Mathematics without Numbers: Towards a Modal-Structural Interpretation

Abstract

Develops a structuralist understanding of mathematics, as an alternative to set‐ or type‐theoretic foundations, that respects classical mathematical truth while minimizing Platonist commitments to abstract entities. Modal logic is combined with notions of part/whole (mereology) enabling a systematic interpretation of ordinary mathematical statements as asserting what would be the case in any (suitable) structure there (logically) might be, e.g. for number theory, functional analysis, algebra, pure geometry, etc. Structures are understood as comprising objects, whatever their nature, standing in suitable relations as given by axioms or defining conditions in mathematics proper. The characterization of structures is aided by the addition of plural quantifiers, e.g. ‘Any objects of sort F’ corresponding to arbitrary collections of Fs, achieving the expressive power of second‐order logic, hence a full logic of relations. (See the author's ‘Structuralism without Structures’, Philosophia Mathematica4 (1996): 100–123.) Claims of absolute existence of structures are replaced by claims of (logical) possibility of enough structurally interrelated objects (modal‐existence postulates). The vast bulk of ordinary mathematics, and scientific applications, can thus be recovered on the basis of the possibility of a countable infinity of atoms. As applied to set theory itself, these ideas lead to a ‘many worlds’—– as opposed to the standard ‘fixed universe’—view, inspired by Zermelo (1930), respecting the unrestricted, indefinite extendability of models of the Zermelo–Fraenkel axioms. Natural motivation for (‘small’) large cardinal axioms is thus provided. In sum, the vast bulk of abstract mathematics is respected as objective, while literal reference to abstracta and related problems with Platonism are eliminated.