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 mathematician, computer scientist, logician, philosopher, professor and editor who made major contributions to mathematical logic and the foundations of theoretical computer scie ...
, Kleene, along with
Rózsa Péter
Rózsa Péter, born 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 born in Budapest, ...
,
Alan Turing
Alan Mathison Turing (; 23 June 1912 – 7 June 1954) was an English mathematician, computer scientist, logician, cryptanalyst, philosopher, and theoretical biologist. Turing was highly influential in the development of theoretical c ...
,
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 Gove ...
, and others, is best known as a founder of the branch of
mathematical logic
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 ...
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 sinc ...
, which subsequently helped to provide the foundations of
theoretical computer science
Theoretical computer science (TCS) is a subset of general computer science and mathematics that focuses on mathematical aspects of computer science such as the theory of computation, lambda calculus, and type theory.
It is difficult to circumsc ...
. Kleene's work grounds the study of
computable function
Computable functions are the basic objects of study in computability theory. Computable functions are the formalized analogue of the intuitive notion of algorithms, in the sense that a function is computable if there exists an algorithm that can d ...
s. A number of mathematical concepts are named after him:
Kleene hierarchy,
Kleene algebra, the
Kleene star
In mathematical logic and computer science, the Kleene star (or Kleene operator or Kleene closure) is a unary operation, either on sets of strings or on sets of symbols or characters. In mathematics,
it is more commonly known as the free monoid ...
(Kleene closure),
Kleene's recursion theorem and the
Kleene fixed-point theorem. He also invented
regular expressions
A regular expression (shortened as regex or regexp; sometimes referred to as rational expression) is a sequence of characters that specifies a search pattern in text. Usually such patterns are used by string-searching algorithms for "find" o ...
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 ...
.
Biography
Kleene was awarded a bachelor's degree from
Amherst College in 1930. He was awarded a Ph.D. in mathematics from
Princeton University
Princeton University is a private research university in Princeton, New Jersey. Founded in 1746 in Elizabeth as the College of New Jersey, Princeton is the fourth-oldest institution of higher education in the United States and one of the n ...
in 1934, where his thesis, entitled ''A Theory of Positive Integers in Formal Logic'', was supervised by
Alonzo Church
Alonzo Church (June 14, 1903 – August 11, 1995) was an American mathematician, computer scientist, logician, philosopher, professor and editor who made major contributions to mathematical logic and the foundations of theoretical computer scie ...
. In the 1930s, he did important work on Church's
lambda calculus
Lambda calculus (also written as ''λ''-calculus) is a formal system in mathematical logic for expressing computation based on function abstraction and application using variable binding and substitution. It is a universal model of computation t ...
. In 1935, he joined the mathematics department at the
University of Wisconsin–Madison
A university () is an institution of higher (or tertiary) education and research which awards academic degrees in several academic disciplines. ''University'' is derived from the Latin phrase ''universitas magistrorum et scholarium'', which ...
, 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), located in Princeton, New Jersey, in the United States, is an independent center for theoretical research and intellectual inquiry. It has served as the academic home of internationally preeminent scholar ...
in Princeton, 1939–1940, he laid the foundation for
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 sinc ...
, 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, often abbreviated as WWII or WW2, was a world war that lasted from 1939 to 1945. It involved the World War II by country, vast majority of the world's countries—including all of the great power ...
, Kleene was a
lieutenant commander
Lieutenant commander (also hyphenated lieutenant-commander and abbreviated Lt Cdr, LtCdr. or LCDR) is a commissioned officer rank in many navies. The rank is superior to a lieutenant and subordinate to a commander. The corresponding rank i ...
in the
United States Navy
The United States Navy (USN) is the maritime service branch of the United States Armed Forces and one of the eight uniformed services of the United States. It is the largest and most powerful navy in the world, with the estimated tonnage ...
. 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 City, the most populous city in the United States, located in the state of New York
* New York (state), a state in the northeastern United States
New York may also refer to:
Film and television
* '' ...
, 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. It was founded in 1923 and conducts basic scientific research, applied research, technological ...
in
Washington, D.C.
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
mathematical logic
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 ...
, 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 phil ...
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, system ...
and 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 ...
.
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 Alonzo Church. The current president of the ASL is ...
, 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 (born March 28, 1942) is an American philosopher, writer, and cognitive scientist whose research centers on the philosophy of mind, philosophy of science, and philosophy of biology, particularly as those fields rel ...
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 and social scienc ...
.
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
Environment most often refers to:
__NOTOC__
* Natural environment, all living and non-living things occurring naturally
* Biophysical environment, the physical and biological factors along with their chemical interactions that affect an organism or ...
, and was active in many
conservation causes.
Legacy
At each conference of the
Symposium on Logic in Computer Science 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.
Introduction to Metamathematics.' New York: Van Nostrand. (Ishi Press: 2009 reprint).
* 1956.
* 1965 (with Richard Eugene Vesley). ''The Foundations of Intuitionistic Mathematics''. North-Holland.
* 1967.
Mathematical Logic.'
John Wiley & Sons
John Wiley & Sons, Inc., commonly known as Wiley (), is an American multinational publishing company founded in 1807 that focuses on academic publishing and instructional materials. The company produces books, journals, and encyclopedias, i ...
. Dover reprint, 2002. .
* 1981. "Origins of Recursive Function Theory" in
Annals of the History of Computing 3'' No. 1.
* 1987.
See also
*
Kleene–Brouwer order In descriptive set theory, the Kleene–Brouwer order or Lusin–Sierpiński order is a linear order on finite sequences over some linearly ordered set (X, <), that differs from the more commonly used Kleene–Rosser paradox
In mathematics, the Kleene–Rosser paradox is a paradox that shows that certain systems of formal logic are inconsistent, in particular the version of Haskell Curry's combinatory logic introduced in 1930, and Alonzo Church's original lambda ...
*
Kleene's O
*
Kleene's T predicate In computability theory, the T predicate, first studied by mathematician Stephen Cole Kleene, is a particular set of triples of natural numbers that is used to represent computable functions within formal theories of arithmetic. Informally, the ...
*
List of pioneers in computer science
Notes
References
External links
*
Biographical memoir– by
Saunders Mac Lane
Saunders Mac Lane (4 August 1909 – 14 April 2005) was an American mathematician who co-founded category theory with Samuel Eilenberg.
Early life and education
Mac Lane was born in Norwich, Connecticut, near where his family lived in Taftvill ...
Kleene bibliography* – Interview with Kleene and
John Barkley Rosser about their experiences at Princeton
*
{{DEFAULTSORT:Kleene, Stephen Cole
American computer scientists
American logicians
Amherst College alumni
Computability theorists
Educators from Hartford, Connecticut
Institute for Advanced Study visiting scholars
Intuitionism
National Medal of Science laureates
Princeton University alumni
University of Wisconsin–Madison faculty
1909 births
1994 deaths
20th-century American mathematicians
Mathematicians from Connecticut