“Mathematical Philosophy and Philosophical Mathematics“ by Timothy Williamson

Title: Mathematical Philosophy and Philosophical Mathematics Abstract: Some of Bertrand Russell’s lectures in China (1920-21) overlap his book Introduction to Mathematical Philosophy (1919). One might expect mathematical philosophy to use mathematical methods in answering philosophical questions, while philosophical mathematics would use philosophical methods in answering mathematical questions. Thus philosophical mathematics poses a threat to the idea that only a mathematical proof can answer a pure mathematical question. However, Russell argues that abductive methods like those of philosophy are needed to determine the first principles of mathematics, so that not even ‘pure’ mathematics can be kept pure of such apparently non-mathematical methods. The lecture will discuss such ‘impure’ aspects of mathematics in relation to unrestricted quantification, higher-order logic, axioms of set theory, and modality.