Development of the conceptual provisions of the qualitative theory

R. R. Mukhin

Čebyševskij sbornik, Tome 23 (2022) no. 5, pp. 269-290.

Résumé

The work is devoted to the study of the evolution of the main provisions of the qualitative theory, under the sign of which the development of all mathematics of the twentieth century took place. In the development of qualitative theory there are several stages with clearly defined trends: the formation of a qualitative theory, when new approaches, a new language and a system of concepts were formed (late 19th – 20s of the 20th century); the next stage is the widespread use of methods of topology and functional analysis, probabilistic representations and the expansion of qualitative theory with the allocation of independent areas (late 1920s – mid-twentieth century); from the middle of the twentieth century to the present – the modern stage. It is distinguished by the fact that the idea of mathematics as a single science was embodied in the qualitative theory. Qualitative theory has absorbed the ideas and methods of various branches (topology, functional analysis, the theory of Lie groups, etc.). The unifying role of a qualitative theory is that it embodies two cultures in mathematics, one of them is aimed at solving problems, and the other – at building and comprehending theories. In this respect, qualitative theory is not just a specific branch, but a peculiar approach to mathematical problems. A feature of the present stage is the still unprecedented convergence with the field of applications, especially with physics. Physics is not just a consumer, it has stimulated fundamental changes in mathematics itself. It becomes difficult to draw a distinguishable boundary between some branches of mathematics and theoretical physics. Qualitative theory has transformed the face of all mathematics and its applications.

Détail

Keywords: qualitative theory, topology, topological invariance, dynamical system, local and global description.

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