For quite awhile I assumed that my forthcoming book The Annotated Turing: A Guided Tour through Alan Turing's Historic Paper on Computability and the Turing Machine would not even mention Dutch mathematician Luitzen Egbertus Jan Brouwer (1881 – 1966). I was seriously mistaken. Brouwer played a major role in the debates in the early decades of the last century involving the foundations of mathematics. Three distinct approaches emerged, called logicism , formalism , and intuitionism . Logicism is mostly closely associated with Alfred North Whitehead and Bertrand Russell's three-volume Principia Mathematica , which carried on the work of Gottlob Frege in attempting to derive all of mathematics from basic principles of logic. Formalism is mostly closely associated with David Hilbert, who tried to treat mathematics in a strictly formal manner as the manipulation of symbols. Most important to Hilbert were the establishment of certain metamathematical characteristics of axiomatic systems, Read More...