Mr. Xu Lizhi, a famous mathematician, advocate and leader of mathematical methodology in China, pointed out: "Methodology is a science that discusses certain development laws and research methods.
Mathematical methods play a key role in promoting the development of mathematics, and the solution of many difficult and important problems often depends on the breakthrough of mathematical concepts and methods. For example, after Descartes founded analytic geometry and mathematicians adopted RMI (Relationship-Mapping-Inversion) method with the help of analytic geometry, the three major problems of ruler drawing in ancient Greece were completely solved. This inspired later mathematicians to solve the problem of relative compatibility between Euclidean geometry and real number theory in a similar way. For another example, under the guidance of Galois group theory, the root solution problem of algebraic equations is successfully solved. Not only that, the thinking method of group theory has also made great changes in the study of algebra, from classical local research to modern systematic structural integrity research.
The early research of mathematical methodology began in17th century. Both Descartes, a French mathematician, and Leibniz, a German mathematician, have discussed this and published monographs. Many famous great mathematicians in history, such as Euler, Gauss, Poincare, Hilbert and others, have also expressed many incisive opinions on the decline of mathematical methodology. However, the systematic study of mathematical methodology has only been carried out in recent decades. He has made outstanding contributions in this respect. As the first mathematician and mathematics educator in the United States, Paulia, in recent decades, modern computer technology has entered the stage of artificial intelligence and simulated thinking, which further promoted the vigorous development of mathematical methodology. The latest research results of information theory, cybernetics, cognitive science and artificial intelligence have been introduced into the field of mathematical methodology. However, it is only 20 years since Mr. Xu Lizhi formally put forward the name of "Mathematical Methodology" and made it an independent discipline.
Mathematical science and mathematical historical materials are the sources of mathematical methodology. At the same time, mathematical methodology also involves philosophy, thinking science, psychology, general scientific methodology, system science and many other fields.
Mathematical methodology can be divided into macro-mathematical methodology and micro-mathematical methodology.
The macro methodology of mathematics studies the law of the emergence, formation and development of the whole mathematics, the structure of mathematical theory and the relationship between mathematics and other sciences. One of the main ways to study macro methodology is to study the history of mathematics. Another main way to study macro methodology is to study the construction of mathematical theory system.
Mathematical micro methodology studies some concrete mathematical methods, especially the methods of mathematical discovery and creation. Including mathematical thinking methods, mathematical problem-solving psychology and mathematical problem-solving theory.
This subject seems not difficult, as long as you read it carefully and understand it yourself, it is easy to master.