Intuitionism in mathematics

In this article, we survey intuitionism as a philosophy of mathematics, with emphasis on the philosophical views endorsed by Brouwer, Heyting, and Dummett. Before we proceed, however, a few general remark are in order. We must stress that intuitionism is not to be regarded as synonymous with constructivism, an umbrella term that roughly refers to any particular form of mathematics that adopts "we can construct" as the appropriate interpretation of the phrase "there exists". However, intuitionism remains one of the most prominent varieties of constructive mathematics in existence. The curious reader can see our related entry constructive mathematics for complementary background. For more on the intuitionistic rejection of actual infinities, see also the entry on the infinite. Finally, because intuitionism advocates a revision of classical mathematics, a certain amount of mathematical knowledge is required to fully appreciate some parts of this article, most importantly Section 1.2 on intuitionistic analysis. Readers can check any introductory textbook on classical real analysis or topology if they are not familiar with our terminology. Some familiarity with basic set theory will also be presupposed in Section 1.1, in particular regarding transfinite ordinals and uncountable cardinalities. Since our focus is intuitionistic mathematics, intuitionistic logic is not presented in this article. But we include a list of notable theorems and non-theorems of intuitionistic logic in the appendix for reference.