Philosophy of Mathematics

Catalog description

After some stage-setting on logic, and an overview of philosophical questions raised by mathematics, we will focus on Logicism: the thesis that arithmetic (understood as the part of mathematics concerned with the numbers in various number-systems) is or is a fragment of higher-order logic. We will approach Logicism historically, through writing by Frege and B. Russell. We will then look two other views of mathematics: Hilbertian formalism and Intuitionism. This will involve discussion of three restrictive projects: predicativism (imposing predicativity restrictions on definitions, comprehension principles and induction), constructivism (restricting logic to intuitionistic logic), and finitistic arithmetic. Since philosophical work in mathematics is enmeshed with mathematical work on the foundations of mathematics, it will be necessary to devote time to fully mathematical material (sets, proofs and models).