Moscow, Russian Federation
Russian Federation
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.
mathematics, the subject of mathematics, the nature of mathematical knowledge, methods of mathematics
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