The Foundational Crisis in Mathematics

source: Philosophical Overdose    2017年1月28日
An introduction to the foundational crisis in mathematics, the 19th-20th century crisis in searching for the proper foundations for mathematics in the face of various paradoxes regarding Non-Euclidean geometries, the infinities of Georg Cantor, Bertrand Russell's discovery of an inconsistency at the very heart of set theory (and hence all mathematics), and Kurt Gödel's proof of the incompleteness of mathematics itself. These issues led to various philosophical schools, including Formalism (formulated by David Hilbert, which conceives of mathematics as the manipulation of meaningless symbols), and versions of Finitism which rejects actual infinities like those of Cantor (e.g. Intuitionism, which is a kind of Constructive Mathematics formulated by L. E. J. Brouwer and inspired by Immanuel Kant, which rejects the law of the excluded middle in logic and conceives of mathematics as a construction of the human mind).