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Mathematics and the Infinite: An Introduction

Lecturer: Arran Stewart

Originally Taught: Summer School 2015

3-5pm | 9-13 Feb

Thinkers in both continental and analytic traditions make liberal use of arguments drawn from the mathematical results of Cantor, Gödel and Turing. Alain Badiou in Being and Event, for instance, makes use of Cantor's theories of the transfinite in his examination of ontology, and the analytic philosopher John Lucas relies on Gödel's proofs of mathematical incompleteness to argue that minds cannot be explained as machines.

To a mathematical outsider, these arguments can seem opaque – do the mathematical results justify all that is claimed? In these lectures we provide an introduction to the concepts needed to illuminate the work of Cantor and later thinkers who have built on his results. The lectures require no mathematical background: in straightforward terms, they will explore mathematical conceptualizations of structure, proof, and countability.

We will touch on some of the contrasting approaches to work at the juncture of philosophy and mathematics, but the primary goal will be to outline the native "mathematical approach", which will give students a foundation for further exploration of their own in the area.

We start with the most basic examples of symbols and structures, and how they can be manipulated, and build gradually up to an elucidation of (one approach to) the infinite in mathematics, explicating Cantor's Theorem and his results on uncountability.

Course schedule

Lecture 1: Symbols, structure and rules
Lecture 2: Cases and contradiction
Lecture 3: Mapping and matching
Lecture 4: Counting and number; Hilbert's Hotel
Lecture 5: Cantor and the Uncountable