Foundations

Abstract

In the twentieth century, there was a great deal of concrete, practical mathematics. Statistics flourished, the computer proved the Four Colour theorem, and number theorists factored 100 digit integers. On the other hand, much twentieth century mathematics was characterised by a degree of abstraction never seen before. It was not the Euclidean plane that was studied, but the vector spaces and topological spaces which are abstractions of it. It was not particular groups that were studied so much as the whole ‘category’ of groups. Much twentieth century mathematics can be classified as philosophical. Set theorists attempted to find the ultimate basis for all mathematics. Set theorists also probed the infinite. Workers in foundations examined the limits of human reason itself, with Kurt Gödel (1906–1978) showing that some mathematical statements are subject neither to proof nor to counterexample. Various logics were put forward in an attempt to elucidate the nature of valid, human thought.