Eco: The I Ching and the Binary Calculus

by Esteban

Diagram_of_I_Ching_hexagrams_owned_by_Gottfried_Wilhelm_Leibniz,_1701

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.