Mathematical Thinking Tools in the History of Mathematics (Based on the Theory of Positive Operators)

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Abstract

The work outlines a not entirely traditional approach to the study of the history of mathematics, associated with distinguishing the systems of mathematical thought tools (symbolic, conceptual, ideal), which allow representing and operating various types of relationships, characteristic of a particular branch of mathematics. With this approach, the main task for the history of mathematics is the analysis of the emergence, evolution, and transformation of the systems of its thinking tools, from the standpoint of expanding the horizons of knowledge and reaching new levels of abstraction, and broadening the array of methods for using such tools. To demonstrate the results of application of this approach, the authors examined one of the sections of functional analysis, the theory of positive operators, in the context of the history of its genesis and initial development from the mid-1900s to the 1960s. On this path, key thinking tools related to the finite-dimensional period of development of the theory (positive matrices, oscillatory matrices, etc.) and expressing the relationships between its significant mathematical entities (Perronʼs theorem) were identified. In addition, the first steps were made towards comprehending further transformation of the said tools (the concept of a cone, the definition of positive functionals and operators) and mathematical relationships (Jentzsch’s and Urysohn’s theorems in the integral and abstract forms, etc.). The results obtained by O. Kellogg, C. Sturm, M. G. Krein, F. R. Gantmakher, M. A. Rutman, M. A. Krasnoselskii, L. A. Ladyzhenskii and other mathematicians are discussed in the article.

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Egor M. Bogatov

Gubkin Branch of the MISIS National Research University of Science and Technology; Stary Oskol Technological Institute of the MISIS National Research University of Science and Technology

Author for correspondence.
Email: embogatov@inbox.ru
Russian Federation, 16, Komsomolskaya St., Gubkin, Belgorod Region, 309516; 42, Makarenko Nbhd., Stary Oskol, Belgorod Region, 309512

Alexey V. Borovskikh

Lomonosov Moscow State University; Scientific and Educational Mathematical Center of the Khetagurov North Ossetian State University

Email: bor.bor@mail.ru
Russian Federation, 1, Leninskie Gory, Moscow, 119991; 16, Tsereteli St., Vladikavkaz, Republic of North Ossetia-Alania, 362025

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2. Fig. 1. Geometric meaning of Rutman's condition - existence of intersection of a concave function with a sloping line

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