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Tag: Ramòn Llull

Eco: Blind Thought, 2

Wittgenstein, Ludwig

Ludwig Wittgenstein (1899-1951), portrait by Moritz Nähr (1859-1945), 1930, held by the Austrian National Library under Accession Number Pf 42.805: C (1). 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 70 years or less. 

“As Leibniz observed in the Accessio ad arithmeticum infinitorum of 1672 (Sämtliche Schriften und Briefen, iii/1, 17), when a person says a million, he does not represent mentally to himself all the units in that number. Nevertheless, calculations performed on the basis of this figure can and must be exact.

Blind thought manipulates signs without being obliged to recognize the corresponding ideas. For this reason, increasing the power of our minds in the manner that the telescope increases the power of our eyes, it does not entail an excessive effort.

“Once this has been done, if ever further controversies should arise, there should be no more reason for disputes between two philosophers than between two calculators. All that will be necessary is that, pen in hand, they sit down together at a table and say to each other (having called, if they so please, a friend) “let us calculate.” (In Gerhardt 1875: VII, 198ff).

Leibniz’s intention was thus to create a logical language, like algebra, which might lead to the discovery of unknown truths simply by applying syntactical rules to symbols. When using this language, it would no more be necessary, moreover, to know at every step what the symbols were referring to than it was necessary to know the quantity represented by algebraic symbols to solve an equation.

Thus for Leibniz, the symbols in the language of logic no longer stood for concrete ideas; instead, they stood in place of them. The characters “not only assist reasoning, they substitute for it.” (Couturat 1901: 101).

Dascal has objected (1978: 213) that Leibniz did not really conceive of his characteristica as a purely formal instrument apparatus, because symbols in his calculus are always assigned an interpretation. In an algebraic calculation, he notes, the letters of the alphabet are used freely; they are not bound to particular arithmetical values.

For Leibniz, however, we have seen that the numerical values of the characteristic numbers were, so to speak, “tailored” to concepts that were already filled with a content–“man,” “animal,” etc.

It is evident that, in order to demonstrate that “man” does not contain “monkey,” the numerical values must be chosen according to a previous semantic decision. It would follow that what Leibniz proposed was really a system both formalized and interpreted.

Now it is true that Leibniz’s posterity elaborated such systems. For instance, Luigi Richer (Algebrae philosophicae in usum artis inveniendi specimen primum, “Melanges de philosophie et de mathématique de la Societé Royale de Turin,” 1761: II/3), in fifteen short and extremely dry pages, outlined a project for the application of algebraic method to philosophy, by drawing up a tabula characteristica containing a series of general concepts (such as aliquid, nihil, contingens, mutabile) and assigning to each a conventional sign.

The system of notation, semicircles orientated in various ways, makes the characters hard to distinguish from one another; still, it was a system of notation that allowed for the representation of philosophical combinations such as “This Possible cannot be Contradictory.”

This language is, however, limited to abstract reasoning, and, like Lull, Richer did not make full use of the possibilities of combination in his system as he wished to reject all combinations lacking scientific utility (p. 55).

Towards the end of the eighteenth century, in a manuscript dating 1793-4, we also find Condorcet toying with the idea of a universal language. His text is an outline of mathematical logic, a langue des calculs, which identifies and distinguishes intellectual processes, expresses real objects, and enunciates the relations between the expressed objects and the intellectual operations which discover the enunciated relations.

The manuscript, moreover, breaks off at precisely the point where it had become necessary to proceed to the identification of the primitive ideas; this testifies that, by now, the search for perfect languages was definitively turning in the direction of a logico-mathematical calculus, in which no one would bother to draw up a list of ideal contents but only to prescribe syntactic rules (Pellerey 1992a: 193ff).

We could say that Leibniz’s characteristica, from which Leibniz had also hoped to derive metaphysical truths, is oscillating between a metaphysical and ontological point of view, and the idea of designing a simple instrument for the construction of deductive systems (cf. Barone 1964: 24).

Moreover, his attempts oscillate between a formal logic (operating upon unbound variables) and what will later be the project of many contemporary semantic theories (and of artificial intelligence as well), where syntactic rules of a mathematical kind are applied to semantic (and therefore interpreted) entities.

But Leibniz ought to be considered the forerunner of the first, rather than of the second, line of thought.

The fundamental intuition that lies behind Leibniz’s proposal was that, even if the numbers were chose arbitrarily, even if it could not be guaranteed that the primitives posited for the same of argument were really primitive at all, what still guaranteed the truth of the calculus was the fact that the form of the proposition mirrored an objective truth.

Leibniz saw an analogy between the order of the world, that is, of truth, and the grammatical order of the symbols in language. Many have seen in this a version of the picture theory of language expounded by Wittgenstein in the Tractatus, according to which “a picture has logico-pictorial form in common with what it depicts” (2.2).

Leibniz was thus the first to recognize that the value of his philosophical language was a function of its formal structure rather than of its terms; syntax, which he called habitudo or propositional structure, was more important than semantics (Land 1974: 139).

“It is thus to be observed that, although the characters are assumed arbitrarily, as long as we observe a certain order and certain rule in their use, they give us results which always agree with each other. (Dialogus in Gerhardt 1875: VII, 190-3).

Something can be called an “expression” of something else whenever the structure [habitudines] subsisting in the expression corresponds to the structure of that which it wishes to express [ . . . ].

From the sole structure of the expression, we can reach the knowledge of the properties of the thing expressed [ . . . ] as long as there is maintained a certain analogy between the two respective structures.” (Quid sit idea in Gerhardt 1875: VII, 263-4).

What other conclusion could the philosopher of preestablished harmony finally have reached?”

Umberto Eco, The Search for the Perfect Language, translated by James Fentress, Blackwell. Oxford, 1995, pp. 281-4.

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: From Leibniz to the Encyclopédie


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: Primitives and Organization of Content, 2

Ramon Llull, Arbor naturalis et logicalis, Liber de logica nova, Valencia, Alonso de Proaza, 1512

Ramon Llull (1232-1315), Arbor naturalis et logicalis, Liber de logica nova, Valencia, 1512. A Porphyrian Tree of logical relations, original c. 1305, logica nova edition 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. 

“Nevertheless, such a dictionary-like structure would not allow us to define the difference between a cat and a tiger, or even between a canine and a feline animal. To do this, it is necessary to insert differences into the classification.

Aristotle, in his studies of definition, said that, in order to define the essence of a thing, we should select such attributes which “although each of them has a wider extension than the subject, all together they have not” (Posterior Analytics II, 96a, 35).

Such a structured representation was known in the Middle Ages as Porphyry’s Tree (because it was derived from the Isagoge of the Neo-Platonic philosopher Porphyry, living in the second-third century AD), and was still taken as a definitional model by the English searchers for a real character.

In a Porphyrian Tree each genus is divided by two differences which constitute a pair of opposites. Each genus, with the addition of one of its divisive differences, produces an underlying species, which is so defined by its genus and its constitutive difference.

Umberto Eco, The Search for the Perfect Language, Figure 10.2, p. 225

Umberto Eco, The Search for the Perfect Language, Figure 10.2, p. 225. 

In figure 10.2, there is an example of how a Porphyrian Tree establishes the difference between human beings and gods (understood as natural forces) and between human beings and beasts.

The terms in upper-case refer to genera and species while those in lower-case refer to differences, that is, to particular accidents which occur only in a given species. We see that the diagram defines a human being as a “rational and mortal animal,” which, in classical terms, is considered a satisfactory definition because there cannot be a rational and mortal animal which is not a human being, and only human beings are so.

Unfortunately this diagram does not tell us anything about the differences between dogs and cats, or horses and wolves, or cats and tigers. In order to obtain new definitions, new differences need to be inserted into the diagram.

Besides this, we can see that, although differences occur in one species, in this tree there are differences, such as “mortal/immortal,” which occur in two different species.

This makes it difficult to know whether or not the same differences will be reproduced at some further point in the tree when it becomes necessary to specify the difference not just between dogs and cats, but also between violets and roses, diamonds and sapphires, and angels and demons.

Even taxonomy as practiced by modern zoology defines through dichotomies. Dogs are distinguished from wolves, and cats from tigers, on the basis of a dichotomy by taxonomic entities known as taxa (figure 10.3).

Umberto Eco, The Search for the Perfect Language, Figure 10.3, p. 226

Umberto Eco, The Search for the Perfect Language, Figure 10.3, p. 226. 

Yet modern zoologists are well aware that a system of classification is not the same as a system of definitions.

Classification does not capture the essence of the thing itself; it simply embeds things in a system of increasingly inclusive classes, where the lower nodes are linked by entailment to the upper ones: if something is a Canis familiaris, it cannot but be, by entailment, a Canis, a canid and a fissiped.

But Canidae and Fissipeda are taken as primitives only in the framework of the classification and are not considered as semantic primitives.

Zoologists know that, within their classification, at the node Canidae they must presuppose a set of properties common to the whole family, and that at the node Carnivora there is a set of properties common to the whole order: in the same vein, “mammal” is not a semantic primitive but a technical name which stands for (more or less) “viviparous animal which nourishes its young by the secretion of milk through its mammary glands.”

The name of a substance can be either designative (thus indicating the genus to which that substance belongs) or diagnostic, that is, transparent and self-definatory.

In Species plantarum by Linnaeus (1753), given the two species, Arundo calamogrostis and Arundo arenaria, their designative names show that they belong to the same genus and establish their difference; however, their properties are then made clearer by a diagnostic description which specifies that the Arundo calamogrostis is “calycibus unifloris, cumulo ramoso,” while the Arundo arenaria is “calycibus unifloris, foliis involutiis, mucronato pungentibus” (see Slaughter 1982: 80).

However, the terms used for this description are no longer pseudo-primitives–like those of the metalanguage of taxas; they are terms of the common natural language used for diagnostic purposes.

By contrast, for the authors of a priori languages, each expression had to express all the properties of the designated thing. We shall see how such a difficulty will affect all the projects discussed in the following chapters.”

Umberto Eco, The Search for the Perfect Language, translated by James Fentress, Blackwell. Oxford, 1995, pp. 225-7.

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: Infinite Songs & Locutions


Giordano Bruno (1548-1600), memory wheel, De Umbris Idearum, 1582, reconstructed by Dame Frances Yates, Warburg Institute. Frances Yates wrote Giordano Bruno and the Hermetic Tradition, Chicago, 1964. 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.  

 “Between Lull and Bruno might be placed the game invented by H.P. Harsdörffer in his Matematische und philosophische Erquickstunden (1651: 516-9). He devises 5 wheels containing 264 units (prefixes, suffixes, letters and syllables).

This apparatus can generate 97,209,600 German words, including many that were still non-existent but available for creative and poetic use (cf. Faust 1981: 367). If this can be done for German, why not invent a device capable of generating all possible languages?

The problem of the art of combination was reconsidered in the commentary In spheram Ioannis de sacro bosco by Clavius in 1607. In his discussion of the four primary qualities (hot, cold, dry and wet), Clavius asked how many pairs they might form.

Mathematically, we know, the answer is six. But some combinations (like “hot and cold,” “dry and wet”) are impossible, and must be discarded, leaving only the four acceptable combinations: “Cold and dry” (earth), “hot and dry” (fire), “hot and wet” (air), “cold and wet” (water).

We seem to be back with the problem of Lull: a conventional cosmology limits the combinations.

Clavius, however, seemed to wish to go beyond these limits. He asked how many dictiones, or terms, might be produced using the 23 letters of the Latin alphabet (u being the same as v), combining them 2, 3, 4 at a time, and so on until 23.

He supplied a number of mathematical formulae for the calculations, yet he soon stopped as he began to see the immensity of the number of possible results–especially as repetitions were permissible.

In 1622, Paul Guldin wrote a Problema arithmeticum de rerum combinationibus (cf. Fichant 1991: 136-8) in which he calculated the number of possible locutions generated by 23 letters. He took into account neither the question of whether the resulting sequences had a sense, nor even that of whether they were capable of being pronounced at all.

The locutions could consist of anything from 2 to 23 letters; he did not allow repetitions. He arrived at a result of more than 70,000 billion billion. To write out all these locutions would require more than a million billion billion letters.

To conceive of the enormity of this figure, he asked the reader to imagine writing all these words in huge notebooks: each of these notebooks had 1,000 pages; each of these pages had 100 lines; each of these lines could accommodate 60 characters.

One would need 257 million billion of these notebooks. Where would you put them all? Guldin then made a careful volumetric study, imagining shelf space and room for circulation in the libraries that might store a consignment of these dimensions.

If you housed the notebooks in large libraries formed by cubes whose sides measured 432 feet, the number of such cubic buildings (hosting 32 million volumes each) would be 8,050,122,350. And where would you put them all? Even exhausting the total available surface space on planet earth, one would still find room for only 7,575,213,799!

In 1636 Father Marin Mersenne, in his Harmonie universelle, asked the same question once again. This time, however, to the dictiones he added “songs,” that is, musical sequences.

With this, the conception of universal language has begun to appear, for Mersenne realizes that the answer would necessarily have to include all the locutions in all possible languages. He marveled that our alphabet was capable of supplying “millions more terms than the earth has grains of sand, yet it is so easy to learn that one hardly needs memory, only a touch of discernment” (letter to Peiresc, c. April 1635; cf. Coumet 1975; Marconi 1992).

In the Harmonie, Mersenne proposed to generate only pronounceable words in French, Greek, Arabic, Chinese and every other language. Even with this limitation one feels the shudder provoked by a sort of Brunian infinity of possible worlds.

The same can be said of the musical sequences that can be generated upon an extension of 3 octaves, comprising 22 notes, without repetitions (shades of future 12-tone compositions!).

Mersenne observed that to write down all these songs would require enough reams of paper to fill in the distance between heaven and earth, even if every sheet contained 720 of these 22-note songs and every ream was so compressed as to be less than an inch thick.

In fact the number of possible songs amounted to 1,124,000,727,777,607,680,000 (Harmonie, 108). By dividing this figure by the 362,880 songs contained in each ream, one would still obtain a 16-digit figure, whilst the number of inches between the center of the earth and the stars is only 28,826,640,000,000 (a 14-digit figure).

Anyone who wished to copy out all these songs, a thousand per day, would have to write for 22,608,896,103 years and 12 days.”

Umberto Eco, The Search for the Perfect Language, translated by James Fentress, Blackwell. Oxford, 1995, pp. 139-41.

Eco: Bruno: Ars Combinatoria & Infinite Worlds, 3


Giordano Bruno (1548-1600), a mnemonic diagram, which appears towards the end of Cantus circaeus (Incantation of Circe), 1582, which also appears on the cover of Opere mnemotecniche, Vol. 1: De umbris idearum, 1582, Rita Sturlese, et al, ed. 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 her critical edition of De umbris (1991), Sturlese gives an interpretation of the use of the wheels that differs sharply from the “magical” interpretation given by Yates (1972). For Yates, the wheels generated syllables by which one memorizes images to be used for magical purposes.

Sturlese inverts this: for her, it is the images that serve to recall the syllables. Thus, for Sturlese, the purpose of the entire mnemonical apparatus was the memorization of an infinite multitude of words through the use of a fixed, and relatively limited, number of images.

If this is true, then it is easy to see that Bruno’s system can no longer be treated as an art where alphabetic combinations lead to images (as if it were a scenario-generating machine); rather, it is a system that leads from combined images to syllables.

Such a system not only aids memorization but, equally, permits the generation of an almost unlimited number of words–be they long and complex like incrassatus or permagnus, or difficult like many Greek, Hebrew, Chaldean, Persian or Arabic terms (De umbris, 169), or rare like scientific names of grass, trees, minerals, seeds or animal genera (De umbris, 152). The system is thus designed to generate languages–at least at the level of nomenclature.

Which interpretation is correct? Does Bruno concatenate the sequence CROCITUS to evoke the image of Pilumnus advancing rapidly on the back of a donkey with a bandage on his arm and a parrot on his head, or has he assembled these images so as to memorize CROCITUS?

In the “Prima Praxis” (De umbris, 168-72) Bruno tells us that it is not indispensable to work with all five wheels because, in most known languages, it is rare to find words containing syllables with four or five letters.

Furthermore, where such syllables do occur (for instance, in words like trans-actum or stu-prans), it is usually eash to devise some artifice that will obviate the necessity of using the fourth and fifth wheel.

We are not interested in the specific short cuts that Bruno used except to say that they cut out several billion possibilities. It is the very existence of such short cuts that seems significant.

If the syllabic sequences were expressing complex images, there should be no limit for the length of the syllables. On the contrary, if the images were expressing syllables, there would be an interest in limiting the length of the words, following the criteria of economy already present in most natural languages (even though there is no formal limit, since Leibniz will later remark that there exists in Greek a thirty-one-letter word).

Besides, if the basic criterion of every art of memory is to recall the unfamiliar through the more familiar, it seems more reasonable that Bruno considered the “Egyptian” traditional images as more familiar than the words of exotic languages.

In this respect, there are some passages in De umbris that are revealing: “Lycas in convivium cathenatus presentabat tibi AAA. . . . Medusa, cum insigni Plutonis presentabit AMO” (“Lycaon enchained in a banquet presents to you AAA . . . Medusa with the sign of Pluto presents AMO”).

Since all these names are in the nominative case, it is evident that they present the letters to the user of the system and not the other way around. This also follows from a number of passages in the Cantus circaeus where Bruno uses perceivable images to represent mathematical or abstract concepts that might not otherwise be imaginable or memorizable (cf. Vasoli 1958: 284ff).

That Bruno bequeathed all this to the Lullian posterity can be seen from further developments of Lullism.”

Umberto Eco, The Search for the Perfect Language, translated by James Fentress, Blackwell. Oxford, 1995, pp. 137-9.

Eco: Bruno: Ars Combinatoria & Infinite Worlds, 2


Ettore Ferrari (1845-1929), Giordano Bruno Burned at the Stake, a bas relief on the plinth of the monument to Bruno in Campo de’Fiori square in Rome. This photo by Giovanni Dall’Orto © 2008. The copyright holder of this photo allows anyone to use for any purpose, provided that the copyright holder is properly attributed. Redistribution, derivative work, commercial use, and all other use is permitted.  


“Thus this language claimed to be so perfect as to furnish the keys to express relations between things, not only of this world, but of any of the other infinite worlds in their mutual concordance and opposition.

Nevertheless, in its semiotic structure, it was little more than an immense lexicon, conveying vague meanings, with a very simplified syntax. It was a language that could be deciphered only by short-circuiting it, and whose decipherment was the privilege only of the exegete able to dominate all its connections, thanks to the furor of Bruno’s truly heroic style.

In any case, even if his techniques were not so different from those of other authors of arts of memory, Bruno (like Lull, Nicholas of Cusa and Postel, and like the reformist mystics of the seventeenth century–at whose dawn he was to be burnt at the stake) was inspired by a grand utopian vision.

His flaming hieroglyphical rhetoric aimed at producing, through an enlargement of human knowledge, a reform, a renovation, maybe a revolution in the consciousness, customs, and even the political order of Europe. Of this ideal, Bruno was the agent and propagandist, in his wandering from court to European court.

Here, however, our interest in Bruno is limited to seeing how he developed Lullian techniques. Certainly, his own metaphysics of infinite worlds pushed him to emphasize the formal and architectonic aspects of Lull’s endeavor.

One of his mnemonic treatises, De lampade combinatoria lulliana ad infinita propositiones et media inveniendi (1586), opens by mentioning the limitless number of propositions that the Ars is capable of generating, and then says: “The properties of the terms themselves are of scant importance; it is only important that they show an order, a texture, an architecture.” (I, ix).

In the De umbris idearum (1582) Bruno described a set of movable, concentric wheels subdivided into 150 sectors. Each wheel contained 30 letters, made up of the 23 letters of the Latin alphabet, plus 7 letters from the Greek and Hebrew alphabets to which no letter corresponded in Latin (while, for instance, A could also stand for Alpha and Alef).

To each of the single letters there corresponded a specific image, representing for each respective wheel a different series of figures, activities, situations, etc. When the wheels were rotated against each other in the manner of a combination lock, sequences of letters were produced which served to generate complex images. We can see this in Bruno’s own example (De umbris, 163):

Giordano Bruno, De umbris, 163

Reproduced from Umberto Eco, The Search for the Perfect Language, James Fentress, trans., Blackwell: Oxford, 1995, p. 136, from Giordano Bruno, De umbris idearum, 1582, p. 163. 

In what Bruno called the “Prima Praxis,” the second wheel was rotated so as to obtain a combination such as CA (“Apollo in a banquet”). Turning the third wheel, he might obtain CAA (“Apollo enchained in a banquet”). We shall see in a moment why Bruno did not think it necessary to add fourth and fifth wheels as he would do for the “Secunda Praxis,” where they would represent, respectively, adstantia and circumstantias.

In his “Secunda Praxis,” by adding the five vowels to each of the 30 letters of his alphabet, Bruno describes 5 concentric wheels, each having 150 alphabetical pairs, like AA, AE, AI, AO, AU, BA, BE, BO, and so on through the entire alphabet.

These 150 pairs are repeated on each of the 5 wheels. As in the “Prima Praxis,” the significance changes with every wheel. On the first wheel, the initial letter signifies a human agent, on the second, an action, on the third, an insignia, on the fourth, a bystander, on the fifth, a set of circumstances.

By moving the wheels it is possible to obtain images such as “a woman riding on a bull, combing her hair while holding a mirror in her left hand, accompanied by an adolescent carrying a green bird in his hand” (De umbris, 212, 10).

Bruno speaks of images “ad omnes formationes possible, adaptabiles” (De umbris, 80), that is, susceptible of every possible permutation. In truth, it is almost impossible to write the number of sequences that can be generated by permutating 150 elements 5 at a time, especially as inversions are allowed (De umbris, 223).

This distinguishes the art of Bruno, which positively thirsts after infinity, from the art of Lull.”

Umberto Eco, The Search for the Perfect Language, translated by James Fentress, Blackwell. Oxford, 1995, pp. 135-7.

Eco: The Concordia Universalis of Nicholas of Cusa


Meister des Marienlebens, Kreuzigung, Passionsalter aus Bernkastel-Kues, 1460. 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 seductive potentiality of Lull’s appeal to the principle of universal concord is revealed by the resumption of his project, two centuries later, by Nicholas of Cusa. Nicholas is famous as the figure who revived Plato during the years between the crisis of scholasticism and the beginning of the Renaissance.

Nicholas also propounded the idea of an infinitely open universe, whose centre was everywhere and whose circumference nowhere. As an infinite being, God transcended all limits and overcame every opposition.

As the diameter of a circle increased, its curvature diminished; so at its limit its circumference became a straight line of infinite length.

Likewise, in God all opposites coincide. If the universe had a centre, it would be limited by another universe. But in the universe, God is both centre and circumference. Thus the earth could not be the centre of the universe.

This was the starting point for a vision of the plurality of worlds, of a reality founded on mathematical principles, which can be submitted to continuous investigation, where the world, if not infinite in a strict sense, was at least capable of assuming an infinite number of guises.

The thought of Nicholas is rich in cosmological metaphors (or models) founded upon the image of the circle and the wheel (De docta ignorantia, II, 11), in which the names of the divine attributes (explicitly borrowed from Lull) form a circle where each supports and confirms the others (I, 21).

The influence of Lull is even more explicitly revealed when Nicholas notes that the names by which the Greeks, Latins, Germans, Turks and Saracens designate the divinity are either all in fundamental accord, or derive from the Hebrew tetragrammaton (see the sermon Dies sanctificatus).

The ideas of Lull had spread to the Veneto towards the close of the fourteenth century. Nicholas probably came into contact with them in Padua. Their diffusion was, in part, a reaction against a scholastic Aristotelianism now in crisis; yet the diffusion also reflected the feverish cultural atmosphere generated by close contacts with the East.

Just as Catalonia and Majorca had been frontier territories in contact with the Muslim and Jewish worlds at the time of Lull, so the Venetian Republic had opened itself to the world of Byzantium and of the Arab countries two centuries later. The emerging currents of Venetian humanism were inspired by a new curiosity and respect for other cultures (cf. Lohr 1988).

It was thus appropriate that in this atmosphere there should have reemerged the thought of a figure whose preaching, whose theological speculations, and whose research on universal language were all conceived with the aim of building an intellectual and religious bridge between the European West and the East.

Lull believed that true authority could not be based on a rigid unity, but rather on the tension between various centers. It was the laws of Moses, the revelations of Christ and the preaching of Mohammed that, taken together, might produce a unified result.

Lull’s doctrine acted as a mystical and philosophical stimulus and seemed an imaginative and poetic alternative to the encyclopedia of Aristotelian scholasticism, but it provided a political inspiration as well.

The works of a writer who had dared to put his doctrine into the vernacular proved congenial to humanists who, on the one hand, had begun to celebrate the dignity of their own native tongues, but, on the other hand, wondered how it was possible to establish a rational discussion which broke the boundaries of national traditions, a philosophy which could reanimate the body of encyclopedic scholasticism by injecting the leaven of exotic new doctrines, expressed in languages still entirely unknown.

In his De pace fidei, Nicholas opened a polemical dialogue with the Muslims. He asked himself Lull’s question: how might the truth of Christian revelation be demonstrated to followers of the two other monotheistic religions?

Perhaps, Nicholas mused, it was a mistake to translate the persons of the Trinity as “Father,” “Son” and “Holy Ghost.” Perhaps they should have been given more philosophical names (better understandable by other cultures).

In his ecumenical fervor, Nicholas even went so far as to propose to the Jews and the Muslims that, if they would accept the Gospels, he would see that all Christians received circumcision. It was a proposal, as he confessed at the end, whose practical realization might present certain difficulties. (De pace fidei, XVI, 60).

Nicholas retained from Lull the spirit of universal peace as well as his metaphysical vision. Yet before the thrilling potential of Nicholas’s own vision of an infinity of worlds could be translated into a new and different version of the art of combination, new ideas would have to fertilize the humanist and Renaissance world.

The rediscovery of the art of combination would have to wait for the rediscovery of Hebrew, for Christian kabbalism, for the spread of Hermeticism, and for a new and positive reassessment of magic.”

Umberto Eco, The Search for the Perfect Language, translated by James Fentress, Blackwell. Oxford, 1995, pp. 69-72.


Eco: The Arbor Scientarium, 2

Ramon Llull, Arbor Scientiae, Rome, 1295

Ramon Llull, Arbor Scientiae, Rome, 1295. 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.  

“Between the first and last versions of his art, Lull’s thought underwent a long process of evolution (described by Carreras y Artau 1939: I, 394), in order to render his art able to deal not only with theology and metaphysics, but also with cosmology, law, medicine, astronomy, geometry and psychology.

Increasingly, the art became a means of treating the entire range of knowledge, drawing suggestions from the numerous medieval encyclopedias, and anticipating the encyclopedic dreams of the Renaissance and the baroque.

All this knowledge, however, needed to be ordered hierarchically. Because they were determinations of the first cause, the dignities could be defined circularly, in reference to themselves; beyond the dignities, however, began the ladder of being. The art was designed to permit a process of reasoning at every step.

The roots of the Tree of Science were the nine dignities and the nine relations. From here, the tree then spread out into sixteen branches, each of which had its own, separate tree. Each one of the sixteen trees, to which there was dedicated a particular representation, was divided into seven parts–roots, trunk, major branches, lesser branches, leaves, fruits and flowers.

Eight of the trees clearly corresponded to eight of the subjects of the tabula generalis: these are the Arbor elementalis, which represents the elementata, that is, objects of the sublunary world, stones, trees and animals composed of the four elements; the Arbor vegetalis;  the Arbor sensualis; the Arbor imaginalis, which represents images that replicate in the mind whatever is represented on the other trees; the Arbor humanalis et moralis (memory, intellect and will, but also the various sciences and arts); the Arbor coelestialis (astronomy and astrology); the Arbor angelicalis; and the Arbor divinalis, which includes the divine dignities.

To this list are added another eight: the Arbor mortalis (virtues and vices); the Arbor eviternalis (life after death); the Arbor maternalis (Mariology); the Arbor Christianalis (Christology); the Arbor imperialis (government); the Arbor apostolicalis (church); the Arbor exemplificalis (the contents of knowledge); and the Arbor quaestionalis, which contains four thousand questions on the various arts.

To understand the structure of these trees, it is enough to look at only one–the Arbor elementalis. Its roots are the nine dignities and nine relations. Its trunk represents the conjoining of these principles, out of which emerges the confused body of primordial chaos which occupies space.

In this are the species of things and their dispositions. The principle branches represent the four elements (earth, air, fire and water) which stretch out into the four masses which are made from them (the seas and the lands).

The leaves are the accidents. The flowers are the instruments, such as hands, feet and eyes. The fruits represent individual things, such as stone, gold, apple, bird.

Calling this a “forest” of trees would be an improper metaphor: the trees overlay one another to rise hierarchically like the peaked roof of a pagoda. The trees at the lower levels participate in those higher up.

The vegetable tree, for example, participates in the tree of elements; the sensual tree participates in the first two; the tree of imagination is built up out of the first three, and it forms the base from which the next tree, the human one, will arise (Llinares 1963: 211-2).

The system of trees reflects the organization of reality itself; it represents the great chain of being the way that it is, and must metaphysically be. This is why the hierarchy constitutes a system of “true” knowledge.

The priority of metaphysical truth over logical validity in Lull’s system also explains why he laid out his art the way he did: he wished his system to produce, for any possible argument, a middle term that would render that argument amenable to syllogistic treatment; having structured the system for this end, however, he proceeded to discard a number of well-formed syllogisms which, though logically valid, did not support the arguments he regarded as metaphysically true.

For Lull, the significance of the middle term of the syllogism was thus not that of scholastic logic. Its middle term served to bind the elements of the chain of being: it was a substantial, not a formal, link.

If the art is a perfect language, it is so only to the extent to which it can speak of a metaphysical reality, of a structure of being which exists independently of it. The art was not a mechanism designed to chart unknown universes.

In the Catalan version of his Logica Algazelis, Lull writes, “De la logic parlam tot breau–car a parlor avem Deu.” (“About logic we will be brief, for it is to talk about God”).

Much has been written about the analogy between Lull’s art and the kabbala. What distinguishes kabbalistic thought from Lull’s is that, in the kabbala, the combination of the letters of the Torah had created the universe rather than merely reflected it.

The reality that the kabbalistic mystic sought behind these letters had not yet been revealed; it could be discovered only through whispering the syllables as the letters whirled.

Lull’s ars combinatoria, by contrast, was a rhetorical instrument; it was designed to demonstrate what was already known, and lock it for ever in the steely cage of the system of trees.

Despite all this, the art might still qualify as a perfect language if those elementary principles, common to all humanity, that it purported to expound really were universal and common to all peoples.

As it was, despite his effort to assimilate ideas from non-Christian and non-European religions, Lull’s desperate endeavor failed through its unconscious ethnocentrism. The content plane, the universe which his art expounded, was the product of the western Christian tradition.

It could not change even though Lull translated it into Arabic or Hebrew. The legend of Lull’s own agony and death is but the emblem of that failure.”

Umberto Eco, The Search for the Perfect Language, translated by James Fentress, Blackwell. Oxford, 1995, pp. 67-9.

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 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.