A definition of a property P is impredicative if it quantifies over a domain to which P belongs. Due to influential arguments by Ramsey and Gödel, impredicative mathematics is often thought to possess special metaphysical commitments. It seems that an impredicative definition of a property P does not have the intended meaning unless P already exists, suggesting that the existence of P cannot depend on its explicit definition. Carnap (1937 [1934], p. 164) argues, however, that accepting impredicative definitions amounts to (...) choosing a "form of language" and is free from metaphysical implications. This paper explains this view in its historical context. I discuss the development of Carnap’s thought on the foundations of mathematics from the mid-1920s to the mid-1930s, concluding with an account of Carnap’s (1937 [1934]) non-Platonistic defense of impredicativity. This discussion is also important for understanding Carnap’s influential views on ontology more generally, since Carnap’s (1937 [1934]) view, according to which accepting impredicative definitions amounts to choosing a "form of language", is an early precursor of the view Carnap presents in "Empiricism, Semantics and Ontology" (1956 [1950]), according to which referring to abstract entities amounts to accepting a "linguistic framework". (shrink)

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 supposedly decisive objections against 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 issues by looking at Carnap's response to the incompleteness theorems. Drawing on arguments put forward by Gödel and Beth, I show 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. (shrink)

Herbert G. Bohnert (1918–84) was a student and lifelong friend of Carnap. He wrote a doctoral thesis on Ramsey sentences and, after wavering between academia and the computer industry, eventually taught at Michigan State University. Bohnert defended Carnapian positions throughout his career and was especially productive in the 1970s. Unfortunately, Carnap’s philosophy was deemed hopelessly out of date during this period, and partly for this reason, Bohnert is almost completely forgotten today. This fate is undeserved. In this article, I reconstruct (...) his career and philosophical development by drawing on publications, as well as diary entries and private correspondence. I then argue that Carnap scholarship would benefit from closer attention to Bohnert’s writings. I support this claim by means of a case study from the philosophy of mathematics: Bohnert’s account of a conversation about set theory with Carnap in 1968 sheds new light on the contested question of whether Carnap held mathematics to be analytic in natural language. (shrink)