History of mathematics

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The history of mathematics is a science that studies the occurrence, development and laws of mathematical science. It not only traces back the evolution and development process of mathematical content, ideas and methods, but also explores various factors that affect this process, as well as the impact of the development of mathematical science on human civilization Impact. Therefore, the research object of mathematical history not only includes specific mathematical content, but also involves history, philosophy, culture, religion, etc social sciences And humanities Content is an interdisciplinary subject.

Chinese name: History of mathematics
Foreign name: History of mathematics
Three crises: 1. Irrational number 2, infinitesimal number 3, Russell paradox
Mathematical embryonic stage: Before 600 BC

Primary Mathematics Period: 600 BC to the middle of the 17th century
Variable mathematics period: From the middle of the 17th century to the 1920s
Modern Mathematics Period: 1920s to World War II
Modern Mathematics Period: Since 1940s
Discipline code: eleven thousand and eleven^[1-2]

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six Chronology of Events
seven Three crises

research meaning

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The history of mathematics belongs to both the field of history and the field of mathematical science. Therefore, the study of the history of mathematics should follow the laws of both history and mathematical science. According to this feature, mathematical analysis can be regarded as a special auxiliary means for the study of the history of mathematics. In the absence of historical data or when historical data are indistinguishable, standing at the height of modern mathematics, mathematical principle analysis can be carried out on the content and methods of ancient mathematics, so as to achieve the purpose of rectifying the source, summarizing the theory and putting forward historical hypotheses. Mathematical analysis is actually a connection between "ancient" and "modern".

The significance of studying the history of mathematics lies in:

1. The scientific significance of the history of mathematics

Every science has its own history of development. As a science in history, it has both its historicity and its realism 。 Its reality is first shown in Scientific concepts And the continuity of methods scientific research To some extent, it is the deepening and development of scientific tradition in history, or the solution of scientific problems in history. Therefore, we can not separate the relationship between scientific reality and scientific history. Mathematical science has a long history. Compared with natural science, mathematics is more cumulative science, and its concepts and methods are more continuous, such as ancient civilization The decimal value system notation and four rules formed in Algorithm , which we still use today, such as Fermat's conjecture 、 Goldbach conjecture Such historical problems have long been the research focus in the field of modern number theory. Mathematical traditions and materials of mathematical history can be developed in real mathematical research. Many famous mathematical masters at home and abroad have profound mathematical history cultivation or research, and are good at drawing nutrients from historical materials to make the past serve the present and bring forth the new. Famous mathematician in China Wu Wenjun Sir, in his early years topology He made outstanding achievements in his research field and began his research in the 1970s History of Chinese Mathematics He created a new situation in the theory and method of the research on the history of Chinese mathematics, especially under the inspiration of Chinese traditional mathematical mechanization thought“ Wu Method ”About Machine Proof of Geometric Theorem His work is worthy of making the past serve the present and revitalizing national culture A model of.

2. The cultural significance of the history of mathematics

M. Klein, an American historian of mathematics, once said: "The general characteristics of an era are closely related to the mathematical activities of this era to a large extent, and this relationship is particularly obvious in our era.". "Mathematics is not only a method, an art or a language, but also a rich content Knowledge system , its contents are natural scientist 、 Social scientist , Philosophers logician And artists are very useful, and at the same time, they influence the theories of politicians and theologians. Mathematics has widely influenced human life and thought, which is the formation of modern culture The main force of. Therefore, the history of mathematics is a reflection of human beings from one side cultural history It is also the most important part of the history of human civilization. many historian Through the mirror of mathematics, learn about other major ancient Cultural characteristics And value orientation 。 Mathematicians in ancient Greece (600-300 BC) emphasized rigorous reasoning and the conclusions drawn from it, so they did not care about the practicality of these results, but taught people to carry out abstract reasoning and inspired people to pursue ideals and beauty. Through the investigation of the history of Greek mathematics, it is easy to understand why ancient Greece has beautiful literature, extremely rational philosophy, and idealized architecture and sculpture that are difficult for later generations to surpass. and Rome The history of mathematics tells us that, Roman culture It was foreign. The Romans lacked originality and paid attention to practicality.

3. Educational significance of the history of mathematics

After learning the history of mathematics, we will naturally feel that the development of mathematics and Illogical In other words, the actual situation of mathematical development and what we have learned mathematics textbooks Very inconsistent. The mathematics we learned in middle school basically belongs to the 17th century Calculus former Elementary mathematics Knowledge, and most of the content of mathematics department in university is in the 17th and 18th century Advanced mathematics 。 these ones here Mathematics textbook It has been tempered and repeatedly compiled under the guidance of the principle of combining scientificity with educational requirements. It is a knowledge system that selects and compiles historical mathematical materials according to a certain logical structure and learning requirements. In this way, it will inevitably abandon the actual background, knowledge background The evolution process and various factors that lead to its evolution, therefore, it is difficult to obtain the original appearance and panorama of mathematics only by learning mathematics textbooks. At the same time, it ignores those mathematical materials and methods that have been eliminated by history but may be useful for real science. The best way to make up for this deficiency is to learn the history of mathematics.

In the eyes of ordinary people, mathematics is a boring subject, so many people regard it as a daunting course. To some extent, this is because our mathematics textbooks often teach some rigid and unchanging mathematical content. If we infiltrate the content of mathematical history into mathematics teaching to make mathematics alive, we can stimulate students' learning interest It also helps students to deepen their understanding of mathematical concepts, methods and principles.

The history of science is a science Interdisciplinary From the perspective of the current education situation, the gap between liberal arts and science has led to the fact that the talents trained by our education have become increasingly unable to adapt to today's natural science and social sciences It is precisely because of the interdisciplinary nature of the history of science that the highly penetrated modern society can show its role in communicating arts and sciences. Through learning the history of mathematics, students in the mathematics department can accept mathematics Professional training At the same time humanities Students of liberal arts or other majors can understand the general picture of mathematics and gain mathematical and physical training through learning the history of mathematics. In history, the achievements and moral character of mathematicians will also play a very important role in the cultivation of teenagers' personality.

Chinese mathematics has a long history. Before the 14th century, it was the most developed country in the world in mathematics. There were many outstanding mathematicians and made many brilliant achievements. Its long standing algorithmic mathematical model centered on calculation, procedural and mechanical geometrical theorem Of Deductive reasoning It reflects the characteristic axiomatic mathematical model and alternately affects the development of world mathematics. Due to various complex reasons, China fell behind after the 16th century, and it gradually merged into the trend of modern mathematics after a long and difficult development process. As a result of educational mistakes, we, who are influenced by modern mathematical civilization, tend to forget our ancestors Traditional science know nothing about. The history of mathematics can help students understand Ancient Chinese Mathematics To understand China Modern mathematics The reason for backwardness, the current situation of modern mathematical research in China, and developed country The gap in mathematics to stimulate students' patriotic enthusiasm and revitalize Ethnic science 。

History

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The task of mathematical history research is to understand the basic historical facts in the process of mathematical development, reproduce its original appearance, and Historical phenomenon For mathematical achievements Theoretical system And Development mode Make scientific and reasonable explanation, explanation and evaluation, and then explore the law and cultural essence of the development of mathematical science. As the basic method and means of mathematical history research, there are often historical textual research, mathematical analysis comparative study Etc.

The duty of historians is to narrate history according to historical data. Realism is the basic principle of historiography. Since the 17th century, western history has been formed textology It appeared earlier in China, especially in the Qianjia period of the Qing Dynasty. It is still the main method of historical research, but with the progress of the times, the textual research method is constantly improving, Scope of application It's just expanding. Of course, it should be recognized that the historical data are true and false, and the process of textual research involves the textual examiners mentality This will inevitably affect the choice of textual research materials and the results of textual research. In other words, the authenticity of the historical research conclusion is relative. At the same time, we should realize that textual research is not the ultimate goal of historical research, and the research of mathematical history cannot be textual research for textual research.

No comparison, no thinking, and all scientific thinking and investigation are indispensable comparison, or comparison is the beginning of knowledge. The development of the world is multipolar and different countries and regions . Cultural exchanges between different nationalities common development Therefore, with diversification History of World Civilization Development and Western centrism With the weakening of ideas, heterogeneous regional civilization has been increasingly valued, thus Mathematical culture The comparison of mathematics and the study of the history of mathematical communication are also increasingly active. The comparative study of mathematical history is often carried out around three aspects: mathematical achievements, mathematical scientific paradigm, and the social background of mathematical development.

The history of mathematics belongs to both the field of history and the field of mathematical science. Therefore, the study of the history of mathematics should follow the laws of both history and mathematical science. According to this feature, mathematical analysis can be used as a special auxiliary means for the study of mathematical history modern mathematics Height of, right Ancient mathematics The contents and methods shall be analyzed by mathematical principles to achieve a thorough reform of the original Theoretical summary And the purpose of putting forward the historical hypothesis. Mathematical analysis is actually a connection between "ancient" and "modern".

Ancient history

① ancient Greek Someone once wrote the History of Geometry, which has not been handed down.

② Proclos in the 5th century Euclid 《 Geometric primitives 》Some data are also retained in the notes to Volume I.

③ Medieval arab countries In some biographical works and mathematical works, the life stories of some mathematicians and other materials related to the history of mathematics are described.

④ In the 12th century, ancient Greece and the Middle Ages Arabic Mathematics Book incoming Western Europe 。 The translation of these works is both Mathematical research It is also the arrangement and preservation of classical mathematical works.

modern history

From the 18th century, J Monticera 、C. Bossuet 、A.C. Kostner At the same time, Montekla published the History of Mathematics in 1758 (supplemented by Laurent from 1799 to 1802) as a representative. Since the end of the 19th century, more and more people have studied the history of mathematics, Dynastic history The research on the history of science and science has also been carried out gradually. After 1945, new developments have taken place. The research on the history of mathematics after the end of the 19th century can be divided into the following aspects.

1. General History Research

The representative works can be cited as M B. Cantor's Lectures on the History of Mathematics (4 volumes, 1880-1908) and C B. Boyer (1894, 1919) D E. Smith (2 volumes, 1923-1925), Loria (3 volumes, 1929-1933), etc. French Bourbaki School I wrote a history of mathematics《 Principles of Mathematics 》。 Represented by Yushkevich Soviet Union There are also scholars and Japanese scholars represented by Yoshihiro Miyong and Juntaro Ito Multivolume The General History of Mathematics was published. Written by M. Klein in 1972《 Ancient and modern mathematical thought 》A book is a masterpiece since the 1970s.

2. History of Ancient Greece

many Ancient Greek Mathematics Jia's works have been translated into modern languages, and J 50. Heiberg, Hulch, T.L. Heath, etc. Loria and Heath also wrote a general history of ancient Greek mathematics. Since the 1930s, the famous Algebra home Van der Waerden stay History of Ancient Greek Mathematics They also made achievements. Since the 1960s Hungary The work of A. Saab is more prominent. He discussed the origin of Euclid's axiom system from the history of philosophy.

3、 Ancient Egypt history

hold Babylon Cuneiform ironing board Book counting and Ancient Egypt Papyrus It is a hard job to translate the books into modern characters. Chase And Archibald and others have translated the papyrus book, and Neugebauer The study of cuneiform characters on clay tablets for decades is even more famous. His Research on the Mathematical Historical Materials of Cuneiform Characters (1935, 1937) and the Mathematical Book of Cuneiform Characters (co authored with Sax, 1945) are authoritative works in this field. His book, Ancient Precise Science (1951), collects the research achievements on the history of mathematics in ancient Egypt and Babylon over the past half century. Van der Waerden "The Awakening of Science" (1954) was added to the history of ancient Greek mathematics and became one of the authoritative works in the history of ancient world mathematics.

4. Dynastic history

Lectures on the History of Mathematical Development in the 19th Century (1926-1927), written by German mathematician (C.) F. Klein, is Chronotype The beginning of modern mathematical history research be published in book form In the 20th century, however, most of the views on mathematics reflected in it were in the 19th century. Until 1978, French mathematician Jean Alexander Eugene Diogenes 1700-1900 Introduction to the History of Mathematics 》Before publication, there were not many monographs on the history of chronological mathematics, but there were famous papers such as "Mathematics for Half a Century" written by (C.H.) H. Weier. For the history of various branches of mathematics probability theory , until manifold Concept Hilbert mathematical problem There are a variety of monographs, and there are many famous writers. Many famous mathematicians participated in the study of the history of mathematics, probably based on (J. -) H. Poincare's belief that "if we want to predict the future of mathematics, the appropriate way is to study the history and current situation of this science", Or as H. Weir said: "If we do not know the concept, method and result of the establishment and development of ancient Greek predecessors, we cannot understand the goal of mathematics in the past 50 years, nor its achievements."

5. Biography of Mathematicians

And their complete works and《 Selections 》This is one of the great works in the study of the history of mathematics. In addition, a variety of Selected Readings of Mathematical Classics appeared, compiling precious fragments of famous works of mathematicians in past dynasties.

6、 Journal of Mathematics

It first appeared at the end of the 19th century. M.B. Cantor (1877~1913, volume 30) and Loria (1898~1922, volume 21) both edited the magazine of mathematical history. The most famous one is the Treasure of Mathematics (1884~1915, volume 30) edited by Ernestlem. Modern is international History of Science The International Journal of the History of Mathematics, edited by the Society for the History of Mathematics.

Chinese history

China is famous in the world for its long historical tradition《 Legal chronicle 》The function and history of mathematics are often discussed in the article of "counting". For example, the earlier Book of the Han Dynasty - Annals of the Law and Calendar said that mathematics was "the deduction of calendar, the generation of laws, the control of instruments, the gauge circle, the square, the weight, the balance, the yardstick Jialiang , explore the hidden, go deep and reach far, don't forget ". Sui Shu · Lv Li Zhi records the history of pi calculation, recording Zu Chongzhi The brilliant achievements of. Official history《 biographies 》Sometimes, biographies of mathematicians are also given. Official《 Classics and records 》There is a mathematical bibliography.

In the preface and postscript of ancient Chinese mathematical books, the content of mathematical history often appears.

as Liu Hui Note《 Chapter Nine Arithmetic 》Preface (263) once talked about the history of the formation of Nine Chapters of Arithmetic; Wang Xiaotong In the "Shangji Ancient Suanjing Table", comments were made on the mathematical work of Liu Hui, Zu Chongzhi and others; Zu Yiwei《 Siyuan Jade Mirror 》Written Preface Described by Tianyuan Skill Developed into Quaternion History of. Song edition《 Numerology 》The appendix later contains "the origin of mathematics", which is the first one used in China and the world Typography The preserved data of mathematical history. Cheng Dawei 《 Algorithm statistics 》(1592) At the end of the book is attached the "Origin of Suanjing", recording the mathematical bibliography between the Song and Ming dynasties.

The above mentioned fragmentary data are correct Ancient Chinese Mathematics The systematic arrangement and research of history is based on qianjia school Under the influence of the late qing dynasty. It mainly includes: ① sorting out and studying ancient calculation books《 Ten Books of Suanjing 》The revision, annotation and publication of (Han Tang Jian Shu) and Song Yuan Jian Shu Dai Zhen （1724～1777）、 Li Huang （?～1811）、 Ruan Yuan (1764-1849), Shen Qinpei (calculated in 1829《 Siyuan Jade Mirror 》）、 Roslin (1789~1853) and others ② edited and published《 Biography 》(Biographies of mathematicians and astronomers), which "originated from the Yellow Emperor and ended in the Zhao (Qing) Dynasty, and all scholars who studied it were handed down by people", was created by Ruan Yuan Li Rui (1795-1799). Later, Roslin wrote an addendum (1840), Zhu Kebao Work《 Three Parts of Chou Ren's Biography 》（1886）， Huang Zhongjun Again《 Part Four of Chou Ren's Biography 》（1898）。 Chou Ren Zhuan is actually a mathematical history of biographies. It has a large number of people, rich information and fair comments, which can be compared with Monticera's history of mathematics.

Use modern Mathematical concept , Yes History of Chinese Mathematics So that the study of the history of Chinese mathematics can be established in modern times scientific method The founder of the above discipline is Li Yan and Qian Baocong 。 Since the May 4th Movement, they began to collect ancient books for textual research, collation and research for more than half a century. Since the 1930s, both of them have published monographs on the history of Chinese mathematics《 History of Chinese Computing 》（1937）、《 Outline of Chinese Mathematics 》（1958）； Qian Baocong wrote the History of Chinese Mathematics (I, 1932) and edited the History of Chinese Mathematics (1964). Qian Baocong proofread《 Ten Books of Suanjing 》(1963), together with the above-mentioned monographs, are authoritative works.

Since the end of the 19th century（ Weilie Yali He Shishen, etc.) published articles on the history of Chinese mathematics in foreign languages. Japanese in the early 20th century yoshio mikami The Development of Mathematics in China and Japan and the 1950s Joseph Needham In his masterpiece《 History of Science and Technology in China 》(Volume III) gives a comprehensive introduction to the history of Chinese mathematics. Some Chinese classical calculation books have been translated into Japanese, English, French, Russian, German and other languages. In Britain, the United States, Japan, Russia, France Belgium Some people in other countries directly use China Classical literature To study the history of mathematics in China and compare it with the history of mathematics in other countries and regions.

Scope of study

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According to the scope of research, it can be divided into internal history and external history.

Internal history: study the history of mathematical development from the internal causes of mathematics (including the relationship with other natural sciences);

External history: from the outside Social causes (including political, economic, philosophical and other reasons) social factors Relationship between.

History of Mathematics and Mathematical research Branches of, and Social History It is closely related to all aspects of cultural history, which indicates that the history of mathematics has Interdisciplinary And comprehensive.

In terms of research materials, archaeological materials, historical archives, historical mathematics Original literature Various historical documents, ethnological materials, cultural history materials, as well as the interview records of mathematicians, are important research objects, among which the original mathematical documents are the most commonly used and the most important first hand research data 。 In terms of research objectives, we can study the evolution history of mathematical ideas, methods, theories and concepts; Can study mathematical science and human society Interactive relationship; Can study the history of the spread and exchange of mathematical ideas; Can study mathematician's life and so on.

research contents

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1. The research contents of the history of mathematics are:

one
On the Methodology of Mathematics History Research
two
History of mathematics General History
three
History of Mathematics
four
The History of Mathematics in Different Countries, Nationalities and Regions and Its Comparison
five
History of Mathematics in Different Periods
six
Biographies of mathematicians
seven
Mathematical thought , Concept Mathematical method History of development
eight
The relationship between mathematical development and other scientific and social phenomena
nine
History of Mathematics Education
ten
History of mathematics philology

2. According to its research scope, it can be divided into internal history and external history:

one
Internal history: study the history of mathematical development from the internal causes of mathematics;
two
External history: to study the relationship between mathematical development and other social factors from external social reasons.

Development stage

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The development of mathematics is phased, so researchers divide the history of mathematics into several periods according to certain principles. Academics usually divide the development of mathematics into the following five periods:

one
The embryonic stage of mathematics (before 600 BC);
two
Elementary mathematics Period (600 BC to the middle of the 17th century);
three
Variable mathematics period (from the middle of the 17th century to the 1920s);
four
Modern mathematics Period (1920s to the Second World War);
five
modern mathematics Period (since 1940s).

Chronology of Events

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It is unknown how many people have worked hard in the development of mathematics and listed the chronology of major events in the history of mathematics:

Recommended Egyptian hieroglyphics around 3000 BC

Early Babylon, 2400-1600 BC Clay tablet For cuneiform characters, the notation of 60 carry value system is adopted. Known Pythagorean theorem

1850-1650 BC Egypt Papyrus （ Moscow papyrus And Rhineland cursive script), using the 10 digit non positional numeration system

The Yin Ruins of China from 1400 BC to 1100 BC Oracle , already in decimal system Numeration

Duke Zhou (11th century BC), Gou San, Gu Si, Xian Wu known in the Shang Dynasty

About 600 BC Greece Thales Started the proof of proposition

About 540 BC Greece Pythagorean school , found Pythagorean theorem, and led to the discovery of incommensurability

About 500 BC, India's Sutra of Rope Method gave a fairly accurate value of √ 2, and knew the Pythagorean Theorem

About 460 BC, the Greek school of Homo sapiens proposed Geometric drawing Three major problems: turning a circle into a square Trisection angle And double cube

About 450 BC Greece Elia School Of Zeno Put forward a paradox

430 BC Greece antiphon propose Exhaustion method

About 380 BC Greece Plato In Athens, he founded a "school park" and advocated the training through geometry learning Logical thinking ability

370 BC Greece Odexsos Establishing the theory of proportion

The History of Geometry by Eudomos, circa 335 BC

In China, decimal value system is adopted

Euclid, Greece, circa 300 BC《 Geometric primitives 》Is the earliest example of establishing deductive mathematical system with axiom method

287-212 BC Greece Archimedes , the area and volume of a large number of complex geometric figures are determined; The upper and lower bounds of pi are given; Propose to speculate the answer of the question with mechanical method, implying the thought of modern integral theory

230 BC Greece Eratoceni Invention“ Sieving method ”

225 BC Greece Apollonius By《 Conic theory 》

About 150 BC, the earliest mathematics book in existence in China《 Arithmetical book 》Completed the book (in Hubei from 1983 to 1984 Jiangling Unearthed)

About 100 BC China《 Zhou Bi Suanjing 》A book describing Pythagorean theorem

The most important mathematical works in ancient China《 Chapter Nine Arithmetic 》It has been basically shaped through generations of additions and revisions (the first time when the book was written was between 50 and 100 AD), including positive and negative arithmetic, fraction Four arithmetic operations 、 Linear equations Solution, proportional calculation and linear interpolation are all important contributions in the history of world mathematics

About 62 A.D. Greece Helen Give a formula to express the area with the length of three sides of a triangle（ Helen formula ）

In about 150 AD, Ptolemy of Greece wrote Astronomy, which developed Trigonometry

About 250 A.D. Greece Difantu "Arithmetic", dealing with a lot of Indefinite equation Question, and a series of abbreviations are introduced. It is the representative work of ancient Greek algebra

In about 263 AD, Liu Hui of China annotated Nine Chapters of Arithmetic, creating Cyclotomy , calculate pi, prove Circle area formula , derivation tetrahedron and Quadrangular pyramid Volume, etc., including Limit thought

About 300 A.D. China《 Sunzi Suanjing 》It is a book that systematically describes the numeration system of planning, Volume II“ Things are unknown ”The question is Sun Tzu's remainder theorem Origin of

320 AD Greece Pappus By《 Mathematical Collection 》, summarizing the research achievements of various ancient Greek schools, and recording“ Pappus's hexagon theorem ”And Revolver Volume calculation method

In A.D. 410, Shhpatiya, the first female mathematician in history, annotated the works of Euclid, Diophantus and others

In 462 AD, Zu Chongzhi of China calculated the pi between 3.1415926 and 3.1415927, and took 22/7 as Approximation rate 355/113 Density ratio (now called Ancestral rate ）

Chinese Zu Chongzhi and His Son Zu The principle of "the same power potential leads to the same product" is proposed, which is now called Zuo principle , equivalent to Western Kavalieri Principle (1635)

499 AD India Ayeport He wrote "Collected Works of Ayeport", summarizing the knowledge of astronomy, arithmetic, algebra and trigonometry in India at that time. Given π=3.1416, try Continued fraction Solving indeterminate equations

In 600 AD, Liu Zhuo of China initiated the second equal spacing Interpolation Equation, then developed unequal interval quadratic Interpolation method (Monk a line , 724) and cubic interpolation（ Guo Shoujing ，1280）

Written by Wang Xiaotong of China in about 625 AD《 Seize the Ancient Suanjing 》, is the first figure Cubic equation Numerical solution 's works

628 AD India Brahmagupta It is known that he wrote the Calculation Book of the Brahmanic Calendar Cyclic Quadrilateral The area calculation method has promoted I Quadratic indeterminate equation Research on

656 A.D. China Li Chunfeng Ten books of Suanjing are annotated, and later called Ten Books of Suanjing

In 820 AD, Arab Hualazimi wrote Algebra Equation solving As the main content, the book was translated into Latin Introduced to Europe

Around 870 AD, India appeared with zero decimal system Digital, later introduced into Arabia and evolved into India today Arabic numerals

C. 1050 China Jia Xian propose Binomial coefficient Table (now known as Jia Xian Triangle and Multiplication opening method ）

1100 Arabia omar khayyam The first two Conic curve To represent the root of the cubic equation

1150 India Boshgalo Second The Collection of Boshgara Writings was written in the Middle Ages Indian Mathematics It gives some special solutions to the binary indefinite equation x 2.=1+py 2. It has some knowledge of negative numbers and uses irrational numbers

In 1202, Italian L. Fibonacci wrote the Abacus Book, which systematically introduced the Indo Arabic numerals and various algorithms of integers and fractions to Europeans

A.D. 1247 China Qin Jiushao By《 Count Nine Chapters 》, creating Dayan Seeks a Skill Summation Higher order equation Numerical solution Of Positive and negative square cutting , equivalent to Western Horner method （1819）

A.D. 1248 China Li Ye By《 Circular borescope 》Is the first book in existence in China that systematically discusses Tianyuanshu

About 1250 A.D. Arabia Nasir al-Din al-Tusi Began to make trigonometry independent from astronomy《 Geometric primitives 》Translated as arabic

1303 China Zhu Shijie By《 Siyuan Jade Mirror 》, promote Tianyuan Technique to four element technique, study Higher order arithmetic sequence Summation problem

In A.D. 1325, T. Bradwarden of England will tangent Cotangent Introducing trigonometry

Abacus was popularized in China in the 14th century

In about 1360 AD, N. Olsim of France wrote the "Proportional Algorithm", introduced the concept of sub index, and studied the change and rate of change in the "On Map Lines" and other works, creating the principle of map lines, that is, longitude and latitude (equivalent to horizontal Ordinate ）Indicate the position of the point and further discuss Function image

1427 Arabia Kashi He wrote The Key to Arithmetic, systematically discussed the principles and methods of arithmetic and algebra, and worked out 17 accurate digits of pi in the Theory of Circumference

In 1464, J Regmontanus On the General Triangle, the first systematic work on trigonometry in Europe, in which Sine law

Euclid's Geometric Elements (Latin translation) was first printed and published in 1482

1489 Czech Republic Weidman first used the symbols+and - to indicate addition and subtraction

In 1545 AD, G. Caldano of Italy published the Great Art, which described the solution of the cubic equation of S. Ferro (1515), N. Tartaglia (1535) and L Ferrari Of (1544) Quartic equation solution

1572 A.D. Italian R Bombelli The publication of Algebra pointed out that for the irreducible case of cubic equation imaginary number Three must be obtained by operation Real root , giving a preliminary theory of imaginary numbers

1585 Netherlands S. Steven create Decimal fraction (Decimal) notation

1591, France F Weida By《 Introduction to Analytical Methods 》, introducing a large number of algebraic symbols, improving the solution of cubic and quartic equations, and pointing out that Relation between root and coefficient , is the symbol Algebra Of Founder

In 1592 AD, Cheng Dawei of China wrote Zhizhi Suanfa Tongzong, detailing Abacus The book was introduced to Japan and Korea in the late Ming Dynasty

In 1606, Xu Guangqi of China cooperated with Matteo Ricci to bring Euclid《 Geometric primitives 》The first six volumes are translated into Chinese

Founded by J. Napier in 1614 logarithm theory

In 1615, Kepler of Germany wrote the New Solid Geometry of Wine Bucket, which has the method of calculating the volume of wine bucket, and is the transition from Archimedes' quadrature method to modern integration method

It was first proposed by Girard of the Netherlands in 1629 Fundamental theorem of algebra

Fermat of France has learned the gist of analytic geometry, and mastered the pursuit of maximum Minimum method

1635 Italy (F.) B Kavalieri Establishment“ Principle of incomposability ”

In 1637, the French R Descartes The publication of Geometry, founded analytic geometry

French Fermat proposed“ Fermat's big theorem ”

In 1639, G. De Zag of France wrote the First Draft on Dealing with the Intersection of Cones and Planes, which is Projective geometry pioneer

In 1640, B. Pascal of France published "On Conic Curve"

In 1642, French B. Pascal invented addition and subtraction Mechanical computer

In 1655, J Wallis By《 Arithmetica infinitorum 》, importing Infinite series And Infinite product , the first infinity symbol ∞

In 1657, C. Huygens of the Netherlands wrote "On the Reasoning of Dice Game", which was introduced Mathematical expectation Concept is an early work of probability theory. Prior to this, B. Pascal, Fermat, etc. had begun to consider probability theory by dealing with gambling problems

In 1665, the British I Newton A manuscript has been circulated numerology This is the earliest literature on calculus. Later, he further developed stream number technology and established it in his works such as Analysis of Infinite Multinomial Equations (written in 1669 and published in 1711), Stream Number Method and Infinite Series (written in 1671 and published in 1736) Basic Theorem of Calculus

In 1666, G W. Leibniz Written as On the Technology of Combination mathematical logic thought

Written by I. Barrow of England in 1670《 Geometrical Lectures 》, Introduction“ Differential triangle ”Concept

In about 1680 AD, Japanese Guan Xiaohe initiated Hejian, which was introduced determinant Concept, creating the study of "circular theory"

In 1684, G W. Leibniz published the first article in Xueyi Differential calculus The first paper was published two years later Integral calculus Thesis, creative Integral sign

In 1687, the British I Newtonian《 Mathematical Principles of Natural Philosophy 》Publication, first published in geometric form

1689 Switzerland John First· Bernoulli The problem of "steepest droop curve" was raised, which led to Variational method Generation of

France G. - F- L'Hopital Published Infinitesimal Analysis, which contains Lopida rule

Published by I. Newton in 1707《 generalized Arithmetic 》, explains algebraic equation theory

Jacobi Bernoulli, Switzerland, 1713《 Guesswork 》Published with Bernoulli's law of large numbers

1715 A.D. British B Taylor Publication《 Positive and Negative Incremental Methods 》, in which he discovered in 1712 that the function was expanded into series Taylor formula

1722 A.D. France A Desmoff Give the formula (cos φ+i sin φ) ^ n=cos n φ+i sin n φ

1730 Scotland J. Sterling published "Differential Method, or a Brief Introduction to Infinite Series", in which he gave! Of Stirling formula

1731 A.D. France A. - C. Clarow wrote Research on Double Curvature Curve, which created Space curve Theory of

In 1736, L. Euler of Switzerland solved the problem of Konesburg Seven bridge problem

In 1742, C Maclaurin Publication《 General theory of stream number 》, trying to establish the theory of flow mathematics with a rigorous method, in which the Marklaurin expansion is given

In 1744, L. Euler of Switzerland wrote "Skills for Seeking Curves with Certain Maximal or Minimal Properties", marking the birth of variational method as a new branch of mathematics

In 1747, J. le R. d'Alembert of France published "Research on String Vibration" and derived Equation of string vibration , Yes partial differential equation The beginning of research

Published by L. Euler of Switzerland in 1748《 Introductio in analysin infinitorum 》, and later published Differential Calculus (1755) and Integral Calculus (1770), based on the concept of function Calculus Theory, giving a large number of important results, marking the development of calculus New stage

In 1750, G Clem Give a solution Linear equations Of Cramer's rule

Published by L. Oula of Switzerland polyhedron Formula: V-E+F=2

In 1770, J. - L. Lagrange Deeply explore algebraic equations Radical Solve the problem, consider Rational function The number of values taken when a variable is permuted becomes the leader of permutation group theory

Germany J H. Created by Lambert Hyperbolic function Comprehensive study of

In 1777, G- 50. L Buffon raised the question of needle injection, yes Geometric probability Early Study of Theory

In 1779, French □. Bezu wrote "General Theory of Algebraic Equations", which systematically discussed Elimination method theory

In 1788, J. - 50. Lagrange's Analytical Mechanics was published, which made mechanics analysis and summarized the achievements of variational methods

1794 A. - France M. Legendre's Fundamentals of Geometry was published as the standard geometry textbook at that time

Established in France Paris Polytechnic School and Paris Normal University

In 1795, G Mengri The loose leaf paper on the application of analysis to geometry was published as Differential geometry pioneer

In 1797, J- 50. By Lagrange《 Analytic function theory 》He advocates the establishment of calculus theory based on the power series expansion of functions

Norway C. Wessel was the first to give the geometric representation of complex numbers

Published by G. Monge of France in 1799《 descriptive geometry Learning, making descriptive geometry a specialized branch of geometry

Germany C F. Gaussian Give the first proof of the basic theorem of algebra

1799-1825 A.D. French P- S. Laplace 5 volumes of the great book "Celestial Mechanics" was published, which contains many important mathematical contributions, such as Laplace equation , potential function, etc

In 1801, Germany C F. Gauss《 Disquisitiones Arithmeticae 》Publication marks the starting point of near algebra theory

In 1802, J E. The History of Mathematics, co authored by Monticera and Lalande, was published in 4 volumes, becoming the earliest systematic work on the history of mathematics

1807, J. - B. - J. Fourier in heat conduction In the study, the Trigonometric series Representation （ Fourier series ）, his Thought summary Published in 1822《 Analytic theory of heat 》Medium

1810 A.D. J. - D. Gergang Established the Annual Journal of Pure and Applied Mathematics, which is the earliest specialized mathematical journal

1812, England Cambridge The Analytical Society was founded

French P. S. Laplace wrote Analytic Theory of Probability, put forward the classical definition of probability, and introduced analytical tools into probability theory

Read by A. - L. Cauchy of France in 1814 Complex function theory The first important paper《 Report on the Theory of Definite Integral 》(Officially published in 1827), initiated the research of complex variable function theory

1817 A.D. Czech B Polzano By《 Proof of pure analysis 》, given for the first time Continuity Proper definition of derivative and general series astringency Of Discriminant criterion

1818 AD France S- D. Poisson export wave equation Solvable“ Poisson formula ”

1821 A.D. France A- 50. Cauchy published Course in Algebraic Analysis, but the introduction may not be analytical expression The function concept of; Independent of B. Boccano, he put forward the definitions of limit, continuity, derivative, etc. and the criteria for the convergence of series, which is the first influential work in the analysis of rigorous motion

In 1822, J. - V. Poncelet By《 On the projective property of figures 》, established Projective geometry Basics

In 1826, Norwegian N H. Abel The book "On a General Property of a Wide Class of Transcendental Functions" initiated the study of elliptic function theory

Germany A L. Crayle Establishment《 Journal of Pure and Applied Mathematics 》

France J- D. Gergang and J- 5. Pencel is established separately Duality principle

In 1827, Germany C F. Gauss wrote "General Research on Surfaces", creating the intrinsic geometry of surfaces

Germany A F. The Calculus of the Center of Gravity, written by Mabius, introduced Homogeneous coordinate , and J. Pruck et al. opened up the algebraic direction of projective geometry

Written by G. Green in 1828《 mathematical analysis Applications in Electromagnetic Theory, Development Geopotential theory

1829 AD Germany C G.J. Jacobian By《 New Foundation of Elliptic Function Theory 》, Yes elliptic function Founding works of theory

Russia Н.И. Lobachevsky First published Non Euclidean geometry On "On the Basis of Geometry"

From 1829 to 1832, E. Galois of France thoroughly solved the roots of algebraic equations Solvability Question, established the basic concept of group theory

Written by G. Picok of England in 1830《 General Theory of Algebra 》, pioneered the establishment of algebra in a deductive way, paving the way for more abstract ideas in algebra

1832 A.D. Hungarian J Bordeaux Publication《 Science of absolute space 》, independent of Н. И. Lobachevsky proposed the idea of non Euclidean geometry

J. Steiner, Switzerland, Systematic Development of Geometric Interdependence, using Projective projection Concept from Simple structure Structured complex structure, developed projective geometry

In 1836, J Liouville establish French Journal of Pure and Applied Mathematics

1837 A.D. German P G. L. Dirichlet's present general function definition (correspondence between variables)

In 1840, French A. - L. Cauchy proved that differential equation Initial value problem Existence of solutions

1841-1856 A.D. Germany K (T.W.) Weierstrass's work on the rigor of analysis advocated that the analysis should be based on the arithmetic concept, and the ε - δ statement and series of limit should be given Uniform convergence Overview

Read; At the same time, the complex variable function theory is established on the basis of power series

In 1843, W R. Hamilton find Quaternion

1844 AD Germany E E. Kummer found Ideal number Concept of

Germany H G. Glassman Publication《 Linear extension theory 》。 The hypercomplex number system of three components is established, and the general concept of three dimensional geometry is proposed

In 1847, K G. C. von Stowe wrote Geometry of Position, which does not rely on the concept of measurement to establish a projective geometry system

A Gloria propose Abstract group concept

In 1851, German (G.F.) B. Riemann wrote General Theoretical Basis of Functions of Single Complex Variable, giving Single value analytic function It is a classic paper on the function theory of complex variable

In 1854, German (G.F.) B. Riemann wrote "Assumptions on the Basis of Geometry", creating the Riemannian geometry

G. Bull published Research on the Law of Thinking, establishing Logic algebra (i.e Boolean algebra ）

In 1855, A. Kelley of England introduced the basic concept of matrix And operation

In 1858, Germany (G.F.) B. Riemann gave ζ function The integral representation of and the Functional equation , proposed Riemann conjecture Germany A F. Mebius found One-sided surface （ Mebius zone ）

1859 China Li Shanlan Algebra《 Generation of micro products 》And《 Geometric primitives 》The publication of the last nine volumes in Chinese is the beginning of the translation of modern western mathematical works

Li Shanlan of China established the famous Combinatorial identity (Li Shanlan identity)

1861, Germany K (T.W.) Weierstrass Berlin Continuous but everywhere in the speech Nondifferentiable function Example of

1863 A.D. German P G. L. Dirichlet published Lectures on Number Theory Analytic number theory Classic literature of

1865 London Mathematics Society It is the first established mathematics association in history

1866, Russia П. л Chebyshev utilize Chebyshev inequality Establish about independence random variable Sequential Law of large numbers , becoming the central topic of probability theory research

In 1868, E. Bertrami of Italy wrote On the Interpretation of Non European Geometry in Pseudosphere Implementation on Lobachevsky geometry This is the first non Euclidean geometric model

German (G.F.) B. Riemann's Representability of Functions by Trigonometric Series was officially published, establishing Riemann integral theory

In 1871, F. Klein of Germany (C.) Projective space It is obtained by properly introducing measurement in Hyperbolic geometry And Elliptic geometry , which is a non Euclidean geometric model obtained without surface

Germany G (F.P.) Cantor introduced the concept of infinite set for the first time in the study of the uniqueness of the representation of trigonometric series, and laid the foundation for set theory Foundation of

Published by German (C.) F. Klein in 1872《 Erlangen Program 》, established that various geometries are regarded as Transformation group Of Invariant theory And unifies geometry based on group theory

Real number theory Establishment of: G (F.P.) Cantor's basic sequence theory; J. W.R. DeDekin's Partition theory ； K.（T. W. Weierstrass's Monotone Sequence Theory

1873, France C Hermit Prove e's Transcendence

1874, Norway M S. The study of Lie Kaichuang continuous transformation group, now called Li Group theory

Published by G. Frege of Germany (F.L.) in 1879《 Conceptual language 》He established the quantifier theory and gave the first rigorous logical axiom system. Later, he published the Foundation of Arithmetic (1884) and other works, trying to build mathematics on the basis of logic

From 1881 to 1884, Germany (C.) F. Klein and France (J. -) H. Poincare founded Automorphic function theory

From 1881 to 1886, French (J. -) H. Poincare's paper on the curve determined by differential equation established the qualitative theory of differential equation

In 1882, German M Pasch Give the first projective geometry Axiomatic system

Germany F.von Linderman Proving the transcendence of π

In 1887, French (J. -) G. Dabu wrote the General Theory of Surfaces, which developed Moving frame method

In 1889, G Piano By《 New Method of Arithmetic Principle 》, giving Natural number Axiomatic system

1894 A.D. T （J.） Stilges Published Research on Continued Fractions and introduced new integrals（ Stieltjes integral ）

Written by H. Poincare of France (J. -) in 1895《 Positional geometry 》, establish the method of studying manifolds by subdivision, and lay the foundation for combinatorial topology

Germany F G. Frobenius Representation theoretic system study

In 1896, H Minkovsky The Geometry of Numbers, establishing systematic geometric theory of numbers

France J （-S.） Adama And Wali Boussin certification Prime theorem

The first session in 1897 International Congress of Mathematicians In Switzerland Zurich hold

Founded by K. Pearson in 1898 Descriptive statistics

In 1899, D. Hilbert of Germany published Geometric Basis, giving the first complete Euclidean geometry Axiom system, which created axiomatic method , and indicates that Fundamentals of Mathematics Of formalism viewpoint

In 1900, D. Hilbert of Germany gave a speech entitled《 mathematical problem 》Report of. Put forward 23 famous mathematical problems

Three crises

Announce

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1. Irrational number

About the 5th century BC, not accessible Approximation 's discovery led to Pythagoras paradox 。 At that time Pythagorean school Attach importance to the study of the constant factors in nature and society, and call geometry, arithmetic, astronomy and music“ Quadrivium ”Pursue the harmony of the universe Regularity 。 They believe that everything in the universe can be reduced to integers or the ratio of integers. One of the major contributions of the Pythagorean School is to prove that Pythagorean theorem , but we also found some right triangle Of hypotenuse It can not be expressed as an integer or the ratio of integers (incommensurability), such as a right triangle with a right angle side length of 1. This paradox directly violated the fundamental creed of the Bishop School and led to the "crisis" of cognition at that time, which led to The first mathematical crisis 。

By 370 BC, this contradiction was Odox It is solved by giving a new definition to the proportion. His method of dealing with incommensurability appears in the fifth volume of Euclid's The Original. The explanation of irrational numbers given by Eudox and Didkin in 1872 is basically consistent with the modern explanation. Middle school geometry textbook Similar triangles The treatment of "" still reflects some difficulties and subtleties caused by incommensurability. The first mathematical crisis had a great impact on the mathematical ideas of ancient Greece. This shows that, geometry Some of the truths of, Geometric quantity It can not be completely expressed by integers and their analogies, but can be expressed by geometric quantities. The authority of integers began to shake, and the identity of geometry rose; The crisis also showed that intuition and experience are not necessarily reliable, and the proof of reasoning is reliable. Since then, the Greeks began to attach importance to deductive reasoning, and thus established Geometric axiom System, which can not but be said to be Mathematical thought Last great revolution!

2、 Infinitesimal

In the 18th century, Differential method and integration method It has been widely and successfully applied in production and practice, and most mathematicians have no doubt about the reliability of this theory.

In 1734, British philosopher and Archbishop Berkeley published "An Analyst or Advice to an Unbelieving Mathematician", aiming at Calculus The problem of infinitesimal, the basis of Berkeley paradox 。 He pointed out that "Newton was seeking

When the derivative of, we first give x an increment of 0, and apply binomial (x+0) n, subtract from it to get the increment, divide by 0 to get the ratio of the increment to the increment of x, and then let 0 disappear to get the final ratio of the increment. Here Newton violated Law of contradiction The procedure of "x" - first assume that x has an increment and then make the increment zero, that is, assume that x has no increment. " He believes that the infinitesimal dx is both equal to zero and not equal to zero. It is absurd to call it and wave it away. "dx is a lost soul". Infinitesimal quantity Are they zero, infinitesimal and their analysis reasonable? This has led to a century and a half long debate in the field of mathematics and even philosophy. Which led to The Second Mathematical Crisis 。

The mathematical thought of the 18th century was indeed imprecise, intuitively emphasizing formal calculation regardless of the reliability of the foundation. In particular, there is no clear concept of infinitesimal, so the concepts of derivative, differential, integral, etc. are not clear, the concept of infinity is not clear, and the summation of divergent series Arbitrariness , symbol is not strictly used, differentiation is carried out without considering continuity, existence of derivative and integral is not considered, and function can be expanded power series wait.

It was not until the 1920s that some mathematicians paid more attention to the strict foundation of calculus. from Polzano , Abel, Cauchy, Diliheli, etc Wells The work of Trass, DeDekin and Cantor ended. After more than half a century, the contradictions were basically solved mathematical analysis Laid a strict foundation.

3、 Russell paradox

The third crisis in the history of mathematics was caused by the sudden impact in 1897. On the whole, it has not been solved to a satisfactory degree. The crisis was caused by the discovery of paradox on the edge of Cantor's general set theory. because Set concept Has penetrated many branches of mathematics, and in fact set theory Became the foundation of mathematics, so the discovery of paradoxes in set theory naturally caused Basic structure Of Effectiveness Doubt.

In 1897, Forty revealed the first paradox in set theory. Two years later, Cantor found a very similar paradox. In 1902, Russell Another paradox is found, which involves no other concept except the concept of set itself. Russell's paradox has been popularized in many forms. The most famous one was given by Russell in 1919, which involved the dilemma of a village barber. The barber announced such a principle: he shaved all people who did not shave themselves, and only shaved people like this in the village. When people try to answer the following questions, they realize the paradoxical nature of this situation: "Does the barber shave himself?"? If he doesn't shave himself, then he should shave himself according to the principle; If he shaves himself, he does not conform to his principles.

Russell paradox The whole mathematical building was shaken. No wonder Frege, after receiving Russell's letter, wrote at the end of Volume 2 of the Basic Rules of Arithmetic he was about to publish: "A scientist will not encounter anything more embarrassing than this, that is, when the work is completed, its foundation collapses. When the book is waiting to be printed, a letter from Mr. Russell puts me in this position". So it ended nearly 12 years of hard research. Acknowledging infinite sets and infinite cardinals is like all disasters have come out. This is The Third Mathematical Crisis The essence of. Although paradoxes can be eliminated and contradictions can be solved, mathematical certainty But they are losing step by step. modern Axiomatic set theory It's hard to say which is true or false, but we can't eliminate them all. They are flesh and blood with the whole mathematics. Therefore, the third crisis has been solved on the surface, but in essence, it continues in other forms more profoundly.

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