Jacques Herbrand (12 February 1908 – 27 July 1931) was a French

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 ...

. 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 Richard Courant (January 8, 1888 – January 27, 1972) was a German American mathematician. He is best known by the general public for the book '' What is Mathematics?'', co-written with Herbert Robbins. His research focused on the areas of real ...

. He worked in

mathematical logic Mathematical logic is the study of logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of for ...

and

class field theory In mathematics, class field theory (CFT) is the fundamental branch of algebraic number theory whose goal is to describe all the abelian Galois extensions of local and global fields using objects associated to the ground field. Hilbert is credit ...

. He introduced recursive functions. ''

Herbrand's theorem 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 ...

'' refers to either of two completely different theorems. One is a result from his doctoral thesis 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 ...

, and the other one half of the

Herbrand–Ribet theorem In mathematics, the Herbrand–Ribet theorem is a result on the class group of certain number fields. It is a strengthening of Ernst Kummer's theorem to the effect that the prime ''p'' divides the ideal class group, class number of the cyclotomic f ...

. The

Herbrand quotient In mathematics, the Herbrand quotient is a quotient of orders of Group cohomology, cohomology groups of a cyclic group. It was invented by Jacques Herbrand. It has an important application in class field theory. Definition If ''G'' is a finite cyc ...

is a type of

Euler characteristic In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space ...

, used in

homological algebra Homological algebra is the branch of mathematics that studies homology (mathematics), homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precurs ...

. He contributed to

Hilbert's program In mathematics, Hilbert's program, formulated by German mathematician David Hilbert in the early part of the 20th century, was a proposed solution to the foundational crisis of mathematics, when early attempts to clarify the foundations of mathema ...

in 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 ...

by providing a

constructive Although the general English usage of the adjective constructive is "helping to develop or improve something; helpful to someone, instead of upsetting and negative," as in the phrase "constructive criticism," in legal writing ''constructive'' has ...

consistency proof In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent i ...

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 Supérieure École may refer to: * an elementary school in the French educational stages normally followed by secondary education establishments (collège and lycée) * École (river), a tributary of the Seine flowing in région Île-de-France * École, Savoi ...

in Paris under

Ernest Vessiot Ernest Vessiot (; 8 March 1865 – 17 October 1952) 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. He was Maître de Conférences at Lille ...

in 1929. He joined the army in October 1929, however, and so did not defend his thesis at the

Sorbonne Sorbonne may refer to: * Sorbonne (building), historic building in Paris, which housed the University of Paris and is now shared among multiple universities. *the University of Paris (c. 1150 – 1970) *one of its components or linked institution, ...

until the following year. He was awarded a

Rockefeller fellowship The Rockefeller Foundation is an American private foundation and philanthropic medical research and arts funding organization based at 420 Fifth Avenue, New York City. The second-oldest major philanthropic institution in America, after the Ca ...

that enabled him to study in

Germany Germany,, officially the Federal Republic of Germany, is a country in Central Europe. It is the second most populous country in Europe after Russia, and the most populous member state of the European Union. Germany is situated betwe ...

in 1931, first with

John von Neumann John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest cove ...

Berlin Berlin ( , ) is the capital and largest city of Germany by both area and population. Its 3.7 million inhabitants make it the European Union's most populous city, according to population within city limits. One of Germany's sixteen constitue ...

, then during June with Emil Artin in

Hamburg (male), (female) en, Hamburger(s), Hamburgian(s) , timezone1 = Central (CET) , utc_offset1 = +1 , timezone1_DST = Central (CEST) , utc_offset1_DST = +2 , postal ...

, and finally with

Emmy Noether Amalie Emmy NoetherEmmy is the '' Rufname'', the second of two official given names, intended for daily use. Cf. for example the résumé submitted by Noether to Erlangen University in 1907 (Erlangen University archive, ''Promotionsakt Emmy Noeth ...

Göttingen Göttingen (, , ; nds, Chöttingen) is a college town, university city in Lower Saxony, central Germany, the Capital (political), capital of Göttingen (district), the eponymous district. The River Leine runs through it. At the end of 2019, t ...

. 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 (general) impossibility of proving the consistency of a theory, using merely the theory itself. Herbrand studied Gödel's essay and wrote an appendix to his own study explaining why Gödel's result did not contradict his own. "On the consistency of arithmetic" was published posthumously.

Death

In July of 1931, Herbrand was mountain-climbing in the French

Alps The Alps () ; german: Alpen ; it, Alpi ; rm, Alps ; sl, Alpe . are the highest and most extensive mountain range system that lies entirely in Europe, stretching approximately across seven Alpine countries (from west to east): France, Sw ...

with two friends when he fell to his death in the granite mountains of

Massif des Écrins The mountains of the Massif des Écrins (; Écrins Massif) form the core of Écrins National Park in Hautes-Alpes and Isère in the French Alps.Bourbaki"'' said French mathematician

Claude Chevalley Claude Chevalley (; 11 February 1909 – 28 June 1984) 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 foun ...

quoted in Michèle Chouchan, ''"Nicolas Bourbaki Faits et légendes"'', Éditions du choix, 1995.

Bibliography

* Primary literature: * 1967.

Jean van Heijenoort Jean Louis Maxime van Heijenoort (; July 23, 1912 – March 29, 1986) was a historian of mathematical logic. He was also a personal secretary to Leon Trotsky from 1932 to 1939, and an American Trotskyist until 1947. Life Van Heijenoort was born ...

(ed.), ''From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931''. Cambridge, Massachusetts: Harvard Univ. Press. ** 1930. "Investigations in proof theory," 525–81. ** 1931. "On the consistency of arithmetic," 618–28. * 1968.

(ed.), ''Jacques Herbrand, Écrits logiques''. Paris: Presses Universitaires de France. * 1971. Warren David Goldfarb (transl., ed.), ''Logical Writings of Jacques Herbrand'' Cambridge, Massachusetts: Harvard University Press.

References

External links

* {{DEFAULTSORT:Herbrand, Jacques 1908 births 1931 deaths French logicians Logicians École Normale Supérieure alumni University of Paris alumni Proof theorists 20th-century French mathematicians French male non-fiction writers Mountaineering deaths Sport deaths in France Scientists from Paris 20th-century French philosophers 20th-century French male writers

Biography

Death

Bibliography

See also

References

External links