The incompleteness of formal systems for arithmetic has been a recognized fact of mathematics. The term “incompleteness” suggests that the formal system in question fails to offer a deduction which it ought to. This chapter focuses on the status of a formal system, Peano Arithmetic, and explores a viewpoint on which Peano Arithmetic occupies an intrinsic, conceptually well-defined region of arithmetical truth. The idea is that it consists of those truths which can be perceived directly from the purely arithmetical content (...) of a categorical conceptual analysis of the notion of natural number. The chapter explores its conceptual stability in light of the apparently destabilizing effect, through extensibility, of the phenomenon of incompleteness. It also offers heuristic and conceptual support for the viewpoint that Peano Arithmetic is the strongest natural first-order system for arithmetic, and that there is a sense in which it is complete with respect to purely arithmetical truth. (shrink)

I see the perspective of Tolerance as enshrining an attitude toward philosophical work that stresses its continuity with the procedures of conceptual clarification through mathematisation found in the sciences. What I have tried to show is that Carnap's understanding of the philosophical foundations of mathematics is inseparable from his understanding of the business of philosophy of empirical science.