Lyons: Jacob Marechal for Simon Vincent, 5 May 1517.
Third edition, the first edited by the Lullist Bernard de Lavinheta, of the Ars Magna, Lull’s greatest work. Lull invented an ‘art of finding truth’ which inspired Leibniz’s dream of a universal algebra four centuries later … The most distinctive characteristic of his Art is clearly its combinatory nature, which led to both the use of complex semi-mechanical techniques that sometimes required figures with separately revolving concentric wheels – ‘volvelles’ – and to the symbolic notation of its alphabet. These features justify its classification among the forerunners of both modern symbolic logic and computer science.
Third edition, the first edited by the Lullist Bernard de Lavinheta (d. ca. 1530), of the Ars Magna, his greatest work, now recognised as perhaps the first work in computer science. “Lull invented an ‘art of finding truth’ which inspired Leibniz's dream of a universal algebra four centuries later … The most distinctive characteristic of [his] Art is clearly its combinatory nature, which led to both the use of complex semi-mechanical techniques that sometimes required figures with separately revolving concentric wheels – ‘volvelles’, in bibliographical parlance – and to the symbolic notation of its alphabet. These features justify its classification among the forerunners of both modern symbolic logic and computer science, with its systematically exhaustive consideration of all possible combinations of the material under examination, reduced to a symbolic coding … The Art’s function as a means of unifying all knowledge into a single system remained viable throughout the Renaissance and well into the seventeenth century. As a system of logical inquiry, its method of proceeding from basic sets of pre-established concepts by the systematic exploration of their combinations – in connection with any question on any conceivable subject – can be succinctly stated in terms taken from the Dissertatio de arte combinatoria (1666) of Leibniz, which was inspired by the Lullian Art: ‘A proposition is made up of subject and predicate; hence all propositions are combinations. Hence the logic of inventing [discovering] propositions involves solving this problem: 1. Given a subject, [finding] the predicates; 2. Given a predicate, finding the subjects [to which it may] apply, whether by way of affirmation or negation’” (DSB). “From today’s point of view Llull made contributions to the following areas: 1. The idea of a formal language, 2. The idea of a logical rule, 3. The computation of combinations, 4. The use of binary and ternary relations, 5. The use of symbols for variables, 6. The idea of substitution for a variable, 7. The use of a machine for logic. He also pioneered the integration of ontology with logic. Leibniz (1646–1716) mentions Llull by name several times and explicitly uses his ideas in [Ars combinatoria]. Leibniz learnt of Llull’s ideas through Kircher [Ars magna sciendi, 1669]. This work led directly into the development of computing machines. So Llull contributed ideas that are fundamental to the modern disciplines of computer science and computer engineering” (Crossley, p. 40). All early editions of this work are of the greatest rarity. ABPC/RBH list, in the last 35 years, no complete copy of the first edition (Venice, 1480), only the Honeyman copy of the second (Barcelona, 1501) (Sotheby's, May 13, 1980, lot 2063, £1,900 = $4,378, “modern vellum, marginally repaired, wormed”), and none of the third.
Running throughout Lull’s immense oeuvre “is a leitmotif that enables one to arrive at an overall, if not unitary, view, that leitmotif being the Ars lulliana or Lullian Art: a philosophico-theological system that makes use of common basic concepts from the three monotheistic religions of its day, subjecting them to discussion with a view to convincing Muslims (and Jews) via rational argument of the truth of the Christian mysteries of faith. By revising his Art and extending it to all fields of human knowledge, Ramon Llull succeeded in creating a universal science, based on the algebraic notation of its basic concepts and their combination by means of mechanical figures …
“From a more abstract point of view, Llull’s combinatorial Art can be described as a process of elementary analysis and of reconstruction. On the one hand, it resolves the historical religions into their most primitive elements; on the other, it represents these elements by letters (from B to K), in order to recombine these letters and the elements of the different religions that they designate until, through these combinations, a vision of the world is reached that is as consistent as possible: this will correspond to truth. Undoubtedly, this process which Llull applied to all kinds of question – not just religious controversies – is a key ingredient of modern thought. One only has to think of Gottfried Wilhelm Leibniz’s characteristica universalis: thus, in his Dissertatio de arte combinatoria, in 1666, the young Leibniz, clearly inspired by Llull, had already outlined the project of a reconstruction of the whole of reality based on a definite number of basic notions. Leibniz criticizes the basic notions of the Lullian “alphabet” as too limited and proposes another alternative and broader alphabet. In contradistinction to Llull, Leibniz does not represent these basic notions with letters but rather uses numbers. Thus, the basic notion of “space” is represented by the number 2, the basic notion of “between” by the number 3, and the basic notion of “the whole” by the number 10. Consequently, according to Leibniz, a complex concept such as, for instance, “interval” can be formulated as 2.3.10, that is, “space between the whole”. Leibniz was convinced that in this way all questions could be reduced to mathematical problems and that, in order to solve any problem, we only have to set about calculating. This is the meaning of Leibniz’s famous “Calculemus!”
“It is through Leibniz that Llull’s influence also became decisive for more recent developments such as formal logic, as developed by Gottlob Frege in the late 19th
century. According to Frege, Leibniz’s characteristica, in its later evolution, limited itself to different fields, such as arithmetic, geometry, chemistry and so on, but did not become universal as Leibniz, in fact, had wished. This is why Frege, in his famous
Begriffsschrift from 1879, intended to create an elementary language that would unify the different formal languages which, after Leibniz, had been established in the different natural sciences. This language developed into the formal logic that until now has dominated the philosophical discourse and which was an important step in the journey towards the creation of computing languages. What characterizes this kind of logic is its formal notation, using variables and symbols to represent the different logical propositions and operations. Based on this notation, Frege developed the so-called logical calculus. Although the language reached by this formal logic differs from that of the Art, Llull can be considered as the forerunner of this project, insofar as in his thought one can already find the idea of an elementary language that follows logical rules and uses variables while operating with the principle of substitution of these variables” (Fidora & Sierra, pp. 1-2).
Lull’s work begins “with an ‘Alphabet’ giving the meaning of nine letters, in which he says, ‘B signifies goodness, difference, whether?, God, justice, and avarice. C signifies …’, and so on, all of which can best be set out in a table. He then sets out the components of the first column in his First Figure, or Figure A (f. iv). Notice first of all, as always with Llull, the letters don’t represent variables, but constants. Here they’re connected by lines to show that in the Divinity these attributes are mutually convertible. That is to say that God’s goodness is great, God’s greatness is good, etc. This, in turn was one of Llull’s definitions of God, because in the created world, as we all know too well, people’s goodness is not always great, nor their greatness particularly good, etc. Now such a system of vertices connected by lines is what, as mathematicians, you will of course recognize as a graph. This might seem to be of purely anecdotal interest, but as we shall see in a moment, the relational nature of Llull’s system is fundamental to his idea of an Ars combinatoria.
The components of the second column are set out in a Second Figure, or Figure T (f. iiv). Here we have a series of relational principles related among themselves in three groups of three, hence the triangular graphs. The first triangle has difference, concordance, and contrariety; the second beginning, middle, and end; and the third majority, equality, and minority. The concentric circles between the triangles and the outer letters show the areas in which these relations can be applied …
“The Third Figure (f. iiiir) combines the first two: Here Llull explains that B C, for instance, implies four concepts: goodness and greatness (from Figure A), and difference and concordance (from Figure T), permitting us to analyze a phrase such as ‘Goodness has great difference and concordance’ in terms of its applicability in the areas of sensual/sensual, sensual/intellectual, and intellectual/intellectual. It furthermore, as he points out, permits us to do this systematically throughout the entire alphabet. This is important, because one of the ways in which Llull conceived his Art as ‘general’ was precisely in its capacity to explore all the possible combinations of its components. Now as mathematicians, you will recognize this figure as a half matrix, and you will also see that, in relation to the graph of the First Figure, it is an adjacency matrix. Because such a matrix is symmetrical (in Llull’s case this means he makes no distinction between B-C and C-B), he saw no reason to reproduce the other half; and because his graph admits no loops (that is, omits relations such as B-B), he could also omit the principal diagonal.
“If the Third Figure explores all possible binary combinations, the Fourth Figure (f. iiiiv) does the same for ternary combinations. In medieval manuscripts, the outside circle is normally drawn on the page, and the two inner ones are separate pieces of parchment or paper held in place on top of it by a little piece of string, permitting them to rotate in relation to each other and to the larger circle” (Bonner, pp. 9-14). In the present printed version of the work, the parts of the volvelle illustrating the ternary combinations were printed on a separate sheet and intended to be cut out and mounted on the Fourth Figure. In this copy the volvelle has never been assembled; the sheet on which the parts of the volvelle are printed is bound at the end of the book.
“Binary relations are worked out more extensively in a section he calls ‘The Evacuation of the Third Figure’. For the ‘compartment’, as he calls it, of B C, he not only uses ‘goodness’ and ‘greatness’ from the First Figure, and ‘difference’ and ‘concordance’ from the Second Figure, but also the first two questions of the third column of the alphabet, those also corresponding to the letters B C, which are ‘whether?’ and ‘what?’. This means that for the combination of ‘goodness’ and ‘greatness’ one has three possibilities, a statement and two questions:
- Goodness is great.
- Whether goodness is great?
- What is great goodness?
and so on for ‘goodness’ and ‘difference’, ‘goodness’ and ‘concordance’, for a total of 12 propositions and 24 questions.
“Ternary relations are worked out in a Table based on the Fourth Figure … the full form of the Ars generalis ultima has 84 [columns]!” (Bonner, p. 14). These ternary relations are listed in a twelve-page table (ff. viiiv – xiiir), and then a short interpretation is given for each entry.
Llull's ideas would be developed further by Giovanni de la Fontana (1395-1455) and Nicholas of Cusa (1401-64) in the 15th century (in his work De conjecturis Nicholas developed his method ars generalis conjecturandi, in which he describes a way of making conjectures, illustrated by wheel charts and symbols that much resemble those of Llull), and Giordano Bruno (1548-1600) in the 16th century (Bruno used the rotating figures of the Lullist system as instruments of a system of artificial memory, and attempted to apply Lullian mnemotechnics to different modes of rhetorical discourse).
Ramon Llull (1232/3–1315/6) “was born on Majorca around 1232, only two or three years after the King of Aragon and Catalonia had recovered the island from the Muslims. This meant that Llull grew up in an island that was still strongly multicultural. Muslims continued to represent perhaps a third of the population, and Jews, although a much smaller minority, were an important economic and cultural force on the island. So when at the age of thirty he was converted from a profligate youth and he decided to devote his life to the service of the Church, it seemed only logical to do so by trying to convert these ‘infidels’, as they were then called. And he decided to do this in three ways: (1) to develop a system that his adversaries would find difficult to refute (which is what we’ll see in a moment), and to try to persuade them of the truth of Christianity instead of just trying to refute their own doctrines, as his predecessors had done; (2) to be willing to risk his life in proselytizing among Muslims and Jews (he in fact made three trips to North Africa); and (3) to try to persuade Kings and Popes of the need for setting up language schools for missionaries, for which purpose he travelled many times throughout France and Italy. He lived to 83 or 84, an incredible age when the average life-span was around 40, dying in 1316” (Bonner, p. 5).
The editor of this edition, the Franciscan Bernard de Lavinheta (d. c. 1530), was the greatest Lullist of the early 16th century. “Almost nothing is known of [his] background, nor even whether he was Spanish or French. We only know that before coming to Paris he taught at Salamanca. The brand of Lullism he brought there was that of the Lullist school of Barcelona and its interest in the Art. He was the first, as a trained theologian, to teach the Art at the University of Paris, thereby giving it the official sanction it had lacked for a century and a half. His publication of Lullian works at Lyon, Paris and Cologne in 1514-18 was very influential throughout Europe” (Bonner, Selected Works of Ramon Llull, vol. 1, p. 80).
Palau 143693; Rogent & Duran 65; Tomash L142. Alexander Fidora & Carles Sierra (eds.), Ramon Llull: From the Ars Magna to Artificial Intelligence, 2011 (http://www.iiia.csic.es/library). Anthony Bonner, ‘What was Llull up to?’, pp. 5-24 in Fidora & Sierra (ibid.). John N. Crossley, ‘Ramon Llull’s contribution to computer science,’ pp. 39-60 in ibid.
Small 4to, ff. , 124; gothic letter; title and last preliminary leaf printed in red and black, title with large woodcut printer's device and strip ornament on two sides, 5 large woodcut diagrams, one with volvelles printed on a separate sheet intended to be cut out and mounted, smaller printer's device at end. 18th century German boards.