The main subjects of mathematics mainly come from the needs of commercial calculation, understanding the relationship between numbers, measuring land and predicting astronomical events. These four requirements are generally related to a wide range of sub-fields in mathematics, such as quantity, structure, space and change (that is, arithmetic, algebra, geometry and analysis). In addition to the above-mentioned main concerns, there are sub-fields used to explore the relationship between the core of mathematics and other fields: to logic, to set theory (foundation), to empirical mathematics in different sciences (applied mathematics), and to the rigorous study of uncertainty in modern times.

amount

The study of quantity begins with numbers, and at the beginning is the familiar arithmetic operations of natural numbers and integers and those described in arithmetic. In number theory, the deeper properties of integers are studied, including famous results, such as Fermat's last theorem. Number theory also includes two unsolved problems that are widely discussed: twin prime conjecture and Goldbach conjecture.

When the number system is further developed, integers are considered as a subset of rational numbers and included in real numbers, and continuous quantities are represented by real numbers. Real numbers can be further extended to complex numbers. Further generalization of numbers can continue to include quaternions and octal numbers. The consideration of natural numbers will also lead to over-limit numbers, which formulizes the concept of counting to infinity. Another research field is its size, which leads to cardinality and another infinite concept: Avery number, which allows meaningful comparison between the sizes of infinite sets.

structure

Many mathematical objects, such as sets of numbers and functions, have internal structures. The structural properties of these objects are discussed in groups, rings, bodies and other abstract systems that are themselves objects. This is the field of abstract algebra. Here is a very important concept, that is, vector, which is extended to vector space and studied in linear algebra. The study of vector combines three basic fields of mathematics: quantity, structure and space. Vector analysis extends it to the fourth basic field, namely change.

space

The study of space comes from geometry, especially Euclidean geometry. Trigonometry combines space and numbers, including the famous Pythagorean theorem. Nowadays, the study of space extends to higher dimensional geometry, non-Euclidean geometry (which plays a central role in general relativity) and topology. Numbers and spaces play an important role in analytic geometry, differential geometry and algebraic geometry. In differential geometry, there are concepts such as fiber bundle and calculation on manifold. Algebraic geometry has the description of geometric objects such as polynomial equation solution set, which combines the concepts of number and space; There is also the study of topological groups, which combines structure and space. Lie groups are used to study space, structure and change. Among its many branches, topology is probably the most advanced field in mathematics in the twentieth century, which includes the long-standing Poincare conjecture and the controversial four-color theorem, which has only been proved by computers and never verified by manpower.

Fundamentals and philosophy

In order to understand the basis of mathematics, mathematical logic and set theory are developed. Georg Cantor (1845- 19 18) initiated the set theory, boldly marched into infinity, provided a solid foundation for all branches of mathematics, and its own content was quite rich, put forward the existence of real infinity, and made inestimable contributions to the future development of mathematics. Cantor's work has brought a revolution to the development of mathematics. Because his theory transcended intuition, it was opposed by some great mathematicians at that time. Even the mathematician Pi Aucar, who is famous for his "profound and creative", compared set theory to an interesting "morbid situation", and even his teacher Kroneck hit back at Cantor as a "mental derangement" and "walked into a hell beyond numbers". Cantor is still full of confidence in these criticisms and accusations. He said: "My theory is as firm as a rock, and anyone who opposes it will shoot himself in the foot." He also pointed out: "The essence of mathematics lies in its freedom, and it is not bound by traditional ideas." This argument lasted for ten years. Cantor suffered from schizophrenia at 1884 because of frequent depression, and finally died in a mental hospital.

However, after all, history has fairly evaluated his creation. Set theory gradually penetrated into all branches of mathematics at the beginning of the 20th century and became an indispensable tool in analytical theory, measurement theory, topology and mathematical science. At the beginning of the 20th century, Hilbert, the greatest mathematician in the world, spread Cantor's thoughts in Germany, calling him "a mathematician's paradise" and "the most amazing product of mathematical thoughts". British philosopher Russell praised Cantor's works as "the greatest works that can be boasted in this era".

Mathematical logic focuses on putting mathematics on a solid axiomatic framework and studying the results of this framework. It is the birthplace of Godel's second incomplete theorem, which is perhaps the most widely spread achievement in logic-there is always a true theorem that cannot be proved. Modern logic is divided into recursion theory, model theory and proof theory, which are closely related to theoretical computer science.

Engels said: "Mathematics is a science that studies the quantitative relationship and spatial form of the existing world."