Abstract
In a previous publication Mancosu (2015a) I have discussed Frege’s approach to definitions by abstraction by paying special attention to its Grassmannian roots and the teaching of elementary geometry in nineteenth century Germany (especially to debates on the concept of “direction”). Moreover, I emphasized the originality of Frege in taking the extension of a concept as the value of the abstraction operator thereby identifying, for instance, the direction of a line a with the extension of the concept “x is parallel to a”. In this paper, I will take for granted the results of that article and I will only flag my use of them by referring to the article at the appropriate junctions. My attention in this paper will focus on the logico-foundational analysis of definitions by abstraction in the Peano school and in Russell. We will see that many of the philosophical debates at the time foreshadowed contemporary debates in the neologicist literature.
