Jacques Herbrand

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Jacques Herbrand
J Herbrand 1931.jpg
Herbrand in summer 1931, as photographed by Natascha Artin-Brunswick
Born(1908-02-12)12 February 1908
Paris, France
Died27 July 1931(1931-07-27) (aged 23)
La Bérarde, Isère, France
NationalityFrench
Alma mater École Normale Supérieure
University of Paris (PhD, 1930)
Known for Herbrand's theorem
Herbrand–Ribet theorem
Herbrand quotient
Scientific career
Fields Mathematical logic, class field theory
Thesis Recherches sur la théorie de la démonstration (Investigations on Proof Theory)  (1930)
Doctoral advisor Ernest Vessiot

Jacques Herbrand (12 February 1908 – 27 July 1931) was a French mathematician. Although he died at age 23, he was already considered one of "the greatest mathematicians of the younger generation" by his professors Helmut Hasse, and Richard Courant. [1]

Helmut Hasse German mathematician

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 class field theory and diophantine geometry, and to local zeta functions.

Richard Courant German American mathematician (1888-1972)

Richard Courant was a German American mathematician. He is best known by the general public for the book What is Mathematics?, co-written with Herbert Robbins.

Contents

He worked in mathematical logic and class field theory. He introduced recursive functions. 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.

Mathematical logic is a subfield of mathematics exploring the applications of formal logic to mathematics. It bears close connections to metamathematics, the foundations of mathematics, and theoretical computer science. The unifying themes in mathematical logic include the study of the expressive power of formal systems and the deductive power of formal proof systems.

In mathematics, class field theory is the branch of algebraic number theory concerned with the abelian extensions of number fields, global fields of positive characteristic, and local fields. The theory had its origins in the proof of quadratic reciprocity by Gauss at the end of 18th century. These ideas were developed over the next century, giving rise to a set of conjectures by Hilbert that were subsequently proved by Takagi and Artin. These conjectures and their proofs constitute the main body of class field theory.

Herbrand's theorem is a fundamental result of mathematical logic obtained by Jacques Herbrand (1930). It essentially allows a certain kind of reduction of first-order logic to propositional logic. Although Herbrand originally proved his theorem for arbitrary formulas of first-order logic, the simpler version shown here, restricted to formulas in prenex form containing only existential quantifiers, became more popular.

Biography

Herbrand finished his doctorate at École Normale Supérieure 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.

Ernest Vessiot French mathematician

Ernest Vessiot was a French mathematician. He was born in Marseille, France and died in La Bauche, Savoie, France. He entered the École Normale Supérieure in 1884.

University of Paris former university in Paris, France from 1896 to 1968

The University of Paris, metonymically known as the Sorbonne, was a university in Paris, France, active 1150–1793, and 1806–1970.

The Rockefeller Foundation is a private foundation based at 420 Fifth Avenue, New York City. It was established by the six-generation Rockefeller family. The Foundation was started by Standard Oil owner John D. Rockefeller ("Senior"), along with his son John D. Rockefeller Jr. ("Junior"), and Senior's principal oil and gas business and philanthropic advisor, Frederick Taylor Gates, in New York State on May 14, 1913, when its charter was formally accepted by the New York State Legislature.

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.

Alps Major mountain range system in Central Europe

The Alps are the highest and most extensive mountain range system that lies entirely in Europe, separating Southern from Central and Western Europe and stretching approximately 1,200 kilometres (750 mi) across eight Alpine countries : France, Switzerland, Monaco, Italy, Liechtenstein, Austria, Germany, and Slovenia. The mountains were formed over tens of millions of years as the African and Eurasian tectonic plates collided. Extreme shortening caused by the event resulted in marine sedimentary rocks rising by thrusting and folding into high mountain peaks such as Mont Blanc and the Matterhorn. Mont Blanc spans the French–Italian border, and at 4,810 m (15,781 ft) is the highest mountain in the Alps. The Alpine region area contains about a hundred peaks higher than 4,000 metres (13,000 ft).

Massif des Écrins mountain range

The mountains of the Massif des Écrins form the core of the Écrins National Park in France.

Quotation

"Jacques Herbrand would have hated Bourbaki" said French mathematician Claude Chevalley quoted in Michèle Chouchan, "Nicolas Bourbaki Faits et légendes", Éditions du choix, 1995.

Nicolas Bourbaki collective pseudonym for a group of (mainly French) 20th-century mathematician

Nicolas Bourbaki is the collective pseudonym of a group of mathematicians. Their aim is to reformulate mathematics on an extremely abstract and formal but self-contained basis in a series of books beginning in 1935. With the goal of grounding all of mathematics on set theory, the group strives for rigour and generality. Their work led to the discovery of several concepts and terminologies still used, and influenced modern branches of mathematics.

Claude Chevalley French mathematician

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. He was a founding member of the Bourbaki group.

Bibliography

Primary literature:

See also

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References

  1. Chevalley, Claude; Lautman, Albert (1971). "Biographical Note on Jacques Herbrand". In Goldfarb, Warren D.; Van Heijenoort, Jean (eds.). Logical Writings. Berlin: Springer. pp. 21–23. ISBN   90-277-0176-8.