From Intuition to Tolerance : The Development of Carnap ’ s Philosophy of Mathematics ∗

I shall consider a sequence of episodes in Carnap’s thinking about mathematics beginning with his doctoral dissertation Der Raum (1922), which represents the most explicitly Kantian episode in his thought. I proceed to developments leading up through the Aufbau (1928), then turn to The Logical Syntax of Language (1934)—where the Principle of Tolerance is first officially formulated—and conclude by considering developments in the following semantic period and beyond. I am not describing a transition from an epistemological grounding of mathematics in some kind of intuition—whether Kantian, Husserlian, Brouwerian, or Gödelian—to a pluralistic approach involving a “tolerant” appreciation of a variety of epistemological strategies. Nothing could be further from the truth. For Carnap is just as tolerant, in this sense, in Der Raum, the main point of which is to argue that logical and formalistic approaches to space and geometry (as in Russell or Hilbert), empirical or physical approaches (as in Helmholtz or Einstein), and intuition-based approaches in the Kantian, neo-Kantian, and Husserlian traditions are all basically correct. It is just that these different and apparently opposing approaches are really talking about different types or “meanings” of space—formal, physical, and intuitive. And, although Carnap abandons the intuitive space of Der Raum by the time of the Aufbau, he preserves important aspects of Kantian and neo-Kantian epistemology in this work and even later. We obtain a better sense of Carnap’s characteristic approach to the philosophy of mathematics—and to philosophy more generally—if we take seriously his claim that, throughout his career, he held no philosophical positions in the usual sense at all. Carnap develops this claim at length in his Intellectual Autobiography (1963a, 17-18). He describes the variety of “philosophical languages” he adopted in speaking with different “philosophically interested” friends (17): “With one friend I might talk in a language that could be characterized as realistic or even materialistic; . . In a talk with another friend, I