Formalised Axiomatic Theories and Bourbaki’s Concept of Species of Set-Theoretical Structure

This paper investigates Bourbaki’s concepts of set-theoretical structure and isomorphism. First, I introduce Bourbaki’s formalism in a modern guise. Second, I show that there is a one-to-one correspondence (up to logical equivalence) between theories consisting of finitely many axioms expressed in some formal (first-order or higher-order) language and t-species of set-theoretical structure—that is, species of structure S which are transportable in the sense that the extension of the predicate “being a structure of species S” is closed under isomorphisms. Third, I examine the significance of the formal definitions and results presented in this paper for the philosophy of mathematics and the philosophy of the empirical sciences. In particular, I argue that: (1) since definitions of predicates “being a structure of species S” are conservative and eliminable relative to ZFC, such predicates can be used to achieve a reduction of (large parts of) mathematics to ZFC; (2) since t-species of structure are transportable, they can provide a basis for a set-theoretical form of structuralism by enabling us to “forget” those aspects of structures that are not isomorphism-invariant; and (3) the correspondence between t-species of structure and formal axiomatic theories shows that there is no deep divide between the so-called semantic and syntactic approaches in the philosophy of science.