Open AccessThe Gap between the Intensional and Extensional in Mathematics
Author(s) -
Vitaliy Valentinovich Tselishchev,
Aleksandr V. Khlebalin
Publication year - 2020
Publication title -
sibirskij filosofskij žurnal
Language(s) - English
Resource type - Journals
ISSN - 2541-7517
DOI - 10.25205/2541-7517-2020-18-2-48-58
Subject(s) - axiom , foundations of mathematics , extensional definition , epistemology , constructive set theory , opposition (politics) , axiom of choice , mathematics , philosophy of mathematics , divergence (linguistics) , mathematical logic , key (lock) , calculus (dental) , philosophy , set theory , computer science , algorithm , political science , linguistics , law , geometry , dentistry , biology , tectonics , paleontology , programming language , medicine , set (abstract data type) , politics , computer security
The paper is devoted to the study of the divergence of the intensionalist and extensionalist traditions in the foundations of mathematics. One of the important manifestations of this discrepancy was the debate on the status of the Axiom of Choice. In particular, we argue that Russell's challenging Axioms of the Choice is connected with his intensionalist philosophy of mathematics and the extensionalist approach of Zermelo. It is shown that the opposition of the intensionalist and extensionalist approaches includes such key problems of the philosophy of mathematics as the epistemological features of theorems and axioms, the nature of logical-philosophical analysis, and the role of logic in mathematics.