Frege's Attack on "Abstraction" and his Defense of the "Applicability" of Arithmetic (as Part of Logic)

• Published in: South African journal of philosophy 2003-01, Vol.22 (1), p.63-80
• Main Author: Strauss, Daniël F M
• Format: Article
• Language: English
• Subjects: History of science and technology; Logic; Logic and calculus; Mathematical sciences and techniques
• ISSN: 0258-0136; 2073-4867
• DOI: 10.4314/sajpem.v22i1.31361

The traditional understanding of abstraction operates on the basis of the assumption that only entities are subject to thought processes in which particulars are disregarded and commonalities are lifted out (the so-called method of genus proximum and differentia specifica). On this basis Frege criticized the notion of abstraction and convincingly argued that (this kind of) "entitary-directed" abstraction can never provide us with any numbers. However, Frege did not consider the alternative of "property-abstraction." In this article an argument for this alternative kind of abstraction is formulated by introducing a notion of the "modal universality" of the arithmetical and by developing it in terms of the distinction between type-laws (laws for entities - applicable to a limited class of entities) and modal laws (obtaining for every possible entity without any restriction). In order to substantiate this argument a case is made for the acceptance of an ontic foundation for the arithmetical (and other modes or functions of reality - with special reference to Cassirer, Bernays, Gödel and Wang), which, in the final section, serves to give an ontological account of (i) the connections between the arithmetical and other aspects of reality and (ii) the applicabality of arithmetic. In the course of the argument the impasse of logicism is briefly highlighted, while a few remarks are made with regard to the logical subject-object relation in connection with Frege's view that number attaches to a concept.