Pure mathematics is the knowledge of studying mathematics itself, not for the purpose of application. Studying the internal relations of mathematical laws abstracted from the objective world can also be said to be studying the laws of mathematics itself.
Pure mathematics is to study the internal relations of mathematical laws abstracted from the objective world, and it can also be said that it is mathematics that studies the laws of mathematics itself.
Classification:
Generally speaking, it can be divided into three categories, namely, the geometric category of studying spatial forms, the algebraic category of studying discrete systems and the analytical category of studying continuous phenomena.
Geometric classes that study spatial forms belong to the first category, such as differential geometry and topology. Differential geometry is the study of smooth curves and surfaces, which uses mathematical analysis and differential geometry as research tools. It is widely used in mechanics and some engineering problems (such as elastic shell structure, gears and so on). Topology is to study the invariant properties of geometric figures under continuous transformation, which is called "topological properties". When the rubber film is deformed but not broken or folded, the closure of the curve and the intersection of the two curves remain unchanged.
Algebras that study discrete systems belong to the second category, such as number theory and modern algebra. Number theory is a subject that studies the properties of integers. According to the different research methods, it can be roughly divided into elementary number theory, algebraic number theory, geometric number theory, analytical number theory and so on. Modern algebra expands the objects of algebra from numbers to vectors and matrices. It studies the laws and properties of more general algebraic operations, and discusses the properties and structures of groups, rings and vector spaces. Modern algebra has such branches as group theory, ring theory and Galois theory. Widely used in analytical mathematics, geometry, physics and other disciplines.
The analysis of continuous phenomena belongs to the third category, such as differential equation, function theory and functional analysis. A differential equation is an equation containing the derivative or partial derivative of an unknown function. If the unknown function is a unary function, it is called an ordinary differential equation; If the unknown function is a multivariate function, it is called a partial differential equation. Function theory is the general name of real function theory (studying real functions in real numbers) and complex variable functions (studying the properties of functions in complex planes). Functional analysis is to study functions, operators and limit theory in infinite vector space (such as function space) by using the viewpoints of function theory, geometry and algebra. It studies not a single function, but a group of functions with some * * * properties. It is widely used in mathematics and physics.
History:
The word "pure mathematics" in19th century comes from the full name of Sadleirian chair (en: Sadleirian chair), which is a professorship established in the middle of19th century. The idea of "pure" mathematics as an independent discipline may have developed from that time. Mathematicians of Gauss generation did not completely distinguish between "purity" and "application". After that, the specialization and specialization, especially Veiershtrass' method of studying mathematical analysis, made the difference between the two more and more big.
In the 20th century, influenced by Hilbert, mathematicians began to use axiomatic systems. Russell established the logical formula of "pure mathematics" in the form of quantitative proposition. With the axiomatization of mathematics, these formulas become more and more abstract, and "strict proof" becomes a simple standard.
In fact, "strictness" is nothing new in "proof". According to bourbaki Group, pure mathematics has been proved.