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Tag: 1971

Eco: The I Ching and the Binary Calculus


The French Jesuit Joachim Bouvet (1656-1730) sent this unattributed diagram of hexagrams to Gottfried Wilhelm Leibniz (1646-1730) circa 1701. The arabic numerals written on the diagram were added by Leibniz. This artifact is held in the Leibniz Archive, Niedersächsische Landesbibliothek, and was published in Franklin Perkins, Leibniz and China: A Commerce of Light, Cambridge,2004, p. 117. 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. 

Leibniz’s tendency to transform his characteristica into a truly blind calculus, anticipating the logic of Boole, is no less shown by his reaction to the discovery of the Chinese book of changes–the I Ching.

Leibniz’s continuing interest in the language and culture of China is amply documented, especially during the final decades of his life. In 1697 he had published Novissima sinica (Dutens 1768: IV, 1), which was a collection of letters and studies by the Jesuit missionaries in China.

It was a work seen by a certain Father Joachim Bouvet, a missionary just returned from China, who responded by sending Leibniz a treatise on the ancient Chinese philosophy which he saw as represented by the 64 hexagrams of the I Ching.

The Book of Changes had for centuries been regarded as a work of millennial antiquity. More recent studies, however, have dated it to the third century BCE. Nevertheless, scholars of the time of Leibniz still attributed the work to a mythical author named Fu Hsi.

As its function was clearly magical and oracular, Bouvet not unnaturally read the hexagrams as laying down the fundamental principles for Chinese traditional culture.

When Leibniz described to Bouvet his own research in binary arithmetic, that is, his calculus by 1 and 0 (of which he also praised the metaphysical ability to represent even the relation between God and nothingness), Bouvet perceived that this arithmetic might admirably explain the structure of the Chinese hexagrams as well.

He sent Leibniz in 1701 (though Leibniz only received the communication in 1703) a letter to which he added a wood-cut showing the disposition of the hexagrams.

In fact, the disposition of the hexagrams in the wood-cut differs from that of the I Ching, nevertheless, this error allowed Leibniz to perceive a signifying sequence which he later illustrated in his Explication de l’arithmétique binaire (1703).

Umberto Eco, The Search for the Perfect Language, figure 14.1, p. 285

Umberto Eco, The Search for the Perfect Language, figure 14.1, p. 285. 

Figure 14.1 shows the central structure of the diagrams seen by Leibniz. The sequence commences, in the upper left hand corner, with six broken lines, then proceeds by gradually substituting unbroken for broken lines.

Leibniz read this sequence as a perfect representation of the progression of binary numbers (000, 001, 010, 110, 101, 011, 111 . . . ). See figure 14.2.

Umberto Eco, The Search for the Perfect Language, figure 14.2, p. 286

Umberto Eco, The Search for the Perfect Language, figure 14.2, p. 286. 

Once again, the inclination of Leibniz was to void the Chinese symbols of whatever meaning was assigned to them by previous interpretations, in order to consider their form and their combinatorial possibilities.

Thus once more we find Leibniz on the track of a system of blind thought in which it was syntactic form alone that yielded truths. Those binary digits 1 and 0 are totally blind symbols which (through a syntactical manipulation) permit discoveries even before the strings into which they are formed are assigned meanings.

In this way, Leibniz’s thought not only anticipates by a century and a half Boole’s mathematical logic, but also anticipates the true and native tongue spoken by a computer–not, that is, the language we speak to it when, working within its various programs, we type expressions out on the keyboard and read responses on the screen, but the machine language programmed into it.

This is the language in which the computer can truly “think” without “knowing” what its own thoughts mean, receiving instructions and re-elaborating them in purely binary terms.

Certainly Leibniz mistook the nature of the I Ching, since “the Chinese interpreted the kua in every manner except mathematically” (Lozano 1971). Nevertheless, the formal structures that he (rightly enough) isolated in these diagrams appeared to him so esoterically marvelous that, in a letter to Father Bouvet, he did not hesitate in identifying the true author of the I Ching as Hermes Trismegistus–and not without reasons, because Fu Hsi was considered in China as the representative of the era of hunting, fishing and cooking, and thus can be considered, as can Hermes, the father of all inventions.”

Umberto Eco, The Search for the Perfect Language, translated by James Fentress, Blackwell. Oxford, 1995, pp. 284-7.

Eco: Primitives and Organization of Content


William Blake (1757-1827), The Tyger, 1794. Scan of a plate printed by the author collected in Songs of Experience, designed after 1789 and printed in 1794. Copies A and B are both held by the British Museum. This work is in the public domain in its country or origin and other countries and areas where the copyright term is the author’s life plus 100 years or less.

“In order to design characters that directly denote notions (if not the things themselves that these notions reflect), two conditions must be fulfilled: (1) the identification of primitive notions; (2) the organization of these primitives into a system which represents the model of the organization of content.

It is for this reason that these languages qualify as philosophical and a priori. Their formulation required individuating and organizing a sort of philosophical “grammar of ideas” that was independent from any natural language, and would therefore need to be postulated a priori.

Only when the content-plane had been organized would it be possible to design the characters that would express the semantic primitives. As Dalgarno was later to put it, the work of the philosopher had to precede that of the linguist.

For the polygraphers, invention was simply the job of assigning numbers to a collection of words from a given natural language. The inventors of philosophic a priori languages needed to invent characters that referred to things or notions: this meant that their first step was to draw up a list of notions and things.

This was not an easy task. Since the lexicon of any natural language is always finite in number, while the number of things, including physically existing objects, rational entities, accidents of all types, is potentially infinite, in order to outline a list of real characters it is necessary to design an inventory which is not only universal: it must also be in some way limited.

It is mandatory to establish which notions are the most universally common, and then to go on by analyzing the derivative notions according to a principle of compositionality by primitive features.

In this way, the entire set of possible contents that the language is able to express has to be articulated as a set of “molecular aggregates” that can be reduced to atomic features.

Suppose we had three semantic atoms such as ANIMAL, CANINE and FELINE. Using them, we might analyze the following four expressions:

Umberto Eco The Search for the Perfect Language p. 222.png

Umberto Eco, The Search for the Perfect Language, p. 222. 

Yet the features that analyze the content of the above expressions ought to be entities totally extraneous to the object language.

The semantic feature CANINE, for example, must not be identifiable with the word canine. The semantic features ought to be extra-linguistic and possibly innate entities. At least they should be postulated as such, as when one provides a computer with a dictionary in which every term of a given language can be split into minor features posited by the program.

In any case, the initial problem is how to identify these primitive and atomic features and set a limit on their number.

If one means by “primitive” a simple concept, it is very difficult to decide whether and when one concept is simpler than another. For the normal speaker, the concept of “man” is simpler–that is, easier to understand–than the one of “mammal.”

By contrast, according to every sort of semantic analysis, “mammal” is a component of (therefore simpler than) “man.” It has been remarked that for a common dictionary it is easier to define terms like infarct than terms like to do (Rey-Debone 1971: 194ff).

We might decide that the primitives depend on our world experience; they would correspond to those that Russell (1940) called “object-words,” whose meanings we learn by ostension, in the same way as a child learns the meaning of the word red by finding it associated with different occurrences of the same chromatic experience.

By contrast, according to Russell, there are “dictionary-words” that can be defined through other words, such as pentagram. Yet Russell remarks, for a child who had grown up in a room decorated with motifs in the form of a pentagram, this word would be an object one.

Another alternative would be to regard primitives as innate Platonic ideas. This solution would be philosophically impeccable; yet not even Plato himself was able to establish what and how many these innate ideas were.

Either there is an idea for every natural kind (for horses, platypuses, fleas, elms and so on–which means an atomic feature for every element of the furnishing of the world), or there are a few abstract ideas (the One, the Many, the Good and mathematical concepts), but through them it would be difficult to define compositionally a horse or a platypus.

Suppose instead we decided to order the system of primitives by dichotomic disjunctions so that, by virtue of the systematic relations obtaining between the terms, they must remain finite in number.

With such a structure we would be able to define by a finite number of atomic primitives a great number of molecular entities. A good example of this alternative is the reciprocally embedded system of hyponyms and hyperonyms used by lexicographers.

It is organized hierarchically in the form of a tree of binary disjunctions: to each opposed pair of hyponyms there corresponds a single hyperonym, which, in its turn, is opposed to another hyperonym to form the next level of hyponyms, to which a further hyperonym will correspond, and so on.

In the end, regardless of how many terms are embedded in the system, the whole structure must finish at its apex in a single patriarch-hyperonym.

Thus the example of the table on p. 222 above would take the following format:

Umberto Eco, The Search for the Perfect Language, Figure 10.1, p. 224

Umberto Eco, The Search for the Perfect Language, Figure 10.1, p. 224.

According to many contemporary authors, this kind of semantic structure would analyze the content in the format of a dictionary (as opposed to an encyclopedia).

In an encyclopedia-like representation one introduces elements of world knowledge (for example that a tiger is a yellow cat with stripes on its fur), and these elements are potentially infinite in number.

In a dictionary-like representation the features are, on the contrary, analytic, in the sense that they are the only and necessary conditions for the definition of a given content: a cat is necessarily a feline and an animal and it would be contradictory to assert that a cat is not an animal, since the feature “animal” is analytically a part of the definition of cat.

In this sense it would be easy to distinguish analytical from synthetical judgments. “A tiger is a feline animal” would be analytical, so uniquely depending on our rigorously organized dictionary competence (which is exclusively linguistic), while “tigers are man-eaters” would depend on our extra-linguistical world knowledge.”

Umberto Eco, The Search for the Perfect Language, translated by James Fentress, Blackwell. Oxford, 1995, pp. 221-5.

Nakamura: Magic Produces Wonder

The Sensuous Metaphysics of Magic: Mutual Constitution and Correspondence

“The representation of a wish is, eo ipso, the representation of its fulfillment. Magic, however, brings a wish to life; it manifests a wish.”

Ludwig Wittgenstein, Remarks on Frazer’s Golden Bough (Miles and Rhees 1971)

“Implicit in Wittgenstein’s aphorism that magic “manifests a wish” is the notion that magic requires concrete demonstration: the fulfillment of the wish made real.

At first glance, magic as both the manifestation of a wish and its fulfillment seems to pose a contradiction in this act of making real. But magic is an exchange that seeks synthesis, and such exchange, “as in any other form of communication, surmounts the contradiction inherent in it” (Levi-Strauss 1987:58).

Mikhail Bakhtin (1984) surmised, “to be means to communicate” (287). And the movement of such exchange presumes a sensuous intimacy between the outside world and ourselves: “to be means to be for another, and through the other, for oneself. A person has no internal sovereign territory, he is wholly and always on the boundary; looking inside himself, he looks into the eyes of another or with the eyes of another” (Bakhtin 1984:287).

This is the human orientation of being amidst the constant flux of the world that provokes our fear as much as desire, and discloses the condition for a way of knowing directly and sensuously.

Giambattista Vico (1999[1744] ), a forward-thinking but marginalized philosopher of his time, implicated bodily sense in a critique of the Cartesian principle of Cogito; in response to the reductive logic of geometric certainty, he formulated the axiom: man can only know what he himself has made — “verum et factum convertuntur” — and to make is to transform oneself by becoming other (Vico 1999[1744]:160).

The implication of this premise posits that human knowledge cannot be exhausted by rationality; it is also sensory and imaginative. Although Vico’s project poses three progressive historical eras of man: the first ruled by the senses, the second by imagination, and the third by reflective reason, we now recognize that all three modalities of knowledge exist throughout human history albeit at different scales and intensities.

From this perspective, magic, which embraces bodily imitation and play, is better viewed as a poetic reinterpretation of the concrete reality of human action rather than the discovery of an objective reality that presumes to regulate it (Böhm 1995:117).

Indeed it is our sensory faculties and not our rational faculties that better apprehend certain complexities of the magical realm: we know when we feel.

In encounters with magic, we apprehend the apparent trickery of bodies, substances, and things. Our reaction to such events often betrays delight, horror, fear, disgust, attraction, and fascination simultaneously, and such disorientation is desired.

Magic produces wonder, and in doing so returns us to a state of apprehending the world that short-circuits those automatic processes of intellection that discipline the senses. And wonder is central to a mode of understanding that is “capable of grasping what, in ourselves and in others precedes and exceeds reason” (Pettigrew 1999:66).

Bodily sense is key here, since it can know something more than words express. The “trick” of magic, then, lies in attaining the unknown by disorganizing all the senses; in effect, it acts to deregulate relationships that are rigorously regulated by normative cultural forms.

The aesthetic experience of magic seeks the recovery of correspondences between people, things, and places in their pre-differentiated unity, a unity that becomes obscured through “habitual modes of perception” (Harrison 1993:180).

In this way, magic aims at the perceptual movements that continually render meaning rather than at meaning itself. In this intercalary register of experience, magic presumes a certain direct engagement with the world; specifically, it recalls a pre-differentiated world as an open possibility of interrelations constantly in flux.”

Carolyn Nakamura, “Mastering matters: magical sense and apotropaic figurine worlds of Neo-Assyria,” Archaeologies of materiality (2005): 24-6.

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