# Stephen Cole Kleene

**Stephen Cole Kleene** (/ˈkleɪni/ *KLAY-nee*;[lower-alpha 1] January 5, 1909 – January 25, 1994) 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 closure), Kleene's recursion theorem and the Kleene fixpoint theorem. He also invented regular expressions in 1951 to describe McCulloch-Pitts neural networks, and made significant contributions to the foundations of mathematical intuitionism.

Stephen Kleene | |
---|---|

Born | |

Died | January 25, 1994 85) | (aged

Residence | United States |

Nationality | American |

Alma mater | Amherst College Princeton University |

Known for | |

Awards | Leroy P. Steele Prize (1983) National Medal of Science (1990) |

Scientific career | |

Fields | Mathematics |

Institutions | University of Wisconsin–Madison |

Doctoral advisor | Alonzo Church |

Doctoral students | Robert Constable Joan Moschovakis Yiannis Moschovakis Nels David Nelson Dick de Jongh |

## Biography

Kleene was awarded the BA degree from Amherst College in 1930. He was awarded the Ph.D. in mathematics from Princeton University in 1934. His thesis, entitled *A Theory of Positive Integers in Formal Logic*, was supervised by Alonzo Church. In the 1930s, he did important work on Church's lambda calculus. In 1935, he joined the mathematics department at the University of Wisconsin–Madison, 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 in Princeton, 1939–40, he laid the foundation for recursion theory, 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, Kleene was a lieutenant commander in the United States Navy. He was an instructor of navigation at the U.S. Naval Reserve's Midshipmen's School in New York, and then a project director at the Naval Research Laboratory in Washington, D.C.

In 1946, Kleene returned to Wisconsin, becoming a full professor in 1948 and the Cyrus C. MacDuffee professor of mathematics in 1964. He was chair of the Department of Mathematics and Computer Science, 1962–63, and Dean of the College of Letters and Science from 1969 to 1974. The latter appointment he took on despite the considerable student unrest of the day, stemming from the Vietnam War. He retired from the University of Wisconsin in 1979. In 1999 the mathematics library at the University of Wisconsin was renamed in his honor.[3]

Kleene's teaching at Wisconsin resulted in three texts in mathematical logic, 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 that enhanced their canonical status and made them easier to teach and understand. Kleene and Vesley (1965) is the classic American introduction to intuitionist logic and mathematics.

... recursive function theory is of central importance in computer science. Kleene is responsible for many of the fundamental results in the area, including the Kleene normal form theorem (1936), the Kleene recursive theorem (1938), the development of the arithmetical and hyper-arithmetical hierarchies in the 1940s and 1950s, the Kleene-Post theory ofdegrees of unsolvability (1954), and higher-type recursion theory. which he began in the late 1950s and returned to in the late 1970s. ... Beginning in the late 1940s, Kleene also worked in a second area, Brouwer's intuitionism. Using tools from recursion theory, he introduced recursive realizability, an important technique for interpreting intuitionistic statements. In the summer of 1951 at the Rand Corporation, he produced a major breakthrough in a third area when he gave an important characterization of events accepted by a finite automaton.[4]

Kleene served as president of the Association for Symbolic Logic, 1956–58, and of the International Union of History and Philosophy of Science,[5] 1961. In 1990, he was awarded the National Medal of Science. The importance of Kleene's work led to the saying that "Kleeneness is next to Gödelness."

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.

## Selected publications

- 1936. « General recursive functions of natural numbers »,
*Math. Ann.*,**112**, 727–742. - 1952.
*Introduction to Metamathematics.*New York: Van Nostrand. (Ishi Press: 2009 reprint).[6] - 1951. S. C. Kleene (15 December 1951). "Representation of Events in Nerve Nets and Finite Automata" (PDF). U. S. Air Force Project Rand Research Memorandum (RM-704). The RAND Corporation.
- 1956. S.C. Kleene (1956). Shannon, Claude; McCarthy, John (eds.).
*Representation of Events in Nerve Nets and Finite Automata*.*Automata Studies*. OCLC 564148.

- 1965 (with Richard Eugene Vesley).
*The Foundations of Intuitionistic Mathematics*. North-Holland.[7]

- 1967.
*Mathematical Logic.*John Wiley. Dover reprint, 2002. ISBN 0-486-42533-9. - 1981. "Origins of Recursive Function Theory" in
*Annals of the History of Computing 3,*No. 1.

## Kleene award

At each conference of the Symposium on Logic in Computer Science the Kleene award, in honour of S.C. Kleene, is given for the best student paper.[8]

## See also

## Notes

- Although his last name is commonly pronounced /ˈkliːni/
*KLEE-nee*or /kliːn/*KLEEN*, Kleene himself pronounced it /ˈkleɪni/*KLAY-nee*.[1] His son, Ken Kleene, wrote: "As far as I am aware this pronunciation is incorrect in all known languages. I believe that this novel pronunciation was invented by my father."[2]

## References

- This article is based on material taken from the
*Free On-line Dictionary of Computing*prior to 1 November 2008 and incorporated under the "relicensing" terms of the GFDL, version 1.3 or later.

- Pace, Eric (January 27, 1994). "Stephen C. Kleene Is Dead at 85; Was Leader in Computer Science".
*The New York Times*. - In Entry "Stephen Kleene" at Free Online Dictionary of Computing.
- "About S.C. Kleene". University of Wisconsin—Madison Libraries. Retrieved 23 April 2016.
- Keisler, H. Jerome (September 1994). "Stephen Cole Kleene 1909–1994".
*Notices of the AMS*.**41**(7): 792. - IUHPS website; also known as "International Union of the History and the Philosophy of Science". A member of ICSU, the International Council for Science (formerly named International Council of Scientific Unions).
- "WorldCat: editions for 'Introduction to metamathematics.'". Retrieved April 2, 2013.
- Bishop, Errett (1965). "Review:
*The foundations of intuitionistic mathematics*, by Stephen Cole Kleene and Richard Eugene Vesley".*Bull. Amer. Math. Soc*.**71**(6): 850–852. doi:10.1090/s0002-9904-1965-11412-4. - http://lics.siglog.org/archive/kleene-award.html

## External links

- O'Connor, John J.; Robertson, Edmund F., "Stephen Cole Kleene",
*MacTutor History of Mathematics archive*, University of St Andrews. - Stephen Cole Kleene at the Mathematics Genealogy Project
- Biographical memoir – by Saunders Mac Lane
- Kleene bibliography
- "The Princeton Mathematics Community in the 1930s - STEPHEN C. KLEENE and J. BARKLEY ROSSER". Archived from the original on 10 March 2005. - Interview with Kleene and John Barkley Rosser about their experiences at Princeton
- Stephen Cole Kleene at DBLP Bibliography Server