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Tag: Ars magna

Eco: Dee’s Magic Language


Florence Estienne Méric Casaubon (1599-1671), A True and Faithful Relation of what Passed for Many Yeers between Dr. John Dee [ . . . ] and Some Spirits, London, 1659. 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 his Apologia compendiaria (1615) Fludd noted that the Rosicrucian brothers practiced that type of kabbalistic magic that enabled them to summon angels. This is reminiscent of the steganography of Trithemius. Yet it is no less reminiscent of the necromancy of John Dee, a man whom many authors considered the true inspirer of Rosicrucian spirituality.

In the course of one of the angelic colloquies recorded in A True and Faithful Relation of what Passed for Many Yeers between Dr. John Dee [ . . . ] and Some Spirits (1659: 92), Dee found himself in the presence of the Archangel Gabriel, who wished to reveal to him something about the nature of holy language.

When questioned, however, Gabriel simply repeated the information that the Hebrew of Adam, the language in which “every word signifieth the quiddity of the substance,” was also the primal language–a notion which, in the Renaissance, was hardly a revelation.

After this, in fact, the text continues, for page after page, to expatiate on the relations between the names of angels, numbers and secrets of the universe–to provide, in short, another example of the pseudo-Hebraic formulae which were the stock in trade of the Renaissance magus.

Yet it is perhaps significant that the 1659 Relation was published by Meric Casaubon, who was later accused of partially retrieving and editing Dee’s documents with the intention of discrediting him.

There is nothing, of course, surprising in the notion that a Renaissance magus invoked spirits; yet, in the case of John Dee, when he gave us an instance of cipher, or mystic language, he used other means.

In 1564, John Dee wrote the work upon which his contemporary fame rested–Monas hieroglyphica, where he speaks of a geometrical alphabet with no connection to Hebrew. It should be remembered that Dee, in his extraordinary library, had many of Lull’s manuscripts, and that many of his kabbalistic experiments with Hebrew characters in fact recall Lull’s use of letters in his art of combination (French 1972: 49ff).

Dee’s Monas is commonly considered a work of alchemy. Despite this, the network of alchemical references with which the book is filled seems rather intended to fulfill a larger purpose–that of explicating the cosmic implications deriving from Dee’s fundamental symbol, the Monad, based upon circles and straight lines, all generated from a single point.


John Dee (1527-1609), Monas hieroglyphica, 1564, held in the Bibliothèque nationale de France. The Monad is the symbol at the heart of the illustration labeled Figure 8.1 in Eco’s  The Search for the Perfect Language, Oxford, 1995, p. 186.

In this symbol (see figure 8.1), the main circle represented the sun that revolves around its central point, the earth, and in its upper part was intersected by a semi-circle representing the moon.

Both sun and moon were supported on an inverted cross which represented both the ternary principle–two straight lines which intersect plus their point of intersection–and the quaternary principle–the four right angles formed at the intersections of the two lines.

The sum of the ternary and quaternary principles constituted a further seven-fold principle, and Dee goes even on to squeeze an eight-fold principle from the diagram.

By adding the first four integers together, he also derives a ten-fold principle. By such a manipulatory vertigo Dee then derives the four composite elements (heat and cold, wet and dry) as well as other astrological revelations.

From here, through 24 theorems, Dee makes his image undergo a variety of rotations, decompositions, inversions and permutations, as if it were drawing anagrams from a series of Hebrew letters.

Sometimes he considers only the initial aspects of his figure, sometimes the final one, sometimes making numerological analyses, submitting his symbol to the kabbalistic techniques of notariqon, gematria, and temurah.

As a consequence, the Monas should permit–as happens with every numerological speculation–the revelation of the whole of the cosmic mysteries.

However, the Monad also generates alphabetic letters. Dee was emphatic about this in the letter of dedication with which he introduced his book. Here he asked all “grammarians” to recognize that his work “would explain the form of the letters, their position and place in the alphabetical order, and the relations between them, along with their numerological values, and many other things concerning the primary Alphabet of the three languages.”

This final reference to “the three languages” reminds us of Postel (whom Dee met personally) and of the Collège des Trois Langues at which Postel was professor. In fact, Postel, to prove that Hebrew was the primal language in his 1553 De originibus, had observed that every “demonstration of the world” comes from point, line and triangle, and that sounds themselves could be reduced to geometry.

In his De Foenicum literis, he further argued that the invention of the alphabet was almost contemporary with the spread of language (on this point see many later kabbalistic speculations over the origins of language, such as Thomas Bang, Caelum orientis, 1657: 10).

What Dee seems to have done is to take the geometrical argument to its logical conclusion. He announced in his dedicatory letter that “this alphabetic literature contains great mysteries,” continuing that “the first Mystic letters of Hebrews, Greeks, and Romans were formed by God and transmitted to mortals [ . . . ] so that all the signs used to represent them were produced by points, straight lines, and circumferences of circles arranged by an art most marvelous and wise.”

When he writes a eulogy of the geometrical properties of the Hebrew Yod, one is tempted to think of the Dantesque I; when he attempts to discover a generative matrix from which language could be derived, one thinks of the Lullian Ars.

Dee celebrates his procedure for generating letters as a “true Kabbalah [ . . . ] more divine than grammar itself.”

These points have been recently developed by Clulee (1988: 77-116), who argues that the Monas should be seen as presenting a system of writing, governed by strict rules, in which each character is associated with a thing.

In this sense, the language of Monas is superior to the kabbala, for the kabbala aims at the interpretation of things only as they are said (or written) in language, whereas the Monas aims directly at the interpretation of things as they are in themselves. Thanks to its universality, moreover, Dee can claim that his language invents or restores the language of Adam.

According to Clulee, Dee’s graphic analysis of the alphabet was suggested by the practice of Renaissance artists of designing alphabetical letters using the compass and set-square.

Thus Dee could have thought of a unique and simple device for generating both concepts and all the alphabets of the world.

Neither traditional grammarians nor kabbalists were able to explain the form of letters and their position within the alphabet; they were unable to discover the origins of signs and characters, and for this reason they were uncapable (sic) to retrieve that universal grammar that stood at the bases of Hebrew, Greek and Latin.

According to Clulee, what Dee seems to have discovered was an idea of language “as a vast, symbolic system through which meanings might be generated by the manipulation of symbols” (1988: 95).

Such an interpretation seems to be confirmed by an author absent from all the bibliographies (appearing, to the best of my knowledge, only in Leibniz’s Epistolica de historia etymologica dissertatio of 1717, which discusses him in some depth).

This author is Johannes Petrus Ericus, who, 1697, published his Anthropoglottogonia sive linguae humanae genesis, in which he tried to demonstrate that all languages, Hebrew included, were derived from Greek.

In 1686, however, he had also published a Principium philologicum in quo vocum, signorum et punctorum tum et literarum massime ac numerorum origo. Here he specifically cited Dee’s Monas Hieroglyphica to derive from that matrix the letters of all alphabets (still giving precedence to Greek) as well as all number systems.

Through a set of extremely complex procedures, Ericus broke down the first signs of the Zodiac to reconstruct them into Dee’s Monad; he assumed that Adam had named each animal by a name that reproduced the sounds that that each emitted; then he elaborated a rather credible phonological theory identifying classes of letters such as “per sibilatione per dentes,” “per tremulatione labrorum,” “per compressione labrorum,” “per contractione palati,” “per respiratione per nares.”

Ericus concluded that Adam used vowels for the names of the beasts of the fields, and mutes for the fish. This rather elementary phonetics also enabled Ericus to deduce the seven notes of the musical scale as well as the seven letters which designate them–these letters being the basic elements of the Monas.

Finally, he demonstrated how by rotating this figure, forming, as it were, visual anagrams, the letters of all other alphabets could be derived.

Thus the magic language of the Rosicrucians (if they existed, and if they were influenced by Dee) could have been a matrix able to generate–at least alphabetically–all languages, and, therefore, all the wisdom of the world.

Such a language would have been more than a universal grammar: it would have been a grammar without syntactic structures, or, as Demonet (1992: 404) suggests, a “grammar without words,” a silent communication, close to the language of angels, or similar to Kircher’s conception of hieroglyphs.

Thus, once again, this perfect language would be based upon a sort of communicative short-circuit, capable of revealing everything, but only if it remained initiatically secret.”

Umberto Eco, The Search for the Perfect Language, translated by James Fentress, Blackwell. Oxford, 1995, pp. 185-90.

Eco: Lullian Kabbalism, 2


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.

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.

Eco: The Alphabet and the Four Figures, 3


Jonathan Swift, Gulliver’s Travels, 1892 George Bell and Sons edition, Project Gutenberg. Also see Jonathan Swift, Gulliver’s Travels, A.J. Rivero, ed., New York: W.W.Norton, 2001, Part III, chapter 5. Cited in Bethany Nowviskie, “Ludic Algorithms,” in Kevin Kee, ed., Pastplay: Teaching and Learning History with Technology, Ann Arbor, MI: University of Michigan Press, 2014. 

“It follows that Lull’s art is not only limited by formal requirements (since it can generate a discovery only if one finds a middle term for the syllogism); it is even more severely limited because the inferences are regulated not by formal rules but rather by the ontological possibility that something can be truly predicated of something else.

The formal rules of the syllogism would allow such arguments as “Greed is different from goodness — God is greedy — Therefore God is different from goodness.” Yet Lull would discard both the premises and the conclusion as false.

The art equally allows the formulation of the premise “Every law is enduring,” but Lull rejects this as well because “when an injury strikes a subject, justice and law are corrupted” (Ars brevis, quae est de inventione mediorum iuris, 4.3a).

Given a proposition, Lull accepts or rejects its logical conversion, without regard to its formal correctness (cf. Johnston 1987: 229).

Nor is this all. The quadruples derived from the fourth figure appear in the columns more than once. In Ars magna the quadruple BCTB, for example, figures seven times in each of the first seven columns.

In V, 1, it is interpreted as “Whether there exists some goodness so great that it is different,” while in XI, 1, applying the rule of logical obversion, it is read as “Whether goodness can be great without being different”–obviously eliciting a positive response in the first case and a negative one in the second.

Yet these reappearances of the same argumentative scheme, to be endowed with different semantic contents, do not bother Lull. On the contrary, he assumes that the same question can be solved either by any of the quadruples from a particular column that generates it, or from any of the other columns!

Such a feature, which Lull takes as one of the virtues of his art, represents in fact its second severe limitation. The 1,680 quadruples do not generate fresh questions, nor do they furnish new proofs.

They generate instead standard answers to an already established set of questions. In principle, the art only furnishes 1,680 different ways of answering a single question whose answer is already known.

It cannot, in consequence, really be considered a logical instrument at all. It is, in reality, a sort of dialectical thesaurus, a mnemonic aid for finding out an array of standard arguments able to demonstrate an already known truth.

As a consequence, any of the 1,680 quadruples, if judiciously interpreted, can yield up the correct answer to the question for which it is adapted.

See, for instance, the question “Whether the world is eternal” (“Utrum mundus sit aeternus“). Lull already knew the answer: negative, because anyone who thought the world eternal would fall into the Averroist error.

Note, however, that the question cannot be generated directly by the art itself; for there is no letter corresponding to world. The question is thus external to the art.

In the art, however, there does appear a term for eternity, that is, D; this provides a starting point.

In the second figure, D is tied to the relative principle contrarietas or opposition, as manifested in the opposition of the sensible to the sensible, of the intellectual to the sensible, and of the intellectual to the intellectual.

The same second figure also shows that D forms a triangle with B and C. The question also began with utrum, which appears at B under the heading Questiones in the tabula generalis. This constitutes a hint that the solution needs to be sought in the column in which appear B, C and D.

Lull says that “the solution to such a question must be found in the first column of the table;” however, he immediately adds that, naturally, “it could be found in other columns as well, as they are all bound to each other.”

At this point, everything depends on definitions, rules, and a certain rhetorical legerdemain in interpreting the letters. Working from the chamber BCDT (and assuming as a premise that goodness is so great as to be eternal), Lull deduces that if the world were eternal, it would also be eternally good, and, consequently, there would be no evil.

“But,” he remarks, “evil does exist in the world as we know by experience. Consequently we must conclude that the world is not eternal.” This negative conclusion, however, is not derived from the logical form of the quadruple (which has, in effect, no real logical form at all), but is merely based on an observation drawn from experience.

The art may have been conceived as the instrument to use universal reason to show the Averroist Muslims the error of their ways; but it is clear that unless they already shared with Lull the “rational” conviction that the world cannot be eternal, they are not going to be persuaded by the art.”

Umberto Eco, The Search for the Perfect Language, translated by James Fentress, Blackwell. Oxford, 1995, pp. 62-4.

Eco: The Alphabet and the Four Figures, 2

Raymond Llull, Combinations, Strasbourg ed 1598

Umberto Eco, The Search for the Perfect Language, 1995, pg. 60. Figure 4.2, a page of combinations from the Strasbourg edition of the Ars Magna of Raymond Llull, 1598. 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.  

Taken in groups of 3, 9 elements generate 84 combinations–BCD, BCE, CDE, etc. If, in his Ars breu and elsewhere, Lull sometimes speaks of 252 (84*3) combinations, it is because to each triple can be assigned three questions, one for each of the letters of the triple (see also the Jesuit Athanasius Kircher, Ars magna sciendi, p. 14.


Athanasius Kircher, Ars Magna Sciendi, Amsterdam, 1669. 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. 

Each triple further generates a column of 20 combinations (giving a table of 20 rows by 84 columns) because Lull transforms the triples into quadruples by inserting the letter T. In this way, he obtains combinations like BCDT, BCTB, BTBC, etc. (See examples in figure 4.2, at the top of this page).

The letter T, however, plays no role in the art; it is rather a mnemonic artifice. It signifies that the letters that precede it are to be read as dignities from the first figure, while those that follow it are to be read as relative principles as defined in the second figure.

Thus, to give an example, the quadruple BCTC must be read: B (= goodness) + C (= greatness) and therefore (switching to the second figure) C (=  concordance).

Looking at the tabula generalis, we further notice that combinations with an initial B take the question utrum, those with an initial C take quid, etc. This produces from BCTC the following reading: “Whether goodness is great inasmuch as it contains in itself concordant things.”

This produces a series of quadruples which seem, at first sight, embarrassing: the series contains repetitions. Had repetitions been permissible, there would have been 729 triples instead of 84.

The best solution to the mystery of these repetitions is that of Platzek (1953-4: 141). He points out that, since, depending on whether it precedes or follows the T, a letter can signify either a dignity or a relation, each letter has, in effect, two values.

Thus–given the sequence BCTB–it should be read as BCb. The letters in upper case would be read as dignities, and the one in lower case as a relation. It follows that, in his 84 columns, Lull was not really listing the combinations for three letters but for six. Six different elements taken three at a time give 20 permutations, exactly as many appear in each column.

The 84 columns of 20 quadruples each yield 1,680 permutations. This is a figure obtained by excluding inversions of order.

At this point, however, a new question arises. Given that all these 1,680 quadruples can express a propositional content, do they all stand for 1,680 valid arguments as well?


Athanasius Kircher, Ars Magna Sciendi sive Combinatoria, Amsterdam, 1669. Frontispiece. 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.  

Not at all, for not every sequence generated by the art is syllogistically valid. Kircher, in his Ars magna sciendi, suggests that one must deal with the resulting sequences as if they were anagrams: one starts by forming a complete list of all the possible arrangements of the letters of a particular word, then discards those that do not correspond to other existing words.

The letters of the Latin word ROMA, for example, can be combined in 24 different orders: certain sequences form acceptable Latin words, such as AMOR, MORA, RAMO; others, however, such as AOMR, OAMR, MRAO, are nonsense, and are, as it were, thrown away.

Lull’s own practice seems to suppose such a criterion. He says, for example, in his Ars magna, segunda pars principalis that in employing the first figure, it is always possible to reverse subject and predicate (“Goodness is great” / “Greatness is good”).

It would not, however, be possible to reverse goodness and angel, for while angel participates in goodness, goodness does not participate in angel, since there are beings other than angels which are good.

In other words, angel entails goodness but not vice versa. Lull also adds that the combination “Greed is good” is inherently unacceptable as well. Whoever wishes to cultivate the art, Lull says, must be able to know what is convertible and what is not.”

Umberto Eco, The Search for the Perfect Language, translated by James Fentress, Blackwell. Oxford, 1995, pp. 60-2.

Eco: The Alphabet and the Four Figures


Raymond Llull (1232-1316), Ars magna, segunda figurageneralis et ultima, 1517, held in the Getty Research Institute and digitized by that institution in collaboration with the Internet Archive, generously posted on 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 ars combinatoria of Lull employs an alphabet of nine letters–B to K, leaving out J–and four figures (see figure 4.1). In a tabula generalis that appears in several of his works, Lull set out a table of six groups of nine entities, one for each of the nine letters.

The first group are the nine absolute principles, or divine dignities, which communicate their natures to each other and spread throughout creation.

After this, there are nine relative principles, nine types of question, nine subjects, nine virtues and nine vices.

Lull specifies (and this is an obvious reference to Aristotle’s list of categories) that the nine dignities are subjects of predication, while the other five series are predicates. We shall see that subject and predicate are sometimes allowed to exchange their roles, while in other cases variations of order are not considered as pertinent.

First figure. This traces all the possible combinations between the dignities, thus allowing predications such as “Goodness [bonitas] is great,” “Greatness [magnitudo] is glorious,” etc.

Since the dignities are treated as nouns when they appear as a predicate, the lines connecting them can be read in both directions. The line connecting magnitudo and bonitas can, for example, be read as both “Greatness is good” and “Goodness is great.” This explains why 36 lines produce 72 combinations.

The first figure is designed to allow regular syllogisms to be inferred. To demonstrate, for example, that goodness can be great, it is necessary to argue that “all that is magnified by greatness is great–but goodness is what is magnified by greatness–therefore goodness is great.”

The first table excludes self-predications, like BB or CC, because, for Lull, there is no possibility of a middle term in an expression of the type “Goodness is good” (in Aristotelian logic, “all As are B–C is an A–therefore C is a B” is a valid syllogism because, following certain rules, the middle term A is so disposed to act as the, as it were, bond between B and C).

Second figure. This serves to connect the relative principles with triples of definitions. They are the relations connecting the divine dignities with the cosmos. Since it is intended merely as a visual mnemonic that helps to fix in the mind the various relations between different types of entity, there is no method of combination associated with the second figure.

For example, difference, concordance and opposition (contrarietas) can each be considered in reference to (1) two sensible entities, such as a plant and a stone, (2) a sensible and an intellectual entity, like body and soul, and (3) two intellectual entities, like the soul and an angel.

Third figure. Here Lull displayed all possible letter pairings. The figure contains 36 pairs inserted in what Lull calls the 36 chambers. The figure makes it seem that he intended to exclude inversions.

Yet, in reality, the figure does contemplate inversions in order, and thus the number of the chambers is virtually 72 since each letter is permitted to function as either subject or predicate (“Goodness is great” also gives “Greatness is good:” Ars magna, VI, 2).

Having established the combinations, Lull proceeds to what he calls the “evacuation of the chambers.” Taking, for example, chamber BC, we read it first according to the first figure, obtaining goodness and greatness (bonitas and magnitudo); then according to the second figure, obtaining difference and concordance, (differentia and concordantia: Ars magna, II, 3).

From these two pairs we derive 12 propositions: “Goodness is great,” “Difference is great,” Goodness is different,” “Goodness is different,” “Difference is good,” “Goodness is concordant,” “Difference is concordant,” “Greatness is good,” “Concordance is good,” “Greatness is different,” “Concordance is different,” “Greatness is concordant,” and “Concordance is great.”

Going back to the tabula generalis in figure 4.1, we find that, under the next heading, Questiones, B and C  are utrum (whether) and quid (what). By combining these 2 questions with the 12 propositions we have just constructed, we obtain 24 questions, like “Whether goodness is great?,” or “What is a great goodness?” (see Ars magna, VI, 1).

In this way, the third figure generates 432 propositions and 864 questions–at least in theory. In reality, there are 10 additional rules to be considered (given in Ars magna, VI, iv).

For the chamber BC, for example, there are the rules B and C. These rules depend on the theological definition of the terms, and on certain argumentative constraints which have nothing to do with the rules of combination.


Quarta figura, fourth figure.

Fourth figure. This is the most famous of the figures, and the one destined to have the greatest influence on subsequent tradition. In this figure, triples generated by the nine elements are considered.

In contrast to the preceding figures, which are simply static diagrams, the fourth figure is mobile. It is a mechanism formed by three concentric circles, of decreasing size, inserted into each other, and held together usually by a knotted cord.

If we recall that in the Sefer Yezirah the combination of the letters was visually represented by a wheel or a spinning disc, it seems probable that Lull, a native of Majorca, has been influenced here by the kabbalistic tradition that flourished in his time in the Iberian peninsula.”

Umberto Eco, The Search for the Perfect Language, translated by James Fentress, Blackwell. Oxford, 1995, pp. 56-60.


Eco: The Elements of the Ars Combinatoria

Ramon_Llull, Ars Magna, Fig_1

Raymond Llull (1232-1316), Figure 1 from Ars magna, 1300. 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. 

“Given a number of different elements n, the number of arrangements that can be made from them, in any order whatever, is expressed by their factorial n!, calculated as 1.*2*3. . . . *n.

This is the method for calculating the possible anagrams of a word of n letters, already encountered as the art of temurah in the kabbala. The Sefer Yezirah informed us that the factorial of 5 was 120.

As n increases, the number of possible arrangements rises exponentially: the possible arrangements for 36 elements, for example, are 371, 993, 326, 789, 901, 217, 467, 999, 448, 150, 835, 200, 000, 000.

If the strings admit repetitions, then those figures grow upwards. For example, the 21 letters of the Italian alphabet can give rise to more than 51 billion billion 21-letter-long sequences (each different from the rest); when, however, it is admitted that some letters are repeated, but the sequences are shorter than the number of elements to be arranged, then the general formula for n elements taken t at a time with repetitions is n1  and the number of strings obtainable for the letters of the Italian alphabet would amount to 5 billion billion billion.

Let us suppose a different problem. There are four people, A. B, C, and D. We want to arrange these four as couples on board an aircraft in which the seats are in rows that are two across; the order is relevant because I want to know who will sit at the the window and who at the aisle.

We are thus facing a problem of permutation, that is, of arranging n elements, taken t at a time, taking the order into account. The formula for finding all the possible permutations is n!/(n-t)In our example the persons can be disposed this way:

AB     AC     AD     BA     CA     DA     BC     BD     CD     CB     DB     DC

 Suppose, however, that the four letters represented four soldiers, and the problem is to calculate how many two-man patrols could be formed from them. In this case the order is irrelevant (AB or BA are always the same patrol). This is a problem of combination, and we solve it with the following formula: n!/t!(n-t)! In this case the possible combinations would be:

AB     AC     AD     BC     BD     CD

Such calculuses are employed in the solution of many technical problems, but they can serve as discovery procedures, that is, procedures for inventing a variety of possible “scenarios.”

In semiotic terms, we are in front of an expression-system (represented both by the symbols and by the syntactic rules establishing how n elements can be arranged t at a time–and where t can coincide with n), so that the arrangement of the expression-items can automatically reveal possible content-systems.

In order to let this logic of combination or permutation work to its fullest extent, however, there should be no restrictions limiting the number of possible content-systems (or worlds) we can conceive of.

As soon as we maintain that certain universes are not possible in respect of what is given in our own past experience, or that they do not correspond to what we hold to be the laws of reason, we are, at this point, invoking external criteria not only to discriminate the results of the ars combinatoria, but also to introduce restrictions within the art itself.

We saw, for example, that, for four people, there were six possible combinations of pairs. If we specify that the pairing is of a matrimonial nature, and if A and B are men while C and D are women, then the possible combinations become four.

If A and C are brother and sister, and we take into the account the prohibition against incest, we have only three possible groupings. Yet matters such as sex, consanguinity, taboos and interdictions have nothing to do with the art itself: they are introduced from outside in order to control and limit the possibilities of the system.”

Umberto Eco, The Search for the Perfect Language, translated by James Fentress, Blackwell. Oxford, 1995, pp. 54-6.

Eco: The Ars Magna of Raymond Lull

Raymond Lull, Tabula Generalis, pg. 57, Eco, Search for a Perfect Language, 1995

Raymond Lull (1232-1316), Tabula Generalis, figure 4.1, Lull’s Alphabet, from Umberto Eco, The Search for a Perfect Language, Blackwell, Oxford, 1995, pg. 57. 

“A near contemporary of Dante, Ramòn Llull (Latinized as Lullus and Anglicized as Lull–and sometimes as Lully) was a Catalan, born in Majorca, who lived probably between 1232 (or 1235) and 1316.

Majorca during this period was a crossroads, an island where Christian, Jewish and Arab cultures all met; each was to play a role in Lull’s development. Most of his 280 known works were written initially in Arabic or Catalan (cf. Ottaviano 1930).

Lull led a carefree early life which ended when he suffered a mystic crisis. As a result, he entered the order of Tertian friars.

It was among the Franciscans that all of the earlier strands converged in his Ars magna, which Lull conceived as a system for a perfect language with which to convert the infidels. The language was to be a universal; it was to be articulated at the level of expression in a universal mathematics of combination; its level of content was to consist of a network of universal ideas, held by all peoples, which Lull himself would devise.

St. Francis had already sought to convert the sultan of Babylonia, and the dream of establishing universal concordance between differing races was becoming a recurrent theme in Franciscan thought. Another of Lull’s contemporaries, the Franciscan Roger Bacon, foresaw that contact with the infidels (not merely Arabs, but also Tartars) would require study of foreign languages.

The problem for him, however, was not that of inventing a new, perfect language, but of learning the languages that the infidels already spoke in order to convert them, or, failing that, at least to enrich Christian culture with a wisdom that the infidels had wrongfully appropriated (“tamquam ab iniustis possessoribus“).

The aims and methods of Lull and Bacon were different; yet both were inspired by ideals of universality and of a new universal crusade based on peaceful dialogue rather than on arms.

In this utopia the question of language played a crucial role (cf. Alessio 1957). According to legend, Lull was to die martyred at the hands of the Saracens, to whom he had appeared, armed with his art, believing it to be an infallible means of persuasion.

Lull was the first European philosopher to write doctrinal works in the vulgar tongue. Some are even in popular verses, so as to reach readers who knew neither Latin nor Arabic: “per tal che hom puscha mostrar / logicar e philosophar / a cels que win saben lati / ni arabichi” (Compendium, 6-9).

His art was universal not merely in that it was designed to serve all peoples, but also in that it used letters and figures in a way (allegedly) comprehensible even to illiterates of any language.”

Umberto Eco, The Search for the Perfect Language, translated by James Fentress, Blackwell. Oxford, 1995, pp. 53-4.

Notes on Theurgy and the Mnemotechnic Metasciences of Raymond Lully.

Excerpts from footnotes in a book that I am reading:

8. “Briefly, theurgy is the art of bringing down celestial beings (angels) through the use of prayers on the one hand, and of ecstatic ascent toward union with God, on the other.

9. Among mnemotechnic metasciences, the Ars magna of Raimundus Lullus is no doubt the most famous.”

–Benedek Lang, Unlocked Books: Manuscripts of Learned Magic in the Medieval Libraries of Central Europe, 2008. Pg. 165.