Mathematics (hanyu pinyin: shùXué；; ; Greek: μ α θ η μ α κ; English: Mathematics comes from the ancient Greek word μ θ η μ α (má th?ma), which means learning, learning and science. Ancient Greek scholars regarded it as the starting point of philosophy and the "foundation of learning". In addition, there is a narrow and technical meaning-"mathematical research". Even in its etymology, its adjective meaning related to learning will be used to refer to mathematics.

Its plural form in English and its plural form in French +es become mathématiques, which can be traced back to the Latin neutral plural (Mathematica) and translated by Cicero from the Greek plural τ α α θ ι α ι κ (tamath ē matiká).

In ancient China, mathematics was called arithmetic, also called arithmetic, and finally it was changed to mathematics. Arithmetic in ancient China was one of the six arts (called "number" in the six arts).

Mathematics originated from the early production activities of human beings, and the ancient Babylonians had accumulated some mathematical knowledge, which could be applied to practical problems. As far as mathematics itself is concerned, their mathematical knowledge is only obtained through observation and experience, and there is no comprehensive conclusion and proof, but their contribution to mathematics should also be fully affirmed.

The knowledge and application of basic mathematics is an indispensable part of individual and group life. The refinement of its basic concepts can be found in ancient mathematical documents of ancient Egypt, Mesopotamia and ancient India. Since then, its development has continued to make small progress. But algebra and geometry at that time were still independent for a long time.

Algebra can be said to be the most widely accepted "mathematics". It can be said that the first mathematics he came into contact with was algebra since everyone began to learn to count when he was a child. Mathematics is a subject that studies numbers, and algebra is also one of the most important parts of mathematics. Geometry is the earliest branch of mathematics studied by people.

Until the Renaissance in16th century, Descartes founded analytic geometry, which linked algebra and geometry which were completely separated at that time. From then on, we can finally prove the theorem of geometry through calculation; At the same time, abstract algebraic equations can also be graphically represented. Then more subtle calculus was developed.

At present, mathematics has included many branches. French Bourbaki School, founded in 1930s, thinks that mathematics, at least pure mathematics, is a theory to study abstract structures. Structure is a deductive system based on initial concepts and axioms. They think that mathematics has three basic parent structures: algebraic structure (group, ring, field, lattice …), ordered structure (partial order, total order …) and topological structure (neighborhood, limit, connectivity, dimension …).

Mathematics is applied in many different fields, including science, engineering, medicine and economics. The application of mathematics in these fields is generally called applied mathematics, which sometimes arouses new mathematical discoveries and promotes the development of new mathematical disciplines. Mathematicians also study pure mathematics, that is, mathematics itself, without any practical application. Although a lot of work started from the study of pure mathematics, it may find a suitable application later.

Specifically, there are sub-fields to explore the relationship between the core of mathematics and other fields: from logic and set theory (mathematical basis), to empirical mathematics in different sciences (applied mathematics), to more modern uncertainty research (chaos and fuzzy mathematics).

As far as verticality is concerned, the exploration in each field of mathematics is getting deeper and deeper.

Definition of mathematics

Aristotle defined mathematics as "quantitative science", which lasted until18th century. /kloc-since the 0/9th century, mathematical research has become more and more rigorous, and it has begun to involve abstract topics such as group theory and projection geometry that have no clear relationship with quantity and measurement. Mathematicians and philosophers have begun to put forward various new definitions. Some of these definitions emphasize the deductive nature of a lot of mathematics, some emphasize its abstraction, and some emphasize some themes in mathematics. Today, even among professionals, the definition of mathematics has not been reached. Whether mathematics is an art or a science has not even been decided. [8] Many professional mathematicians are not interested in the definition of mathematics, or think it is undefined. Some just said, "Mathematics is done by mathematicians."

The three main mathematical definitions are called logicians, intuitionists and formalists, each of which reflects a different school of philosophical thought. Everyone has serious problems, no one generally accepts it, and no reconciliation seems feasible.

The early definition of mathematical logic is Benjamin Peirce's Science of Drawing Inevitable Conclusions (1870). In Principles of Mathematics, Bertrand Russell and alfred north whitehead put forward a philosophical program called logicism, trying to prove that all mathematical concepts, statements and principles can be defined and proved by symbolic logic. The logical definition of mathematics is Russell's "All mathematics is symbolic logic" (1903).

Intuitionism equates mathematics with some mental phenomena from the mathematician L.E.J. Brouwer. An example of the definition of intuitionism is that "mathematics is a psychological activity constructed one after another". Intuitionism is characterized by rejecting some mathematical ideas that are considered effective according to other definitions. In particular, although other mathematical philosophies allow objects that can be proved to exist, even if they cannot be constructed, intuitionism only allows mathematical objects that can actually be constructed.

Formalism defines mathematics through mathematical symbols and operational rules. Haskell Curry simply defined mathematics as "formal system science". [33] A formal system is a set of symbols, or symbols, and there are some rules that tell how the symbols are combined into formulas. In the formal system, the word axiom has a special meaning, which is different from the ordinary meaning of "self-evident truth" in the formal system. Axiom is a combination of symbols contained in a given formal system, without using the rules of the system to deduce it. [2]

On the structure of mathematics

Many mathematical objects, such as numbers, functions and geometry, reflect the internal structure of continuous operations or the relationships defined in them. Mathematics studies the properties of these structures, for example, number theory studies how integers are represented under arithmetic operations. In addition, things with similar properties often occur in different structures, which makes it possible for a class of structures to describe their state through further abstraction and then axioms. What needs to be studied is to find out the structures that satisfy these axioms among all structures. Therefore, we can learn abstract systems such as groups, rings and domains. These studies (structures defined by algebraic operations) can form the field of abstract algebra. Because abstract algebra has great universality, it can often be applied to some seemingly unrelated problems. For example, some problems of drawing rulers and rulers in ancient times were finally solved by Galois theory, which involved domain theory and group theory. Another example of algebraic theory is linear algebra, which makes a general study of vector spaces with quantitative and directional elements. These phenomena show that geometry and algebra, which were originally considered irrelevant, actually have a strong correlation. Combinatorial mathematics studies the method of enumerating several objects satisfying a given structure.