The Binary Book of Changes

The following is the first chapter from my recently published book Leibniz, Einstein, and China.

The Chinese, often described as inscrutable, do not excel in explaining their culture to the world. Their culture has been shaped by a worldview that can be traced back to the Book of Changes, the ancient classic that suggests nature and the cosmos as a whole is an interplay of polar opposites — plus and minus, active and passive, and life and death, generically known as yin and yang. Over the centuries, the Chinese internalized the yin-yang principle to the extent that they find it difficult to articulate how it informs their worldview. It is like asking fish to describe water.

On the other hand, people in the West have not always excelled at listening. James Legge, the Scottish sinologist who first translated the Book of Changes into English, ridiculed claims by the Chinese that their ancient book anticipated Western science, including (electro) magnetism. In the introduction to his translation of the Book of Changes, Legge wrote:

“Chinese scholars and gentlemen… who have got some little acquaintance with western science, are fond of saying that all the truths of electricity, heat, light and other branches of European physics are in the Eight Trigrams (the constituent parts of the 64 hexagrams). When asked how, then, they and their countrymen have been, and are, ignorant of those truths, they say that they have to learn them first from western books, and, then, looking into the Yi (Book of Changes), they see that they were all known to Confucius more than 2000 years ago. The vain assumption thus manifested is childish; and, until the Chinese drop their hallucination about the Yi as containing all things that have ever been dreamt of in all philosophies, it will prove a stumbling block to them, and keep them from entering the true path of science.”

A patronizing attitude toward the Chinese was not unusual in Legge’s days. The Book of Changes didn’t explain the “truths of electricity and other branches of European physics.” Rather, it asserts that the universe operates on a two-fold principle. The Chinese expressed this notion in their view of Creation: “When the yin and the yang, initially united, separated forever, the mountains poured forth water.” Magnetism was a primary manifestation of the mutually dependent opposites. The Chinese invented the magnetic compass after they discovered magnetic stones, a “physical” manifestation of yin and yang.

The assumption that the universe is an interplay of two opposite forces emerged in China’s animistic age and led to the notion of Tao, the “source” of yin and yang. Tao as a concept is a sophisticated form of animism; everything is related but differentiated and animated by the interaction of polar opposites.

Legge probably meant to say that the generic types of yin and yang are too broad and too vague to have enabled the development of modern science. More important was that the Chinese did not develop the scientific concept of proof. Joseph Needham, in his monumental book series Science and Civilisation in China, argued that the Chinese did not take the empirical method far enough. On the other hand, the generic nature of the yin-yang principle was key in the development of Chinese civilization. Moreover, it accommodates virtually all aspects of reality, from human psychology to the nature of reality. This explains why the Book of Changes has had such enduring and widespread appeal.

Nearly all aspects of Chinese civilization — whether cosmology, natural philosophy, sociology, or art — can be traced to the Book of Changes. Confucius appropriated the Eight Trigrams to develop his hierarchical social structure by associating the eight family members with the eight trigrams. The Book of Changes informed the development of Chinese architecture, medicine, and warfare.

Fig. 1–1 The Eight Trigrams that form the basis of the 64 hexagrams, the foundation of the Book of Changes.

The Chinese typically wrapped their inventions in mythical tales. They attribute the Eight Trigrams to Fu-Xi, one of China’s five mythological Emperors. Fu-Xi is regarded as the founder of the Chinese Empire. His name is typically placed at the top of the chronological table of Chinese history. The Eight Trigrams represent eight natural phenomena — Heaven (pure yang), Earth (pure yin), and six “gradations” having various degrees of yin-yang “content” or “tendencies.” Their interplay created our terrestrial environment. As we shall see, trigrams and hexagrams resemble Boolean classes, part of discrete mathematics that is central to modern, binary computers.

The Eight Trigrams are composed of yin (broken) and yang (unbroken) lines in various combinations. The 64 hexagrams in the Book of Changes each consist of two of the eight trigrams in all possible combinations. This created 64 further “binary steps” of yin and yang. The attributes given to the Eight Trigrams all derived from natural processes or tendencies carried over to the 64 hexagrams. The trigram for wind (also associated with wood), placed on top of the trigram for mountain, produced the hexagram “Gradual Advance.”

When we think of the effect of wind on a mountain or the growth of wood, the attribute of gradual advance makes sense. It is an attribute or tendency of nature. The same is true of the trigram for lake, combined with the trigram for mountain. It produced the hexagram “Mutual Influence.” All other hexagrams convey similar natural qualities and tendencies, each the sum of its two constituent trigrams. Given the binary nature of the yin and yang, the changes between the hexagrams are discrete (binary) steps, rather than continuous (analog).

Fig. 1–2 Hexagrams from pure yang to pure yin in discrete, binary steps. The hexagrams in between are primarily yin or yang.

The Book of Changes is a manual, or guide, to the bipolar universe. Consulting the book can help us resolve conflicting thoughts. We may have doubts about the wisdom of starting a new venture, getting married, or moving to a new location. The hexagrams in the Book of Changes confront us with a constellation of archetypal opposites — favorable/unfavorable, gain/loss, advancing/retreating. Using the binary logic of yes and no, pro and con, we mentally resolve binary choices. The Book of Changes assumes we already know the answer in our subconscious, the reason why Carl Jung was fascinated by this psychological dimension of the book. Transpersonal psychologist Marysol Sterling Gonzalez referred to the Book of Changes as a psychological computer, and philosopher Will Buckingham called it the “uncertainty machine.”

The psychological dimension of the Book of Changes hinges on the distinction between nature and human nature. The human notion of Tao starts with the premise that nature is based on (yin-yang) opposites, both concrete and abstract, real and perceived. Tao, itself, does not know the difference between nature and human nature, physics and metaphysics, matter and spirit. That means that we have to resolve these opposites in our own mind. Doing so enables us to insert ourselves in the binary universe with the least amount of friction.

Leibniz and the 64 hexagrams

In 1679, Gottfried Leibniz published “De Progressione Dyadica,” the paper that first described his binary code of 0 and 1. Twenty-four years later, in 1703, Leibniz published a second paper entitled, “Explication de l’Arithmétique Binaire,” in which he detailed the mathematical functions of addition, subtraction, multiplication, and division for the binary code. In this paper, Leibniz made a remarkable claim. He called the binary a “rediscovery” of an ancient system of symbols invented thousands of years ago by the Chinese. Leibniz argued that although the 64 hexagrams used in the Book of Changes use different symbols — broken and unbroken lines, instead of 0 and 1 — the principle was the same.

Fig. 1–3 Leibniz’s diagram showing the correlation between the trigrams and the binary code.

What happened between Leibniz’s initial paper on the binary code, and his second paper published more than 20 years later, in which he made his claim? Leibniz corresponded with Father Joachim Bouvet, a Jesuit missionary working in Beijing. In 1697, Leibniz had sent Father Bouvet an outline of his binary arithmetic, believing it could help the Jesuit Father with his missionary work. European science was an important tool in the Jesuit attempts to convert China to Christianity. In his paper, “Explication de l’Arithmétique Binaire,” Leibniz recounted his exchange with Father Bouvet. He got some of the historical facts wrong, but he pointed in the right direction:

“What is astounding in this (binary) reckoning is that this arithmetic by 0 and 1 happens to contain the secret of the lines of an ancient king and philosopher named Fohy [Fu-Xi], who is believed to have lived more than 4000 years ago, and whom the Chinese regard as the founder of their empire and their sciences… The Chinese lost the significance of these Cova of Lineations [the hexagrams] of Fohy, perhaps more than 1000 years ago… It is hardly more than two years ago that I sent to R.P. Bouvet, the celebrated French Jesuit, who died in Peking, my method of counting by 0 and 1. He needed nothing further to make the observation that this was the key to the Figures of Fohy. When he thus wrote me on November 14, 1701, he sent me this princely philosopher’s Grand Figure [the 64 hexagrams], which is such that this Father had deciphered the Enigma of Fohy with the aid of that which I had communicated to him. Since these Figures are perhaps the most ancient monuments of science which exist on this earth, this restitution of their meaning, after so long an interval of time, would seem most curious.”

Leibniz’s invention of the binary code was part of his attempt to construct a mechanical calculating machine. He realized that the design and construction of such a machine would be simplified if the number of different parts required for its construction was reduced to a minimum. Leibniz realized the numbers 0 and 1 could express any quantity if they were used in longer strings. Arab numerals, the basis of the decimal system, require 10 different numerals, and the binary code only two.

Leibniz failed to build a mechanical calculator, and after his death, the binary code was largely forgotten and ignored by other mathematicians. But, in the 19th century, the English mathematician George Boole recognized the value of Leibniz’s binary logic. Boole invented a new form of mathematics that carries his name: Boolean algebra, or “an algebra of classes.”

Boolean algebra represented a revolution in mathematics and logic. It enabled mathematicians to perform mathematical operations on classes that had previously not been regarded as mathematical objects. Boolean classes can be objects, processes, or even abstract thoughts and ideas that share certain characteristics, features, or qualities. We can create a class of green apples and a class of red apples, then create a third group that accommodates both: fruit. A textbook example of Boolean logic goes like this: If the symbol x represents the class of all “white” objects, and the symbol y, the class of all “round objects,” the compound symbol xy represents the class of objects that are simultaneously white and round.

Note that Boolean classes are based on the same principle as the trigrams and hexagrams used in the Book of Changes. Trigrams are Boolean classes of objects, processes, and tendencies that are predominantly yin or yang. The Book of Changes uses the “logical operators” of true or false and favorable or unfavorable, while Boole used logical true-false operators, like if, then, nor, etc. Boolean classes and the logical operators to perform mathematical functions on classes would become crucial to the development of modern binary computers.

Fig. 1–4 Boolean algebra of classes based on the true-false logic and conditional statement if/then/or, etc.

One more step was required to complete the circle and shed light on Leibniz’s intuition that the Book of Changes foreshadowed the binary code. In 1937, the American mathematician and computer science pioneer Claude Shannon published his landmark thesis, A Symbolic Analysis of Relay and Switching Circuits. Shannon demonstrated that Leibniz’s binary code was perfectly suited for the implementation of Boolean logic in electrical circuits. The binary number 1 would denote “true” (yes, inside a class), and 0 would be “false” (no, outside a class). Shannon was primarily concerned with solving mathematical problems originating from increasingly complex telephone switching circuits, but his paper provided the basis on which nearly all modern computers would be built.

The second point Shannon made in his paper earned him the epithet “Father of the Information Age.” Shannon realized that a code was just a code. It can be given any arbitrary attribute, as long as the meaning is agreed upon. In his paper, Shannon pointed out that a binary number or a string of binary numbers can stand for letters, music, images, or any other attribute we decide to give it. Giving binary numbers arbitrary attributes is at the heart of modern computing. Punch the letter A on a keyboard and the computer reads 100 0001. The attribute is arbitrary, but part of the international standard for alpha-numerical keyboards known as the ASCII code.

Fig. 1–5 Boolean logic implemented with Leibniz’s binary code is at the heart of modern digital computing.

The Chinese had applied the same principle to the broken and unbroken lines that made up the Eight Trigrams. The attributes given to the trigrams are arbitrary. Their meaning is full of symbolism and open to interpretations, but the underlying principle is the binary code, and the Chinese yin-yang code is the same: an agreement on their meaning so that it becomes a standard.

British Sinologist Joseph Needham recounted the story of Leibniz’s encounter with China in Science and Civilisation of China. Needham explained how Leibniz’s distant vision, the “mechanization of thought,” had come one step closer in the 1940s, when an interdisciplinary team of scientists led by Norbert Wiener developed cybernetics. Needham wrote:

“It [the binary code] has been found to be, as Wiener points out in his important book on cybernetics (the study of self-regulating systems whether animal or mechanical), the most suitable system for the great computing machines of the present day. It has been found convenient to build them on a binary basis, using only “on” or “off” positions, whether of switches in electrical circuits or of thermonic valves, and the type of algorithm followed is, therefore, the Boolean algebra of classes, which gives only the choice of “yes” or “no”, of being either inside a class or outside. It is, therefore, no coincidence that Leibniz, besides developing the binary arithmetic, was also the founder of modern mathematical logic and a pioneer in the construction of calculating machines. As we may see later, Chinese influence was responsible, at least in part, for his conception of an algebraic or mathematical logic, just as the system of order in the Book of Changes foreshadowed the binary arithmetic.”

Cybernetics would lead to the development of artificial intelligence, a self-learning advancement on cybernetics that promised to “mechanize thought” in many different fields of science, as Leibniz had envisioned.

From: Leibniz, Einstein, and China, Olive Press, Amsterdam