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Gerhard Karl Erich Gentzen (24 November 1909 – 4 August 1945) was a German
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, structure, space, models, and change. History On ...
and logician. He made major contributions to the
foundations of mathematics Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the nature of mathe ...
,
proof theory Proof theory is a major branchAccording to Wang (1981), pp. 3–4, proof theory is one of four domains mathematical logic, together with model theory, axiomatic set theory, and recursion theory. Barwise (1978) consists of four corresponding part ...
, especially on
natural deduction In logic and proof theory, natural deduction is a kind of proof calculus in which logical reasoning is expressed by inference rules closely related to the "natural" way of reasoning. This contrasts with Hilbert-style systems, which instead use ax ...
and
sequent calculus In mathematical logic, sequent calculus is a style of formal logical argumentation in which every line of a proof is a conditional tautology (called a sequent by Gerhard Gentzen) instead of an unconditional tautology. Each conditional tautology i ...
. He died of starvation in a Soviet prison camp in
Prague Prague ( ; cs, Praha ; german: Prag, ; la, Praga) is the capital and largest city in the Czech Republic, and the historical capital of Bohemia. On the Vltava river, Prague is home to about 1.3 million people. The city has a temperate ...
in 1945, having been interned as a German national after the
Second World War 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 vast majority of the world's countries—including all of the great powers—forming two opposi ...
.


Life and career

Gentzen was a student of
Paul Bernays Paul Isaac Bernays (17 October 1888 – 18 September 1977) was a Swiss mathematician who made significant contributions to mathematical logic, axiomatic set theory, and the philosophy of mathematics. He was an assistant and close collaborator of ...
at the
University of Göttingen The University of Göttingen, officially the Georg August University of Göttingen, (german: Georg-August-Universität Göttingen, known informally as Georgia Augusta) is a public research university in the city of Göttingen, Germany. Founded ...
. Bernays was fired as "non-
Aryan Aryan or Arya (, Indo-Iranian *''arya'') is a term originally used as an ethnocultural self-designation by Indo-Iranians in ancient times, in contrast to the nearby outsiders known as 'non-Aryan' (*''an-arya''). In Ancient India, the term ...
" in April 1933 and therefore
Hermann Weyl Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is asso ...
formally acted as his supervisor. Gentzen joined the
Sturmabteilung The (; SA; literally "Storm Detachment") was the original paramilitary wing of the Nazi Party. It played a significant role in Adolf Hitler's rise to power in the 1920s and 1930s. Its primary purposes were providing protection for Nazi ralli ...
in November 1933 although he was by no means compelled to do so. Nevertheless he kept in contact with Bernays until the beginning of the
Second World War 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 vast majority of the world's countries—including all of the great powers—forming two opposi ...
. In 1935, he corresponded with Abraham Fraenkel in Jerusalem and was implicated by the Nazi teachers' union as one who "keeps contacts to the
Chosen People Throughout history, various groups of people have considered themselves to be the chosen people of a deity, for a particular purpose. The phenomenon of a "chosen people" is well known among the Israelites and Jews, where the term ( he, עם ס� ...
." In 1935 and 1936,
Hermann Weyl Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is asso ...
, head of the Göttingen mathematics department in 1933 until his resignation under Nazi pressure, made strong efforts to bring him to 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 schola ...
in Princeton. Between November 1935 and 1939 he was an assistant of
David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many ...
in Göttingen. Gentzen joined the
Nazi Party The Nazi Party, officially the National Socialist German Workers' Party (german: Nationalsozialistische Deutsche Arbeiterpartei or NSDAP), was a far-right political party in Germany active between 1920 and 1945 that created and supported t ...
in 1937. In April 1939 Gentzen swore the
oath of loyalty An oath of allegiance is an oath whereby a subject or citizen acknowledges a duty of allegiance and swears loyalty to a monarch or a country. In modern republics, oaths are sworn to the country in general, or to the country's constitution. For ...
to
Adolf Hitler Adolf Hitler (; 20 April 188930 April 1945) was an Austrian-born German politician who was dictator of Germany from 1933 until his death in 1945. He rose to power as the leader of the Nazi Party, becoming the chancellor in 1933 and the ...
as part of his academic appointment. From 1943 he was a teacher at the German Charles-Ferdinand University of Prague. Under a contract from the SS, Gentzen worked for the
V-2 The V-2 (german: Vergeltungswaffe 2, lit=Retaliation Weapon 2), with the technical name ''Aggregat 4'' (A-4), was the world’s first long-range guided ballistic missile. The missile, powered by a liquid-propellant rocket engine, was develope ...
project. Gentzen was arrested during the citizens uprising against the occupying German forces on 5 May 1945. He, along with the rest of the staff of the German University in
Prague Prague ( ; cs, Praha ; german: Prag, ; la, Praga) is the capital and largest city in the Czech Republic, and the historical capital of Bohemia. On the Vltava river, Prague is home to about 1.3 million people. The city has a temperate ...
was subsequently handed over to Soviet forces. Because of his past association with the SA, NSDAP and NSD Dozentenbund, Gentzen was detained in a prison camp, where he died of starvation on 4 August 1945.


Work

Gentzen's main work was on the foundations of
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, in
proof theory Proof theory is a major branchAccording to Wang (1981), pp. 3–4, proof theory is one of four domains mathematical logic, together with model theory, axiomatic set theory, and recursion theory. Barwise (1978) consists of four corresponding part ...
, specifically
natural deduction In logic and proof theory, natural deduction is a kind of proof calculus in which logical reasoning is expressed by inference rules closely related to the "natural" way of reasoning. This contrasts with Hilbert-style systems, which instead use ax ...
and the
sequent calculus In mathematical logic, sequent calculus is a style of formal logical argumentation in which every line of a proof is a conditional tautology (called a sequent by Gerhard Gentzen) instead of an unconditional tautology. Each conditional tautology i ...
. His
cut-elimination theorem The cut-elimination theorem (or Gentzen's ''Hauptsatz'') is the central result establishing the significance of the sequent calculus. It was originally proved by Gerhard Gentzen in his landmark 1934 paper "Investigations in Logical Deduction" for ...
is the cornerstone of proof-theoretic semantics, and some philosophical remarks in his "Investigations into Logical Deduction", together with
Ludwig Wittgenstein Ludwig Josef Johann Wittgenstein ( ; ; 26 April 1889 – 29 April 1951) was an Austrian- British philosopher who worked primarily in logic, the philosophy of mathematics, the philosophy of mind, and the philosophy of language. He is consi ...
's later work, constitute the starting point for
inferential role semantics Inferential role semantics (also conceptual role semantics, functional role semantics, procedural semantics, semantic inferentialism) is an approach to the theory of meaning that identifies the meaning of an expression with its relationship to other ...
. One of Gentzen's papers had a second publication in the ideological ''
Deutsche Mathematik ''Deutsche Mathematik'' (German Mathematics) was a mathematics journal founded in 1936 by Ludwig Bieberbach and Theodor Vahlen. Vahlen was publisher on behalf of the German Research Foundation (DFG), and Bieberbach was chief editor. Other editor ...
'' that was founded by
Ludwig Bieberbach Ludwig Georg Elias Moses Bieberbach (; 4 December 1886 – 1 September 1982) was a German mathematician and Nazi. Biography Born in Goddelau, near Darmstadt, he studied at Heidelberg and under Felix Klein at Göttingen, receiving his doct ...
who promoted "Aryan" mathematics. Gentzen proved the consistency of the
Peano axioms In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearly ...
in a paper published in 1936. In his ''Habilitationsschrift'', finished in 1939, he determined the proof-theoretical strength of Peano arithmetic. This was done by a direct proof of the unprovability of the principle of transfinite induction, used in his 1936 proof of consistency, within Peano arithmetic. The principle can, however, be expressed in arithmetic, so that a direct proof of Gödel's incompleteness theorem followed. Gödel used a coding procedure to construct an unprovable formula of arithmetic. Gentzen's proof was published in 1943 and marked the beginning of ordinal proof theory.


Publications

* * * * * * (Lecture hold in Münster at the institute of Heinrich Scholz on 27 June 1936) * * * *


Posthumous

* * – Published by
Paul Bernays Paul Isaac Bernays (17 October 1888 – 18 September 1977) was a Swiss mathematician who made significant contributions to mathematical logic, axiomatic set theory, and the philosophy of mathematics. He was an assistant and close collaborator of ...
. * – Published by
Paul Bernays Paul Isaac Bernays (17 October 1888 – 18 September 1977) was a Swiss mathematician who made significant contributions to mathematical logic, axiomatic set theory, and the philosophy of mathematics. He was an assistant and close collaborator of ...
.


See also

*
Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, a ...


Notes


References

* - (English translation). * * - an English translation.


External links

* * {{DEFAULTSORT:Gentzen, Gerhard 1909 births 1945 deaths People from Greifswald University of Göttingen alumni Nazi Party members Charles University faculty 20th-century German mathematicians German logicians People from the Province of Pomerania 20th-century German philosophers Deaths by starvation Nazis who died in prison custody German people who died in Soviet detention