A non reductionist logicism with explicit definitions
Article [Version of Record]
Is part ofLes Cahiers d'Ithaque ; 2013
Publisher(s)Société Philosophique Ithaque
This paper introduces and examines the logicist construction of Peano Arithmetic that can be performed into Leśniewski’s logical calculus of names called Ontology. Against neo-Fregeans, it is argued that a logicist program cannot be based on implicit definitions of the mathematical concepts. Using only explicit definitions, the construction to be presented here constitutes a real reduction of arithmetic to Leśniewski’s logic with the addition of an axiom of infinity. I argue however that such a program is not reductionist, for it only provides what I will call a picture of arithmetic, that is to say a specific interpretation of arithmetic in which purely logical entities play the role of natural numbers. The reduction does not show that arithmetic is simply a part of logic. The process is not of ontological significance, for numbers are not shown to be logical entities. This neo-logicist program nevertheless shows the existence of a purely analytical route to the knowledge of arithmetical laws.
Joray, P. (2013). A non reductionist logicism with explicit definitions. Dans Fradet et Lepage (dir.), "La crise des fondements : quelle crise?". Montréal, Québec : Les Cahiers d'Ithaque.