by Umberto Eco
There is a reason why steganographies act as propagators of a Llullism that goes beyond Llull. The steganographer is not interested in the content (and therefore in the truth) of the combinations he produces. The elementary system requires only that elements of the steganographic expression (combinations of letters or other symbols) may be freely correlated (in ever different ways, so that their encoding is unpredictable) to elements of the expression to be encoded. They are merely symbols that take the place of other symbols. The steganographer, then, is encouraged to attempt more complex combinations, of a purely formal nature, in which all that matters is a syntax of the expression that is ever more vertiginous, and every combination is an unconstrained variable.
Thus, we have Gustavus Selenus,20 in his 1624 Cryptometrices et Cryptographiae, going so far as to construct a wheel of twenty-five concentric circles combining twenty-five series of twenty-four doublets each. And, before you know it, he presents us with a series of tables that record circa 30,000 doublets. The possible combinations become astronomical (see Figure 10.8).
If we are going to have combinations, why stop at 1,680 propositions, as Llull did? Formally, we can say everything.
It is with Agrippa that the possibility is first glimpsed of borrowing from both the Kabbalah and from Llullism the simple technique of combining the letters, and of using that technique to construct an encyclopedia that was not an image of the finite medieval cosmos but of a cosmos that was open and expanding, or of different possible worlds.
His In artem brevis R. Llulli (which appears along with the other works of Llull in the Strasburg edition of 1598) appears at first sight to be a fairly faithful summary of the principles of the Ars, but we are immediately struck by the fact that, in the tables that are supposed to illustrate Llull’s fourth figure, the number of combinations becomes far greater, since repetitions are not avoided.
As Vasoli (1958: 161) remarks,
Agrippa uses this alphabet and these illustrations only as the basis for a series of far more complex operations obtained through the systematic combination and progressive expansion of Llull’s typical figures and, above all, through the practically infinite expansion of the elementa. In this way the subjects are multiplied, defining them within their species or tracing them back to their genera, placing them in relation with terms that are similar, different, contrary, anterior or posterior, or again, referring them to their causes, effects, actions, passions, relations, etc. All of which, naturally, makes feasible a practically infinite use of the Ars.
The Carreras y Artau brothers (1939: 220–221) observe that in this way Agrippa’s art is inferior to Llull’s because it is not based on a theology. But, at least from our point of view and from that of the future development of combinatory systems, this constitutes a strong point rather than a weakness. With Agrippa, Llullism is liberated from theology.
Figure 10.8
Rather, if we must speak of a limit, it is clear that, for Agrippa too, the point is not to lay the foundations for a logic of discovery, but instead for a wide-ranging rhetoric, at most to complicate the list of disciplines configured by his encyclopedia, but always in such a way as to provide—as is the case with a mnemonic technique—notions that can be manipulated by the proficient orator.
Llull was timid with respect to the form of the content. Agrippa broadens the possibilities of the form of the expression in an attempt to articulate vaster structures of content, but he does not go all the way. If he had applied the combinatory system to the description of the inexhaustible network of cosmic relations outlined in the De occulta philosophia he would have taken a decisive step forward. He did not.
Bruno, on the other hand, will try to make his version of Llull’s Ars tell everything and more. Given an infinite universe whose circumference (as Nicholas of Cusa already asserted) was nowhere and its center everywhere, from whatever point the observer contemplates it in its infinity and substantial unity, the variety of forms to be discovered and spoken of is no longer limited. The ruling idea of the infinity of worlds is compounded with the idea that each entity in the world can serve at the same time as a Platonic shadow of other ideal aspects of the universe, as sign, reference, image, emblem, hieroglyphic, seal. By way of contrast too, naturally, because the image of something can also lead us back to unity through its opposite.
The images of his combinatory system, which Bruno finds in the repertory of the hermetic tradition, or even constructs for himself from his fevered phantasy, are not merely intended, as was the case with previous mnemonic techniques, for remembering, but also for envisaging and discovering the essence of things and their relationships.
They will connect with the same visionary energy with which Pico disassembled and reassembled the first word of the sacred text. A thing can represent another thing by phonetic similarity (the horse, in Latin equus, can represent the man who is aequus or just), by putting the concrete for the abstract (a Roman warrior for Rome), by the coincidence of their initial syllables (asinus for asyllum), by proceeding from the antecedent to the consequence, from the accident to the subject and vice versa, from the insignia to the one who wears it. Or, once again, by recurring to Kabbalistic techniques and using the evocative power of the anagram and of paronomasia (palatio for Latio, cf. Vasoli 1958: 285–286).
The combinatory technique becomes a language capable of expressing, not just the events and relationships of this world, but of all of the infinite worlds, in their mutual harmony with one another.
Where are the constraints imposed by a metaphysics of the Great Chain of Being now? The title of one of Bruno’s mnemotechnical treatises, De lampade combinatoria Lulliana continues ad infinitas propositiones et media invenienda.21 The reference to the infinity of propositions that can be generated is unequivocal.
The problem of combinatorial techniques will be taken up by other authors, though in an openly anti-Kabbalistic key, with the express purpose of displaying skepticism in the face of the proliferation of mystical tendencies, of demonstrating the weakness and the approximative nature of the Rabbinical calculus, and of bringing the technique back to a purely formal mathematical calculus (indifferent to meaning) but nevertheless capable of predicting how many new expressions and how many new languages could be produced using only the letters of the Latin alphabet.
In German Jesuit Christopher Clavius’s In Sphaerum Ioannis de Sacro Bosco,22 the author considers how many dictiones, or how many terms, could be produced with the twenty-three letters of the Latin alphabet (at the time there was no difference between u and v or i and j, and no k or y), combining them two by two, three by three, and so on, up to words made up of twenty-three letters. Clavius supplies the mathematical formulas for this calculus, but he stops short at a certain point before the immensity of the possible results, especially if repetitions were to be included.
In 1622, Pierre Guldin composed his Problema arithmeticum de rerum combinationibus (cf. Fichant 1991: 136–138), in which he calculates all the dictions that can be generated with twenty-three letters, regardless of whether they make sense or can be pronounced, but not including repetitions. He establishes that the number of words (of variable length from two to twenty-three letters) would be more than 70,000 billion billion (to write them out would require more than a million billion billion letters). To have an idea of the implications of this number, think of writing all of these words in registers of 1,000 pages, with 100 lines per page and sixty characters per line. They would fill 257 million billion such registers. And if we wished to house them in a library—Guldin studies point by point its arrangement, its extension, how one would navigate within it, if we had at our disposal cubic structures measuring 432 feet per side, each of them capable of holding 32 million volumes, 8,052,122,350 such bookcases would be required. But what realm could accommodate so many structures? Calculating the surface available throughout the entire planet, we could accommodate only 7,575,213,799 of them!
Marin Mersenne, in various of his writings
(cf. Coumet 1975), wonders how many names it would take if we were to give a different name to each individual. And not only that: to every individual hair on the head of every human being. Maybe he was echoing the traditional medieval lament for the penuria nominum or penury of names, according to which there are more things in need of a name than there are names to go around. With the appropriate formula (and the calculations Mersenne engages in are dizzying), it would be possible to generate copious lexicons for all languages.
In addition to the alphabetical dictiones, Mersenne also takes into consideration the canti or musical sequences that can be produced without repetition over the space of three octaves (we may have here an initial allusion to the notion of the dodecaphonic series), and he observes that to record all these canti would require more reams of paper than, if they were piled on top of one another, would cover the distance from earth to the heavens, even if each sheet were to contain 720 canti each with 22 notes and every ream were compressed so as to measure less than an inch: because the canti that can be produced on the basis of 22 notes are 1,124,000,727,607,680,000, and dividing them by the 362,880 that will fit on a ream, the result would still be a number of 16 figures, while the distance from the center of the earth to the stars is only 28,826,640,000,000 inches (14 figures). And if we were to write down all these canti, at the rate of 1,000 a day, it would take 22,608,896,103 years and 12 days.
There is in all this giddy rapture a consciousness of the infinite perfectibility of knowledge, for which mankind, the new Adam, has the possibility in the course of the centuries to name everything that the first Adam did not find time to baptize. In this way, the combinations aspire to compete with that ability to know the individual that belongs solely to God (whose impossibility will be sanctioned by Leibniz). Mersenne had done battle against Kabbalah and occultism, but the vertiginous gyrations of the Kabbalah had evidently seduced him, and here he is spinning the Llullian wheels for all he’s worth, no longer capable of distinguishing between divine omnipotence and the possible omnipotence of a perfect combinatorial language manipulated by man, to the point that in his Quaestiones super Genesim (cols. 49 and 52) he sees in the presence in man of the infinite a manifest proof of the existence of God.
But this ability to imagine the infinite possibilities of the combinatory technique manifests itself because Mersenne, like Clavius, Guldin, and others (the theme returns, for example, in Comenius, Linguarum methodus novissima III, 19),23 is no longer calculating with concepts (as Llull did) but with alphabetical sequences, mere elements of expression, uncontrolled by any orthodoxy that is not that of the numbers. Without realizing it, these authors are already approaching that notion of “blind thought” that will be brought to fruition, with greater critical awareness, by Leibniz, the inaugurator of modern formal logic.
In his Dissertatio de arte combinatoria, the same Leibniz, after complaining (correctly) that Llull’s whole method was concerned more with the art of improvising a discussion than with acquiring complete knowledge of a given subject, entertained himself by calculating how many possible combinations Llull’s Ars really consented, if all of the mathematical possibilities permitted by nine elements were exploited; and he came up with the number (theoretical of course) 17,804,320,388,674,561.
But, to exploit these possibilities, one had to do the opposite from what Llull had done and to take seriously the combinatory incontinence of people like Guldin and Mersenne. If Llull had invented an extremely flexible syntax and then handicapped it with a very rigid semantics, what was needed was a syntax that was not hampered by any semantic limitations. The combinatory process ought to generate empty symbolic forms, not yet bound to any content. The Ars thus became a calculus with meaningless symbols.
This is a state of affairs that shows how much progress Llullism has made, providing tools for our contemporary theoreticians of artificial and computerized languages, while betraying the pious intentions of Ramon Llull. And that to reread Llull today as if he had had an inkling of computer science (apart from the obvious anachronism) would be to betray his intentions.
All Llull had in mind was speaking of God and convincing the infidel to accept the principles of the Christian faith, hypnotizing them with his whirling wheels. So the legend that claims he died a martyr’s death in Muslim territory, though it may not be true, is nonetheless a good story.
A fusion of the following articles: “La lingua universale di Ramón Llull” (Eco 1991); “Pourquoi Llulle n’était pas un kabbaliste” (Eco 1992c); and “I rapporti tra Revolutio Alphabetaria e Lullismo” (Eco 1997a). These same themes are taken up in Eco (1993) [English trans. (1995)].
1. Burgonovus, Cabalisticarum selectiora, obscvrioraque dogmata (Venice: Apud Franciscum Franciscium Senensem, 1569); Paulus Scalichius, Encyclopedia seu orbis disciplinarum tam sacrarum quam prophanarum Epistemon (Basel, 1559); Jean Belot, Les Oeuvres de M. Jean Belot cure de Milmonts, professeur aux Sciences divines et célestes. Contenant la chiromence, physionomie, l’Art de Mémoire de Raymond Llulle, traité des divinations, augures et songes, les sciences stéganographiques etc. (Rouen: Jean Berthelin, 1669).
2. Disquisitionum magicarum libri sex (Mainz: König, 1593).
3. Arithmologia, Sive De abditis Numerorum Mysterijs (Rome: Ex Typographia Varesij, 1665).
4. The number of possible permutations is given by the factorial of n (n!) which is calculated: 1*2*3* … *n. For example, three elements ABC can be combined in six triplets (ABC, ACB, BAC, BCA, CAB, CBA), distinguished only by the order of their elements.
5. The formula is n! / (n - t)!. For example, given four elements ABCD, they can be arranged into twelve possible duplets.
6. The formula is n! / t! (n - t)!. Given the four elements ABCD, they can be combined into six possible duplets.
7. The woodcuts that follow are taken from Bernardus de Lavinheta, Practica compendiosa artis Raymundi Llulli (Lyon, 1523).
8. It will be seen that, by “middle term,” Llull means something different from what was understood by Scholastic syllogists. However that may be, excluded from this first table are self-predicatory combinations like BB or CC, because for Llull the premise “Goodness is good” does not permit us to come up with a middle term (cf. Johnston 1987: 234).
9. Our references to Llull’s texts are to the Zetzner edition (Strasburg, 1598), since it is on the basis of this edition that the Llullian tradition is transmitted to later centuries. Therefore, by Ars magna we mean the Ars generalis ultima, which in the 1598 edition is entitled Ars magna et ultima.
10. Athanasius Kircher, Ars magna sciendi (Amsterdam: Jannson, 1669).
11. Sefer Yetzira, University Press of America, 2010.
12. That the emanative or participative process goes from the root to the leaves is simply a question of iconographic convention. Note how Kircher, in his Ars magna sciendi, constructs his tree of the sciences, on a model related to the Porphyrian tree, with the Dignities at the top. As for Llull, in works like the Liber de ascensu et descensu intellectus (1304), the hierarchy of beings is represented as a ladder on which the artist proceeds from the effects to the causes, from the sensitive to the intellectual, and vice versa.
13. “We are … a thousand leagues away from modern formal logic. What we have here is a logic that is material in the highest degree, and therefore a kind of Topics or art of invention” (Platzeck 1953: 579). And again: “truth or logical correctness is never formally appreciated for its own sake, but always with reference to gnoseological truth” (Platzeck 1954: 151).
14. See Johnston (1987), chapter 15, entitled “Natural Middle,” in which these points are persuasively and searchingly discussed. “[The Ars] does not require systematic coherence of a deductive nature among its arguments; it is endlessly capable of offering yet another analogical explanation of the same idea or concept, or of restating the same truth in different terms. This explains both the volume and exhaustively repetitive character of nearly all of Llull’s 240 extant writings” (Johnston 1987: 7).
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15. On the other hand, Agrippa’s point of departure is the principle that “although all the demons or intelligences speak the language of the nation over which they preside, they make exclusive use of Hebrew when they interact with those who understand this mother tongue.… These names … though of unknown sound and meaning, must have, in the work of magic … greater power than significant names, when the spirit, dumbfounded by their enigma … fully convinced that it is acting under some divine influence, pronounces them in a reverent manner, even though it does not understand them, to the greater glory of the divinity” (De occulta philosophia libri III [Paris: Ex Officina Jacobi Dupuys, 1567], III:23–26). John Dee evokes angels of dubious celestiality with invocations such as Zizop, Zchis, Esiasch, Od, Iaod (cf. A True and Faithful Relation (London, printed by D. Maxwell for T. Garthwait, 1659).
16. Hillgarth (1971: 283) states that Pico, more interested in Kabbalism than in the Ars of Llull, cited Llull because he was better known than the Hebrew Kabbalah. For a subtle difference of opinion on this point, see Zambelli (1995[1965]: 59, n. 14).
17. La piazza universale di tutte le professioni del mondo, Nuovamente Ristampata & posta in luce, da Thomaso Garzoni di Bagnacavallo. Aggiuntovi in questa nuova Impressione alcune bellissime Annotazioni a discorso per discorso (Venice: Appresso Roberto Maietti, 1599).
18. Artis kabbalisticae, sive sapientiae divinae academia: in novem classes amicissima cum breuitate tum claritate digesta (Paris: Apus Melchiorem Nondiere, 1621).
19. Traité des chiffres, Ou Secretes Manieres d’Escrire (Paris: Chez Abel L’Angelier, 1587).
20. Cryptomenytices et cryptographiae libri ix (Lüneburg: Excriptum typis Johannis Henrici Fratrum, 1624).
