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
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ...
. He made major contributions to the
foundations of mathematics
Foundations of mathematics is the study of the philosophy, 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 natu ...
,
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. Jon Barwise, Barwise (1978) consists of four correspo ...
, 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 axiom ...
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 opposin ...
.
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 assoc ...
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 ral ...
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 opposin ...
. In 1935, he corresponded with
Abraham Fraenkel
Abraham Fraenkel ( he, אברהם הלוי (אדולף) פרנקל; February 17, 1891 – October 15, 1965) was a German-born Israeli mathematician. He was an early Zionist and the first Dean of Mathematics at the Hebrew University of Jerusalem. ...
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 assoc ...
, 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 scholar ...
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 a ...
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 politics, far-right political party in Germany active between 1920 and 1945 that crea ...
in 1937. In April 1939 Gentzen swore the
oath of loyalty to
Adolf Hitler
Adolf Hitler (; 20 April 188930 April 1945) was an Austrian-born German politician who was dictator of Nazi Germany, Germany from 1933 until Death of Adolf Hitler, his death in 1945. Adolf Hitler's rise to power, He rose to power as the le ...
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. Jon Barwise, Barwise (1978) consists of four correspo ...
, 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 axiom ...
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
Proof-theoretic semantics is an approach to the semantics of logic that attempts to locate the meaning of propositions and logical connectives not in terms of interpretations, as in Tarskian approaches to semantics, but in the role that the propo ...
, 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 considere ...
'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 editors w ...
'' 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 doctorate ...
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, ...
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