The Gap between the Intensional and Extensional in Mathematics

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.