Fri Oct 17, 2008
60 Evans Hall, 4:10–6 PM
Peter Koellner (Harvard)
Truth in Mathematics: The Question of Pluralism
In his classic work “The Logical Syntax of Language” (1934) Carnap defends three distinctive claims: (1) The thesis that logic and pure mathematics are analytic and hence without content and purely formal. (2) A radical pluralist conception of pure mathematics (embodied in his Principle of Tolerance) according to which we have great freedom in choosing the fundamental postulates. (3) A minimalist conception of philosophy on which most traditional questions are rejected as pseudo-questions and philosophy is identified with the study of the logical syntax of the language of science. Carnap’s discussion is quite sophisticated metamathematically and his position is quite subtle. Indeed I think it is the most sophisticated defense of pluralism in mathematics that has appeared to date and that is my reason for concentrating on it. I will begin by criticizing both Carnap’s radical form of pluralism and his minimalist conception of philosophy. I will then turn to the question of what it would take to establish pluralism in mathematics. My focus will be on approaches where (in contrast to the approach of Carnap and many recent approaches) the question of pluralism is sensitive to actual developments in mathematics. This will involve describing various “bifurcation scenarios” in set theory that are made available by some recent results (joint with Hugh Woodin). These scenarios are sensitive to the Omega Conjecture.