Stephen Cole Kleene ( ; January 5, 1909 – January 25, 1994) was an American

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...

. One of the students of

Alonzo Church Alonzo Church (June 14, 1903 – August 11, 1995) 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 bes ...

, Kleene, along with

Rózsa Péter Rózsa Péter, until January 1934 Rózsa Politzer, (17 February 1905 – 16 February 1977) was a Hungarian mathematician and logician. She is best known as the "founding mother of recursion theory". Early life and education Péter was bor ...

Alan Turing Alan Mathison Turing (; 23 June 1912 – 7 June 1954) was an English mathematician, computer scientist, logician, cryptanalyst, philosopher and theoretical biologist. He was highly influential in the development of theoretical computer ...

Emil Post Emil Leon Post (; February 11, 1897 – April 21, 1954) was an American mathematician and logician. He is best known for his work in the field that eventually became known as computability theory. Life Post was born in Augustów, Suwałki Govern ...

, and others, is best known as a founder of the branch of

mathematical logic Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic com ...

known as

recursion theory Computability theory, also known as recursion theory, is a branch of mathematical logic, computer science, and the theory of computation that originated in the 1930s with the study of computable functions and Turing degrees. The field has since ex ...

, which subsequently helped to provide the foundations of

theoretical computer science Theoretical computer science is a subfield of computer science and mathematics that focuses on the Abstraction, abstract and mathematical foundations of computation. It is difficult to circumscribe the theoretical areas precisely. The Associati ...

. Kleene's work grounds the study of

computable function Computable functions are the basic objects of study in computability theory. Informally, a function is ''computable'' if there is an algorithm that computes the value of the function for every value of its argument. Because of the lack of a precis ...

s. A number of mathematical concepts are named after him: Kleene hierarchy,

Kleene algebra In mathematics and theoretical computer science, a Kleene algebra ( ; named after Stephen Cole Kleene) is a semiring that generalizes the theory of regular expressions: it consists of a set supporting union (addition), concatenation (multiplicati ...

, the

Kleene star In mathematical logic and theoretical computer science, the Kleene star (or Kleene operator or Kleene closure) is a unary operation on a Set (mathematics), set to generate a set of all finite-length strings that are composed of zero or more repe ...

(Kleene closure),

Kleene's recursion theorem In computability theory, Kleene's recursion theorems are a pair of fundamental results about the application of computable functions to their own descriptions. The theorems were first proved by Stephen Kleene in 1938 and appear in his 1952 ...

and the

Kleene fixed-point theorem In the mathematical areas of order and lattice theory, the Kleene fixed-point theorem, named after American mathematician Stephen Cole Kleene, states the following: :Kleene Fixed-Point Theorem. Suppose (L, \sqsubseteq) is a directed-complete pa ...

. He also invented

regular expressions A regular expression (shortened as regex or regexp), sometimes referred to as rational expression, is a sequence of character (computing), characters that specifies a pattern matching, match pattern in string (computer science), text. Usually ...

in 1951 to describe McCulloch-Pitts neural networks, and made significant contributions to the foundations of mathematical

intuitionism In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of fu ...

Biography

Kleene was awarded a bachelor's degree from

Amherst College Amherst College ( ) is a Private college, private Liberal arts colleges in the United States, liberal arts college in Amherst, Massachusetts, United States. Founded in 1821 as an attempt to relocate Williams College by its then-president Zepha ...

in 1930. He was awarded a Ph.D. in mathematics from

Princeton University Princeton University is a private university, private Ivy League research university in Princeton, New Jersey, United States. Founded in 1746 in Elizabeth, New Jersey, Elizabeth as the College of New Jersey, Princeton is the List of Colonial ...

in 1934, where his thesis, entitled ''A Theory of Positive Integers in Formal Logic'', was supervised by

. In the 1930s, he did important work on Church's

lambda calculus In mathematical logic, the lambda calculus (also written as ''λ''-calculus) is a formal system for expressing computability, computation based on function Abstraction (computer science), abstraction and function application, application using var ...

. In 1935, he joined the mathematics department at the

University of Wisconsin–Madison The University of Wisconsin–Madison (University of Wisconsin, Wisconsin, UW, UW–Madison, or simply Madison) is a public land-grant research university in Madison, Wisconsin, United States. It was founded in 1848 when Wisconsin achieved st ...

, where he spent nearly all of his career. After two years as an instructor, he was appointed assistant professor in 1937. While a visiting scholar at the

Institute for Advanced Study The Institute for Advanced Study (IAS) is an independent center for theoretical research and intellectual inquiry located in Princeton, New Jersey. It has served as the academic home of internationally preeminent scholars, including Albert Ein ...

in Princeton, 1939–1940, he laid the foundation for

, an area that would be his lifelong research interest. In 1941, he returned to Amherst College, where he spent one year as an associate professor of mathematics. During

World War II World War II or the Second World War (1 September 1939 – 2 September 1945) was a World war, global conflict between two coalitions: the Allies of World War II, Allies and the Axis powers. World War II by country, Nearly all of the wo ...

, Kleene was a lieutenant commander in the

United States Navy The United States Navy (USN) is the naval warfare, maritime military branch, service branch of the United States Department of Defense. It is the world's most powerful navy with the largest Displacement (ship), displacement, at 4.5 millio ...

. He was an instructor of navigation at the U.S. Naval Reserve's Midshipmen's School in

New York New York most commonly refers to: * New York (state), a state in the northeastern United States * New York City, the most populous city in the United States, located in the state of New York New York may also refer to: Places United Kingdom * ...

, and then a project director at the

Naval Research Laboratory The United States Naval Research Laboratory (NRL) is the corporate research laboratory for the United States Navy and the United States Marine Corps. Located in Washington, DC, it was founded in 1923 and conducts basic scientific research, appl ...

Washington, D.C. Washington, D.C., formally the District of Columbia and commonly known as Washington or D.C., is the capital city and federal district of the United States. The city is on the Potomac River, across from Virginia, and shares land borders with ...

In 1946, Kleene returned to the University of Wisconsin-Madison, becoming a full professor in 1948 and the Cyrus C. MacDuffee professor of mathematics in 1964. He served two terms as the Chair of the Department of Mathematics and one term as the Chair of the Department of Numerical Analysis (later renamed the Department of Computer Science). He also served as Dean of the College of Letters and Science in 1969–1974. During his years at the University of Wisconsin he was thesis advisor to 13 Ph.D. students. He retired from the University of Wisconsin in 1979. In 1999 the mathematics library at the University of Wisconsin was renamed in his honor. Kleene's teaching at Wisconsin resulted in three texts in

, Kleene (1952, 1967) and Kleene and Vesley (1965). The first two are often cited and still in print. Kleene (1952) wrote alternative proofs to the

Gödel's incompleteness theorems Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of in formal axiomatic theories. These results, published by Kurt Gödel in 1931, are important both in mathematical logic and in the phi ...

that enhanced their canonical status and made them easier to teach and understand. Kleene and Vesley (1965) is the classic American introduction to

intuitionistic logic 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 ...

and mathematical

. Kleene served as president of the

Association for Symbolic Logic The Association for Symbolic Logic (ASL) is an international organization of specialists in mathematical logic and philosophical logic. The ASL was founded in 1936, and its first president was Curt John Ducasse. The current president of the ASL ...

, 1956–1958, and of the International Union of History and Philosophy of Science, 1961. The importance of Kleene's work led to

Daniel Dennett Daniel Clement Dennett III (March 28, 1942 – April 19, 2024) was an American philosopher and cognitive scientist. His research centered on the philosophy of mind, the philosophy of science, and the philosophy of biology, particularly as those ...

coining the saying, published in 1978, that "Kleeneness is next to Gödelness." In 1990, he was awarded the

National Medal of Science The National Medal of Science is an honor bestowed by the President of the United States to individuals in science and engineering who have made important contributions to the advancement of knowledge in the fields of behavioral science, behavior ...

. Kleene and his wife Nancy Elliott had four children. He had a lifelong devotion to the family farm in Maine. An avid mountain climber, he had a strong interest in nature and the environment, and was active in many conservation causes.

Legacy

At each conference of the

Symposium on Logic in Computer Science The ACM–IEEE Symposium on Logic in Computer Science (LICS) is an annual academic conference on the theory and practice of computer science in relation to mathematical logic. Extended versions of selected papers of each year's conference appear i ...

the Kleene Award, in honour of Stephen Cole Kleene, is given for the best student paper.

Selected publications

* 1935. * 1935. * 1935. * 1936. * 1936. * 1938. * 1943. * 1951. * 1952. * 1956. * 1965 * 1967.
Mathematical Logic.
'

John Wiley & Sons John Wiley & Sons, Inc., commonly known as Wiley (), is an American Multinational corporation, multinational Publishing, publishing company that focuses on academic publishing and instructional materials. The company was founded in 1807 and pr ...

. Dover reprint, 2002. . * 1981. "Origins of Recursive Function Theory" in
Annals of the History of Computing 3
'' No. 1. * 1987.

Notes

References

External links

*
Biographical memoir
– by

Saunders Mac Lane Saunders Mac Lane (August 4, 1909 – April 14, 2005), born Leslie Saunders MacLane, was an American mathematician who co-founded category theory with Samuel Eilenberg. Early life and education Mac Lane was born in Norwich, Connecticut, near w ...