Editing Jacques Herbrand

{{Short description|French mathematician, 1908-1931}}
{{Use dmy dates|date=February 2023}}
{{Infobox scientist
|name              = Jacques Herbrand
|image             = J Herbrand 1931.jpg
|image_size        = 
|caption           = Herbrand in summer 1931, as photographed by [[Natascha Artin Brunswick]]
|birth_date        = {{birth date|1908|02|12|df=y}}
|birth_place       = Paris, France
|death_date        = {{death date and age|1931|07|27|1908|02|12|df=y}}
|death_place       = [[La Bérarde]], Isère, France
|nationality       = French
|thesis_title = Recherches sur la théorie de la démonstration (Investigations on Proof Theory)
|thesis_url   = https://eudml.org/doc/192791
|thesis_year  = 1930
|fields            = [[Mathematical logic]], [[class field theory]]
|alma_mater        = [[École Normale Supérieure]]<br/>[[University of Paris]] (PhD, 1930)
|doctoral_advisor  = [[Ernest Vessiot]]
|known_for         = [[Herbrand's theorem]]<br/>[[Herbrand–Ribet theorem]]<br/>[[Herbrand quotient]]
}}
'''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]].<ref>{{Cite book |last1=Chevalley |first1=Claude |last2=Lautman |first2=Albert |title=Logical Writings |chapter=Biographical Note on Jacques Herbrand |editor-last=Goldfarb |editor-first=Warren D. |location=Berlin |publisher=Springer |year=1971 |pages=21–23 |isbn=90-277-0176-8 }}</ref>

He worked in [[mathematical logic]] and [[class field theory]]. He introduced [[recursion#Functional recursion|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 mathematics|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 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 [[University of Paris|Sorbonne]] until the following year. He was awarded a [[Rockefeller Foundation|Rockefeller fellowship]] that enabled him to study in [[Germany]] in 1930-1931, first with [[John von Neumann]] in [[Berlin]], then during June with [[Emil Artin]] in [[Hamburg]], and finally with [[Emmy Noether]] in [[Göttingen]].

In Berlin, Herbrand followed a course on Hilbert's proof theory given by von Neumann. During the course, von Neumann explained Gödel's first incompleteness theorem and found, independently of Gödel, the second incompleteness theorem that he also presented in the lectures. A letter of Herbrand's of 5 December 1930 to his friend Claude Chevalley contains a description of von Neumann's idea. An earlier letter to Vessiot, of 28 November, explained Gödel's first incompleteness theorem in the form of failure of omega-consistency.<ref>{{Cite book |last=von Plato |first=Jan |title=The Great Formal Machinery Works|publisher=Princeton |year=2017 |pages=250-251 |isbn=978-0-691-17417-4 }}</ref>

Herbrand's last paper was titled "Sur la non-contradiction de l'arithmétique" (On the consistency of arithmetic). It contains a consistency proof for a restricted system of arithmetic, similar to a result of von Neumann's. Herbrand had studied Gödel's incompleteness article in Easter 1931 through the page proofs [[Paul Bernays]] had lent him. In the last section of his paper, Herbrand makes a comparison of his restricted result to that of Gödel's. The paper was received by the editors the very same day Herbrand lost his life, 27 July, and published posthumously.<ref>{{Cite book |title=Logical Writings |editor-last=Goldfarb |editor-first=Warren D.|location=Berlin |publisher=Springer |year=1971 |page=282 |isbn=90-277-0176-8 }}</ref>

== Death ==
In July 1931, Herbrand was mountain-climbing in the French [[Alps]] with two friends when he fell to his death in the granite mountains of [[Massif des Écrins]]. <ref>{{MacTutor Biography|id=Herbrand}} </ref><ref>{{cite news |date=30 July 1931 |title=Le Temps |language=FR |page=4 |publisher=Bibliothèque nationale de France |issue=25541 |type=Print newspaper |url=https://gallica.bnf.fr/ark:/12148/bpt6k248444p |access-date=28 June 2023 |archive-url= |archive-date= |ref=Herbrand-Death}}</ref>

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

==Bibliography==
* {{cite report | author=Claus-Peter Wirth and Jörg Siekmann and Christoph Benzmüller and Serge Autexier | title=Lectures on Jacques Herbrand as a Logician | institution=[[DFKI]] | type=SEKI Report | number=SR-2009-01 | year=2009 | arxiv=0902.4682 }}

Primary literature:
* 1967. [[Jean van Heijenoort]] (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. [[Jean van Heijenoort]] (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.

==See also==
* [[Herbrand interpretation]]
* [[Herbrand structure]]
* [[Herbrand Award]] – by the Conference on Automated Deduction, for automated deduction
* [[Prix Jacques Herbrand]] – by the French Academy of Sciences, for mathematics and physics
* [[Herbrandization]] – a validity-preserving normal form of a formula, dual to [[Skolemization]]
* [[Herbrand's theorem on ramification groups]]
* [[Rollo Davidson]] (1944–1970) – another mathematician who died in a mountain climbing accident
* [[Church–Turing thesis|(Gödel-Herbrand) computability thesis]]: before Church and Turing, in 1933 with [[Kurt Gödel]], they created a formal definition of a class called [[general recursive function]]s.

==References==
{{Reflist}}

==External links==
* {{MacTutor Biography|id=Herbrand}}

{{Authority control}}

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