It's all the same thing. Of course it's equal.
In addition, there are some broader understandings of mathematics. For example, some people think that "mathematics is a cultural system" and "mathematics is a language", and mathematical activities are social. It is a highly intelligent crystallization of human beings' understanding, adaptation and transformation of nature, self-improvement and social improvement in the historical process of human civilization development. Mathematics has a key influence on the way of human thinking. Some people think that mathematics is an art. "Compared with mathematics as a discipline, I almost prefer to regard it as an art, because the persistent creative activities of mathematicians under the guidance of the rational world (although uncontrolled) are similar to those of artists, such as painters. This is real, not imaginary. Mathematicians' strict deductive reasoning can be compared to special attention skills here. Just as one can't be a painter without certain skills, one can't be a mathematician without a certain level of precise reasoning ability. These qualities are the most basic. Together with other more subtle qualities, they constitute the quality of an excellent artist or an excellent mathematician. In both cases, the most important thing is imagination. " "Mathematics is the music of reasoning" and "Music is the mathematics of images". This is to discuss the essence of mathematics from the process of mathematical research and the qualities that mathematicians should have. Some people regard mathematics as a basic attitude and method to treat things, a kind of spirit and concept, that is, mathematical spirit, mathematical concept and attitude. In the article "Mathematics in Society", Mogens Nice believes that mathematics is a discipline. "In the epistemological sense, it is a science, and its goal is to establish, describe and understand objects, phenomena, relationships and mechanisms in certain fields. If this field is composed of what we usually think of as mathematical entities, then mathematics plays the role of pure science. In this case, mathematics aims at internal self-development and self-understanding and is independent of the outside world. On the other hand, if the field under consideration exists outside mathematics, mathematics plays a role in the use of science. The difference between these two aspects of mathematics is not the problem of mathematics content itself, but the focus of people's attention. Whether it is pure theory or application, mathematics as a science is helpful to produce knowledge and insight. Mathematics is also a system of tools, products and processes, which helps us to make decisions and actions related to mastering practical fields other than mathematics. Mathematics is an aesthetic field, which can provide beauty, pleasure and excitement for many people who are addicted to it. As a discipline, mathematics needs a new generation of people to master its dissemination and development. Mathematics learning will not be carried out automatically at the same time, and it needs to be taught. Therefore, mathematics is also a teaching subject in our social education system. "
As can be seen from the above, people speak from the inside of mathematics (and from the perspectives of mathematical content, expression and research process). The relationship between mathematics and society, mathematics and other disciplines, mathematics and human development is discussed. They all reflect the essential characteristics of mathematics from one side and provide a perspective for us to fully understand the essence of mathematics.
Based on the above understanding of the essential characteristics of mathematics, people have also discussed the specific characteristics of mathematics from different aspects. The general view is that mathematics has the characteristics of abstraction, accuracy and wide application, in which abstraction is the most essential feature. One, 20%. Alexander Love said, "Even superficial mathematical knowledge can easily perceive these characteristics of mathematics: first, it is abstract; Second, accuracy, or better, he said is the rigor of logic and the certainty of conclusions; Finally, it is the extreme universality of its application. " Wang Zikun said, "The characteristics of mathematics are: abstract content, extensive application, rigorous reasoning and certainty of conclusions." This view is mainly based on the content of mathematics. In addition, from the process of mathematical research and the relationship between mathematics and other disciplines, mathematics is also vivid, realistic and quasi-empirical. The characteristics of "falsifiability". The understanding of mathematical characteristics also has the characteristics of the times. For example, the rigor of mathematics has different standards in each historical development period of mathematics. From Euclid's geometry to Luo Baltscheffskij's geometry to Hilbert's axiomatic system, the evaluation criteria for rigor are quite different. Especially after Godel put forward and proved the "Incompleteness Theorem ……", people found that even axiomatization, a rigorous scientific method that was once highly respected, was flawed. Therefore, the rigor of mathematics is reflected in the history of mathematics development and has relativity. Regarding the paradox of mathematics, Paulia pointed out in his "Mathematics and Conjecture" that "mathematics is regarded as an argument science. However, this is only one aspect. The final form of stereotyped mathematics seems to be pure demonstration materials, only proof. However, the process of creating mathematics is the same as any other knowledge. Before proving a mathematical theorem, you should guess the content of this theorem. Before you make a detailed proof, you have to guess the idea of proof. You have to synthesize the observed results and make an analogy. You must do it again and again. The result of mathematicians' creative work is demonstration, that is, proof; But this proof was discovered through reasonable reasoning and conjecture. As long as the learning process of mathematics can slightly reflect the process of mathematical invention, then conjecture and reasonable reasoning should occupy an appropriate position. "It is from this perspective that we say that the certainty of mathematics is relative and conditional, and it is vivid, realistic and quasi-empirical for mathematics. The emphasis on the characteristics of "falsifiability" actually highlights the observation, experiment and analysis in mathematical research. The importance of thinking processes such as comparison, analogy, induction and association.
research contents
Humans have been dealing with natural numbers since they learned to count. Later, due to the need of practice, the concept of number was further expanded. Natural numbers are called positive integers, while their opposites are called negative integers, and neutral numbers between positive and negative integers are called 0. They add up to an integer.
For integers, four operations can be performed: addition, subtraction, multiplication and division, which are called four operations. Among them, addition, subtraction, multiplication and division can be carried out in an integer range without obstacles. That is to say, any two or more integers are added, subtracted and multiplied, and their sum, difference and product are still an integer. However, the division between integers may not be carried out smoothly within the integer range.
In the application and research of integer operation, people are gradually familiar with the characteristics of integers. For example, integers can be divided into two categories-odd and even (usually called odd and even) and so on. Using some basic properties of integers, we can further explore many interesting and complex mathematical laws. It is the charm of these characteristics that has attracted many mathematicians to study and explore continuously throughout the ages.
The subject of number theory begins with the study of integers, so it is called integer theory. Later, the theory of integers was further developed and called number theory. To be exact, number theory is a subject that studies the properties of integers.
Brief introduction to the development of number theory
Since ancient times, mathematicians have always attached great importance to the study of integer properties, but until the19th century, these research results were only recorded in arithmetic works of various periods in isolation, that is to say, a complete and unified discipline has not yet been formed.
Since ancient China, many famous mathematical works have discussed the content of number theory, such as finding the greatest common divisor, pythagorean array, integer solutions of some indefinite equations and so on. Abroad, mathematicians in ancient Greece have systematically studied one of the most basic problems in number theory-divisibility, and a series of concepts such as prime number, sum number, divisor and multiple have also been put forward and applied. Mathematicians of past dynasties have also made great contributions to the study of integer properties, and gradually improved the basic theory of number theory.
In the study of the properties of integers, it is found that prime numbers are the basic "materials" that constitute positive integers. In order to study the properties of integers in depth, it is necessary to study the properties of prime numbers. Therefore, some problems about the properties of prime numbers have always been concerned by mathematicians.
By the end of18th century, the scattered knowledge about the properties of integers accumulated by mathematicians in past dynasties was very rich, and the conditions for sorting it out and processing it into a systematic discipline were completely mature. Gauss, a German mathematician, concentrated the achievements of his predecessors and wrote a book called Arithmetic Discussion, which was sent to the French Academy of Sciences in 1800, but the French Academy of Sciences rejected Gauss's masterpiece, so Gauss had to publish it himself in 180 1 year. This book initiated a new era of modern number theory.
In On Arithmetic, Gauss standardized the symbols used to study the properties of integers in the past, systematized and summarized the existing theorems at that time, classified the problems to be studied and the methods of will, and introduced new methods.
The basic content of number theory
After number theory became an independent discipline, with the development of other branches of mathematics, the methods of studying number theory are also developing. According to the research method, it can be divided into four parts: elementary number theory, analytic number theory, algebraic number theory and geometric number theory.
Elementary number theory is a branch of number theory, which studies the properties of integers only by elementary methods without the help of other mathematical disciplines. For example, the famous "China's Remainder Theorem" in ancient China is a very important content in elementary number theory.
Analytic number theory is a branch of solving number theory problems with mathematical analysis as a tool. Mathematical analysis is a mathematical discipline based on the concept of limit and taking function as the research object. Solving number theory problems by mathematical analysis was laid by Euler, and Russian mathematician Chebyshev also contributed to its development. Analytic number theory is a powerful tool to solve the problem of number theory. For example, for the proposition with infinite prime numbers, Euler gave the proof of analytical method, which used some knowledge about infinite series in mathematical analysis. In 1930s, vinogradov, a Soviet mathematician, creatively put forward the "triangle sum method", which played an important role in solving some difficult problems in number theory. Chen Jingrun, a mathematician in China, solved Goldbach's conjecture with the screening method in analytic number theory.
Algebraic number theory is a branch that extends the concept of integer to algebraic integer. Mathematicians extended the concept of integer to the general algebraic number field, and accordingly established the concepts of prime number and divisibility.
German mathematician and physicist Minkowski founded and laid the foundation of geometric number theory. The basic object of geometric number theory research is "spatial grid". What is a spatial grid? In a given rectangular coordinate system, the point whose coordinates are all integers is called the whole point; A set of all points is called a spatial grid. Spatial grid is of great significance to geometry and crystallography. Because of the complexity of the problems involved in geometric number theory, it needs a considerable mathematical foundation to study it in depth.
Number theory is a highly abstract mathematical subject. For a long time, its development has been in the state of pure theoretical research, which has played a positive role in the development of mathematical theory. But for most people, its practical significance is not clear.
With the development of modern computer science and applied mathematics, number theory has been widely used. For example, many research results within the scope of elementary number theory are widely used in calculation methods, algebraic coding, combinatorial theory, etc. It is also reported in the literature that some countries now use the "Sun Tzu Theorem" to measure the distance and calculate the discrete Fourier transform with the original root and the original exponent. In addition, many profound research results of number theory have also been applied in approximate analysis, difference set, rapid transformation and so on. Especially due to the development of computer, it is possible to approximate continuous quantity with the calculation of discrete quantity and achieve the required accuracy.
The position of number theory in mathematics is unique. Gauss once said, "Mathematics is the queen of science, and number theory is the crown in mathematics". So mathematicians like to call some unsolved problems in number theory "crown jewels" to encourage people to "choose". Here are some "pearls": Fermat's last theorem, twin prime number problem, Goldbach conjecture, integer problem in a circle, and perfect number problem. ...
In modern China, number theory was also one of the earliest branches of mathematics. Since the 1930s, he has made great contributions to analytic number theory, complexity equation and even distribution, and first-class number theory experts such as Hua, Min Sihe and Ke Zhao have emerged. Among them, Professor Hua is most famous for his research on trigonometric sum assignment and heap prime theory. After 1949, the study of number theory has made great progress. Especially in the research of "screening method" and "Goldbach conjecture", it has made outstanding achievements in the world.
Especially after Chen Jingrun proved in 1966 that "a big even number can be expressed as the sum of the products of a prime number and no more than two prime numbers", it aroused strong repercussions in the international mathematics community, praising Chen Jingrun's paper as a masterpiece of analytical mathematics and the glorious culmination of screening method. So far, this is still the best result of Goldbach's conjecture.
Humans have been dealing with natural numbers since they learned to count. Later, due to the need of practice, the concept of number was further expanded. Natural numbers are called positive integers, while their opposites are called negative integers, and neutral numbers between positive and negative integers are called 0. They add up to an integer.
For integers, four operations can be performed: addition, subtraction, multiplication and division, which are called four operations. Among them, addition, subtraction, multiplication and division can be carried out in an integer range without obstacles. That is to say, any two or more integers are added, subtracted and multiplied, and their sum, difference and product are still an integer. However, the division between integers may not be carried out smoothly within the integer range.
In the application and research of integer operation, people are gradually familiar with the characteristics of integers. For example, integers can be divided into two categories-odd and even (usually called odd and even) and so on. Using some basic properties of integers, we can further explore many interesting and complex mathematical laws. It is the charm of these characteristics that has attracted many mathematicians to study and explore continuously throughout the ages.
The subject of number theory begins with the study of integers, so it is called integer theory. Later, the theory of integers was further developed and called number theory. To be exact, number theory is a subject that studies the properties of integers.
Brief introduction to the development of number theory
Since ancient times, mathematicians have always attached great importance to the study of integer properties, but until the19th century, these research results were only recorded in arithmetic works of various periods in isolation, that is to say, a complete and unified discipline has not yet been formed.
Since ancient China, many famous mathematical works have discussed the content of number theory, such as finding the greatest common divisor, pythagorean array, integer solutions of some indefinite equations and so on. Abroad, mathematicians in ancient Greece have systematically studied one of the most basic problems in number theory-divisibility, and a series of concepts such as prime number, sum number, divisor and multiple have also been put forward and applied. Mathematicians of past dynasties have also made great contributions to the study of integer properties, and gradually improved the basic theory of number theory.
In the study of the properties of integers, it is found that prime numbers are the basic "materials" that constitute positive integers. In order to study the properties of integers in depth, it is necessary to study the properties of prime numbers. Therefore, some problems about the properties of prime numbers have always been concerned by mathematicians.
By the end of18th century, the scattered knowledge about the properties of integers accumulated by mathematicians in past dynasties was very rich, and the conditions for sorting it out and processing it into a systematic discipline were completely mature. Gauss, a German mathematician, concentrated the achievements of his predecessors and wrote a book called Arithmetic Discussion, which was sent to the French Academy of Sciences in 1800, but the French Academy of Sciences rejected Gauss's masterpiece, so Gauss had to publish it himself in 180 1 year. This book initiated a new era of modern number theory.
In On Arithmetic, Gauss standardized the symbols used to study the properties of integers in the past, systematized and summarized the existing theorems at that time, classified the problems to be studied and the methods of will, and introduced new methods.
The basic content of number theory
After number theory became an independent discipline, with the development of other branches of mathematics, the methods of studying number theory are also developing. According to the research method, it can be divided into four parts: elementary number theory, analytic number theory, algebraic number theory and geometric number theory.
Elementary number theory is a branch of number theory, which studies the properties of integers only by elementary methods without the help of other mathematical disciplines. For example, the famous "China's Remainder Theorem" in ancient China is a very important content in elementary number theory.
Analytic number theory is a branch of solving number theory problems with mathematical analysis as a tool. Mathematical analysis is a mathematical discipline based on the concept of limit and taking function as the research object. Solving number theory problems by mathematical analysis was laid by Euler, and Russian mathematician Chebyshev also contributed to its development. Analytic number theory is a powerful tool to solve the problem of number theory. For example, for the proposition with infinite prime numbers, Euler gave the proof of analytical method, which used some knowledge about infinite series in mathematical analysis. In 1930s, vinogradov, a Soviet mathematician, creatively put forward the "triangle sum method", which played an important role in solving some difficult problems in number theory. Chen Jingrun, a mathematician in China, also solved Goldbach's conjecture with analytic number theory.
Algebraic number theory is a branch that extends the concept of integer to algebraic integer. Mathematicians extended the concept of integer to the general algebraic number field, and accordingly established the concepts of prime number and divisibility.
German mathematician and physicist Minkowski founded and laid the foundation of geometric number theory. The basic object of geometric number theory research is "spatial grid". What is a spatial grid? In a given rectangular coordinate system, the point whose coordinates are all integers is called the whole point; A set of all points is called a spatial grid. Spatial grid is of great significance to geometry and crystallography. Because of the complexity of the problems involved in geometric number theory, it needs a considerable mathematical foundation to study it in depth.
Number theory is a highly abstract mathematical subject. For a long time, its development has been in the state of pure theoretical research, which has played a positive role in the development of mathematical theory. But for most people, its practical significance is not clear.
With the development of modern computer science and applied mathematics, number theory has been widely used. For example, many research results within the scope of elementary number theory are widely used in calculation methods, algebraic coding, combinatorial theory, etc. It is also reported in the literature that some countries now use the "Sun Tzu Theorem" to measure the distance and calculate the discrete Fourier transform with the original root and the original exponent. In addition, many profound research results of number theory have also been applied in approximate analysis, difference set, rapid transformation and so on. Especially due to the development of computer, it is possible to approximate continuous quantity with the calculation of discrete quantity and achieve the required accuracy.
The position of number theory in mathematics is unique. Gauss once said, "Mathematics is the queen of science, and number theory is the crown in mathematics". So mathematicians like to call some unsolved problems in number theory "crown jewels" to encourage people to "choose". Here are some "pearls": Fermat's last theorem, twin prime number problem, Goldbach conjecture, integer problem in a circle, and perfect number problem. ...
In modern China, number theory was also one of the earliest branches of mathematics. Since the 1930s, he has made great contributions to analytic number theory, complexity equation and even distribution, and first-class number theory experts such as Hua, Min Sihe and Ke Zhao have emerged. Among them, Professor Hua is most famous for his research on trigonometric sum assignment and heap prime theory. After 1949, the study of number theory has made great progress. Especially in the research of "screening method" and "Goldbach conjecture", it has made outstanding achievements in the world.
Especially after Chen Jingrun proved in 1966 that "a big even number can be expressed as the sum of the products of a prime number and no more than two prime numbers", it aroused strong repercussions in the international mathematics community, praising Chen Jingrun's paper as a masterpiece of analytical mathematics and the glorious culmination of screening method. So far, this is still the best result of Goldbach's conjecture.
Definition of mathematics
Definition 1:
More than a hundred years ago, Engels defined mathematics as "the science of studying the quantitative relationship and spatial form of the objective world", and spatial form refers to geometry.
From: The Idea of Geometry Teaching Reform in Normal Universities Journal of Chuxiong Normal University 200 1 Chen Ping
Abstract: Based on the reflection on the present situation of geometry teaching in normal universities, some suggestions on geometry teaching reform are put forward.
Definition 2:
Mathematical definition is a generalization and summary of the development of mathematics. It must have its stages and limitations, and there is no eternal mathematical definition suitable for any period. 3. The period of modern mathematics (from the end of 19) began when 1873 G Cantor founded the set theory.
From: Talking about Mathematics and Mathematics Education from What is Mathematics Xiao Jiahong, Journal of Lingling College, 2004.
Abstract: What is mathematics? This is an admittedly difficult question to answer. 194 1 year, American mathematicians R. Courant and H. Robbins wrote a book called What is Mathematics? Why doesn't the book take "What is Mathematics" as the topic? I don't know if there is any difference between the two, "what is mathematics?"