Mathematicians look for patterns [9][ 10] and use them to form new conjectures. Mathematicians solve the truth of conjecture through mathematical proof. When mathematical structure is a good model of real phenomena, then mathematical reasoning can provide insight or prediction of nature. By using abstraction and logic, mathematics develops from counting, calculation, measurement and systematic study of the shape and motion of physical objects. Practical mathematics has been a human activity since it was recorded in writing. The research needed to solve mathematical problems may take years or even hundreds of years of continuous exploration.
Strict argument first appeared in Greek mathematics, the most famous of which was Euclid's Elements of Geometry. Since Peano (1858- 1932), David Hilbert (1862- 1943) and others' pioneering work on axiomatic systems in the late19th century, people have become accustomed to regard mathematical research as a way to learn from properly selected axioms and definitions. The development speed of mathematics was relatively slow, until the Renaissance, the interaction between mathematical innovation and new scientific discoveries led to a rapid increase in the speed of mathematical discovery, which continues to this day. [ 1 1]
Galileo galilei (1564–1642) said: "Unless we learn the language and are familiar with the writing of the universe, we cannot understand the universe. It is written in mathematical language, and the letters are triangles, circles and other geometric figures. Without these, it means that it is impossible for human beings to understand a word. Without these, a person will wander in the dark maze. " C.F.Gauss (1777–1855) called mathematics "the queen of science". [13] benjamin pierce (1809–1880) called mathematics "the science of drawing necessary conclusions". David Hilbert said of mathematics: "We are not talking about arbitrariness in any sense here. Mathematics is not like a game, its task is determined by arbitrary rules. On the contrary, it is a conceptual system with inherent inevitability, which can only be the case, but never the opposite. " [15] Albert Einstein (1879–1955) stated that "as far as the reality referred to by the laws of mathematics is concerned, they are uncertain; As far as they are certain, they don't mean reality. " [ 16]
Mathematics is essential in many fields, including natural science, engineering, medicine, finance and social science. Applied mathematics has led to brand-new mathematics disciplines, such as statistics and game theory. Mathematicians are also engaged in pure mathematics, or mathematics for the sake of mathematics, without any application in their minds. There is no clear boundary between pure mathematics and applied mathematics, and the practical application of pure mathematics is often found. [ 17]