Neo-logicism? An ontological reduction of mathematics to metaphysics
In this article the author sets out and defends a new form of logicism: mathematical objects are reducible to (or just are) the abstract objects of a certain axiomatic, mathematics-free metaphysical theory. According to the author, the traditional form of a logicist thesis is based on what he calls classical reduction: a logic \(L\) is defined by a set of analytically true axioms and rules of inference. A mathematical theory \(T\) is classically reducible to \(L\) just in case the primitive terms and predicates of \(T\) are definable in the language of \(L\) and the translations of theorems of \(T\) in \(L\) are logical theorems of \(L\). This classical view of reduction is opposed by metaphysical reduction, advocated by the author. This type of reduction is based on an axiomatic, mathematics-free metaphysical theory of abstract objects \(\Theta \). Given \(\Theta \), the reduction consists in showing, for an arbitrary mathematical theory \(T\), that there is a precise interpretation of the terms and predicates of \(T\) which first analyzes these expressions as denoting abstract objects in \( \Theta \) and second defines a sense in which the theorems of \(T\) are true. Given the identification of the referents of the well-defined expressions of \(T\) in \(\Theta \), the second subtask amounts to specifying in \(\Theta \) the compositional truth conditions of the translations of the theorems of \(T\) in \(\Theta \) in terms of the abstract objects that have been identified as the referents of the (translations of) terms and predicates of \(T\) in \(\Theta \). The theoretical principles of the metaphysical reduction are defined in section 3. In the following, main section it is shown how the objects of an arbitrary mathematical theory can be reduced to the abstract objects of \( \Theta \). Section 5 outlines some consequences of the reduction. This is followed by an analysis of ordinary mathematical statements in \(\Theta \). In the closing section the author answers some objections to this new kind of logicism and addresses more philosophical questions like, e.g., that of mathematical objectivity.
scientific article; zbMATH DE number 1568541