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
