Félix Édouard Justin Émile Borel (; 7 January 1871 – 3 February 1956) 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, mathematical structure, structure, space, Mathematica ...

and

politician A politician is a person who participates in Public policy, policy-making processes, usually holding an elective position in government. Politicians represent the people, make decisions, and influence the formulation of public policy. The roles ...

. As a mathematician, he was known for his founding work in the areas of

measure theory In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude (mathematics), magnitude, mass, and probability of events. These seemingl ...

and

probability Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an e ...

Biography

Borel was born in

Saint-Affrique Saint-Affrique (; Languedocien: ''Sant Africa'') is a commune in the Aveyron department in Southern France. History Saint-Affrique grew in the 6th century around the tomb of St. Africain, bishop of Comminges. In the 12th century a fortre ...

Aveyron Aveyron (; ) is a Departments of France, department in the Regions of France, region of Occitania (administrative region), Occitania, Southern France. It was named after the river Aveyron (river), Aveyron. Its inhabitants are known as ''Aveyro ...

, the son of a

Protestant Protestantism is a branch of Christianity that emphasizes Justification (theology), justification of sinners Sola fide, through faith alone, the teaching that Salvation in Christianity, salvation comes by unmerited Grace in Christianity, divin ...

pastor. He studied at the

Collège Sainte-Barbe The Collège Sainte-Barbe () is a former college in the 5th arrondissement of Paris, France. The Collège Sainte-Barbe was founded in 1460 on Montagne Sainte-Geneviève ( Latin Quarter, Paris). It was until its closure in June 1999 the "oldest ...

and

Lycée Louis-le-Grand The Lycée Louis-le-Grand (), also referred to simply as Louis-le-Grand or by its acronym LLG, is a public Lycée (French secondary school, also known as sixth form college) located on Rue Saint-Jacques (Paris), rue Saint-Jacques in central Par ...

before applying to both the

École normale supérieure École or Ecole may refer to: * an elementary school in the French educational stages normally followed by Secondary education in France, secondary education establishments (collège and lycée) * École (river), a tributary of the Seine flowing i ...

and the

École Polytechnique (, ; also known as Polytechnique or l'X ) is a ''grande école'' located in Palaiseau, France. It specializes in science and engineering and is a founding member of the Polytechnic Institute of Paris. The school was founded in 1794 by mat ...

. He qualified in the first position for both and chose to attend the former institution in 1889. That year he also won the

concours général In France, the Concours Général (), created in 1747, is the most prestigious academic competition held every year between students of ''Première'' (11th grade) and ''Terminale'' (12th and final grade) in almost all subjects taught in both genera ...

, an annual national mathematics competition. After graduating in 1892, he placed first in the

agrégation In France, the () is the most competitive and prestigious examination for civil service in the French public education A state school, public school, or government school is a primary school, primary or secondary school that educates all stu ...

, a competitive civil service examination leading to the position of professeur agrégé. His thesis, published in 1893, was titled ''Sur quelques points de la théorie des fonctions'' ("On some points in the theory of functions"). That year, Borel started a four-year stint as a lecturer at the University of Lille, during which time he published 22 research papers. He returned to the École normale supérieure in 1897, and was appointed to the chair of theory of functions, which he held until 1941. In 1901, Borel married 17-year-old Marguerite, the daughter of colleague Paul Émile Appel; she later wrote more than 30 novels under the pseudonym Camille Marbo. Émile Borel died in Paris on 3 February 1956.

Work

Along with René-Louis Baire and

Henri Lebesgue Henri Léon Lebesgue (; ; June 28, 1875 – July 26, 1941) was a French mathematician known for his Lebesgue integration, theory of integration, which was a generalization of the 17th-century concept of integration—summing the area between an ...

, Émile Borel was among the pioneers of

and its application to

probability theory Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...

. The concept of a

Borel set In mathematics, a Borel set is any subset of a topological space that can be formed from its open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets ...

is named in his honor. One of his books on probability introduced the amusing

thought experiment A thought experiment is an imaginary scenario that is meant to elucidate or test an argument or theory. It is often an experiment that would be hard, impossible, or unethical to actually perform. It can also be an abstract hypothetical that is ...

that entered popular culture under the name

infinite monkey theorem The infinite monkey theorem states that a monkey hitting keys independently and at randomness, random on a typewriter keyboard for an infinity, infinite amount of time will almost surely type any given text, including the complete works of Willi ...

or the like. He also published a series of papers (1921–1927) that first defined

games of strategy A strategy game or strategic game is a game in which the players' uncoerced, and often autonomous, decision-making skills have a high significance in determining the outcome. Almost all strategy games require internal decision tree-style think ...

. John von Neumann objected to this assignment of priority in a letter to ''Econometrica'' published in 1953 where he asserted that Borel could not have defined games of strategy because he rejected the minimax theorem. With the development of

statistical hypothesis testing A statistical hypothesis test is a method of statistical inference used to decide whether the data provide sufficient evidence to reject a particular hypothesis. A statistical hypothesis test typically involves a calculation of a test statistic. T ...

in the early 1900s various tests for

randomness In common usage, randomness is the apparent or actual lack of definite pattern or predictability in information. A random sequence of events, symbols or steps often has no order and does not follow an intelligible pattern or combination. ...

were proposed. Sometimes these were claimed to have some kind of general significance, but mostly they were just viewed as simple practical methods. In 1909, Borel formulated the notion that numbers picked randomly on the basis of their value are

almost always In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (with respect to the probability measure). In other words, the set of outcomes on which the event does not occur ha ...

normal, and with explicit constructions in terms of digits, it is quite straightforward to get numbers that are normal. In 1913 and 1914 he bridged the gap between

hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or János Bolyai, Bolyai–Nikolai Lobachevsky, Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For a ...

and

special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory of the relationship between Spacetime, space and time. In Albert Einstein's 1905 paper, Annus Mirabilis papers#Special relativity, "On the Ele ...

with expository work. For instance, his book ''Introduction Géométrique à quelques Théories Physiques'' described

hyperbolic rotation In linear algebra, a squeeze mapping, also called a squeeze transformation, is a type of linear map that preserves Euclidean area of regions in the Cartesian plane, but is ''not'' a rotation or shear mapping. For a fixed positive real number , th ...

s as transformations that leave a hyperbola

stable A stable is a building in which working animals are kept, especially horses or oxen. The building is usually divided into stalls, and may include storage for equipment and feed. Styles There are many different types of stables in use tod ...

just as a circle around a rotational center is stable. In 1915 he was appointed head of the technical cabinet for the newly formed Directorate of Inventions for National Defense which aimed to coordinate French laboratories for the war effort in the

First World War World War I or the First World War (28 July 1914 – 11 November 1918), also known as the Great War, was a World war, global conflict between two coalitions: the Allies of World War I, Allies (or Entente) and the Central Powers. Fighting to ...

. In 1922, he founded the Paris Institute of Statistics, the oldest French school for statistics; then in 1928 he co-founded the

Institut Henri Poincaré The Henri Poincaré Institute (or IHP for ''Institut Henri Poincaré'') is a mathematics research institute part of Sorbonne University, in association with the Centre national de la recherche scientifique (CNRS). It is located in the 5th arrondi ...

in Paris.

Political career

In the 1920s, 1930s, and 1940s, he was active in politics. From 1924 to 1936, he was a member of the

Chamber of Deputies The chamber of deputies is the lower house in many bicameral legislatures and the sole house in some unicameral legislatures. Description Historically, French Chamber of Deputies was the lower house of the French Parliament during the Bourb ...

. In 1925, he was Minister of the Navy in the cabinet of fellow mathematician Paul Painlevé. During the

Second World War World War II or the Second World War (1 September 1939 – 2 September 1945) was a World war, global conflict between two coalitions: the Allies of World War II, Allies and the Axis powers. World War II by country, Nearly all of the wo ...

, he was a member of the

French Resistance The French Resistance ( ) was a collection of groups that fought the German military administration in occupied France during World War II, Nazi occupation and the Collaboration with Nazi Germany and Fascist Italy#France, collaborationist Vic ...

Honors

Besides the ''Centre Émile Borel'' at the

in Paris and a crater on the Moon, the following mathematical notions are named after him: *

Borel algebra In mathematics, a Borel set is any subset of a topological space that can be formed from its open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are ...

* Borel's lemma * Borel's law of large numbers *

Borel measure In mathematics, specifically in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets (and thus on all Borel sets). Some authors require additional restrictions on the measure, as described below. ...

* Borel–Kolmogorov paradox *

Borel–Cantelli lemma In probability theory, the Borel–Cantelli lemma is a theorem about sequences of events. In general, it is a result in measure theory. It is named after Émile Borel and Francesco Paolo Cantelli, who gave statement to the lemma in the first d ...

* Borel–Carathéodory theorem *

Heine–Borel theorem In real analysis, the Heine–Borel theorem, named after Eduard Heine and Émile Borel, states: For a subset S of Euclidean space \mathbb^n, the following two statements are equivalent: *S is compact, that is, every open cover of S has a finite s ...

* Borel determinacy theorem * Borel right process *

Borel summation In mathematics, Borel summation is a summation method for divergent series, introduced by . It is particularly useful for summing divergent asymptotic series, and in some sense gives the best possible sum for such series. There are several vari ...

Borel distribution The Borel distribution is a discrete probability distribution, arising in contexts including branching processes and queueing theory. It is named after the French mathematician Émile Borel. If the number of offspring that an organism has is Poi ...

* Borel's conjecture about strong measure zero sets (not to be confused with Borel conjecture, named for Armand Borel). Borel also described a

poker Poker is a family of Card game#Comparing games, comparing card games in which Card player, players betting (poker), wager over which poker hand, hand is best according to that specific game's rules. It is played worldwide, with varying rules i ...

model that he coins ''La Relance'' in his 1938 book ''Applications de la théorie des probabilités aux Jeux de Hasard''.Émile Borel and Jean Ville. Applications de la théorie des probabilités aux jeux de hasard. Gauthier-Vilars, 1938 Borel was awarded the Resistance Medal in 1950.

Works

* ''On a few points about the theory of functions'' (PhD thesis, 1894) * ''Introduction to the study of

number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...

and superior algebra'' (1895) * ''A course on the theory of functions'' (1898) * ''A course on

power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''a_n'' represents the coefficient of the ''n''th term and ''c'' is a co ...

'' (1900) * ''A course on

divergent series In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit. If a series converges, the individual terms of the series mus ...

'' (1901) * ''A course on positive terms series'' (1902) * ''A course on

meromorphic functions In the mathematical field of complex analysis, a meromorphic function on an open subset ''D'' of the complex plane is a function that is holomorphic on all of ''D'' ''except'' for a set of isolated points, which are ''poles'' of the function. ...

'' (1903) * ''A course on growth theory at the Paris faculty of sciences'' (1910) * ''A course on functions of a real variable and polynomial serial developments'' (1905) * ''Chance'' (1914) * ''Geometrical introduction to some physical theories'' (1914) * ''A course on complex variable uniform monogenic functions'' (1917) * ''On the method in sciences'' (1919) * ''Space and time'' (1921) * ''

Game theory Game theory is the study of mathematical models of strategic interactions. It has applications in many fields of social science, and is used extensively in economics, logic, systems science and computer science. Initially, game theory addressed ...

and left symmetric core integral equations'' (1921) * ''Methods and problems of the theory of functions'' (1922) * ''Space and time'' (1922) * ''A treatise on probability calculation and its applications'' (1924–1934) * ''Application of

games of chance A game of chance is in contrast with a game of skill. It is a game whose outcome is strongly influenced by some randomizing device. Common devices used include dice, spinning tops, playing cards, roulette wheels, numbered balls, or in the case ...

'' (1938) * ''Principles and classical formulas for probability calculation'' (1925) * ''Practical and philosophical values of probabilities'' (1939) * ''Mathematical theory of contract bridge for everyone'' (1940) * ''Game, luck and contemporary scientific theories'' (1941) * ''Probabilities and life'' (1943) * ''Evolution of

mechanics Mechanics () is the area of physics concerned with the relationships between force, matter, and motion among Physical object, physical objects. Forces applied to objects may result in Displacement (vector), displacements, which are changes of ...

'' (1943) * ''Paradoxes of the infinite'' (1946) * ''Elements of

set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...

'' (1949) * ''Probability and certainty'' (1950) * ''Inaccessible numbers'' (1952) * ''Imaginary and real in mathematics and physics'' (1952) * '' Emile Borel complete works'' (1972)

Articles

*
"La science est-elle responsable de la crise mondiale?"
'' Scientia : rivista internazionale di sintesi scientifica'', 51, 1932, pp. 99–106. *
"La science dans une société socialiste"
''Scientia : rivista internazionale di sintesi scientifica'', 31, 1922, pp. 223–228. *
"Le continu mathématique et le continu physique"
'' Rivista di scienza'', 6, 1909, pp. 21–35.

References

* Michel Pinault, ''Emile Borel, une carrière intellectuelle sous la 3ème République'', Paris, L'Harmattan, 2017. Voir : michel-pinault.over-blog.com

External links

* * * * * *
Author profile
in the database

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{{DEFAULTSORT:Borel, Emile 1871 births 1956 deaths People from Saint-Affrique French Calvinist and Reformed Christians Radical Party (France) politicians Republican-Socialist Party politicians Ministers of marine Members of the 13th Chamber of Deputies of the French Third Republic Members of the 14th Chamber of Deputies of the French Third Republic Members of the 15th Chamber of Deputies of the French Third Republic Members of Parliament for Aveyron 19th-century French mathematicians 20th-century French mathematicians French probability theorists Measure theorists Intuitionism Lycée Louis-le-Grand alumni École Normale Supérieure alumni Academic staff of the Lille University of Science and Technology Members of the French Academy of Sciences French military personnel of World War I French Resistance members Grand Cross of the Legion of Honour Recipients of the Croix de Guerre 1939–1945 (France) Recipients of the Resistance Medal