Carnap’s Untersuchungen: Logicism, Formal Axiomatics, and Metatheory

Abstract

This paper discusses Carnap’s attempts in the late 1920s to provide a formal reconstruction of modern axiomatics.¹ One interpretive theme addressed in recent scholarly literature concerns Carnap’s underlying logicism in his philosophy of mathematics from that time, more specifically, his attempt to “reconcile” the logicist approach of reducing mathematics to logic with the formal axiomatic method. For instance, Awodey & Carus (2007) characterize Carnap’s manuscript Untersuchungen zur allgemeinen Axiomatik from 1928 as a “large-scale project to reconcile axiomatic definitions with logicism, and transform implicit into explicit definitions.” (ibid., 29) It is argued that Carnap’s central idea was to balance a Fregean (or Russellian) foundational stance with the modern model-theoretic viewpoint introduced in Hilbert’s Grundlagen der Geometrie (see (Reck 2004)). It was also shown in recent literature that Carnap’s attempt to provide a logicist reconstruction of axiomatics is limited in several ways.² No closer attention, however, has so far been dedicated to some of the details of his proposed reconciliation.