Emil Leon Post

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Emil Leon Post
Emil Leon Post.jpg
Born February 11, 1897
Augustów, Suwałki Governorate, Russian Empire
(now Poland)
Died April 21, 1954(1954-04-21) (aged 57)
New York City, U.S.
Alma mater City College of New York (B.S., Mathematics, 1917)
Columbia University [1]
Known for Formulation 1,
Post correspondence problem,
completeness-proof of Principia's propositional calculus
Scientific career
Fields Mathematics
Thesis Introduction to a General Theory of Elementary Propositions (1920)
Doctoral advisor Cassius Jackson Keyser

Emil Leon Post ( /pst/ ; 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.

Mathematician person with an extensive knowledge of mathematics

A mathematician is someone who uses an extensive knowledge of mathematics in his or her work, typically to solve mathematical problems.

Computability theory, also known as recursion theory, is a branch of mathematical logic, of computer science, and of the theory of computation that originated in the 1930s with the study of computable functions and Turing degrees. The field has since expanded to include the study of generalized computability and definability. In these areas, recursion theory overlaps with proof theory and effective descriptive set theory.

Contents

Life

Post was born in Augustów, Suwałki Governorate, Russian Empire (now Poland) into a Polish-Jewish family that immigrated to New York City in May 1904. His parents were Arnold and Pearl Post. [2]

Augustów Place in Podlaskie, Poland

Augustów, formerly known in English as Augustovo or Augustowo, is a city in north-eastern Poland with 30,802 inhabitants (2011). It lies on the Netta River and the Augustów Canal. It is situated in the Podlaskie Voivodeship, having previously been in Suwałki Voivodeship (1975–1998). It is the seat of Augustów County and of Gmina Augustów.

Suwałki Governorate governorate of the Russian Empire

Suwałki Governorate was an administrative unit (guberniya) of the Kingdom of Poland with seat in Suwałki. It covered a territory of about 12,300 km².

Russian Empire Former country, 1721–1917

The Russian Empire, also known as Imperial Russia or simply Russia, was an empire that existed across Eurasia and North America from 1721, following the end of the Great Northern War, until the Republic was proclaimed by the Provisional Government that took power after the February Revolution of 1917.

Post had been interested in astronomy, but at the age of twelve lost his left arm in a car accident. This loss was a significant obstacle to being a professional astronomer. He decided to pursue mathematics, rather than astronomy. [3]

Post attended the Townsend Harris High School and continued on to graduate from City College of New York in 1917 with a B.S. in Mathematics. [1]

Townsend Harris High School

Townsend Harris High School is a public magnet high school for the humanities in the borough of Queens in New York City. Students and alumni often refer to themselves as "Harrisites." Townsend Harris consistently ranks as among the top 100 High Schools in the United States. Its most recent U.S. News and World Report ranking is #40 in the nation, and it was named #1 high school in New York City by the New York Post in 2010.

City College of New York senior college of the City University of New York (CUNY) in New York City

The City College of the City University of New York is a public senior college of the City University of New York (CUNY) in New York City.

After completing his Ph.D. in mathematics at Columbia University, supervised by Cassius Jackson Keyser, he did a post-doctorate at Princeton University in the 1920–1921 academic year. Post then became a high school mathematics teacher in New York City.

Columbia University Private Ivy League research university in New York City

Columbia University is a private Ivy League research university in Upper Manhattan, New York City. Established in 1754, Columbia is the oldest institution of higher education in New York and the fifth-oldest institution of higher learning in the United States. It is one of nine colonial colleges founded prior to the Declaration of Independence, seven of which belong to the Ivy League. It has been ranked by numerous major education publications as among the top ten universities in the world.

Cassius Jackson Keyser American mathematician and journalist of pronounced philosophical inclinations

Cassius Jackson Keyser was an American mathematician of pronounced philosophical inclinations.

Princeton University University in Princeton, New Jersey

Princeton University is a private Ivy League 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 nine colonial colleges chartered before the American Revolution. The institution moved to Newark in 1747, then to the current site nine years later, and renamed itself Princeton University in 1896.

Post married Gertrude Singer in 1929, with whom he had a daughter, Phyllis Goodman. Post spent at most three hours a day on research on the advice of his doctor in order to avoid manic attacks, which he had been experiencing since his year at Princeton. [4]

In 1936, he was appointed to the mathematics department at the City College of New York. He died in 1954 of a heart attack following electroshock treatment for depression; [4] [5] he was 57.

Early work

In his doctoral thesis, later shortened and published as "Introduction to a General Theory of Elementary Propositions" (1921), Post proved, among other things, that the propositional calculus of Principia Mathematica was complete: all tautologies are theorems, given the Principia axioms and the rules of substitution and modus ponens. Post also devised truth tables independently of Wittgenstein and C. S. Peirce and put them to good mathematical use. Jean van Heijenoort's well-known source book on mathematical logic (1966) reprinted Post's classic article setting out these results.

While at Princeton, Post came very close to discovering the incompleteness of Principia Mathematica, which Kurt Gödel proved in 1931. Post initially failed to publish his ideas as he believed he needed a 'complete analysis' for them to be accepted. [2]

Recursion theory

In 1936, Post developed, independently of Alan Turing, a mathematical model of computation that was essentially equivalent to the Turing machine model. Intending this as the first of a series of models of equivalent power but increasing complexity, he titled his paper Formulation 1. This model is sometimes called "Post's machine" or a Post–Turing machine, but is not to be confused with Post's tag machines or other special kinds of Post canonical system, a computational model using string rewriting and developed by Post in the 1920s but first published in 1943. Post's rewrite technique is now ubiquitous in programming language specification and design, and so with Church's lambda-calculus is a salient influence of classical modern logic on practical computing. Post devised a method of 'auxiliary symbols' by which he could canonically represent any Post-generative language, and indeed any computable function or set at all.

Correspondence systems were introduced by Post in 1945 to give simple examples of undecidability. He showed that the Post Correspondence Problem (PCP) of satisfying their constraints is, in general, undecidable. With 2 string pairs, PCP was shown to be decidable in 1981. It is known to be undecidable when 9 pairs are used, but Stephen Wolfram suggested that it is also undecidable with just 3 pairs. [6] The undecidability of his Post Correspondence Problem turned out to be exactly what was needed to obtain undecidability results in the theory of formal languages.

In an influential address to the American Mathematical Society in 1944, he raised the question of the existence of an uncomputable recursively enumerable set whose Turing degree is less than that of the halting problem. This question, which became known as Post's problem, stimulated much research. It was solved in the affirmative in the 1950s by the introduction of the powerful priority method in recursion theory.

Polyadic groups

Post made a fundamental and still-influential contribution to the theory of polyadic, or n-ary, groups in a long paper published in 1940. His major theorem showed that a polyadic group is the iterated multiplication of elements of a normal subgroup of a group, such that the quotient group is cyclic of order n 1. He also demonstrated that a polyadic group operation on a set can be expressed in terms of a group operation on the same set. The paper contains many other important results.

Selected papers

See also

Notes

  1. 1 2 Urquhart (2008)
  2. 1 2 O'Connor, John J.; Robertson, Edmund F., "Emil Leon Post", MacTutor History of Mathematics archive , University of St Andrews .
  3. Urquhart (2008), p. 429.
  4. 1 2 Urquhart (2008), p. 430.
  5. Baaz, Matthias, ed. (2011). Kurt Gödel and the Foundations of Mathematics: Horizons of Truth (1st ed.). Cambridge University Press. ISBN   9781139498432.
  6. Wolfram, Stephen (2002). A New Kind of Science. Wolfram Media, Inc. p. 1139. ISBN   1-57955-008-8.

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Further reading