On the role of the existence of paradoxes in the program of the philosophy of mathematics of neologicism

Abstract

The article discusses aspects of the philosophy of mathematics of neologiсism related to the existence of paradoxes. The problem of paradoxes is not sufficiently themed in the neologism programme. However, the use of the methodology of introducing concepts by abstraction principles is limited by the existence of abstraction principles that lead to a contradiction. The main problem of neologicism, The Bad Company Objection relates with the discussion of such principles. The author of the article shows that the search for a uniform criterion for the demarcation of acceptable and unacceptable principles of abstraction did not lead to the necessary result.