In the philosophy of mathematics, intuitionism, or neointuitionism, is an approach where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of fundamental principles claimed to exist in an objective reality. That is, logic and mathematics are not considered analytic activities wherein deep properties of objective reality are revealed and applied, but are instead considered the application of internally consistent methods used to realize more complex mental constructs, regardless of their possible independent existence in an objective reality.
Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics.
Stephen Cole Kleene was an American mathematician. 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'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.
Alonzo Church was an American computer scientist, mathematician, logician, and philosopher who made major contributions to mathematical logic and the foundations of theoretical computer science. He is best known for the lambda calculus, the Church–Turing thesis, proving the unsolvability of the Entscheidungsproblem, the Frege–Church ontology, and the Church–Rosser theorem. Alongside his doctoral student Alan Turing, Church is considered one of the founders of computer science.
Sir Michael Anthony Eardley Dummett was an English academic described as "among the most significant British philosophers of the last century and a leading campaigner for racial tolerance and equality." He was, until 1992, Wykeham Professor of Logic at the University of Oxford. He wrote on the history of analytic philosophy, notably as an interpreter of Frege, and made original contributions particularly in the philosophies of mathematics, logic, language and metaphysics.
Solomon Feferman was an American philosopher and mathematician who worked in mathematical logic. In addition to his prolific technical work in proof theory, computability theory, and set theory, he was known for his contributions to the history of logic and as a vocal proponent of the philosophy of mathematics known as predicativism, notably from an anti-platonist stance.
Kenneth Jon Barwise was an American mathematician, philosopher and logician who proposed some fundamental revisions to the way that logic is understood and used.
In mathematical logic, the disjunction and existence properties are the "hallmarks" of constructive theories such as Heyting arithmetic and constructive set theories (Rathjen 2005).
In mathematical logic, projective determinacy is the special case of the axiom of determinacy applying only to projective sets.
William Alvin Howard is a proof theorist best known for his work demonstrating formal similarity between intuitionistic logic and the simply typed lambda calculus that has come to be known as the Curry–Howard correspondence. He has also been active in the theory of proof-theoretic ordinals. He earned his Ph.D. at the University of Chicago in 1956 for his dissertation "k-fold recursion and well-ordering". He was a student of Saunders Mac Lane.
The Cabal was, or perhaps is, a set of set theorists in Southern California, particularly at UCLA and Caltech, but also at UC Irvine. Organization and procedures range from informal to nonexistent, so it is difficult to say whether it still exists or exactly who has been a member, but it has included such notable figures as Donald A. Martin, Yiannis N. Moschovakis, John R. Steel, and Alexander S. Kechris. Others who have published in the proceedings of the Cabal seminar include Robert M. Solovay, W. Hugh Woodin, Matthew Foreman, and Steve Jackson.
Yiannis Nicholas Moschovakis is a set theorist, descriptive set theorist, and recursion (computability) theorist, at UCLA.
In mathematical logic, algebraic logic is the reasoning obtained by manipulating equations with free variables.
Anne Sjerp Troelstra was a professor of pure mathematics and foundations of mathematics at the Institute for Logic, Language and Computation (ILLC) of the University of Amsterdam.
(Nels) David Nelson, an American mathematician and logician, was born on January 2, 1918, in Cape Girardeau, Missouri. Upon graduation from the Ph.D. program at the University of Wisconsin-Madison, Nelson relocated to Washington, D.C. Nelson remained in Washington, D.C. as a Professor of Mathematics at The George Washington University until his death on August 22, 2003.
Dirk van Dalen is a Dutch mathematician and historian of science.
John Charles Chenoweth McKinsey, usually cited as J. C. C. McKinsey, was an American mathematician known for his work on game theory and mathematical logic, particularly, modal logic.
Sergei Nikolaevich Artemov is a Russian-American researcher in logic and its applications. He currently holds the title of Distinguished Professor at the Graduate Center of the City University of New York where he is the founder and head of its research laboratory for logic and computation. His research interests include proof theory and logic in computer science, optimal control and hybrid systems, automated deduction and verification, epistemology, and epistemic game theory. He is best known for his invention of logics of proofs and justifications.
The following is a list of works by philosopher Graham Priest.
Dutch philosophy is a broad branch of philosophy that discusses the contributions of Dutch philosophers to the discourse of Western philosophy and Renaissance philosophy. The philosophy, as its own entity, arose in the 16th and 17th centuries through the philosophical studies of Desiderius Erasmus and Baruch Spinoza. The adoption of the humanistic perspective by Erasmus, despite his Christian background, and rational but theocentric perspective expounded by Spinoza, supported each of these philosopher's works. In general, the philosophy revolved around acknowledging the reality of human self-determination and rational thought rather than focusing on traditional ideals of fatalism and virtue raised in Christianity. The roots of philosophical frameworks like the mind-body dualism and monism debate can also be traced to Dutch philosophy, which is attributed to 17th century philosopher René Descartes. Descartes was both a mathematician and philosopher during the Dutch Golden Age, despite being from the Kingdom of France. Modern Dutch philosophers like D.H. Th. Vollenhoven provided critical analyses on the dichotomy between dualism and monism.
