The period of modern mathematics refers to the period from 65438 to the 1920s. During this period, mathematics mainly studied the most general quantitative relations and spatial forms. Number and quantity are only very special cases, and the usual geometric images in one-dimensional, two-dimensional and three-dimensional spaces are only special cases. Abstract algebra, topology and functional analysis are the main parts of modern mathematical science. They are courses for college mathematics majors, and non-mathematics majors should also know about them. During the period of variable mathematics, many new disciplines are developing vigorously, and their contents and methods are constantly enriched, expanded and deepened.
At the turn of 18 and 19 century, mathematics reached a rich and intensive situation. It seems that the treasure of mathematics has been exhausted and there is not much room for development. However, this is only the calm before the storm. 19 In the 1920s, the wave of mathematical revolution finally came, and mathematics began a series of essential changes. Since then, mathematics has entered a new period-the modern mathematics period.
/kloc-In the first half of the 9th century, two revolutionary discoveries appeared in mathematics-non-Euclidean geometry and noncommutative algebra.
Around 1826, people discovered non-Euclidean geometry, which is different from the usual Euclidean geometry but also correct. This was first proposed by Robachevsky and Rier. The appearance of non-Euclidean geometry has changed people's view that Euclidean geometry is only a matter of course. His revolutionary thought not only paved the way for new geometry, but also was the prelude and preparation for the emergence of the theory of relativity in the 20th century.
Later, it was proved that the ideological emancipation caused by non-Euclidean geometry was of great significance to modern mathematics and science, because human beings finally began to break through the limitations of senses and go deep into nature. In this sense, Lobachevsky, who contributed his whole life to the establishment and development of non-Euclidean geometry, deserves to be regarded as a pioneer of modern science.
1854, Riemann popularized the concept of space and created a broader field of geometry-Riemann geometry. The discovery of non-Euclidean geometry also promotes the in-depth discussion of axiomatic methods, studies the concepts and principles that can be used as the basis, and analyzes the completeness, compatibility and independence of axioms. From 65438 to 0899, Hilbert made great contributions to this.
1843, Hamilton discovered an algebra-quaternion algebra, in which the multiplicative commutative law does not hold. The appearance of noncommutative algebra has changed people's view that it is unthinkable to have an algebra different from ordinary arithmetic algebra. His revolutionary ideas opened the door to modern algebra.
On the other hand, the concept of group is introduced because of the exploration of the conditions for finding the roots of unary equations. From the 1920s to 1930s, Abel and Galois initiated the study of modern algebra. Modern algebra is relative to classical algebra, and the content of classical algebra is centered on discussing the solutions of equations. After group theory, various algebraic systems (rings, fields, lattices, Boolean algebras, linear spaces, etc. ) all set up. At this time, the research object of algebra expanded to vectors, matrices and so on, and gradually turned to the study of algebraic system structure itself.
The above two events and their development are called the liberation of geometry and algebra.
/kloc-in the 0/9th century, the third far-reaching mathematical event occurred: the arithmeticization of analysis. In 1874, Wilstrass put forward a striking example, asking people to have a deeper understanding of the analysis basis. He put forward a famous idea called "the arithmetic of analysis". The real number system itself should be strict first, and then all the concepts of analysis should be deduced from this number system. He and his successors basically realized this idea, so that all the analysis today can be logically deduced from a postulate set showing the characteristics of real number system.
The research of modern mathematicians goes far beyond the assumption that the real number system is the basis of analysis. Euclidean geometry can also be placed in the real number system through its analytical interpretation; If Euclidean geometry is compatible, then most branches of geometry are compatible. Real number system (or some part) can be used to solve many branches of group algebra; It can make many algebraic compatibility depend on the compatibility of real number system. In fact, it can be said that if the real number system is compatible, all existing mathematics are also compatible.
/kloc-In the late 20th century, due to the work of Dedekind, Cantor and piano, these mathematical foundations have been established on a simpler and more basic natural number system. In other words, they proved that the real number system (from which many kinds of mathematics are derived) can be derived from the postulate set that establishes the natural number system. At the beginning of the 20th century, it was proved that natural numbers can be defined by the concept of set theory, so all kinds of mathematics can be described on the basis of set theory.
Topology was originally a branch of geometry, but it didn't become popular until the second1/4th century of the 20th century. Topology can be roughly defined as continuous mathematical research. Scientists realize that any group of things, whether it is a set of points, numbers, algebraic entities, functions or non-mathematical objects, can form a topological space in a certain sense. The concepts and theories of topology have been successfully applied to the study of electromagnetism and physics.
In the 20th century, many mathematical works devoted themselves to a careful study of the logical basis and structure of mathematics, which led to the emergence of axioms, that is, the study of postulate sets and their properties. Many mathematical concepts have undergone great changes and popularization, and profound basic disciplines such as set theory, modern mathematics and topology have also been widely developed. Some paradoxes caused by general (or abstract) set theory are far-reaching and puzzling, and need to be dealt with urgently. Logic itself, as a tool to draw conclusions on the premise of cognition in mathematics, has been carefully investigated, thus producing mathematical logic. The relationship between logic and philosophy leads to the emergence of different schools of mathematical philosophy.
From the 1940s to 1950s, three earth-shattering events occurred in the history of world science, namely, the utilization of atomic energy, the invention of electronic computers and the rise of space technology. In addition, many new situations have emerged, which have caused drastic changes in mathematics. These situations are: the research object of modern science and technology is more and more beyond the scope of human senses, and it is developing in the direction of high temperature, high pressure, high speed, high intensity, long distance and automation. Take the unit of length as an example, ranging from 1 dust (femto meters, that is, 10- 15 meters) to 1 million parsec (3.258 million light years). These measurements and studies cannot rely on the direct experience of the senses, but more and more rely on the guidance of theoretical calculation. Secondly, the scale of scientific experiments is unprecedented, and a large-scale experiment will consume a lot of manpower and material resources. In order to reduce waste and avoid blindness, accurate theoretical extension and design are urgently needed. Thirdly, modern science and technology tend to be quantitative, and all fields of science and technology need mathematical tools. Mathematics has penetrated into almost all scientific departments, thus forming many marginal mathematics disciplines, such as biomathematics, biostatistics, mathematical biology, mathematical linguistics and so on.
The above situation makes the development of mathematics show some obvious characteristics, which can be simply summarized into three aspects: the formation of computer science, the emergence of many new branches of applied mathematics, and some major breakthroughs in pure mathematics.
1945 after the birth of the first electronic computer, a huge science naturally formed around it because of its wide application and great influence. Roughly speaking, computer science is a science that explores and theoretically studies computer systems, software and some special applications. Computational mathematics can be classified as computer science, but it can also be regarded as an applied mathematics.
Most of the work of computer design and manufacturing is usually computer engineering or electronic engineering. Software refers to programs, programming languages, programming methods, etc. that solve problems. Research software needs to use mathematical logic, algebra, mathematical linguistics, combinatorial theory, graph theory, calculation methods and other mathematical tools. At present, there are thousands of applications of electronic computers, and there is an increasing trend. But only some special applications are classified as computer science, such as machine translation, artificial intelligence, machine proofing, pattern recognition, image processing and so on.
There has never been a strict boundary between applied mathematics and pure mathematics (or basic theory). Generally speaking, pure mathematics is this part of mathematics, and it is not considered to be directly applied to other knowledge fields or production practice for the time being. It indirectly promoted the development of related disciplines or found its direct application after several years. Applied mathematics can be said to be a bridge between pure mathematics and science and technology.
After the 1940s, a large number of new applied mathematics disciplines appeared, and their contents, applications and types were unprecedented. Such as game theory, planning theory, queuing theory, optimization method, operational research, information theory, cybernetics, system analysis, reliability theory and so on. It is difficult to draw a clear line between the research scope and relationship between these branches. Some of them can be regarded as new applications or branches of probability statistics, because they use many tools of probability statistics, and some of them can be classified as computer science and so on.
After the 1940s, the basic theory also developed rapidly, and many breakthrough works appeared, which solved some fundamental problems. In this process, new concepts and methods are introduced, which promotes the development of the whole mathematics. For example, 1990 Hilbert raised 23 outstanding problems at the international conference of educators, and now some of them have been solved. Since the 1960s, some new branches of mathematics have emerged, such as nonstandard analysis, fuzzy mathematics and catastrophe theory. In addition, great progress has been made in classical mathematics in recent decades, such as probability theory, mathematical statistics, analytic number theory, differential geometry, algebraic geometry, differential equations, factor theory, functional analysis, mathematical logic and so on.
The research results of contemporary mathematics are about to explode. Before the end of 17, only 17 magazines published mathematical papers (originally from1665); 10 species in the 8th century; /kloc-there were 950 species in the 0/9th century. Statistics in the 20th century increased even more. At the beginning of this century, only 1000 mathematical papers were published every year; By 1960, the number of abstracts published in American Mathematical Review was 7824, by 1973, it was 204 10, and by 1979, it reached 528 12, showing an exponential growth trend. The three characteristics of mathematics-high abstraction, wide application and rigorous system-are more obvious.
Today, almost every country has its own mathematical society, and many countries also have groups dedicated to mathematics education at all levels. They have become one of the powerful factors to promote the development of mathematics. At present, mathematics has a trend of accelerated development, which is incomparable at any time in the past.
Although modern mathematics presents a colorful situation, its main characteristics can be summarized as follows: (1) The object and content of mathematics have developed greatly in depth and breadth, the ideas, theories and methods of analysis, algebra and geometry have changed greatly, and the trend of continuous differentiation and synthesis of mathematics is strengthening. (2) The entry of electronic computers into the field of mathematics has had a great and far-reaching impact. (3) Mathematics has penetrated into almost all scientific fields and is playing an increasingly important role. Pure mathematics has been developing in depth, and mathematical logic and mathematical foundation have become the foundation of the whole mathematics building.
The above briefly introduces the situation of mathematics in three main development periods: ancient, modern and modern. If the study of mathematics is compared to the study of "flying", then the first phase is mainly to study a few photos of birds (still and unchanged); The second phase mainly studies several films (movements and variables) of birds; The third phase mainly studies the general properties (abstraction and assembly) of birds, planes and spaceships.
This is a development process from simple to complex, from concrete to abstract, from low to advanced, from special to general. From the geometric point of view, Euclidean geometry, analytic geometry and non-Euclidean geometry can be regarded as the representative achievements of the three major development periods of mathematics. Euclid, Descartes and Lobachevsky can be regarded as the representatives of each period.