The Synthesis of Logicism and Formalism in Carnap’s Logical Syntax of Language

Abstract

One important achievement Rudolf Carnap claimed for his book, The Logical Syntax of Language, was that it effected a synthesis of two seemingly antithetical philosophies of mathematics, logicism and formalism. (Carnap 1937, Sec. 84) Reconciling these widely divergent conceptions had been a goal of Carnap’s for several years. But in the years in which Carnap’s synthesis evolved, important intellectual developments influenced the direction of his efforts and, ultimately, the final outcome. These developments were, first of all, the epoch-making theorems proved by Kurt Gödel, which required the abandonment of several theses central to the aims of logicism and formalism. Of far greater significance, in the present context, are the changes in Carnap’s own philosophical outlook, brought about not only by Gödel’s theorems but concurrent discussions within the Vienna Circle as well as his own researches. Consequently, the exact sense in which Carnap attempted the synthesis of logicism and formalism in the Logical Syntax requires careful examination. In what follows below, the evolution of Carnap’s synthesis will be traced, from the first reconciliation he proposed (in Königsberg in 1930), through the synthesis that appeared with the publication of The Logical Syntax of Language. The aim is to determine which modifications of Carnap’s synthesis were required by Gödel’s theorems, and which were motivated by changes in his own thinking. Although the characteristic theses of both logicism and formalism required profound modifications because of Gödel’s theorems, the philosophical impulses that originally fueled their programs retained much of their former virulence. But the changes in Carnap’s thought that ocurred in the years he was developing his synthesis especially affected his appreciation of the philosophical motivations underwriting the logicist approach, so that much of the philosophical insight that inspired it is lost, and Carnap’s combination of logicism and formalism is a putative synthesis at best.