ars combinatoria | Samizdat

July 10, 2016

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 24³²words 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 24^{365,000,000,001}– 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.

May 2, 2016

Eco: The Arbor Scientarium, 2

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.

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May 1, 2016

Eco: The Alphabet and the Four Figures

Illuminati_sacre_pagine_p.fessoris_amplissimi_magistri_Raymundi_Lull._Ars_magna,_generalis_et_vltima_-_quarucunq3_artium_(et)_scientiarum_ipsius_Lull._assecutrix_et_clauigera_-_(et)_ad_eas_aditum_(14591005828)

Raymond Llull (1232-1316), Ars magna, segunda figura, generalis et ultima, 1517, held in the Getty Research Institute and digitized by that institution in collaboration with the Internet Archive, generously posted on archive.org. This work is in the public domain in its country of origin and other countries and areas where the copyright term is the author’s life plus 100 years or less.

“The ars combinatoria of Lull employs an alphabet of nine letters–B to K, leaving out J–and four figures (see figure 4.1). In a tabula generalis that appears in several of his works, Lull set out a table of six groups of nine entities, one for each of the nine letters.

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

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

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

Prima Figura.

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

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

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

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

Segunda figura.

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

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

Tertia Figura.

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

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

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

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

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

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

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

Quarta figura, fourth figure.

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

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

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

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

May 1, 2016

Eco: The Elements of the Ars Combinatoria

Raymond Llull (1232-1316), Figure 1 from Ars magna, 1300. This work is in the public domain in its country of origin and other countries and areas where the copyright term is the author’s life plus 100 years or less.

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

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

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

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

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

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

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

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

AB AC AD BC BD CD

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

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

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

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

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

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

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

Tag: ars combinatoria

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