Syntax and Intuition: On Gödel's Critique of Carnap

Abstract

The syntactic interpretation of mathematics (SIM) refers to the view that interprets mathematical propositions as true expressions solely by virtue of the syntactical conventions governing the use of those expressions and not by corresponding to objective mathematical facts. This view, exemplified by the Vienna Circle and particularly Rudolf Carnap, is the target of Gödel's sustained critique and his defence of mathematical realism and intuition. I analyse Gödel's central arguments against SIM: the arbitrariness and asymmetry arguments, which challenge the notion that mathematics is contentless, and the circularity argument, which concerns the role of consistency proofs. I also examine two contemporary defences of Carnap: one appealing to dependable rather than provable consistency, and another recasting Carnap's project as transforming justificatory questions into pragmatic choices of linguistic frameworks. I contend that both defences fail to fulfil the original philosophical ambitions of the syntactic view, and that this failure indirectly bolsters the case for mathematical realism.