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I Ching Oracle Guide What Leibniz recognized in the hexagrams, and what he missed

What Leibniz recognized in the hexagrams, and what he missed

Gottfried Wilhelm Leibniz (1646-1716) already had binary arithmetic when he first saw a hexagram diagram. That is the core of this story, and the point at which the story, in China too, keeps being retold out of order.

In the spring of 1703, in a woodcut sent from Peking, he read the sixty-four hexagrams of the I Ching as the numbers 0 through 63. The formal match can be shown. The conclusions he, his Jesuit correspondent, and later public memory fastened to it mostly cannot.

Twenty years before the diagram

Leibniz was a jurist by profession. In his spare time he worked on mathematics, philosophy, logic and history. Around 1672-1676 he was already studying number systems that did not run on ten. After encountering proposals for a base-four system, he arrived at the idea that two symbols, 0 and 1, suffice to write any number.

On 15 March 1679 he wrote the essay De l'arithmétique binaire (On binary arithmetic): addition with 0 and 1, set beside a comparison with the decimal system. In 1701 he sent the piece to the Paris Academy, but asked that it not yet be published. He wanted to develop the number theory further, and at that moment he saw no practical use for it. The binary system was therefore in place a good twenty years before the hexagrams.

That chronology has been set down most sharply in China itself by 孙小礼 (Sūn Xiǎolǐ), a professor at the Center for Science and Society, Peking University (北京大学, Běijīng Dàxué). The article 莱布尼茨对中国文化的两大发现 (Láibùnící duì Zhōngguó wénhuà de liǎng dà fāxiàn, Leibniz's two great discoveries of Chinese culture) appeared in 1995 in the 北京大学学报 (Běijīng Dàxué Xuébào, Journal of Peking University), no. 3. In 1999, in the journal of the University of the Chinese Academy of Sciences (中国科学院大学, UCAS), Sun corrected a misunderstanding widespread in China: the idea that Leibniz saw the Zhouyi (周易), took inspiration from the trigram symbols, and only then invented binary arithmetic. The order was the reverse. The system was already there. The diagram came later.

Bouvet, Figurism and the letter of 1701

Joachim Bouvet was born on 18 July 1656 in Le Mans and died on 28 June 1730 in Peking. He entered the Jesuits in 1673. In 1687 he travelled to China as one of the first five "royal mathematicians" of Louis XIV, and became mathematician and astronomer at the court of the Kangxi emperor. He was the leading figure of Figurism, the belief that the ancient Chinese already knew the core of Christian truth, recoverable from the classics. His most explicit Figurist texts remained unpublished until the mid-19th century.

From the 1680s-1690s Leibniz was already corresponding with Jesuits in China, among them Claudio Filippo Grimaldi (闵明我, Mǐn Míngwǒ). In a letter of 1689 and again in 1692 he asked for the fastest way to learn Chinese and for help in collecting Chinese texts. According to Sherwin Doroudi, drawing on Cook and Rosemont, that interest also had a language-philosophical side. Inspired by a theory of the Dutch orientalist Jacob Gohl, Leibniz asked whether Chinese writing could serve as a model for a universal, ideographic "character language" for philosophers. He concluded that the script itself would not do: too many ambiguous signs. Set that conclusion next to 1703, and what stands out is that the hexagrams succeeded for him where the script had failed. That is our reading, not a claim Leibniz made.

On 15 February 1701 Leibniz sent his own binary number table, using only 0 and 1, to Bouvet in Peking. Bouvet replied on 4 November 1701. He saw at once a relation with the symbols of the 64-hexagram diagram and enclosed a woodcut of the arrangement attributed to 伏羲 (Fú Xī). In fact it was the "先天" (Xiāntiān, Early Heaven) arrangement composed by the Song scholar 邵雍 (Shào Yōng, 1011-1077). By a detour through England the letter took more than a year to reach Leibniz. It arrived on 2 April 1703.

He studied the diagram at once. With a broken line (yin) as 0 and an unbroken line (yang) as 1, the sixty-four hexagrams in that arrangement read as the binary numbers 0 through 63 in a row. Leibniz was beside himself with joy. In a letter to Bouvet he called the diagram, in Sun Xiaoli's rendering, "the oldest monument of science in the universe", which for more than 4000 years no one had understood. Without his own invention of binary arithmetic, he wrote, he would have been equally unable to see what the system of 64 hexagrams was about.

It follows, we think, that the recognition in Peking was prepared from Europe. Bouvet had received the table first; only then did he see the diagram as binary. Leibniz guessed that the legendary Fu Xi had once known binary arithmetic, and that the knowledge had later been lost. That same spring he expanded his 1679 paper and published it as Explication de l'arithmétique binaire, qui se sert des seuls caractères 0 et 1, avec des remarques sur son utilité et sur ce qu'elle donne le sens des anciennes figures chinoises de Fohy [Fu Xi]. The article appeared in the Mémoires of the Académie Royale des Sciences, in the volume dated 1703, which was in fact printed only in 1705, on pages 85-89. In 1701 he had held the paper back because he saw no use for it. In 1703 the title points to the ancient Chinese figures of Fu Xi. That suggests the diagram gave him no mathematics he did not already have, but a reason to put that mathematics into the world.

Shao Yong, not Fu Xi

Leibniz was right about a purely formal match: the order of hexagrams in Shao Yong's Xiantian diagram is binary counting from 0 to 63. He then read the pattern inside his own theology. Building everything from 0 and 1 confirmed, for him, that God had created the universe "from nothing" (0) into being (1), and that the ancient Chinese had held a comparable idea of God. Our reading: both frames brought their conclusion with them. Figurism looks for Christian truth in the Chinese classics, and Leibniz read 0 and 1 as nothing and creation. The woodcut supplied the pattern, not the theology. The mathematical pattern remains. The God-layer is theirs.

He took over, without testing it, Bouvet's assumption that the diagram was Fu Xi's own. The I Ching is the product of centuries of layered work by many generations of scholars. The Xiantian arrangement is Shao Yong's, in the 11th century, not Fu Xi's. Bouvet did not know that layered history. Leibniz took the error on faith. Sun Xiaoli sums it up as 邵《易》非古: Shao's Yi is not ancient.

In China Shao Yong belongs to the stream of 象数 (xiàngshù, image-and-number), beside 义理 (yìlǐ, meaning-and-principle). He used a consistent two-valued method of construction: 一分为二、二分为四, one divides into two, two into four. That is still not arithmetic with binary numbers in Leibniz's sense. According to the majority of scholars writing now, among them J.A. Ryan and Sherwin Doroudi, there is no other evidence that this system was ever used in China for addition or multiplication. The I Ching case stands alone. A likeness in how symbols are built is not a shared mathematical insight.

Inventor or discoverer

Whether China "invented" the binary system is a dispute in Chinese scholarship, though one view is clearly the mainstream. The two sides share the dates. They do not share the definition of invention.

The mainstream view, named as such by 盛邦和 (Shèng Bānghé) himself and represented by 焦树安 (Jiāo Shù'ān), Sun Xiaoli and 李存山 (Lǐ Cúnshān), is that Leibniz developed his binary mathematics independently in 1679, years before he knew anything of the I Ching hexagrams. The later match with Shao Yong's diagram is striking, but a number system with addition and multiplication differs in kind from a symbolic arrangement of trigrams that was not used as arithmetic.

Sheng Banghe argues the other side in 二进制中国发明权辩识 (Èrjìnzhì Zhōngguó fāmíngquán biànshí, Disputing China's right of invention of the binary system). Fu Xi's trigrams already contain, on his account, the seed of binary thought; Shao Yong completed a fully worked binary number system in his 皇极经世 (Huángjí Jīngshì). Bouvet, Sheng writes, told Leibniz not to treat the binary system as something new, because Fu Xi had invented it long before; Leibniz would later have granted the point. In that reading Leibniz is a discoverer of something already there, not an inventor. That wording is known only from Sheng's paraphrase; we could not trace it in a primary source, the letter itself. The title speaks of 发明权, the right of invention. It is a position in a debate, not a settled fact.

Our reading: both camps accept that the 1679 paper was there before the 1703 diagram. The dispute is not chronological. Anyone who equates yin/yang doubling with addition in 0 and 1 can speak of Chinese priority. Anyone who requires arithmetic operations cannot.

According to Sun Xiaoli, Leibniz's second discovery concerning China was his regard for Chinese writing. Against a then common European view that derived Chinese signs from Egyptian hieroglyphs, he held that they were philosophically considered and based on relations among concepts. The Japanese scholar 五来欣造 (Gorai Kinzō) treated, again according to Sun Xiaoli, Leibniz's binary arithmetic and the I Ching as "two hands": East and West approaching each other through abstract, symbolic language. The letters, the diagram, and Leibniz's handwritten binary digits beside the hexagrams are still held in German libraries, collected in Leibniz Korrespondiert mit China (Frankfurt, 1990).

The computer that was not predicted

After the Explication Leibniz was celebrated as the (re)discoverer of the binary system, which three centuries later would become the basis of all modern digital computers. Alongside it runs the line that the I Ching predicted the computer, or that the ancient Chinese already knew the digital machine.

What matches is only the abstract, two-valued pattern of counting and arrangement in one specific diagram. There is no evidence that old or imperial China ever performed arithmetic operations with this pattern, let alone entertained an idea of mechanical or electronic calculation. According to a 2020 survey on the Taiwanese science site PanSci, the binary pattern in the I Ching itself had no arithmetical application.

Two other myths remain worth naming. Leibniz did not develop the binary system thanks to the I Ching: 1679 precedes 1703 by more than twenty years. And the diagram was not Fu Xi's; it was Shao Yong's Xiantian arrangement. Neither Bouvet nor Leibniz knew that.

The subject is still alive in China, down to technical blogs that retell Shao Yong's 先天易 and the binary link with Leibniz for a wider public. Academically the core chronology stands. The debate is about meaning, not about the order of the years. Our reading: that resembles other disputes around the I Ching and the book in China today, where the data are shared and the interpretation is not. Our reading is that three uses of two values stand side by side here: Shao Yong's doubling of yin and yang, Leibniz's arithmetic in 0 and 1, and the circuits of the computer, not a prediction of one by another.

Sources

The old ritual Cast the yarrow stalks Ask your question and build your hexagram line by line, with the classical odds.

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