From Intuition to Tolerance in Carnap's Philosophy of Mathematics

Michael Friedman (Stanford) gives a talk at the MCMP workshop "Influences on the Aufbau" (1-3 July, 2013) titled "From Intuition to Tolerance in Carnap's Philosophy of Mathematics". Abstract: I plan to discuss the evolution of Carnap's views on arithmetic and geometry from Der Raum, through the Aufbau period, to Logical Syntax and the semantic period. I will concentrate on the way in which he consistently distinguished the two cases -- where geometry, in Der Raum, is explicitly tied to spatial intuition, and then, even though it remains distinguished from arithmetic, its ties to spatial intui- tion become empirical rather than a priori. The character of Carnap's generally tolerant attitude towards philosophical disputes in the foundations of mathematics thereby changes as well. Carnap's attitude towards intuition in the arithmetical case, finally, becomes clarified in his exchange with E. W. Beth in the Schilpp volume -- and this clarifies his application of the principle of tolerance in this case as well.