Carnap's philosophy of mathematics

For several decades Carnap’s philosophy of mathematics used to be either dismissed or ignored. It was perceived as a form of linguistic conventionalism and thus taken to rely on the bankrupt notion of truth by convention. However, recent scholarship has revealed a more subtle picture. It has been forcefully argued that Carnap is not a linguistic conventionalist in any straightforward sense, and that apparently decisive objections to his position target a straw man. This raises two questions. First, how exactly should we characterise Carnap’s actual philosophy of mathematics? Secondly, is his position an attractive alternative to established views? I will tackle these by looking at Carnap’s response to the incompleteness theorems. Drawing on objections put forward by Gödel and Beth, I argue that some crucial aspects of Carnap’s positive account have remained underdeveloped. Suggestions on what a full evaluation of Carnap’s position requires are made.