibniz first made hints about the combinational possibilities of a binary calculus.

In the Consilium de Encyclopedia nova conscribenda methodo inventoria (Gensini 1990: 110-20) he outlined a system of knowledge to be subjected to a mathematical treatment through rigorously conceived propositions. He proceeded to draw up a plan of how the sciences and other bodies of knowledge would then be ordered: from grammar, logic, mnemonics topics (sic) and so on to morals and to the science of incorporeal things.

In a later text on the Termini simpliciores from 1680-4 (Grua 1948: 2, 542), however, we find him falling back to a list of elementary terms, such as “entity,” “substance” and “attribute,” reminiscent of Aristotle’s categories, plus relations such as “anterior” and “posterior.”

In the Historia et commendatio linguae characteristicae we find Leibniz recalling a time when he had aspired after “an alphabet of human thoughts” such that “from the combination of the letters of this alphabet, and from the analysis of the vocables formed by these letters, things might be discovered and judged.”

It had been his hope, he added, that in this way humanity might acquire a tool which would augment the power of the mind more than telescopes and microscopes had enlarged the power of sight.

Waxing lyrical over the possibilities of such a tool, he ended with an invocation for the conversion of the entire human race, convinced, as Lull had been, that if missionaries were able to induce the idolators to reason on the basis of the calculus they would soon see that the truths of our faith concord with the truths of reason.

Immediately after this almost mystical dream, however, Leibniz acknowledged that such an alphabet had yet to be formulated. Yet he also alluded to an “elegant artifice:”

“I pretend that these marvelous characteristic numbers are already given, and, having observed certain of their general properties, I imagine any other set of numbers having similar properties, and, by using these numbers, I am able to prove all the rules of logic with an admirable order, and to show in what way certain arguments can be recognized as valid by regarding their form alone.” (Historia et commendatio, Gerhardt 1875: VII, 184ff).

In other words, Leibniz is arguing that the primitives need only be postulated as such for ease of calculation; it was not necessary that they truly be final, atomic and unanalyzable.

In fact, Leibniz was to advance a number of important philosophical considerations that led him to conclude that an alphabet of primitive thought could never be formulated. It seemed self-evident that there could be no way to guarantee that a putatively primitive term, obtained through the process of decomposition, could not be subjected to further decomposition.

This was a thought that could hardly have seemed strange to the inventor of the infinitesimal calculus:

There is not an atom, indeed there is no such thing as a body so small that it cannot be subdivided [ . . . ] It follows that there is contained in every particle of the universe a world of infinite creatures [ . . . ] There can be no determined number of things, because no such number could satisfy the need for an infinity of impressions.” (Verità prime, untitled essay in Couturat 1903: 518-23).

If no one conception of things could ever count as final, Leibniz concluded that we must use the conceptions which are most general for us, and which we can consider as prime terms only within the framework of a specific calculus.

With this, Leibniz’s characteristica breaks its link with the research into a definitive alphabet of thought. Commenting on the letter to Mersenne in which Descartes described the alphabet of thoughts as a utopia, Leibniz noted:

“Even though such a language depends upon a true philosophy, it does not depend upon its perfection. This is to say: the language can still be constructed despite the fact that the philosophy itself is still imperfect.

As the science of mankind will improve, so its language will improve as well. In the meantime, it will continue to perform an admirable service by helping us retain what we know, showing what we lack, and inventing means to fill that lack.

Most of all, it will serve to avoid those disputes in the sciences that are based on argumentation. For the language will make argument and calculation the same thing.” (Couturat 1903: 27-8).

This was not only a matter of convention. The identification of primitives cannot precede the formulation of the lingua characteristica because such a language would not be a docile instrument for the expression of thought; it is rather the calculating apparatus through which those thoughts must be found.”

Umberto Eco, The Search for the Perfect Language, translated by James Fentress, Blackwell. Oxford, 1995, pp. 275-7.

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: First Attempts at a Content Organization


Athanasius Kircher (1602-80), Frontispiece of Obeliscus Pamphilius, Obeliscus Pamphilius: Hoc est Interpretatio nova & hucusque intenta obelisci Hieroglyphici, eBook courtesy of GoogleBooks, published by Lud. Grignani 1650, held by Ghent University. 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. 

“Probably in 1660, three years before the publication of the Polygraphia, Kircher wrote a manuscript bearing the title Novum hoc inventum quo omnia mundi idiomata ad unum reducuntur (Mss. Chigiani I, vi, 225, Biblioteca Apostolica Vaticana; cf. Marrone 1986).

Schott says that Kircher kept his system a secret at the express wish of the emperor, who had requested that his polygraphy be reserved for his exclusive use alone.

The Novum inventum was still tentative and incomplete; it contained an extremely elementary grammar plus a lexicon of 1,620 words. However, the project looks more interesting that the later one because it provides a list of 54 fundamental categories, each represented by an icon.

These icons are reminiscent of those that one might find today in airports and railway stations: some were schematically representative (like a small chalice for drinking); others were strictly geometrical (rectangles, triangles, circles).

Some were furthermore superficially derived from Egyptian hieroglyphics. They were functionally equivalent to the Roman numbers in the Polygraphia (in both texts, Arabic numbers referred to particular items).

Thus, for example, the square representing the four elements plus the numeral 4 meant water as an element; water as something to drink was instead expressed by a chalice (meaning the class of drinkable things) followed by the numeral 3.

There are two interesting features in this project. The first is that Kircher tried to merge a polygraphy with a sort of hieroglyphical lexicon, so that his language could be used (at least in the author’s intention) without translating it into a natural language.

Seeing a “square + 4,” the readers should immediately understand that the named thing is an element, and seeing “chalice + 3” they should understand that one is referring to something to drink.

The difficulty was due to the fact that, while both Kircher’s Polygraphia and Becher’s Character allow a translating operator (be it a human being or a machine) to work independently of any knowledge of the meaning of the linguistic items, the Novum inventum requires a non-mechanical and quasi-philosophical knowledge: in order to encode the word aqua as “square + 4,” one should previously know that it is the name of an element–information that the term of a natural language does not provide.

Sir Thomas Urquhart, who published two volumes describing a sort of polygraphy (Ekskubalauron, 1652, and Logopandecteision, 1653), noted that, arbitrary as the order of the alphabet might be, it was still easier to look things up in alphabetical order than in a categorical order.

The second interesting feature of Kircher’s initial project is certainly given by the effort to make the fundamental concepts independent of any existing natural language.

Its weakness is due to the fact that the list of the 54 categories was notably incongruous: it included divine entities, angelic and heavenly, elements, human beings, animals, vegetables, minerals, the dignities and other abstract concepts deriving from the Lullian Ars, things to drink, clothes, weights, numbers, hours, cities, food, family, actions such as seeing or giving, adjectives, adverbs, months of the year.

It was perhaps the lack of internal coherency in this system of concepts that induced Kircher to abandon this line of research, and devote himself to the more modest and mechanical method used in the Polygraphia.

Kircher’s incongruous classification had a precedent. Although he regarded Kircher as the pioneer in the art of polygraphy, in his Technica curiosa (as well as in his Jocoseriorum naturae et artiis sive magiae naturalis centuriae tres) Gaspar Schott gave an extended description of a 1653 project that was certainly earlier than Kircher’s (the Novum inventum is dedicated to Pope Alexander VII, who ascended the pontifical throne only in 1655).

The project was due to another Jesuit, a Spaniard (“whose name I have forgotten,” as Schott says on p. 483), who had presented in Rome (on a single folio) an Artificium, or an Arithmeticus nomenclator, mundi omnes nationes ad linguarum et sermonis unitatem invitans (“Artificial Glossary, inviting all the nations of the world to unity of languages and speech”).

Schott says that the anonymous author wrote a pasigraphy because he was a mute. As a matter of fact the subtitle of the Artificium also reads Authore linguae (quod mirere) Hispano quodam, vere, ut dicitur, muto (“The author of this language–a marvelous thing–being a Spaniard, truly, it is said, dumb”).

According to Ceñal (1946) the author was a certain Pedro Bermudo, and the subtitle of the manuscript would represent a word play since, in Castilian, “Bermudo” must be pronounced almost as Ver-mudo.

It is difficult to judge how reliable the accounts of Schott are; when he described Becher’s system, he improved it, adding details that he derived from the works of Kircher. Be that as it may, Schott described the Artificium as having divided the lexicon of the various languages into 44 fundamental classes, each of which contained between 20 and 30 numbered items.

Here too a Roman number referred to the class and an Arabic number referred to the item itself. Schott noted that the system provided for the use of signs other than numbers, but gave his opinion that numbers comprised the most convenient method of reference since anyone from any nation could easily learn their use.

The Artificium envisioned a system of designating endings, (marking number, tense or case) as complex as that of Becher. An Arabic number followed by an acute accent was the sign of the plural; followed by a grave accent, it became the nota possessionis.

Numbers with a dot above signified verbs in the present; numbers followed by a dot signified the genitive. In order to distinguish between vocative and dative, it was necessary to count, in one case, five, and, in the other, six, dots trailing after the number.

Crocodile was written “XVI.2” (class of animals + crocodile), but should one have occasion to address an assembly of crocodiles (“O Crocodiles!”), it would be necessary to write (and then read) “XVI.2′ . . . . . ‘.

It was almost impossible not to muddle the points behind one word with the points in front of another, or with full stops, or with the various other orthographic conventions that the system established.

In short, it was just as impracticable as all of the others. Still, what is interesting about it is the list of 44 classes. It is worth listing them all, giving, in parenthesis, only some examples of the elements each contained.

  1. Elements (fire, wind, smoke, ashes, Hell, Purgatory, centre of the earth).
  2. Celestial entities (stars, lightning, bolts, rainbows . . .).
  3. Intellectual entities (God, jesus, discourse, opinion, suspicion, soul, stratagems, or ghosts).
  4. Secular statuses (emperor, barons, plebs).
  5. Ecclesiastical states.
  6. Artificers (painters, sailors).
  7. Instruments.
  8. Affections (love, justice, lechery).
  9. Religion.
  10. Sacramental confession.
  11. Tribunal.
  12. Army.
  13. Medicine (doctor, hunger, enema).
  14. Brute animals.
  15. Birds.
  16. Fish and reptiles.
  17. Parts of animals.
  18. Furnishings.
  19. Foodstuffs.
  20. Beverages and liquids (wine, beer, water, butter, wax, and resin).
  21. Clothes.
  22. Silken fabrics.
  23. Wool.
  24. Homespun and other spun goods.
  25. Nautical and aromas (ship, cinnamon, anchor, chocolate).
  26. Metal and coin.
  27. Various artifacts.
  28. Stone.
  29. Jewels.
  30. Trees and fruits.
  31. Public places.
  32. Weights and measures.
  33. Numerals.
  34. Time.
  35. Nouns.
  36. Adjectives.
  37. Verbs.
  38. Undesignated grammatical category.
  39. Undesignated grammatical category.
  40. Undesignated grammatical category.
  41. Undesignated grammatical category.
  42. Undesignated grammatical category.
  43. Persons (pronouns and appellations such as Most Eminent Cardinal).
  44. Vehicular (hay, road, footpad).

The young Leibniz would criticize the absurdity of arrangements such as this in his Dissertatio de arte combinatoria, 1666.

This sort of incongruity will affect as a secret flaw even the projects of a philosophically more sophisticated nature–such as the a priori philosophic languages we will look at in the next chapter.

This did not escape Jorge Luis Borges. Reading Wilkins, at second hand as he admits (in Other Inquisitions, “The analytical idiom of John Wilkins“), he was instantly struck by the lack of a logical order in the categorical divisions (he discusses explicitly the subdivisions of stones), and this inspired his invention of the Chinese classification which Foucault posed at the head of his The Order of Things.

In this imaginary Chinese encyclopedia bearing the title Celestial Emporium of Benevolent  Recognition, “animals are divided into: (a) belonging to the emperor, (b) embalmed, (c) tame, (d) sucking pigs, (e) sirens (f) fabulous, (g) stray dogs. (h) included in the present classification, (i) frenzied, (j) innumerable, (k) drawn with a very fine camelhair brush, (l) et cetera, (m) having just broken the water pitcher, (n) that from a long way off look like flies.”).

Borge’s conclusion was that there is no classification of the universe that is not arbitrary and conjectural. At the end of our panorama of philosophical languages, we shall see that, in the end, even Leibniz was forced to acknowledge this bitter conclusion.”

Umberto Eco, The Search for the Perfect Language, translated by James Fentress, Blackwell. Oxford, 1995, pp. 203-8.

Eco: The Arbor Scientarium

Ramon Llull, Liber de ascensu et decensu intellectus, 1304, first published 1512

Ramon Llull, Liber de ascensu et decensu intellectus, 1304, first published 1512. 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 Lullian art was destined to seduce later generations who imagined that they had found in it a mechanism to explore the numberless possible connections between dignities and principles, principles and questions, questions and virtues or vices.

Why not even construct a blasphemous combination stating that goodness implies an evil God, or eternity a different envy? Such a free and uncontrolled working of combinations and permutations would be able to produce any theology whatsoever.

Yet the principles of faith, and the belief in a well-ordered cosmos, demanded that such forms of combinatorial incontinence be kept repressed.

Lull’s logic is a logic of first, rather than second, intentions; that is, it is a logic of our immediate apprehension of things rather than of our conceptions of them. Lull repeats in various places that if metaphysics considers things as they exist outside our minds, and if logic treats them in their mental being, the art can treat them from both points of view.

Consequently, the art could lead to more secure conclusions than logic alone, “and for this reason the artist of this art can learn more in a month than a logician can in a year.” (Ars magna, X, 101).

What this audacious claim reveals, however, is that, contrary to what some later supposed, Lull’s art is not really a formal method.

The art must reflect the natural movement of reality; it is therefore based on a notion of truth that is neither defined in the terms of the art itself, nor derived from it logically. It must be a conception that simply reflects things as they actually are.

Lull was a realist, believing in the existence of universals outside the mind. Not only did he accept the real existence of genera and species, he believed in the objective existence of accidental forms as well.

Thus Lull could manipulate not only genera and species, but also virtues, vices and every other sort of differentia as well; at the same time, however, all those substances and accidents could not be freely combined because their connections were determined by a rigid hierarchy of beings (cf. Rossi 1960: 68).

In his Dissertatio de arte combinatoria of 1666, Leibniz wondered why Lull had limited himself to a restricted number of elements. In many of his works, Lull had, in truth, also proposed systems based on 10, 16, 12 or 20 elements, finally settling on 9. But the real question ought to be not why Lull fixed upon this or that number, but why the number of elements should be fixed at all.

In respect of Lull’s own intentions, however, the question is beside the point; Lull never considered his to be an art where the combination of the elements of expression was free rather than precisely bound in content.

Had it not been so, the art would not have appeared to Lull as a perfect language, capable of illustrating a divine reality which he assumed from the outset as self-evident and revealed.

The art was the instrument to convert the infidels, and Lull had devoted years to the study of the doctrines of the Jews and Arabs. In his Compendium artis demonstrativa (“De fine hujus libri“) Lull was quite explicit: he had borrowed his terms from the Arabs.

Lull was searching for a set of elementary and primary notions that Christians held in common with the infidels. This explains, incidentally, why the number of absolute principles is reduced to nine (the tenth principle, the missing letter A, being excluded from the system, as it represented perfection or divine unity).

One is tempted to see in Lull’s series the ten Sefirot of the kabbala, but Plazteck observes (1953-4: 583) that a similar list of dignities is to be found in the Koran. Yates (1960) identified the thought of John Scot Erigene as a direct source, but Lull might have discovered analogous lists in various other medieval Neo-Platonic texts–the commentaries of pseudo-Dionysius, the Augustinian tradition, or the medieval doctrine of the transcendental properties of being (cf. Eco 1956).

The elements of the art are nine (plus one) because Lull thought that the transcendental entities recognized by every monotheistic theology were ten.

Lull took these elementary principles and inserted them into a system which was already closed and defined, a system, in fact, which was rigidly hierarchical–the system of the Tree of Science.

To put this in other terms, according to the rules of Aristotelian logic, the syllogism “all flowers are vegetables, X is a flower, therefore X is a vegetable” is valid as a piece of formal reasoning independent of the actual nature of X.

For Lull, it mattered very much whether X was a rose or a horse. If X were a horse, the argument must be rejected, since it is not true that a horse is a vegetable. The example is perhaps a bit crude; nevertheless, it captures very well the idea of the great chain of being (cf. Lovejoy 1936) upon which Lull based his Arbor scientiae (1296).”

Umberto Eco, The Search for the Perfect Language, translated by James Fentress, Blackwell. Oxford, 1995, pp. 64-7.

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