Wittgenstein, Carnap, and Turing: Contrasting Notions of Analysis

Carnap never once mentions Turing in his published writings. From an historical point of view this is unsurprising. For the implications of Turing's analysis of computability for the foundations of mathematics and physics, artificial intelligence, and the very architecture of science were to be developed and widely appreciated only after Carnap's death. Perhaps more significant, once Carnap had developed the position articulated in The Logical Syntax of Language his principle of tolerance licensed a form of conciliatory pluralism about positions in the foundations of mathematics. Carnap's pluralism construed debates over infinitary reasoning, impredicavity, logicism, and intuitionism as rationally tractable, but not through direct reasoning on behalf of truth claims. Rather, he proposed the development of formal axiomatizations of languages and pragmatic assessments of these. In subsequent work Carnap was not inclined to view these particular foundational debates as the primary arena for the articulation of his philosophy1. Instead, he broadened his conception of explication to account for the distinction between analytic and synthetic truth in all areas of science (Carus 2007a).KeywordsFormal SystemTuring MachineDefinite MethodIncompleteness TheoremDeduction TheoremThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.