Microsoft Word - 30.doc


Mathematical Problems of Computer Science 38, 73, 2012. 

73 
 

 
 
 

On a Neo-logicistic Conception of Natural Number 
 

Hakob Madoyan 
PhD in Theoretical Philosophy, Logic and Philosophy of Science 

Armenian State Pedagogical University Chair of Philosophy and Logic 

Email: hasama@inbox.ru 

 
   
 

In the report we discuss a neo-logicistic conception of natural number formulated by 

British philosopher Crispin Wright in his book Frege’s Conception of Numbers as Objects 

(1983). 

For appropriate understanding of the differences between logicistic and neo-logicistic 

conceptions of natural numbers, we recall Bertrand Russell’s criticism of formal axiomatic 

systems containing Peano axioms. Russell argued that such systems with Peano axioms could 

not serve as real foundation for arithmetic, for they define not exactly our ordinary arithmetic but 

a group of systems sharing the same structure. Russell’s argument can be generalized into a 

principle according to which a foundation of arithmetic may be called logicistic only if in 

addition to well-known principles of logicism it also retains our ordinary intuition of numbers. 

We recall also Gottlob Frege’s informal definition of natural number in his Grundlagen der 

Arithmetik (1884) and its formalization in the Grundgesetze der Arithmetik (1893, 1903). 

In the light of Russell’s argument we outline the differences between Frege’s initial 

conception and Wright’s neo-logicistic conception of natural number.