Intuitionism in the context of "Constructivism (math)"

Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems of intuitionistic logic do not assume the law of excluded middle and double negation elimination, which are fundamental inference rules in classical logic.

Formalized intuitionistic logic was originally developed by Arend Heyting to provide a formal basis for L. E. J. Brouwer's programme of intuitionism. From a proof-theoretic perspective, Heyting’s calculus is a restriction of classical logic in which the law of excluded middle and double negation elimination have been removed. Excluded middle and double negation elimination can still be proved for some propositions on a case by case basis, however, but do not hold universally as they do with classical logic. The standard explanation of intuitionistic logic is the BHK interpretation.

Luitzen Egbertus Jan "Bertus" Brouwer (27 February 1881 – 2 December 1966) was a Dutch mathematician and philosopher who worked in topology, set theory, measure theory and complex analysis. Regarded as one of the greatest mathematicians of the 20th century, he is known as one of the founders of modern topology, particularly for establishing his fixed-point theorem and the topological invariance of dimension.

Brouwer also became a major figure in the philosophy of intuitionism, a constructivist school of mathematics which argues that math is a cognitive construct rather than a type of objective truth. This position led to the Brouwer–Hilbert controversy, in which Brouwer sparred with his formalist colleague David Hilbert. Brouwer's ideas were subsequently taken up by his student Arend Heyting and Hilbert's former student Hermann Weyl. In addition to his mathematical work, Brouwer also published the short philosophical tract Life, Art, and Mysticism (1905).

Giovanni Papini (9 January 1881 – 8 July 1956) was an Italian journalist, essayist, novelist, short story writer, poet, literary critic, and philosopher. A controversial literary figure of the early and mid-twentieth century, he was the earliest and most enthusiastic representative and promoter of Italian pragmatism. Papini was admired for his writing style and engaged in heated polemics. Involved with avant-garde movements such as futurism and post-decadentism, he moved from one political and philosophical position to another, always dissatisfied and uneasy: he converted from anti-clericalism and atheism to Catholicism, and went from convinced interventionism – before 1915 – to an aversion to war. In the 1930s, after moving from individualism to conservatism, he finally became a fascist, while maintaining an aversion to Nazism.

As one of the founders of the journals Leonardo (1903) and Lacerba (1913), he conceived literature as "action" and gave his writings an oratory and irreverent tone. Though self-educated, he was an influential iconoclastic editor and writer, with a leading role in Italian futurism and the early literary movements of youth. Working in Florence, he actively participated in foreign literary philosophical and political movements such as the French intuitionism of Bergson and the Anglo-American pragmatism of Peirce and James. Promoting the development of Italian culture and life with an individualistic and dreamy conception of life and art, he acted as a spokesman for Roman Catholic religious beliefs.

The Brouwer–Hilbert controversy (German: Grundlagenstreit, lit. 'foundational debate') was a debate in twentieth-century mathematics over fundamental questions about the consistency of axioms and the role of semantics and syntax in mathematics. L. E. J. Brouwer, a proponent of the constructivist school of intuitionism, opposed David Hilbert, a proponent of formalism. Much of the controversy took place while both were involved with Mathematische Annalen, the leading mathematical journal of the time, with Hilbert as editor-in-chief and Brouwer as a member of its editorial board. In 1928, Hilbert had Brouwer removed from the editorial board of Mathematische Annalen.

Stephen Cole Kleene (/ˈkleɪni/ KLAY-nee; January 5, 1909 – January 25, 1994) was an American mathematician and logician. One of the students of Alonzo Church, Kleene, along with Rózsa Péter, Alan Turing, Emil Post, and others, is best known as a founder of the branch of mathematical logic known as recursion theory, which subsequently helped to provide the foundations of theoretical computer science. Kleene's work grounds the study of computable functions. A number of mathematical concepts are named after him: Kleene hierarchy, Kleene algebra, the Kleene star (Kleene closure), Kleene's recursion theorem and the Kleene fixed-point theorem. He also invented regular expressions in 1951 to describe McCulloch-Pitts neural networks, and made significant contributions to the foundations of mathematical intuitionism.