/kloc-in the second half of the 0/7th century, European science and technology developed rapidly. Due to the improvement of productivity and the urgent needs of all aspects of society, through the efforts of scientists from all over the world and the accumulation of history, calculus theory based on function and limit concept came into being. The idea of calculus can be traced back to the method of calculating area and volume proposed by Archimedes and others in Greece. Newton founded calculus in 1665, and Leibniz also published his works on calculus in 1673- 1676. In the past, differential and integral were studied as two mathematical operations and two mathematical problems respectively. Cavalieri, Barrow, Wallis and others have obtained a series of important results of finding area (integral) and tangent slope (derivative), but these results are isolated and incoherent. Only Leibniz and Newton really communicated integral and differential, and clearly found the internal direct relationship between them: differential and integral are two reciprocal operations. And this is the key to the establishment of calculus. Only by establishing this basic relationship can we establish systematic calculus. And from the differential and quadrature formulas of various functions, the algorithm program of * * * is summarized, which makes the calculus method universal and develops into a symbolic calculus algorithm. Therefore, calculus "was mostly done by Newton and Leibniz, not invented by them". However, there has been a heated debate in the history of mathematics about the order in which calculus was founded. In fact, Newton's research on calculus was earlier than Leibniz's, but Leibniz's results were published earlier than Newton's. Leibniz's paper "Finding a Wonderful Computing Type of Minimax" published in Teacher's Magazine on June 1684 is the earliest calculus document. This six-page paper is not rich in content and vague in reasoning, but it is of epoch-making significance. Newton wrote in the first and second editions of Mathematical Principles of Natural Philosophy published three years later, namely 1687: "Ten years ago, in my correspondence with Leibniz, the most outstanding geometer, I indicated that I already knew the method of determining the maximum and minimum, the tangent method and similar methods, but I concealed this method in my correspondence ... The most outstanding scientist wrote back. He also described his method, which is almost no different from mine except for words and symbols "(but this passage was deleted in the third edition and later editions). So it was later recognized that Newton and Leibniz created calculus independently. Newton started from physics and studied calculus by set method. His application is more combined with kinematics, and his accomplishments are higher than Leibniz's. Leibniz, on the other hand, started from geometric problems, introduced the concept of calculus by analytical method, and got an algorithm, which was more rigorous and systematic than Newton's algorithm. Leibniz realized that good mathematical symbols can save thinking labor, and the skill of using symbols is one of the keys to the success of mathematics. Therefore, the symbols of calculus he created are far superior to Newton's symbols, which has a great influence on the development of calculus. 17 13, Leibniz published the article "History and Origin of Calculus", summed up his thought of establishing calculus, and expounded the independence of his achievements.

Bagua Fiona Fang Diagram and Binary Star System

There are many textual researches on the relationship between Leibniz's binary system and China's gossip, but whether Leibniz invented the binary system under the influence of gossip or invented it alone seems inconclusive. Dewll Hu and Li Changduo's book Leibniz-Binary and Fuxi Bagua Map gives more credible materials, which shows that Leibniz's binary system is at least inspired by Bagua Map to some extent. According to Leibniz's own account, he invented the binary arithmetic before 1679, but it was not until 1, 1703 that he received the Fuxi eight diagrams sent by Jesuit Bai Jin that he began to formally study the eight diagrams symbols and found the consistency between his binary system and Fuxi eight diagrams. A few days later, he wrote a paper on binary arithmetic-on the usage of only 0 and 1 and the meaning of the numbers used by Fu, which was published in the Journal of the Royal Academy of Sciences in France. According to Leibniz's own statement, many researchers believe that Leibniz didn't invent the binary system according to Fuxi's eight diagrams. However, in the book "Leibniz-Binary and Fuxi Eight Diagrams" by Dewll Hu and Li Changduo, it is proved that although Leibniz didn't see Fuxi Eight Diagrams brought to him by Bai Jin until 1703, it doesn't mean that this is the first time he saw Fuxi Eight Diagrams, but as early as 1687. 1687, Jesuit Bai Yingli published Confucius, a philosopher in China, in which 13 introduced Fuxi gossip, with Fuxi gossip sequence diagram, Fuxi gossip orientation diagram and Wang Wen's sixty-four hexagrams. It is worth mentioning that in the sequence diagram of Fuxi Eight Diagrams, the orientation diagram of Fuxi Eight Diagrams and Wang Wen's sixty-four diagrams, the corresponding diagrams are marked with Arabic numerals 1 to 64. In Leibniz's binary system, all numbers can be represented by 0 and the extension of 1, such as 000,001,0 10, 01,100, which respectively represent 0. In the Eight Diagrams of the Book of Changes, the universal principle of the universe can be expressed by the extension of Yin and Yang. If Yin is regarded as 0 and Yang as 1, then all hexagrams can be regarded as the combination of 0 and 1. For example, Kungua is 000000, Gangua is 1 1 11,and Judaica is11and so on. The sixty-four hexagrams in Fuxi diagram can also be regarded as numbers from 0 to 63 in binary arithmetic. Leibniz read Confucius, a philosopher in China, which was just published. In a letter to my friend von Hessen-Reinfeldt, I introduced him that I had read this book. In this letter, the word "Buddha's happiness" also appeared, which translates into "Fuxi" in Chinese. Through these facts, it is not difficult to prove that Leibniz has seen Fuxi gossip sequence diagram, Fuxi gossip orientation diagram and Wang Wen's sixty-four diagrams. However, Leibniz claimed in a letter dated May 1698 that he had been thinking about binary for more than twenty years. Letters from 1703 to 18 in May also indicated that he invented the binary system more than 20 years ago. Its museum also has binary mathematics published by 1679. According to this situation, the content about Yi Tu in China Philosopher Confucius by Bai Yingli should have no influence on his invention of binary system. And Dewll Hu and Li Changduo's book Leibniz-Binary and Fuxi Gossip also has material proof. As early as 1679, that is, before he invented binary, there were books about gossip published in Europe, and Leibniz also saw Yi Tu before 1679. This paper introduces the book Leibniz ―― Binary System and Fuxi Eight Diagrams by Dewll Hu and Li Changduo. 1660, Bisell, a scholar, published an analysis of China's literature and history in the Netherlands, in which the Book of Changes was recorded. Bisell has a close relationship with Leibniz. This book is Leibniz's reference book for understanding China. The Book of Changes is introduced in two parts, including the theory that Dragon and Horse draw negative pictures to show the river, the theory that Fuxi draws pictures to do gossip, and the theory of Taiji Yin and Yang gossip. In addition, from the book China Literature and History Review, we can see that before 1660, Bisell's references to China's cultural documents included History of China published by Jesuit Martino Martini 1658 and Empire of China published by Zeng Dezhao 1642. The theory of yin and yang gossip is only briefly introduced in Empire of China, but it is very detailed in the ancient history of China. The book introduces in detail the evolution process of Taiji Bagua. Yin and Yang give birth to two instruments, two instruments give birth to four images, and four images give birth to gossip. Some scholars believe that China's ancient history may have first introduced sixty-four hexagrams into Europe and influenced Leibniz. Many Achievements of Higher Mathematics Leibniz's achievements in mathematics are enormous, and his research and achievements have penetrated into many fields of higher mathematics. His series of important mathematical theories laid the foundation for later mathematical theories. Leibniz once discussed the properties of negative numbers and complex numbers, and concluded that the logarithm of complex numbers does not exist, and the sum of * * * conjugate complex numbers is a real number. In later research, Leibniz proved that his conclusion was correct. He also studied linear equations and discussed the elimination method in theory. First, he introduced the concept of determinant and put forward some theories of determinant. In addition, Leibniz also founded the basic concept of symbolic logic. Contribution of Computer Science 1673, Leibniz went to Paris to make a computer that can perform addition, subtraction, multiplication, division and square root operations. This is another progress of computing tools after Pascal adder. After Pascal's death, Leibniz found a piece of "adder" paper written by Pascal himself, which aroused his strong desire for invention and determined to extend the function of this machine to multiplication and division. Leibniz had a hard life in his early years. After getting an opportunity to go to France, I created an opportunity to realize my long-cherished dream of being a computer. In Paris, Leibniz hired some famous mechanical experts and craftsmen to assist in the work, and finally built a more perfect mechanical computer in 1674. The machine invented by Leibniz is called "multiplier" and is about 1 meter long. A series of gear mechanisms are installed inside. In addition to the large size, the basic principles are inherited from Pascal. However, Leibniz added a device called "stepping wheel" to the computer. The stepping wheel is a long cylinder with nine teeth, and the nine teeth are distributed on the surface of the cylinder in turn; There is also a pinion that can move along the axial direction to mesh with the stepping wheels one by one. Every time the pinion rotates, the stepping wheel can rotate110 and 2/ 10 respectively according to the number of teeth engaged with the pinion until 9/ 10. In this way, it can add and subtract continuously and repeatedly, and in the process of turning the handle, this repeated addition and subtraction can be converted into multiplication and division. Leibniz's contribution to the computer is not just a multiplier. Around AD 1700, Leibniz was inspired by China's "Yi Tu" (gossip) given to him by his friends, and finally realized the true meaning of binary numbers. Although Leibniz's multiplier still uses decimal system, he first proposed the binary algorithm of computer design, which laid a solid foundation for the modern development of computers. Rich achievements in physics Leibniz's achievements in physics are also extraordinary. 167 1 year, Leibniz published the article "New Hypothesis of Physics", and put forward the concrete principle of motion and the abstract principle of motion, holding that a moving object, no matter how small, will move with the part of the object in a completely static state. He also seriously discussed the conservation principle of momentum put forward by Descartes, put forward the rudiment of the conservation principle of energy, and published a brief proof of Descartes and others' obvious mistakes in the laws of nature in Teacher's Magazine, put forward the problem of the quantity of motion, proved that momentum cannot be used as the unit of measurement of motion, and introduced the concept of kinetic energy, which was considered as a universal physical principle for the first time. He also fully proved that perpetual motion machine is impossible. He also opposed Newton's absolute concept of time and space, and thought that "there is no space without matter, and space itself is not an absolute reality", and that "the difference between space and matter is just like the difference between time and motion, but although these things are different, they are inseparable". This idea later attracted the attention of Mach, Einstein and others. 1684, Leibniz pointed out in the article "New Analysis and Proof of Solid Force" that fiber can be stretched, and its tension is proportional to the elongation, so he proposed to apply Hooke's Law to a single fiber. This hypothesis was later called marriott-Leibniz theory in material mechanics. In optics, Leibniz has also made some achievements. He deduced the law of refraction by using the extreme value method in calculus, and tried to explain the basic laws of optics by using the extreme value method. It can be said that Leibniz's research on physics has been moving towards the goal of establishing an axiomatic system similar to Euclidean geometry for physics. Philosophical contribution of Monadologie's works by modern German philosopher G.W. Leibniz to monism. The original monologue is in French, without a title. 1720, Kohler published a German translation of this article, 172 1 year, and Tang Di translated it into Latin according to the German translation. 1840, J.E. Altman found the original text in Leibniz's manuscript, included it in the Complete Works of Leibniz's Philosophy, and added a title. This article is the work of Leibniz, which highly concentrates the main points he expounded in many philosophical works. Although the space is short, the content is rich. The full text consists of 90 sections, which are roughly divided into two parts: Section 1 ~ 48 mainly discusses the properties of all entities, including that entities should be the last unit that constitutes compound words, and they have no parts themselves, which is a simple thing, that is, a spiritual list; The entity itself should have the inherent dynamic principle and so on. Sections 49 ~ 90 mainly discuss the relationship between entities, including the theory that harmony is predetermined and that this world is "the best world in all possible worlds" and so on. Leibniz's monism is an objective idealism system, which tends to compromise with religious theology, but it also contains some reasonable dialectical factors, such as the idea that everything moves on its own. Leibniz's epistemology is in the same strain as his monism. Starting from the ranking of the list, he belittled sensibility and raised rationality, and regarded perceptual knowledge as pure animal knowledge. Because of this, he opposed empiricism, especially Locke's empiricism. His new theory of human reason was written specifically to oppose Locke's theory of human reason. He believes that empiricism only grasps individual things, but cannot grasp universal and inevitable things. According to experience, this is animal behavior. He said: "Animals only rely on experience and rely on examples to guide themselves." "Animal associations are like purely empirical associations. They think that everything that happened before will happen again in situations that make them feel similar, and they can't judge whether the same reason still holds. This is why people are so easy to catch animals, while pure empiricists are so easy to make mistakes. " Leibniz criticized empiricism from the perspective of the distinction between the universal inevitability of scientific knowledge and individual accidental sensory experience, which is profound. As a rationalist, Leibniz tried to reconcile empiricism and rationalism, trying to find a middle way between Descartes and Locke's theories. After a detailed analysis of Locke's empiricism, he wrote: "I have always been and still agree with the natural concept of God advocated by Descartes, so I also believe that there are other natural concepts that cannot come from feeling. Now I have gone further according to this new system; I even think that all thoughts and actions of our soul come from within ourselves and cannot be given to it by feeling. " Leibniz inherited Descartes' rationalism and advocated the theory of "natural concept". In his view, the list has no "window" for external things to come in and out, and it cannot accept the influence of any external things. Therefore, there can be no objective source of knowledge, only natural. However, Leibniz does not agree with Descartes that the human mind is born with a clear concept of talent. He said: "We can't imagine that in the soul, we can read the eternal law of reason like an open book, just like reading the judge's decree on the bulletin board, without any difficulty and exploration." In his view, the concept of talent was not always clear from the beginning (except in God), but gradually developed from the vague perception of talent. "Thought and truth are endowed in our hearts as natural tendencies, talents, habits or potential abilities, rather than as practical functions." It can be seen that Leibniz emphasized the "potential" of the concept of talent when promoting it. When describing how to turn potential things into reality, he emphasized the role of sensory experience. He said: "As long as we focus on the opportunities provided by feelings, we can find these laws in our hearts." It can be seen that Leibniz does not deny that feeling plays a certain role in the cognitive process. However, Leibniz believes that although the perceptual experience of individual things is necessary, they can only provide us with some special individual examples and cannot provide universal truth. He implies here that since the knowledge of universality and inevitability does not come from sensory experience, it can only be possessed by the mind. Locke pursues the principle of materialism and empiricism, that is, "everything in reason is in feeling". He asserted that all our knowledge is based on experience, and our mind is like a whiteboard, on which experience can write its own symbols. Leibniz proved that thinking is essentially active. Therefore, he supplemented Locke's principle in "A New Theory of Human Reason", that is, "everything in the mind first exists in the feeling, except the reason itself". He believes that since the mind is active, it will not be a whiteboard. He compared the human mind to marble with lines. Although the grain on marble is not a ready-made image, the existing grain of marble determines what kind of image it is suitable for carving. "If there are some lines on this stone, indicating that the image of Hercules is better than other images, this stone will be more determined to carve this image. The image of Hercules can be said to be a gift on this stone to some extent." It can be seen that on the one hand, Leibniz, like Descartes, advocates the theory of "natural concept"; On the other hand, unlike Descartes, he thinks that ideas have potential talent, that is, ideas are initially fuzzy perceptions hidden in people's minds, which need to be processed and pondered rationally before they can become clear ideas. Here, Leibniz expounds the valuable dialectical thought that cognition is a development process, and also shows that his rationalism is a rationalism that gives in to empiricism. As to whether the subject of knowledge is material or spiritual entity, Leibniz opposes Locke's view that God may endow material ideological power. He believes that the mental list is the only entity, and without material entities, the subject of cognition cannot be material entities. He said, "What can be felt or thought can't be mechanical." "Matter can feel and think, which is not a natural thing. It can do this only in two ways. One way is for God to connect it with another entity that can think naturally, and the other way is for God to put ideas into matter in a miraculous way. " On the view of truth, Leibniz thinks that there are two kinds of truth: "the truth of reasoning" and "the truth of fact". When he determined the difference between them, he described their principles. One is the principle of contradiction, because of this principle, we believe that what hides contradiction is fallacy, and the opposite of fallacy is truth. In his New Theory of Human Reason, he described the principle of contradiction in detail: any judgment is either true or false. This includes two true judgments: 1. A judgment cannot be both true and false; 2. Judgment cannot be neither true nor false. So he said: "the truth of reasoning is inevitable, and their opposites are impossible." For example, knowledge derived from mathematical axioms and logical rules is the truth of reasoning and must be reliable. The other is the principle of sufficient reason. Because of this principle, we can see that no phenomenon can be true or realistic, and no judgment can be fair-because there is no sufficient reason to explain why things are like this, not like that. He said: "If anything is true or true, if any statement is true, then there must be a good reason why it is this way and not that way, although these reasons are often unknown to us." Therefore, "the truth of facts is accidental, and their opposite is possible." He believes that perceptual perception provides "the truth", which is accidental and unreliable. He wrote: "The original proof of the inevitable truth only comes from reason, and other truths come from the observation of experience or feeling. Our mind can know two truths, but it is the source of the former truth; For a universal truth, no matter how much special experience we have, if we don't rely on reason to understand its inevitability, we will never be able to determine it by induction. " It can be seen that Leibniz advocates the rationalist view of truth and thinks that the truth of reasoning is based on the mind, not the correct reflection of objective things. In Leibniz's view, in the understanding of practical things, because human beings can't find the final and sufficient reason, these things and the truth consistent with them are accidental. Leibniz emphasized that "parity" is only for us human beings, because human beings cannot find sufficient reasons for truth. For God, all truth is intuitive, and there is no difference between the truth of reasoning and the truth of fact, because the world and its principles are created by God, and God can see everything. In a word, Leibniz held high the truth of reasoning and showed the tendency of rationalism, but after all, he admitted the truth of facts and thought it was provided by perceptual perception, which reflected his concession to empiricism. However, in Leibniz's view, "the truth of facts" does not mean the facts that exist objectively. In his view, everything is the sum of lists, but many lists have only unconscious perception besides God's absolutely clear and definite perception, which Leibniz called the "negative activity" of lists. In his view, matter or object is a combination of these lists. Matter and objects do not exist. They are just vague and unconscious perceptions. They are just phenomena. However, these phenomena must abide by certain laws. These laws are not the laws of material nature, but the laws of appearance, which is the highest appearance-God's arrangement of purpose. In this way, teleology is based on the explanation of natural mechanism.