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Philosophy of Sciences

2021 year, number 1


Vitaliy Valentinovich Tselishchev, Aleksandr Valerievich Khlebalin
Institute of Philosophy and Law, Siberian Branch of the Russian Academy of Sciences, 8 Nikolaeva str., Novosibirsk, 630090, Russia
Keywords: intensional, extensional, set-theoretical foundations, axiom of choice


The article considers the gap between the extensional and intensional elements of mathematical discourse in the development of programs concerning the foundations of mathematics. It is shown that F. Ramsey's desire to separate the intensional and extensional elements of discourse through the typology of paradoxes turned into an inversion of his project. The substantiation is produced that the development of the set-theoretic foundations of mathematics presupposed an extensional interpretation of mathematics, but nevertheless faced the need to assume the intensional element, for example, in relation to the axiom of choice. It is shown that the assumption of the intensional element of mathematical discourse is connected with philosophical programs, while the extensional interpretation is motivated by mathematical principles.