This article is "The Third Mathematical Culture Essay Contest."
Leibniz, Binary and Fu xi gua chart
Author: Zhang Xiaoping, Jiang Hui, Ma Hongyun
Artwork no.: 001
Submission date: 2021.5.8
Leibniz was an encyclopedic scientist who lived in an era that overlapped with the Qing Dynasty's Shunzhi Three Years (1646) - Kangxi Fifty-Five Years (1716). He was very concerned about Chinese history and culture, and he mentioned the Fuxi Guatu in a paper on binary, and mentioned the binary and Fuxi Guatu many times in his correspondence with others. Two scholars, Hu Yang and Li Changduo, believe that the Fu Xi Gua Tu is binary, and Leibniz founded the binary by the Fu Xi Gua Tu 1. These conclusions are debatable.
I. The circulation of the I Ching in Europe
The I Ching is an ancient Chinese book of Bu Zheng, composed of two parts, the Zhou Yi and the Yi Chuan, written in the Western Zhou Dynasty, the author is unknown, and the words and sentences are ancient. The Zhou Yi consisted of the Gua Yan symbol and the Gua Yan ci, and later, Confucian scholars elaborated on the Gua Yan ci from the point of view of righteousness, and wrote the "Yi Chuan", which was formed by the hook and shen examination of successive generations of scholars.
Father Nielas TriBauit of France, who came to China twice in 1610 and 1620, translated the I Ching in Latin in Hangzhou, but the translation had no effect in Europe.
Father Martino Martini of Italy, proficient in mathematics, astronomy and measurement technology, came to China twice and made important contributions to the scientific and cultural exchange between China and the West. In 1658, he published the Ancient History of China in Munich, which contained a discussion of the I Ching, containing a sixty-four trigrams, marked as a painting by Fuxi, which was the first Fuxi guatu introduced to Europe. Wei Kuangguo introduced that the basic symbols in the gua chart are "yin" and "yang", and yin represents hidden and incomplete things. Impotence represents the open, complete thing. The "three-line symbol" composed of them is the Bagua, which represents the heaven, earth, water, fire, thunder, mountains, zee, and wind in natural phenomena. On this basis, the combination of "three-line symbols" can form sixty-four "six-line symbols", that is, sixty-four gua.
In 1660, the scholar Gottlied Spizel's "Commentary on the History of Chinese Literature" was published in Leiden, the Netherlands, and some sources cite the Self-Defending Kuangguo's "Ancient History of China", introducing the contents of the "I Ching" in great detail, containing the sixty-four gua diagrams of Fuxi, and also appeared the phrase "binarium multiplicatis", which refers to the way in which fuxi gua diagrams are generated according to the principle of 2 squares. Spisel was Leibniz's academic friend, and there were several letters between the two discussing philosophical issues. After 1660, Leibniz had read the book "Commentary on Chinese Literature and History" without doubt, and it was around this time that he learned about the I Ching and the Fu Xi Gua Tu.
Father Philippe Couplef of Belgium came to China in 1659. He devoted himself to the dissemination of Chinese culture, and together with several priests translated the University, the Zhongyong, the Analects, and the I Ching into the Latin Four Books of the Four Books of the Western Language, which contained the Order diagram of the Fuxi Bagua and the Azimuth Chart of the Fuxi Bagua, as well as the Sixty-Four GuaTu of the Zhou Dynasty, on which the Arabic numerals 1, 2, 3, 4, 5, 6, 7, 8, and 64 were used to mark the order numbers. The work was published in Paris in 1687 under the title Confucius the Chinese Philosopher. On December 19, 1687, Leibniz wrote a letter to a gentleman named Von Hessen Rheinfels describing his joy at reading the book Confucius the Chinese Philosopher. Leibniz also saw the Fu Xi Gua Tu in the book.
There is a school in the field of Yixue called the Xiangshu School, which mainly explores the psychology of people to avoid evil through the Fu Xi Gua Tu, and prays to control the laws of change in nature, and its theory has no lack of mystical overtones. The northern Song Dynasty philosopher Shao Yong was a master of the Xiangshu school, and he took a unique path, bypassing the shackles of the I Ching, repackaging the I Ching with esoteric terminology and mysterious schemas, and creatively drawing the "Fuxi Sixty-Four Gua Order Diagram" and the "Fuxi Xiantian Sixty-four Gua Direction Map", referred to as the order map and the innate map. The biggest difference between the order diagram and the congenital diagram and the previous FuXi Gua diagram is that they are arranged according to the binary ordinal number. The Southern Song Dynasty master of science Zhu Xi's Zhou Yi Benyi contains these trigrams. It should be pointed out that until 1687, leibniz saw all kinds of fuxi diagrams arranged in a philosophical sense, without highlighting the binary ordinal characteristics.
Second, Leibniz really knew the process of Fu Xi Gua Tu
The person who let Leibniz know more about the Fu Xi Gua Tu was Bai Jin. Bai Jin, a French priest who had been a mathematics teacher in Kangxi, returned to Europe at Kangxi's behest to recruit scientific talents, arrived in Paris in 1697, where he gave a lecture on Yixue, criticizing some people for viewing the I Ching as a superstitious book, saying that the I Ching contained the philosophical ideas of Fu Xi, the founder of the Chinese monarchy, and was as reasonable and perfect as Plato and Aristotle. Soon, Bai Jin read Leibniz's book "Recent Events in China", and the two began to communicate with each other. They survived as many as fifteen letters2, dating from October 18, 1697, and contained many discussions of binary and fuxi diagrams.
On February 28, 1698, Bai Jin wrote to Leibniz for the first time introducing the Fuxi Guatu: "The original Chinese characters were composed of dotted or solid lines, which are said to have been created by Fuxi. I think the real secret to learning it has been found. Father Pak Yingli lists these Chinese characters in the preface to the Chinese philosopher Confucius. The table of Chinese characters mentioned by Bai Jin is the Fu Xi Gua diagram in Bai Yingli's book, not the innate diagram drawn by Shao Yong, and it is not related to binary. He believes that this is just a Chinese character created in a simple and natural way.
After returning to China, Bai Jin had a clearer understanding of the I Ching. On November 8, 1700, Bai Jin wrote a letter to Leibniz from Beijing, saying that Fuxi was the earliest codified figure of mankind, and that Fuxi Guatu was the most primitive symbol of Chinese culture, with a complete metaphysical system, these symbols had both arithmetic and linguistic functions, and the language of expressing ideas could be analyzed through mathematical precision. Bai Jin specifically said that in the Fu Xi Gua chart, if you replace the dotted line with 0 and the solid line with 1, sixty-four gua is the perfect number. "The mystical numbers in the I Ching are identical to those in Pythagoras, Plato, and Egyptian Jewish philosophy, both derived from the mysterious revelation of the Creator." Bai Jin was proficient in the theory of elephant numbers, and was familiar with the Chinese way of representing numbers with arithmetic chips, which was very similar to the overlapping images of yin and yang, and naturally associated with converting the symbols in the gua chart into numbers. At this time, although Bai Jin associated the Fu Xi Gua chart with the numbers, he did not yet understand binary arithmetic and did not consider binary numbers.
Leibniz has always tried to create the idea of "universal characters", using special symbols to represent general concepts, so that the thought process is like geometric reasoning, using logical calculus to discover and invent truth. Leibniz considered the relationship between the Fu Xi Gua chart and the "universal character", so he did not find that the number converted by Bai Jin was a binary number, and lost an opportunity to discover that the Fu Xi Gua Chart contained binary, which delayed a major scientific discovery for a while.
Leibniz wrote back to Bai Jin on February 15, 1701, introducing the idea of "universal characters", and he also mentioned his invention of calculus and binary. Regarding binary, he explained to Bai Jin: "Just as decimal uses ten digits from 0 to 9, it is enough to use only two numbers, 0 and 1. The letter mentions the binary algorithm, listing a table of binary numbers from 0 to 31. Leibniz also suggested that Bai Jin introduce binary to Kangxi.
Leibniz's manuscript to Bai Jinxin on February 15, 1701
The reason why Leibniz introduced binary to Bai Jin was because he had just become a member of the Royal French Academy of Sciences, and he planned to submit a paper on binary to the Academy of Sciences, "The New Science of Numbers", and by the way, he told Bai Jin the content of the paper. On February 26, Leibniz submitted his paper to the Academy of Sciences. On April 30, De Fontenelle, secretary general of the Academy of Sciences, wrote to Leibniz stating that it could not be published in the journal of the Academy of Sciences because the practical value of binary was not reflected in the paper.
In fact, Leibniz was always looking for the practical value of binary. He had the idea of using the binary principle to make a calculator, but it was not realized. He also tried to introduce binary into the field of theology, using binary arithmetic to prove the existence of God. In the library of the Gouta Palace in Turingen, Germany, there is a manuscript of Leibniz with the title: "1 and 0, the magical source of all numbers." This is the mysterious and wonderful model of creation, for everything comes from God. ”
In May 1696, Leibniz visited Duke Ruddph August of Hanover, and while talking about theological culture, he introduced the duke to binary arithmetic. The Duke believed that interpreting the meaning of binary from a theological point of view could provide a scientific explanation for God's genesis. On New Year's Day 1697, Leibniz wrote a letter to the Duke detailing the "from nothing" idea of genesis contained in binary. In his letter, Leibniz designed a "Creation Chart" medallion with a table of binary numbers from 0 to 16 and examples of addition and multiplication, and the words "Nothingness Is Born One" and "One Creates Everything" are written.
On December 20, 1696, Leibniz wrote to Father Min Mingmei, the Qing Dynasty's Qin Tianjian Supervisor, detailing binary arithmetic, listing tables of binary numbers from 0 to 31, and illustrating the algorithms of addition and multiplication. Min Ming I taught Kangxi mathematics, and Leibniz hoped that Min Ming would teach Kangxi binary so that Kangxi could understand the superiority of Christian culture.
Leibniz's "Creation Chart" medallion
In Leibniz's letter to Bai Jin, a strong theological complex was also revealed. The greatness of binary, he writes, is that it simulates God's creative process. The world began with only two states, God and nothingness. God represents perfection, and nothingness represents imperfection. Everything in the world was created by God from nothingness. At this time, Leibniz still did not know the relationship between binary and fuxi diagrams, and the so-called theological application of binary was not suitable for writing in scientific papers.
Manuscript of Leibniz's binary thesis
Bai Jin learned the knowledge of binary from Leibniz's letter, and immediately discovered the relationship between binary and the innate chart, and wrote to Leibniz on November 4, 1701, sending a congenital diagram with the letter, clearly stating that only the solid line needs to be replaced with 1, the dotted line is replaced by 0, and each gua corresponds to a binary number, and the congenital diagram is arranged according to the binary ordinal number. It was Bai Jin's unique insight that unveiled the mystery of the Fu Xi Gua map. The letter reads: "You should not think of binary as a new science, because the Chinese Fuxi has already invented it. ”
Bai Jin sent Leibniz a congenital diagram
The letter passed into Leibniz's hands on 2 April 1703. He immediately studied the innate chart, marked each trigram with a corresponding Arabic numeral, and confirmed that the arrangement of the trigram was consistent with the binary ordinal numbers. He fully agreed with Bai Jin's point of view. As a practical example of binary, he incorporated Bai Jin's discoveries into his binary papers, as well as the apothemastic diagram, entitled "On the Simple Use of Binary Arithmetic of 0 and 1—On Binary Uses and the Meaning of the Ancient Chinese Fuxi Symbol," which was sent to the Royal Academy of Sciences of France on May 5, 1703, and later published in the Annals of the Royal Academy of Sciences in 1703.
Manuscript of Leibniz's letter to Bai Jin of 18 May 1703
After Leibniz finished processing the paper, he replied to Bai Jin and sent it out on May 18. Leibniz writes: "This picture is the oldest scientific artifact in the world, incomprehensible for thousands of years, but so consistent with binary arithmetic." When you explain these symbols to me, I happen to introduce you to binary arithmetic, and they are surprisingly coincidental. If I hadn't invented binary arithmetic, I wouldn't have been able to understand it even if I studied the Fu Xi Gua diagram in depth. I started thinking about binary twenty years ago and realized that numbers expressed in 0 and 1 were perfect and easy to calculate. Because the previous Fuxi Gua charts were not arranged according to binary ordinals, no one has discovered this secret, and the congenital charts are strictly arranged according to ordinal numbers, which was discovered by Bai Jin and Leibniz. Therefore, Leibniz questioned why the traditional gua diagram is not arranged according to the binary ordinal number like the xiantian diagram, and he asked Bai Jin in the letter why the Fuxi Gua diagram in the Chinese philosopher Confucius is different from the xiantian diagram.
Sixty-four diagrams in the book of Bai Yingli
From then on, Leibniz no longer claimed to have invented binary, but only said that he had rediscovered fuxi's knowledge. In his "On the Natural Philosophy of Chinese", there is a section "On the Writing and Symbols Used in Binary Arithmetic of FuXi, the Founder of the Chinese Empire", which is a summary of the content of his correspondence with Bai Jin and represents their common point of view. The text reads:
"Father Bai Jin and I discovered the original meaning of the GuaTu created by Fu Xi, the founder of this empire, which consisted of some dotted and solid lines, with a total of sixty-four symbols, which is the oldest and simplest script in China. In the centuries after Fuxi, King Wen of Zhou, his son Duke Zhou, and Confucius, five centuries later, explored philosophy in the Guatu, and some people wanted to derive things like feng shui and superstition from it. In fact, the sixty-four gua chart is the binary arithmetic founded by the great legislator Fu Xi, and after thousands of years, I rediscovered 3. ”
Although Bai Jin had a lot of experience in the study of Yixue, he still lacked in-depth understanding of the field of Yixue, and he conveyed a lot of wrong information to Leibniz, taking myths and legends as historical materials, and making false praise for Fuxi, without explaining that the xiantian map was drawn by Shao Yong, so that Leibniz mistakenly believed that the xiantian map was an ancient cultural relic. Bai Jin did not even tell the names of yin and yang, which led Leibniz to call yin as a dotted line and a solid line in the manuscript.
Leibniz was not a god, and his influence on Bai Jin and his worship of easy-to-learn culture led him to believe that Fuxi had created the binary, and he did not hesitate to praise it with praise. However, this reflects Leibniz's lack of ambition to plunder people's beauty, and he has a very weak view of his discovery of binary.
Third, fu xi gua chart and binary
During the Shang Dynasty, China already had a relatively complete decimal notation method. According to the common sense of the history of the development of human civilization, "Zhou Yi" as a work of the source of civilization, its content and ideas are in the embryonic state of culture, the arithmetic knowledge in the book is extremely simple, there is no binary content at all. Reading through the "Zhou Yi" shows that the so-called arithmetic knowledge is nothing more than counting, which is expressed in decimal. There are many types of Fuxi Gua Diagrams in Yixue, including Bagua Charts and Sixty-four Gua Diagrams. The ordering of the Bagua chart is generally symmetrical, and the sixty-four gua chart is sorted according to the words of the Bu Zheng. The popular Zhou Yi and the Book of Zhou Yi excavated from the Mawangdui Han Tomb in Changsha are not in the same order, but neither is arranged according to the binary ordinal number.
There is a saying in the field of Yixue research that yin and yang have numerical characteristics. From some of the information left on pottery, oracle bones and bamboo janes excavated by Yin Shang, it is known that yin yao evolved from even numbers, and yang evolved from odd numbers. After the formation of these symbols, although the philosophical significance of them was highlighted, the blurred images of numbers were also preserved, so mathematicians related the I Ching to mathematics. The initiator was Liu Hui of the Wei and Jin dynasties, who wrote in the "Notes on the Arithmetic of the Nine Chapters": "In the past, Bao Yishi began to draw gossip, with the virtue of the gods, with the love of all things, to make the number of nine nines, to merge the changes of the six masters." "Bao Is an alias for Fuxi. Liu Hui's views influenced later mathematicians. Qin Jiushao of the Northern Song Dynasty said in the Nine Chapters of the Book of Numbers: Mathematics "is the Book of Tuluo from the River, the Mysteries of MinFa, the Nine Domains of Bagua, the Intricate and Subtle, and the Extreme for the Use of the Great Yan Emperor." Cheng Dawei of the Ming Dynasty contains an illustration of Fu Xi as a guatu in the "Algorithm Tongzong", and the book writes: "What is the number of Zhao? It's from books! Fu Xi got it to paint, Da Yu got it to order it, and Li Sheng got it to open things to do business. All the heavenly officials, the earthmen, the legal calendar, the military endowments, and the slenderness of the clouds, if there are no numbers, they are not based on the "Easy". ”
Shao Yong uses the deduction of elephant numbers to replace philosophical thinking, giving people the feeling of performing mathematical operations. He discussed in the Outer Part of the View: "There must be words in the intention, there must be images in words, and there must be numbers in elephants." Counting is like life, like life, saying is manifest, and saying is manifest. Elephant number is also hoofed. This means that thoughts can be expressed in words, language can be expressed in images, and images can be expressed in numbers. vice versa. Therefore, the elephant number is a tool for expressing ideas. This is quite consistent with Leibniz's idea of the "universal character".
In the innate diagram drawn by Shao Yong, as long as the yin is treated as 0 and the yang is treated as 1, its arrangement is exactly the same as the binary ordinal, which is undoubtedly a binary model. However, this binary model is only the result of inadvertent willow planting, and it cannot be assumed that Shao Yong created binary, but can only be said to be mathematical ideas that unified the innate diagram into the binary theoretical system.
The common sense of the history of the development of mathematics tells us that any mathematical achievement is achieved for no more than two reasons, one is the progress of mathematical internal theoretical research, and the other is the requirements of the development and progress of mathematics external society. First of all, Shao Yong is not a mathematician, he has not made any achievements in mathematics, has not written mathematical treatises, and has not seen any mathematician with whom he has had academic contacts. Shao Yong did not have any fragmentary remarks on binary, nor did he understand binary theory at all, he did not clearly define and name the mathematical concept of binary, nor did he scientifically express the importance and significance of binary, nor did he improve the logical relationship between binary and other mathematical concepts. From Shao Yong's treatises, it is not clear that he has a mathematical literacy beyond ordinary people, and Shao Yong's creation of binary is like today's "civil mathematicians" to solve the Goldbach conjecture, which is absolutely impossible. Not to mention that none of his contemporaries, even mathematicians of his generation, had ever dabbled in the field of binary studies. The Gua Tu symbol has long existed in the Zhou Yi, and before the Song Dynasty, there was no binary information at all in the Zhou Yi and Fu Xi Gua Tu. Shao Yong reordered these symbols in the innate diagram, not actively according to the binary principle, but just happened to sort out the order of binary numbers. Therefore, we can only be cautious to say that there are binary buds growing in the innate diagram.
Some people have suggested that shao Yong's "doubling method" is the law of "every two into one", which is a completely baseless speculation. Understood only in terms of word meaning, they are not the same thing. "Doubling" means doubling, that is, multiplying by 2, refers to the generation process of the Fu Xi Gua Chart, and if each gua at the previous level is increased by one stroke in turn, the number of gua will be doubled. That is, "One is divided into two, two is divided into four, four is divided into eight, eight is divided into sixteen, sixteen is divided into thirty-two, and thirty-two is divided into sixty-four." Therefore, it is said that the yin is divided into yang, and the soft rigidity is used, and the yi six are also 4. This is a completely decimal statement and has nothing to do with the binary "two-in-one" argument.
From the original meaning of the concept, the guatu symbols composed of yang and yin represent abstract philosophical things, even if they are related to numbers, they are also 1-64 in the decimal system, including Shao Yong, Song, Yuan, Ming, and Qing yi scholars and mathematicians, none of them proposed that they can be expressed by binary numbers. In the process of drawing the innate map, shao Yong used decimal to solve the problem whenever it came to counting. For example, when it comes to the ordering of gossip, his expressions are: Qianyi, Hui II, Li III, Zhen IV, Xun V, Kan VI, Gen VII, kun VIII. If Shao Yong understood the binary principle, then the order of the guatu symbols would be very clear, and it would be easy to be remembered without the help of any method, but until the Southern Song Dynasty, Zhu Xi also wrote memory tips based on the intuitive image of the Bagua symbols: Qian Sanlian (☰ ☷), Kun Liu (), Zhen Yang Lu (☳), Gong Fu Bowl (☶), Away from the Middle Void ☲ (), Kan Zhong man (☵), Du Shang Deficiency (☱), Xun Xia Short (☴). This reflects from one side that neither Shao Yong nor Zhu Xi was aware of the close relationship between the innate chart and the binary. During the Qianjia period of the Qing Dynasty, there was a famous mathematician Wang Lai, whose "Sutra of Reference and Two Calculations" was a work devoted to the theory of the position system, and he did not point out that the innate map was binary.
So, how does the innate graph happen to be the same as the binary ordinal number? This question has puzzled many people, and even some scholars have used the method of probability theory to interpret that there are 64 out of 64 elements! To find the exact same order as the binary numbers in this astronomical arrangement, this is almost impossible! Furthermore, if Shao Yong is not familiar with the binary principle, how can he find this arrangement?
In fact, both the innate graph and the binary are represented by two basic symbols, which is actually a combination of the repeatable arrangement of elements. Take 3 of the 2 symbols at a time and arrange them in a row, for a total of 23 =8 sorts, to get the gossip in Zhou Yi and the first eight numbers in binary. Take 6 of the 2 symbols each time and arrange them in a row, for a total of 26 = 64 arrangements, to get the sixty-four gua in the Zhou Yi and the first sixty-four numbers in the binary.
The order of the trigram in the innate diagram is the same as the ordinal of the binary, which is actually a necessary event. Shao Yong did not need to know binary knowledge, and binary was not a necessary condition for drawing a congenital map. The fact is that Shao Yong creatively used another mathematical method, using the "tree diagram" to naturally generate a binary "mathematical tree", which is the "Fuxi Sixty-Four Gua Sequence Diagram".
Fuxi Sixty-four Gua Order Diagram
According to the painting method of "tree diagram", Shao Yong painted the yin yao black and the yang ye white. First there is Tai Chi, from the bottom up, according to the "doubling method", first draw yin, then draw yang, and paint each other. From Tai Chi, the two instruments, the four elephants, the bagua, the sixteen gua, the thirty-two gua and the sixty-four gua are generated in turn, and eventually become the sixty-four gua sequence diagram, which can be drawn indefinitely. Since the order diagram is generated strictly according to the "tree diagram" method, it is read from the bottom up to form sixty-four gua chart symbols. From the left kun gua in order to the right qiangua, it naturally satisfies the order of the first sixty-four numbers in binary.
Understanding the structure of the sequence diagram, it is easy to draw a congenital diagram. First look at the circle diagram outside it, as long as the right half circle is straightened, it is the left part of the symmetry axis of the sequence diagram. Straightening the left semicircle is the part on the right side of the axis of symmetry in the sequence diagram. Looking at the internal square chart, the law is more obvious, just need to arrange each trigram in the sequence diagram in the order from left to right, every eight as a paragraph, and arrange eight lines in turn. It should be noted that the innate diagram thus constructed still maintains symmetry, but the axisymity in the sequence diagram becomes the central symmetry. It is easy to prove mathematically that in a circle diagram, the two trigrams about the symmetry of the center of the circle are also symmetrical in the sequence diagram. In the square diagram, the two trigrams about the symmetry of the center of the circle are also symmetrical in the sequence diagram.
Since the innate graph and binary are algebraically isomorphic, and the order and symmetrical structure of the primordial graph are no longer secret, it is easy to understand that the mathematical properties of binary arithmetic can also be extended to the innate graph in general. This does not mean that Shao Yong discovered a lot of modern mathematical knowledge in that era, but that mathematical ideas unified the simple binary factors contained in the innate diagram.
Leibniz and binary
Leibniz wrote to Bai Jin on May 18, 1703, that he invented the binary more than twenty years ago, during the period when he founded calculus in Paris. In a letter to D. Bourguet of 15 December 1707, Leibniz said: "When I first founded binary arithmetic, I did not know much about the I Ching. Leibniz had a manuscript, The Interpretation of Binary Arithmetic, written on March 15, 1679, which had not been published and had been shelved for more than twenty years.
But all mathematical innovations and discoveries follow the path opened up by predecessors, and in the process of surpassing predecessors, it is not guaranteed that all enlightening ideological achievements can be taken care of. Before Leibniz invented the binary, he only indirectly understood the I Ching from Wei Kuangguo's "Ancient History of China" and Spishel's "Commentary on chinese Literature and History", which were not mathematical works, although Wei Kuangguo called the "I Ching" a mathematical work, it was only a kind of speculation. The Fuxi Guatu that Leibniz saw was not an innate diagram drawn by Shao Yong, and the order of the Gua diagram was arranged according to philosophical ideas, not in binary order. Leibniz's academic interest was to develop the idea of "universal characters", he was concerned with the linguistic and logical meaning of the symbols, and did not consider the problem from a mathematical point of view, so Leibniz had no way of knowing binary information in the easy-to-learn literature. If Leibniz had been inspired when he saw the Fu xi gua tu, his binary paper would have been published more than twenty years in advance.
In addition, Leibniz was inspired by the fact that the phrase "binarium multiplicatis" mentioned in The Introduction to the Fuxi Guatu in Spisher's writings means binary. This is speculation that ignores historical facts. At the time of Sbisser's writing, the concept of binary had not yet been explicitly articulated. The term "binary" was introduced to academia by Leibniz, who had not yet thought about the problem of binary. Spiesser was not a mathematician, and he had no way of knowing what binary was. This phrase also does not mean binary, but refers to the multiplication of 2, which is the way in which the Fu Xi Gua chart is generated.
In fact, Leibniz's opportunity to discover binary is very simple, it is a completely natural and instantaneous result. Considering the laws of mathematical cognition, people with basic mathematical literacy only need to be familiar with the theory of the carry system, and any position system proposed is an ordinary inference. In fact, any natural number larger than 1 can be used as a cardinal of a progressive system, and in theory it is possible to construct an infinite number of input systems, which is a very simple common sense in mathematics.
Leibniz went to the University of Jena in the summer of 1663 to study mathematics under the tutelage of Professor Erhard Weigel. Wegelius was very experienced in the study of ancient Greek mathematical thought, advocating the mathematical views of Pythagoras and Plato, believing that the harmony of the material world conformed to mathematical laws. Leibniz was deeply inspired by the teacher's thoughts. In 1672, Viglius published the article "Structure of the Holy Ten" in the journal "Joham Meyer" of the University of Jena, which systematically proposed the concept of quaternary, using 0, 1, 2, and 3 to represent all numbers, "full of three into one", symbolizing that "three" is complete. Soon, Leibniz wrote a manuscript of The Interpretation of Binary Arithmetic. There is no doubt that Leibniz is familiar with the theory of the centroid system, whether from the teacher's classroom or from the teacher's thesis.
From Leibniz's On the Natural Philosophy of Chinese, we know that Leibniz was very familiar with the history of the decimal system, and he mentioned that the ancient Romans had used arithmetic with a mixture of pentadal and decimal, mentioning that quadrums and decimal systems had appeared in history, and he wrote explicitly: It was Weglius's quaternary that "gave me an opportunity to propose that all numbers could be written in binary 0s and 1s." ”
It can be seen that Leibniz invented the binary because he was inspired by his teacher and had nothing to do with the Fu Xi Gua Chart. Some people criticize Leibniz for questioning that he deliberately concealed that he was inspired by the Fu Xi Gua Chart to discover binary, which is not based on facts. Leibniz never claimed the right to invent binary for himself. Instead, he touted that Fuxi had invented the binary four thousand years earlier, and he attributed this great discovery to Bai Jin.
In fact, various modes of counting in the position system have long existed in human social activities. In the birthplace of world civilization, the ancient Babylonians were the first to invent the bit value system, using the hexadecimal system, the Maya used the twentieth system, and china invented the bit value system alone, which was the first to use the decimal system. In the Shang Dynasty oracle bone bu ci, there was a number of 1-9 and the notation of the bit value system, and in the Warring States period, there was a decimal calculation number, which represented 0 with emptiness, which was very advanced. The tribes of The island of Manareva in the Pacific Ocean used binary as early as 1450, and some indigenous peoples in Polynesia and Australia still use binary 5 to this day, in fact, the so-called invention of mathematicians, that is, the mathematical treatment of counting methods in human secular life, so the invention of binary is not a great mathematical achievement.
In fact, leibniz's contemporary, Y. Lobkowitz, also discussed decimal and binary in his 1670 bifacial mathematics. Leibniz may not know that in Shakespeare's time there was a brilliant mathematician in England, Thomas Harriot,6 whose manuscripts contained a great deal of mathematical and physical originality. Because there were no scientific journals at the time, there was nowhere to publish these results. Harriot's 1603 manuscript "Mathematical Calculations and Annotations" has a detailed discussion of binary arithmetic, the theoretical structure is almost the same as Leibniz's, using 0 and 1 as the basic counting symbols, naming binary binary binary numeration, proposing addition and subtraction and multiplication arithmetic, and also discussing the related problems of representing binary in the form of continuous fractions.
exegesis
1. Hu Yang and Li Changduo, Leibniz Binary and Fuxi Bagua Tukao, Shanghai, Shanghai People's Publishing House, 2006
2、见R.Widmaier编:《Leibniz corresponds with China: The correspondence with the Jesuit missionaries (1689—1714)》 Frankfurt am Main:Klostermann. 1990
3. Chen Lemin, ed. Leibniz Reader, Nanjing, Jiangsu Education Publishing House, November 2005
4. Shao Yong, "Huangji JingshiShu", Zhengzhou Zhongzhou Ancient Books Publishing House, January 2007
5. See Bashmakova et al., ed. Liu Shaozu, Translation of The Complete Book of Elementary Mathematics (Volume I) Arithmetic (Volume I), Beijing Higher Education Press, June 1959
6. Harriot's story is covered by Robyn Arianrhod, Thomas Harriot: A Life In Science, Cambridge University Press, 2019