Boolean Algebra and the Yi Jing

1. Introduction

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This paper is concerned with the logical and structural properties of the gua, interpreted as the symbolic representation of situations. I begin with some background discussion that will hopefully give the reader some indication of my own particular approach to this great work.

The Nature of this Essay

I am aware that my preoccupation with the structural aspects leaves one half of the Yi untouched. The Yi Jing is the probably the richest, most articulated combination of the intuitive and the rational aspects of the human heart/mind ever to have been formulated. However, this essay focuses almost completely on the rational, structural aspects of the work.

This is a deliberate choice on my part and stems from two reasons.

Firstly, I believe that this aspect of the work has received little serious attention in the west, especially in the context of contemporary information theory. Most of the written material available is either historical, tracing the conceptual and textual development of the book; or presents translations and interpretations of the classical texts associated with the gua.

Secondly, pragmatically, this is the aspect of the Yi that I am best suited to address.

The Yi Jing in the Information Age

There would seem to be a significant degree of interest in the Yi Jing from computer literate people. The number of web sites devoted to the changes is substantial (see the web site listed as [ICh98] in the References Section for a comprehensive selection of cross-references) and there are a wide range of Yi Jing consultation programs available, as freeware, shareware and commercial software. There is also a lively e-mail discussion list called Hexagram-8 devoted to the topic with an extensive archive of past discussions [Hex8] .

We could ask why this is the case. Kirk McElhearn has referred to the Yi Jing as the first example of hypertext. There is a lot of truth to this analogy: if each hexagram is a node, then the changing lines are links between those nodes. Thus, we might almost expect its natural home to be the world wide web. An excellent example of a web-based Yi Jing is Chuck Polisher's I Ching Lexicon which uses the facilities of HTML to link a verbatim translation of the book's text to the definitions of the individual characters [Pol98] . However, if we look closely, the Yi's embodiment of hypertext linking is richer than that currently encodable using standard HTML, and no sites that I am aware of yet attempt to capture this aspect of the changes in their on-line representations.

In addition to its hypertext nature, the Yi is a system of symbols based on a binary representation. Leibniz was the first to realize this connection in the west when the Jesuit missionary Bouvet (himself a mathematician) sent him a copy of Shao Yong's "Prior Heaven" arrangement of the hexagrams in 1702 (see [ deF97 , pp156-158]). There are clear resonances between the symbols of the Yi and Leibniz's binary calculus, and thus with today's digital technology. It is this obvious connection that lead me to explore the application of Boolean algebra and computation tools to the structures of the gua.

Situations

We are interested in situations; in their internal dynamics and in the dynamics of change from one situation to another. Or rather, given that the variety of concrete situations is endless, we are interested in types of situations; the significant features that remain, in some way, invariant across that variety. We are, therefore, interested in the abstract features of our manifest reality.[1]

There has been a growing trend in modern formal logic to look beyond the simplistic mechanisms of the last century. Barwise and Perry make the following comment [B&P83, p7]:

Reality consists of situations - individuals having properties and standing in relations at various spatiotemporal locations. We are always in situations; we can see them, cause them to come about, and have attitudes towards them.

They go on to develop a formal theory of situations that encompasses these entities and facilitates reasoning about them and the connections between them.

I contend that the Yi Jing provides us with a system of notation for situations. It provides us with a finite set of primitive elements (the bigrams, trigrams and hexagrams) whose properties, through study and reflection, we may hope to grasp. These elements can be combined, and the resulting interaction of their properties provide the dynamic description we seek. This essay provides an algebraic basis for investigating the Yi and, in this sense, is more in the tradition of the Image and Number school of interpretation than the Meaning and Principle school. The latter school saw the actual texts of the Yi Jing as the primary path to understanding, whilst for Shao Yong, a prominent exponent of the former school, "mathematics were the purest form of representing the process by which the phenomenal universe emerged from a unitary state of Being." [deF97 p129].

The Notational Engineering Laboratory [NEL98] say that "notational systems do not merely represent abstractions, they discover and then tokenize them." That is, the language of abstraction that we use to describe our reality can give us new insights into that reality. Mathematics is the primary example of this in the west. It has proven itself as the cornerstone of the physical sciences. It is my belief that the Yi is about exactly this. The opening chapters of the Dazhuan speak to this theme directly: "the holy sages instituted the hexagrams, so that phenomena might be perceived therein." (Ta Chuan, Chapter II, verse 1; [Wil83, p287])

In support of this approach to the Yi, de Fancourt says "the cosmology described in the Great Treatise is based on the trigrams and hexagrams, not the text of the Changes. The ancient sages created these `images' so that the mysterious processes of the universe could be fathomed; the text is merely appended to elucidate their meaning." [deF97, pp65-66]

On a similar theme, McElhearn's excellent paper "The Key to the Yi Jing" talks about the role of schemata in understanding the Yi, although he is mostly concerned with developing a deep understanding of the existing, original texts by reconstructing the linguistic and cultural schemata of those who created them. He says "each situation in our lives corresponds to one or more schemata, and each of the hexagrams corresponds to schemata also." [McE98 , p9] The notation-based, algebraic system presented here is really only the logical consequence of that view point. Hexagrams are schemata, and those schemata should be amenable to direct comprehension. It is in the tradition of a direct structural representation of the situations. Further, it fits into an emerging contemporary trend of applying modern analytical techniques to this ancient system (see especially the work of Higgins [Hig98] , and also Goldenberg [Gol75] ).

Part of the purpose of this paper is to attempt to provide the beginnings of a notational understanding of the structures of the hexagrams that will facilitate the dynamic construction of schemata. The long term goal of the project is that there should be some kind of algebra that provides a descriptive mechanism for understanding the structure of the hexagram directly. That understanding, coupled with the imagination, can then provide the information to generate a relevant schema and apply it to the situation at hand.

Names and Notation

Before starting in on the material, I must say a word about the names and notation I have used for the gua.

Firstly, the names of the gua. Where I am quoting from a particular source, then I will use the name that the author being quoted has used in that instance. However, when I am writing from my own perspective I will, for the convenience of the reader, use the translations by Wilhelm [Wil83]. The major, consistent exception to the use of the Wilhelm names is in the section devoted to correctness and correspondence. Because that section makes extensive reference to Cleary's work, I use his translations of the names.

Secondly then, the notation for the gua. Because of the need for this essay to exist through time in a number of different formats, I have had to adopt a notation that enables me to write the gua without any recourse to particular graphical resources. This representation is linear and textual. Yin is written as 0 and yang as 1. Further, whilst the gua are read from the bottom upwards, this notation is read from left to right to fit with the flow of the text. Thus, the trigram Arousing will be written 100.

Further, I shall need to introduce a notation for talking about the gua as an ordered sequence of lines. Informally, in the text, I shall write a hexagram, for example, The Well, as an undifferentiated linear figure 011010. However, the reader should understand this as shorthand for the more explicit form [0,1,1,0,1,0] where each line is properly distinguished from the others in an ordered linear structure.

In addition, when I wish to indicate the extra information concerning changing lines, I shall use the usual numerical designations. Thus, 698 would represent an initial trigram of 010, the Abyss, changing into 100, the Arousing.

All material on this page copyright Dr Andreas Schöter
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