Jacques Herbrand
Encyclopedia
Jacques Herbrand was a French
mathematician
who was born in Paris
, France
and died in La Bérarde, Isère
, France. Although he died at only 23 years of age, he was already considered one of "the greatest mathematicians of the younger generation" by his professors Helmut Hasse
, and Richard Courant
.
He worked in mathematical logic
and class field theory
. He introduced recursive function
s. Herbrand's theorem refers to either of two completely different theorems. One is a result from his doctoral thesis in proof theory
, and the other one half of the Herbrand–Ribet theorem
. The Herbrand quotient
is a type of Euler characteristic
, used in homological algebra
. He contributed to Hilbert's program
in the foundations of mathematics by providing a constructive consistency proof for a weak system of arithmetic. The proof uses the above mentioned, proof-theoretic Herbrand's Theorem.
in Paris under Ernest Vessiot
in 1929. He joined the army in October 1929, however, and so did not defend his thesis at the Sorbonne
until the following year. He was awarded a Rockefeller fellowship
that enabled him to study in Germany
in 1931, first with John von Neumann
in Berlin
, then during June with Emil Artin
in Hamburg
, and finally with Emmy Noether
in Göttingen
.
He submitted his principal study of proof theory and general recursive functions "On the consistency of arithmetic" early in 1931. While the essay was under consideration, Gödel
's "On formally undecidable sentences of Principia Mathematica and related systems I" announced the impossibility of formalizing within a theory that theory's consistency proof. Herbrand studied Gödel's essay and wrote an appendix to his own study explaining why Gödel's result did not contradict his own. In July of that year he was mountain-climbing in the French Alps
with two friends when he fell to his death in the granite mountains of Massif des Écrins
. "On the consistency of arithmetic" was published posthumously.
quoted in Michèle Chouchan "Nicolas Bourbaki Faits et légendes" Edition du choix, 1995.
French people
The French are a nation that share a common French culture and speak the French language as a mother tongue. Historically, the French population are descended from peoples of Celtic, Latin and Germanic origin, and are today a mixture of several ethnic groups...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
who was born in Paris
Paris
Paris is the capital and largest city in France, situated on the river Seine, in northern France, at the heart of the Île-de-France region...
, France
France
The French Republic , The French Republic , The French Republic , (commonly known as France , is a unitary semi-presidential republic in Western Europe with several overseas territories and islands located on other continents and in the Indian, Pacific, and Atlantic oceans. Metropolitan France...
and died in La Bérarde, Isère
Isère
Isère is a department in the Rhône-Alpes region in the east of France named after the river Isère.- History :Isère is one of the original 83 departments created during the French Revolution on March 4, 1790. It was created from part of the former province of Dauphiné...
, France. Although he died at only 23 years of age, he was already considered one of "the greatest mathematicians of the younger generation" by his professors Helmut Hasse
Helmut Hasse
Helmut Hasse was a German mathematician working in algebraic number theory, known for fundamental contributions to class field theory, the application of p-adic numbers to local classfield theory and diophantine geometry , and to local zeta functions.-Life:He was born in Kassel, and died in...
, and Richard Courant
Richard Courant
Richard Courant was a German American mathematician.- Life :Courant was born in Lublinitz in the German Empire's Prussian Province of Silesia. During his youth, his parents had to move quite often, to Glatz, Breslau, and in 1905 to Berlin. He stayed in Breslau and entered the university there...
.
He worked in mathematical logic
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
and class field theory
Class field theory
In mathematics, class field theory is a major branch of algebraic number theory that studies abelian extensions of number fields.Most of the central results in this area were proved in the period between 1900 and 1950...
. He introduced recursive function
Recursive function
Recursive function may refer to:*Recursion , a procedure or subroutine, implemented in a programming language, whose implementation references itself*A total computable function, a function which is defined for all possible inputs...
s. Herbrand's theorem refers to either of two completely different theorems. One is a result from his doctoral thesis in proof theory
Proof theory
Proof theory is a branch of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as plain lists, boxed lists, or trees, which are constructed...
, and the other one half of the Herbrand–Ribet theorem
Herbrand–Ribet theorem
In mathematics, the Herbrand–Ribet theorem is a result on the class number of certain number fields. It is a strengthening of Ernst Kummer's theorem to the effect that the prime p divides the class number of the cyclotomic field of p-th roots of unity if and only if p divides the numerator of the...
. The Herbrand quotient
Herbrand quotient
In mathematics, the Herbrand quotient is a quotient of orders of cohomology groups of a cyclic group. It was invented by Jacques Herbrand. It has an important application in class field theory.-Definition:...
is a type of Euler characteristic
Euler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent...
, used in homological algebra
Homological algebra
Homological algebra is the branch of mathematics which studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and...
. He contributed to Hilbert's program
Hilbert's program
In mathematics, Hilbert's program, formulated by German mathematician David Hilbert, was a proposed solution to the foundational crisis of mathematics, when early attempts to clarify the foundations of mathematics were found to suffer from paradoxes and inconsistencies...
in the foundations of mathematics by providing a constructive consistency proof for a weak system of arithmetic. The proof uses the above mentioned, proof-theoretic Herbrand's Theorem.
Biography
Herbrand finished his doctorate at École Normale SupérieureÉcole Normale Supérieure
The École normale supérieure is one of the most prestigious French grandes écoles...
in Paris under Ernest Vessiot
Ernest Vessiot
Ernest Vessiot was a French mathematician. He was born in Marseille, France and died in La Bauche, Savoie, France...
in 1929. He joined the army in October 1929, however, and so did not defend his thesis at the Sorbonne
University of Paris
The University of Paris was a university located in Paris, France and one of the earliest to be established in Europe. It was founded in the mid 12th century, and officially recognized as a university probably between 1160 and 1250...
until the following year. He was awarded a Rockefeller fellowship
Rockefeller Foundation
The Rockefeller Foundation is a prominent philanthropic organization and private foundation based at 420 Fifth Avenue, New York City. The preeminent institution established by the six-generation Rockefeller family, it was founded by John D. Rockefeller , along with his son John D. Rockefeller, Jr...
that enabled him to study in Germany
Germany
Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...
in 1931, first with John von Neumann
John von Neumann
John von Neumann was a Hungarian-American mathematician and polymath who made major contributions to a vast number of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics and game theory, computer science, numerical analysis,...
in Berlin
Berlin
Berlin is the capital city of Germany and is one of the 16 states of Germany. With a population of 3.45 million people, Berlin is Germany's largest city. It is the second most populous city proper and the seventh most populous urban area in the European Union...
, then during June with Emil Artin
Emil Artin
Emil Artin was an Austrian-American mathematician of Armenian descent.-Parents:Emil Artin was born in Vienna to parents Emma Maria, née Laura , a soubrette on the operetta stages of Austria and Germany, and Emil Hadochadus Maria Artin, Austrian-born of Armenian descent...
in Hamburg
Hamburg
-History:The first historic name for the city was, according to Claudius Ptolemy's reports, Treva.But the city takes its modern name, Hamburg, from the first permanent building on the site, a castle whose construction was ordered by the Emperor Charlemagne in AD 808...
, and finally with Emmy Noether
Emmy Noether
Amalie Emmy Noether was an influential German mathematician known for her groundbreaking contributions to abstract algebra and theoretical physics. Described by David Hilbert, Albert Einstein and others as the most important woman in the history of mathematics, she revolutionized the theories of...
in Göttingen
Göttingen
Göttingen is a university town in Lower Saxony, Germany. It is the capital of the district of Göttingen. The Leine river runs through the town. In 2006 the population was 129,686.-General information:...
.
He submitted his principal study of proof theory and general recursive functions "On the consistency of arithmetic" early in 1931. While the essay was under consideration, Gödel
Godel
Godel or similar can mean:*Kurt Gödel , an Austrian logician, mathematician and philosopher*Gödel...
's "On formally undecidable sentences of Principia Mathematica and related systems I" announced the impossibility of formalizing within a theory that theory's consistency proof. Herbrand studied Gödel's essay and wrote an appendix to his own study explaining why Gödel's result did not contradict his own. In July of that year he was mountain-climbing in the French Alps
Alps
The Alps is one of the great mountain range systems of Europe, stretching from Austria and Slovenia in the east through Italy, Switzerland, Liechtenstein and Germany to France in the west....
with two friends when he fell to his death in the granite mountains of Massif des Écrins
Massif des Écrins
The mountains of the Massif des Écrins form the core of the Écrins National Park in France.-Mountains:*Barre des Écrins 4102 m*La Meije 3983 m*Ailefroide 3954 m*Mont Pelvoux 3946 m*Pic Sans Nom 3913 m*Pic Gaspard 3883 m*Le Râteau 3809 m...
. "On the consistency of arithmetic" was published posthumously.
Quotation
"Jacques Herbrand would have hated Bourbaki" said French mathematician Claude ChevalleyClaude Chevalley
Claude Chevalley was a French mathematician who made important contributions to number theory, algebraic geometry, class field theory, finite group theory, and the theory of algebraic groups...
quoted in Michèle Chouchan "Nicolas Bourbaki Faits et légendes" Edition du choix, 1995.
See also
- Herbrand universe
- Herbrand base
- Herbrand interpretationHerbrand interpretationIn mathematical logic, a Herbrand interpretation is an interpretation in which all constants and function symbols are assigned very simple meanings. Specifically, every constant is interpreted as itself, and every function symbol is interpreted as the function that applies it...
- Herbrand structure
- Herbrand theory
- Herbrand–Ribet theoremHerbrand–Ribet theoremIn mathematics, the Herbrand–Ribet theorem is a result on the class number of certain number fields. It is a strengthening of Ernst Kummer's theorem to the effect that the prime p divides the class number of the cyclotomic field of p-th roots of unity if and only if p divides the numerator of the...
- Herbrand quotientHerbrand quotientIn mathematics, the Herbrand quotient is a quotient of orders of cohomology groups of a cyclic group. It was invented by Jacques Herbrand. It has an important application in class field theory.-Definition:...