Mathematical knowledge and its methods

S. Lebedev; A. Polyanskiy

UDC 1

MATHEMATICAL KNOWLEDGE AND ITS METHODS Mathematical knowledge and its methods

АА Аа

Published in Journal of Philosophical Research · Volume 10, Issue 4 · Pages 19–27 · Rubric: Ontology and theory of knowledge

Received: 04.10.2024 Accepted: 01.11.2024 Published: 17.12.2024

Authors

S. Lebedev

A. Polyanskiy

S. Lebedev

Academic degree: doctor of philosophical sciences

Lomonosov Moscow State University Chief Researcher · Moscow

Bauman Moscow State Technical University professor · Moscow

A. Polyanskiy

Bauman Moscow State Technical University Moscow

ABSTRACT

The philosophy of mathematics is a branch of the philosophy of science that explores the philosophical foundations and problems of mathematics: ontological, epistemological, methodological, logical, axiological prerequisites of mathematics in general, its various directions, disciplines and theories. The most important philosophical problems of mathematics include: the subject of mathematics, the nature of mathematical knowledge, the place of mathematics in science and culture. There are alternative approaches to solving them. For example, when solving the problem of the subject of mathematics, it is, on the one hand, objectivism, both idealistic (Plato, Descartes, Kant, Cantor, Whitehead) and materialistic in nature (Archimedes, Newton, Kolmogorov); and, on the other, subjectivism: logicism and intuitionism (Russell, Brower, Geiting, etc.). When solving the question of the nature of mathematical knowledge (how is mathematical knowledge possible) These are concepts such as mathematical empiricism (Aristotle, Ostrogradsky, Helmholtz, etc.) and mathematical a priori (Plato, Descartes, Leibniz, Weil et al.). In solving the problem of substantiation of mathematics in the 20th century, four main directions were clearly formed: quasi-empiricism (Arnold, Lakatos), logicism (Russell, Frege, Carnap), formalism (Hilbert, Godel, Bernays), intuitionism (Brower, Geiting, etc.) and constructivism (Poincare, Bourbaki, Markov, etc.). There is also no unity among mathematicians and philosophers on the role of mathematical knowledge in culture: from understanding mathematics as a service tool and language of other specific sciences (natural, engineering, social) to understanding it as a self-sufficient, universal and objective science (Gauss, Hilbert, Riemann, Hermit, etc.). This pluralism in the philosophy of mathematics has a generally positive meaning for it, characterizing it as a complex and open cognitive system capable of development.

KEYWORDS

mathematics, the subject of mathematics, the nature of mathematical knowledge, methods of mathematics

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Lebedev S. A., Polyanskiy A. G. Mathematical knowledge and its methods // Journal of Philosophical Research. 2024. no. 4. pp. 19-27. URL: https://naukaru.ru/en/nauka/publications/article/6a9c159f32807fdbab86c65f/mathematical-knowledge-and-its-methods/view. (Date of access 24.09.2026).

MLA

S. Lebedev, Polyanskiy A. G.. "Mathematical knowledge and its methods." Journal of Philosophical Research 10.4 (2024): 19-27. URL: https://naukaru.ru/en/nauka/publications/article/6a9c159f32807fdbab86c65f/mathematical-knowledge-and-its-methods/view.

APA

Lebedev S. A., Polyanskiy A. G. (2024). Mathematical knowledge and its methods. Journal of Philosophical Research, 10(4), 19-27. URL: https://naukaru.ru/en/nauka/publications/article/6a9c159f32807fdbab86c65f/mathematical-knowledge-and-its-methods/view

Chicago

S. Lebedev, Polyanskiy A. G.. "Mathematical knowledge and its methods." Journal of Philosophical Research 10, no. 4 (2024): 19-27. URL: https://naukaru.ru/en/nauka/publications/article/6a9c159f32807fdbab86c65f/mathematical-knowledge-and-its-methods/view.

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{{cite journal | author = Lebedev S. A., Polyanskiy A. G. | title = Mathematical knowledge and its methods | journal = Journal of Philosophical Research | year = 2024 | volume = 10 | issue = 4 | pages = 19-27 | url = https://naukaru.ru/en/nauka/publications/article/6a9c159f32807fdbab86c65f/mathematical-knowledge-and-its-methods/view }}

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MATHEMATICAL KNOWLEDGE AND ITS METHODS Mathematical knowledge and its methods

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