ABSTRACT
Intuitionism is a viewpoint in the philosophy of mathematics, developed by the Dutch mathematician L.E.J. Brouwer, which rejects many standard (“classical”) mathematical results and methods, and in particular rejects as invalid some principles of standard (“classical,” two-valued) elementary logic. It is held by a minority of mathematicians (a minority even of those willing to go public with philosophical statements) but has inspired much mathematical research: the term ‘intuitionism’ is also used for the body of mathematics developed in accordance with the strictures of the philosophy, a body by now of sufficient mathematical richness to be of interest even to researchers not concerned with philosophical issues. The term ‘constructivism’ is also used for similar philosophical viewpoints and mathematical methods: when a contrast is made between them, ‘intuitionism’ refers to a viewpoint or a set of mathematical methods more like those of Brouwer himself and ‘constructivism’ more generically to these and related ones. ‘Intuitionistic (or ‘constructive’) logic’ refers to a specific version of propositional and first-order logic first formalized by Brouwer’s younger colleague Arend Heyting. There doesn’t seem to be any useful analogy with the views known as intuitionistic in such areas of philosophy as meta-ethics, nor is constructive mathematics connected to what is called ‘social constructivism’ about mathematics and mathematics education.
