Carnap on model structures and invariants

Georg Schiemer

University of Vienna

Rudolf Carnap’s thinking about mathematics in the 1920s is best characterized as an early form of structuralism. This is most explicit in his work on general axiomatics in the posthumously published manuscript Untersuchungen zur allgemeinen Axiomatik (Carnap 2000). In spite of the recent increase in scholarly attention to Carnap’s philosophy of mathematics, surprisingly little has so far been said about the specific form of his structuralism. How did he conceive of abstract structures? What role did the notion play in his logical reconstruction of mathematical theories? This talk will address these issues by exploring two distinct attempts to characterize the structural content of axiomatic theories present in Carnap’s work. The first is based on the method of abstraction, i.e. the construction of new and abstract objects by taking equivalence classes. In Untersuchungen, “model structures” are explicated by Carnap in a type-theoretic framework as the isomorphism classes of models of a particular theory. The informal idea underlying this approach is to specify structures as those entities gained by abstracting from all the non-structural properties of the objects considered. This way of defining structures via abstraction clearly echos a central structuralist motive in 19th- and early 20th-century mathematics, namely to identify objects up to isomorphism. The second approach, also documented in Carnap’s work from the time, aims at a direct demarcation between structural and non-structural properties of mathematical objects. The underlying principle here is not abstraction, however, but invariance under (structurepreserving) transformations. Structural properties are explicitly defined by Carnap as the “invariants under isomorphic transformations” (Carnap 2000, 74). Similar remarks on invariants can be found in his Logischer Aufbau (Carnap 1928) as well as in the logic manual Abriss der Logistik (Carnap 1929). This idea of invariance-based structures also has direct mathematical roots, in particular Felix Klein’s Erlangen program of classifying geometries algebraically in terms of their underlying invariants, i.e. in terms of properties that remain invariant under a certain group of transformations. The talk will be devoted to a closer discussion of these two related structuralist ideas and their significance for Carnap’s philosophy of mathematics. Specifically, the aim here will be twofold: first, to retrace Carnap’s thinking on the methods of structure abstraction and invariance as documented in his work from the 1920s. This includes his early contributions to the philosophy of geometry, specifically his dissertation Der Raum (Carnap 1922), as well as his subsequent logical work on general axiomatics and mathematical invariants. Second, to analyze how Carnap’s structuralist framework—in particular his formal notions of model structure and structural property—is applied by him in the logical reconstruction of mathematical theories. This concerns first and foremost his attempt to explicate several metatheoretic concepts in terms of this structuralist terminology in Carnap (2000) and in the unpublished second part of the Untersuchungen manuscript. By drawing on material from Carnap’s Nachlass, the talk will survey Carnap’s explication of different notions of completeness for axiomatic theories in terms of model structures, as well as his use of structural properties in the formulation of two important metatheorems.