Hugo Steinhaus
Encyclopedia
Władysław Hugo Dionizy Steinhaus (January 14, 1887 – February 25, 1972) was a Polish
mathematician
and educator
. Steinhaus obtained his PhD under David Hilbert
at Göttingen University in 1911 and later became a professor at the University of Lwów, where he helped establish what later became known as the Lwów School of Mathematics
. He is credited with "discovering" mathematician Stefan Banach
, with whom he gave a notable contribution to functional analysis
through the Banach-Steinhaus theorem. After World War II
Steinhaus played an important part in the establishment of the mathematics department at Wrocław University and in the revival of Polish mathematics from the destruction of the war.
Author of around 170 scientific articles and books, Steinhaus has left its legacy and contribution on many branches of mathematics, such as functional analysis, geometry
, mathematical logic
, and trigonometry
. Notably he is regarded as one of the early founders of the game theory
and the probability theory
preceding in his studies, later, more comprehensive approaches, by other scholars.
(now in modern Poland
) to a family with Jewish roots. His father, Bogusław, was a local industrialist, owner of a brick factory and a merchant. His mother was Ewalina, née Lipschitz. Hugo's uncle, Ignacy Steinhaus, was an activist in the Koło Polskie (Polish Circle), and a deputy to the Galician Diet
, the regional assembly
of the Kingdom of Galicia and Lodomeria
.
Hugo finished his studies at the gymnasium
in Jasło in 1905. His family wanted him to become an engineer
but he was drawn to abstract mathematics
and began to study the works of famous contemporary mathematicians on his own. In the same year he began studying philosophy and mathematics at the University of Lwów. In 1906 he transferred to Göttingen University. At that University he received his Ph.D. in 1911, having written his doctoral dissertation under the supervision of David Hilbert
. The title of his thesis was Neue Anwendungen des Dirichlet'schen Prinzips ("New applications to Dirichlet's principle").
At the start of World War I
Steinhaus returned to Poland and served in Józef Piłsudski's Polish Legion
, after which he lived in Kraków
.
, which occurred in 1918, Steinhaus worked in Kraków for the Ministry of the Interior
in the ephemeral puppet state of Kingdom of Poland
.
In 1917 he started to work at the University of Lwów
(at the time known as Jan Kazimierz University) and acquired his habilitation
qualification in 1920. In 1921 he became a profesor nadzwyczajny (associate professor) and in 1925 profesor zwyczajny (full professor) at the same university. During this time he taught a course on the then cutting edge theory of Lebesgue integration
, one of the first such courses offered outside of France
.
While in Lwów, Steinhaus co-founded the Lwów School of Mathematics
and was active in the circle of mathematicians associated with the Scottish cafe
, although, according to Stanislaw Ulam, for the circle's gatherings, Steinhaus would have generally preferred a more upscale tea shop down the street.
In September 1939 after Nazi Germany and Soviet Union
both invaded and occupied Poland, as a fulfillment of the Molotov-Ribbentrop Pact they had signed earlier, Lwów initially came under Soviet occupation. Steinhaus considered escaping to Hungary
but ultimately decided to remain in Lwów. The Soviets reorganized the university to give it a more Ukrainian
character, but they did appoint Stefan Banach
(Steinhaus's student) as the dean of the mathematics department and Steinhaus resumed teaching there. The faculty of the department at the school were also strengthened by several Polish refugees from German occupied Poland. According to Steinhaus, during the experience of this period, he "acquired a insurmountable physical disgust in regard to all sorts of Soviet administrators, politicians and commissars"
During the interwar period and the time of the Soviet occupation, Steinhaus contributed ten problems to the famous Scottish Book
, including the last one
, recorded shortly before Lwów was captured by the Nazis in 1941, during Operation Barbarossa
.
Steinhaus, because of his Jewish background, spent the Nazi occupation in hiding, first among friends in Lwów, then in the small towns of Osiczyna
, near Zamość
and Berdechów
, near Kraków. The Polish anti-Nazi resistance provided him with false documents of a forest ranger who had died sometime earlier, by the name of Grzegorz Krochmalny. Under this name he taught clandestine classes (higher education was forbidden for Poles under the German occupation
). Worried about the possibility of imminent death if captured by Germans, Steinhaus, without access to any scholarly material, reconstructed from memory and recorded all the mathematics he knew, in addition to writing other voluminous memoirs, of which only a little part has been published.
Also while in hiding, and cut off from reliable news on the course of the war, Steinhaus devised a statistical means of estimating for himself the German casualties at the front based on sporadic obituaries published in the local press. The method relied on the relative frequency with which the obituaries stated that the soldier who died was someone's son, someone's "second son", someone's "third son" and so on.
According to his student and biographer, Marek Kac, Steinhaus told him that the happiest day of his life were the twenty four hours between the time that the Germans left occupied Poland and the Soviets had not yet arrived ("They had left, and they had not yet come").
(Lwów became part of Soviet Ukraine). Although initially he had doubts, he turned down offers for faculty positions in Łódź and Lublin
and made his way to the city where he began teaching at Wrocław University. While there, he revived the idea behind the Scottish Book from Lwów, where prominent and aspiring mathematicians would write down problems of interest along with prizes to be awarded for their solution, by starting the New Scottish Book. It was also most likely Steinhaus who preserved the original Scottish Book from Lwów throughout the war and subsequently sent it to Stanisław Ulam, who translated it into English.
With Steinhaus' help, Wrocław University became renowned for mathematics, much as the University of Lwów had been.
Later, in the 1960s, Steinhaus served as a visiting professor at the University of Notre Dame
(1961–62) and the University of Sussex
(1966).
, Steinhaus made contributions to a wide range of mathematical sub-disciplines, including geometry
, probability theory
, functional analysis, theory of trigonometric and Fourier series
as well as mathematical logic
. He also wrote in the area of applied mathematics
and enthusiastically collaborated with engineers, geologists, economists, physicians, biologists and, in Kac's words, "even lawyers".
Probably his most notable contribution to functional analysis
was the 1927 proof of the Banach-Steinhaus theorem, given along with Stefan Banach, which is now one of the fundamental tools in this branch of mathematics.
His interest in games
led him to propose an early formal definition of a strategy
, anticipating John von Neumann
's more complete treatment of few years later. As a result he is regarded as one of the early founders of modern game theory
. As a result of his work on infinite games Steinhaus, together with another of his students', Jan Mycielski
, proposed the Axiom of determinacy
.
Steinhaus was also among the early contributors and founders of the probability theory, which at the time was still in its infancy and was not even considered to be an actual part of mathematics. He provided the first axiom
atic measure-theoretic description of coin-tossing, which was to influence the full of axiomatization of probability by the Russian mathematician Andrey Kolmogorov
a decade later. He was also the first to offer precise definitions of what it means for two events to be "independent", as well as for what it means for a random variable
to be "uniformly distributed
".
While in hiding during World War II Steinhaus worked on the so-called "cake cutting problem"; how to divide a given resource in a manner which is "fair" according to precisely defined criteria, such as proportionality
and envy-free
.
Steinhaus was also the first person to conjecture the Ham sandwich theorem
, and one of the first to propose the method of k-means clustering
Steinhaus is said to have "discovered" the Polish mathematician Stefan Banach in 1916, after he overheard someone utter the words "Lebesgue integral" while in a Kraków park (Steinhaus referred to Banach as his "greatest mathematical discovery"). Together with Banach and the other participant of the park discussion, Otto Nikodym, Steinhaus started the Mathematical Society of Kraków, which later evolved into the Polish Mathematical Society
. He was a member of PAU (the Polish Academy of Learning
) and PAN (the Polish Academy of Sciences
), PTM (the Polish Mathematical Society
), the Wrocławskie Towarzystwo Naukowe (Wrocław Scientific Society) as well as many international scientific societies and science academies.
Steinhaus also published one of the first articles in Fundamenta Mathematicae
, in 1921. He also co-founded Studia Mathematica
along with Stefan Banach
(1929), and Zastosowania matematyki (Applications of Mathematics, 1953), Colloquium Mathematicum, and Monografie Matematyczne (Mathematical Monographs).
He received honorary doctorate degrees from Warsaw University (1958), Wrocław Medical Academy (1961), Poznań University (1963) and Wrocław University (1965).
Steinhaus had full command of several foreign languages and was interestingly, known, for his aphorism
s, to the point that a booklet of his most famous ones in Polish
, French
and Latin has been published posthumously.
In 2002, the Polish Academy of Sciences and Wrocław University sponsored "2002, The Year of Hugo Steinhaus", to celebrate his contributions to Polish and world science.
Notable mathematician Mark Kac
, Steinhaus's student, wrote:
Poland
Poland , officially the Republic of Poland , is a country in Central Europe bordered by Germany to the west; the Czech Republic and Slovakia to the south; Ukraine, Belarus and Lithuania to the east; and the Baltic Sea and Kaliningrad Oblast, a Russian exclave, to the north...
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....
and educator
Teacher
A teacher or schoolteacher is a person who provides education for pupils and students . The role of teacher is often formal and ongoing, carried out at a school or other place of formal education. In many countries, a person who wishes to become a teacher must first obtain specified professional...
. Steinhaus obtained his PhD under David Hilbert
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
at Göttingen University in 1911 and later became a professor at the University of Lwów, where he helped establish what later became known as the Lwów School of Mathematics
Lwów School of Mathematics
The Lwów School of Mathematics was a group of mathematicians who worked between the two World Wars in Lviv, then known as Lwów and located in Poland, but now located in western Ukraine. The mathematicians often met at the famous Scottish Café to discuss mathematical problems, and published in the...
. He is credited with "discovering" mathematician Stefan Banach
Stefan Banach
Stefan Banach was a Polish mathematician who worked in interwar Poland and in Soviet Ukraine. He is generally considered to have been one of the 20th century's most important and influential mathematicians....
, with whom he gave a notable contribution to functional analysis
Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
through the Banach-Steinhaus theorem. After World War II
World War II
World War II, or the Second World War , was a global conflict lasting from 1939 to 1945, involving most of the world's nations—including all of the great powers—eventually forming two opposing military alliances: the Allies and the Axis...
Steinhaus played an important part in the establishment of the mathematics department at Wrocław University and in the revival of Polish mathematics from the destruction of the war.
Author of around 170 scientific articles and books, Steinhaus has left its legacy and contribution on many branches of mathematics, such as functional analysis, geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, 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 trigonometry
Trigonometry
Trigonometry is a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves...
. Notably he is regarded as one of the early founders of the game theory
Game theory
Game theory is a mathematical method for analyzing calculated circumstances, such as in games, where a person’s success is based upon the choices of others...
and the probability theory
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
preceding in his studies, later, more comprehensive approaches, by other scholars.
Early life and studies
Steinhaus was born on January 14, 1887 in Jasło, Austria-HungaryAustria-Hungary
Austria-Hungary , more formally known as the Kingdoms and Lands Represented in the Imperial Council and the Lands of the Holy Hungarian Crown of Saint Stephen, was a constitutional monarchic union between the crowns of the Austrian Empire and the Kingdom of Hungary in...
(now in modern Poland
Poland
Poland , officially the Republic of Poland , is a country in Central Europe bordered by Germany to the west; the Czech Republic and Slovakia to the south; Ukraine, Belarus and Lithuania to the east; and the Baltic Sea and Kaliningrad Oblast, a Russian exclave, to the north...
) to a family with Jewish roots. His father, Bogusław, was a local industrialist, owner of a brick factory and a merchant. His mother was Ewalina, née Lipschitz. Hugo's uncle, Ignacy Steinhaus, was an activist in the Koło Polskie (Polish Circle), and a deputy to the Galician Diet
Diet of Galicia
The Diet of Galicia was the regional assembly of the Kingdom of Galicia and Lodomeria, which was part of Austro-Hungary. The Galician diet was a unicameral assembly composed of 150 deputies, which was presided over by a marshal or a vice-marshal that were appointed by the emperor. The...
, the regional assembly
Legislative Assembly
Legislative Assembly is the name given in some countries to either a legislature, or to one of its branch.The name is used by a number of member-states of the Commonwealth of Nations, as well as a number of Latin American countries....
of the Kingdom of Galicia and Lodomeria
Kingdom of Galicia and Lodomeria
The Kingdom of Galicia and Lodomeria was a crownland of the Habsburg Monarchy, the Austrian Empire, and Austria–Hungary from 1772 to 1918 .This historical region in eastern Central Europe is currently divided between Poland and Ukraine...
.
Hugo finished his studies at the gymnasium
Gymnasium (school)
A gymnasium is a type of school providing secondary education in some parts of Europe, comparable to English grammar schools or sixth form colleges and U.S. college preparatory high schools. The word γυμνάσιον was used in Ancient Greece, meaning a locality for both physical and intellectual...
in Jasło in 1905. His family wanted him to become an engineer
Engineer
An engineer is a professional practitioner of engineering, concerned with applying scientific knowledge, mathematics and ingenuity to develop solutions for technical problems. Engineers design materials, structures, machines and systems while considering the limitations imposed by practicality,...
but he was drawn to abstract mathematics
Pure mathematics
Broadly speaking, pure mathematics is mathematics which studies entirely abstract concepts. From the eighteenth century onwards, this was a recognized category of mathematical activity, sometimes characterized as speculative mathematics, and at variance with the trend towards meeting the needs of...
and began to study the works of famous contemporary mathematicians on his own. In the same year he began studying philosophy and mathematics at the University of Lwów. In 1906 he transferred to Göttingen University. At that University he received his Ph.D. in 1911, having written his doctoral dissertation under the supervision of David Hilbert
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
. The title of his thesis was Neue Anwendungen des Dirichlet'schen Prinzips ("New applications to Dirichlet's principle").
At the start of World War I
World War I
World War I , which was predominantly called the World War or the Great War from its occurrence until 1939, and the First World War or World War I thereafter, was a major war centred in Europe that began on 28 July 1914 and lasted until 11 November 1918...
Steinhaus returned to Poland and served in Józef Piłsudski's Polish Legion
Polish Legions in World War I
Polish Legions was the name of Polish armed forces created in August 1914 in Galicia. Thanks to the efforts of KSSN and the Polish members of the Austrian parliament, the unit became an independent formation of the Austro-Hungarian Army...
, after which he lived in Kraków
Kraków
Kraków also Krakow, or Cracow , is the second largest and one of the oldest cities in Poland. Situated on the Vistula River in the Lesser Poland region, the city dates back to the 7th century. Kraków has traditionally been one of the leading centres of Polish academic, cultural, and artistic life...
.
Interwar Poland
During the 1916-1917 period and before Poland had regained its full independencePolish Independence Day
National Independence Day is a public holiday in Poland celebrated every year on 11 November to commemorate the anniversary of Poland's assumption of independent statehood in 1918 after 123 years of partition by Russia, Prussia and Austria....
, which occurred in 1918, Steinhaus worked in Kraków for the Ministry of the Interior
Ministry of Interior and Administration of the Republic of Poland
Ministry of the Interior and Administration is an administration structure controlling main administration and security branches of the Polish government. Current Minister is Jerzy Miller.-History and function:...
in the ephemeral puppet state of Kingdom of Poland
Kingdom of Poland (1916–1918)
The Kingdom of Poland, also informally called the Regency Kingdom of Poland , was a proposed puppet state during World War I by Germany and Austria-Hungary in 1916 after their conquest of the former Congress Poland from Russia...
.
In 1917 he started to work at the University of Lwów
Lviv University
The Lviv University or officially the Ivan Franko National University of Lviv is the oldest continuously operating university in Ukraine...
(at the time known as Jan Kazimierz University) and acquired his habilitation
Habilitation
Habilitation is the highest academic qualification a scholar can achieve by his or her own pursuit in several European and Asian countries. Earned after obtaining a research doctorate, such as a PhD, habilitation requires the candidate to write a professorial thesis based on independent...
qualification in 1920. In 1921 he became a profesor nadzwyczajny (associate professor) and in 1925 profesor zwyczajny (full professor) at the same university. During this time he taught a course on the then cutting edge theory of Lebesgue integration
Lebesgue integration
In mathematics, Lebesgue integration, named after French mathematician Henri Lebesgue , refers to both the general theory of integration of a function with respect to a general measure, and to the specific case of integration of a function defined on a subset of the real line or a higher...
, one of the first such courses offered outside of 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...
.
While in Lwów, Steinhaus co-founded the Lwów School of Mathematics
Lwów School of Mathematics
The Lwów School of Mathematics was a group of mathematicians who worked between the two World Wars in Lviv, then known as Lwów and located in Poland, but now located in western Ukraine. The mathematicians often met at the famous Scottish Café to discuss mathematical problems, and published in the...
and was active in the circle of mathematicians associated with the Scottish cafe
Scottish Café
The Scottish Café was the café in Lwów where, in the 1930s and 1940s, mathematicians from the Lwów School collaboratively discussed research problems, particularly in functional analysis and topology....
, although, according to Stanislaw Ulam, for the circle's gatherings, Steinhaus would have generally preferred a more upscale tea shop down the street.
World War II
Soviet invasion of Poland
Soviet invasion of Poland can refer to:* the second phase of the Polish-Soviet War of 1920 when Soviet armies marched on Warsaw, Poland* Soviet invasion of Poland of 1939 when Soviet Union allied with Nazi Germany attacked Second Polish Republic...
both invaded and occupied Poland, as a fulfillment of the Molotov-Ribbentrop Pact they had signed earlier, Lwów initially came under Soviet occupation. Steinhaus considered escaping to Hungary
Hungary
Hungary , officially the Republic of Hungary , is a landlocked country in Central Europe. It is situated in the Carpathian Basin and is bordered by Slovakia to the north, Ukraine and Romania to the east, Serbia and Croatia to the south, Slovenia to the southwest and Austria to the west. The...
but ultimately decided to remain in Lwów. The Soviets reorganized the university to give it a more Ukrainian
Ukrainians
Ukrainians are an East Slavic ethnic group native to Ukraine, which is the sixth-largest nation in Europe. The Constitution of Ukraine applies the term 'Ukrainians' to all its citizens...
character, but they did appoint Stefan Banach
Stefan Banach
Stefan Banach was a Polish mathematician who worked in interwar Poland and in Soviet Ukraine. He is generally considered to have been one of the 20th century's most important and influential mathematicians....
(Steinhaus's student) as the dean of the mathematics department and Steinhaus resumed teaching there. The faculty of the department at the school were also strengthened by several Polish refugees from German occupied Poland. According to Steinhaus, during the experience of this period, he "acquired a insurmountable physical disgust in regard to all sorts of Soviet administrators, politicians and commissars"
During the interwar period and the time of the Soviet occupation, Steinhaus contributed ten problems to the famous Scottish Book
Scottish Book
The Scottish Book was a thick notebook used by mathematicians of the Lwow School of Mathematics for jotting down problems meant to be solved. The notebook was named after the "Scottish Café" where it was kept....
, including the last one
Banach's matchbox problem
Banach's match problem is a classic problem in probability attributed to Stefan Banach. Feller says that the problem was inspired by a humorous reference to Banach's smoking habit in a speech honouring him by H...
, recorded shortly before Lwów was captured by the Nazis in 1941, during Operation Barbarossa
Operation Barbarossa
Operation Barbarossa was the code name for Germany's invasion of the Soviet Union during World War II that began on 22 June 1941. Over 4.5 million troops of the Axis powers invaded the USSR along a front., the largest invasion in the history of warfare...
.
Steinhaus, because of his Jewish background, spent the Nazi occupation in hiding, first among friends in Lwów, then in the small towns of Osiczyna
Osiczyna
Osiczyna is a village in the administrative district of Gmina Skierbieszów, within Zamość County, Lublin Voivodeship, in eastern Poland. It lies approximately east of Skierbieszów, north-east of Zamość, and south-east of the regional capital Lublin....
, near Zamość
Zamosc
Zamość ukr. Замостя is a town in southeastern Poland with 66,633 inhabitants , situated in the south-western part of Lublin Voivodeship , about from Lublin, from Warsaw and from the border with Ukraine...
and Berdechów
Berdechów
Berdechów is a village in the administrative district of Gmina Bobowa, within Gorlice County, Lesser Poland Voivodeship, in southern Poland. It lies approximately north of Bobowa, north-west of Gorlice, and south-east of the regional capital Kraków....
, near Kraków. The Polish anti-Nazi resistance provided him with false documents of a forest ranger who had died sometime earlier, by the name of Grzegorz Krochmalny. Under this name he taught clandestine classes (higher education was forbidden for Poles under the German occupation
Education in Poland during World War II
This article covers the topic of underground education in Poland during World War II. Secret learning prepared new cadres for the post-war reconstruction of Poland and countered the German and Soviet threat to exterminate the Polish culture....
). Worried about the possibility of imminent death if captured by Germans, Steinhaus, without access to any scholarly material, reconstructed from memory and recorded all the mathematics he knew, in addition to writing other voluminous memoirs, of which only a little part has been published.
Also while in hiding, and cut off from reliable news on the course of the war, Steinhaus devised a statistical means of estimating for himself the German casualties at the front based on sporadic obituaries published in the local press. The method relied on the relative frequency with which the obituaries stated that the soldier who died was someone's son, someone's "second son", someone's "third son" and so on.
According to his student and biographer, Marek Kac, Steinhaus told him that the happiest day of his life were the twenty four hours between the time that the Germans left occupied Poland and the Soviets had not yet arrived ("They had left, and they had not yet come").
After World War II
In the last days of World War II Steinhaus, still in hiding, heard a rumor that University of Lwów was to be transferred to the city of Wrocław, which Poland was to acquire as a result of the Potsdam agreementPotsdam Agreement
The Potsdam Agreement was the Allied plan of tripartite military occupation and reconstruction of Germany—referring to the German Reich with its pre-war 1937 borders including the former eastern territories—and the entire European Theatre of War territory...
(Lwów became part of Soviet Ukraine). Although initially he had doubts, he turned down offers for faculty positions in Łódź and Lublin
Lublin
Lublin is the ninth largest city in Poland. It is the capital of Lublin Voivodeship with a population of 350,392 . Lublin is also the largest Polish city east of the Vistula river...
and made his way to the city where he began teaching at Wrocław University. While there, he revived the idea behind the Scottish Book from Lwów, where prominent and aspiring mathematicians would write down problems of interest along with prizes to be awarded for their solution, by starting the New Scottish Book. It was also most likely Steinhaus who preserved the original Scottish Book from Lwów throughout the war and subsequently sent it to Stanisław Ulam, who translated it into English.
With Steinhaus' help, Wrocław University became renowned for mathematics, much as the University of Lwów had been.
Later, in the 1960s, Steinhaus served as a visiting professor at the University of Notre Dame
University of Notre Dame
The University of Notre Dame du Lac is a Catholic research university located in Notre Dame, an unincorporated community north of the city of South Bend, in St. Joseph County, Indiana, United States...
(1961–62) and the University of Sussex
University of Sussex
The University of Sussex is an English public research university situated next to the East Sussex village of Falmer, within the city of Brighton and Hove. The University received its Royal Charter in August 1961....
(1966).
Mathematical contributions
Steinhaus authored over 170 works. Unlike his student, Stefan Banach, who tended to specialize narrowly in the field of functional analysisFunctional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
, Steinhaus made contributions to a wide range of mathematical sub-disciplines, including geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, probability theory
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
, functional analysis, theory of trigonometric and Fourier series
Fourier series
In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...
as well as 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...
. He also wrote in the area of applied mathematics
Applied mathematics
Applied mathematics is a branch of mathematics that concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, "applied mathematics" is a mathematical science with specialized knowledge...
and enthusiastically collaborated with engineers, geologists, economists, physicians, biologists and, in Kac's words, "even lawyers".
Probably his most notable contribution to functional analysis
Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
was the 1927 proof of the Banach-Steinhaus theorem, given along with Stefan Banach, which is now one of the fundamental tools in this branch of mathematics.
His interest in games
Games
The term games is the plural of gameIt may refer to:* GamesTM magazine* Games * Games, an electronic music album by Larry Fast* Games , an album by Leo Ku* Games * "Games" , an episode of the TV series House...
led him to propose an early formal definition of a strategy
Strategy (game theory)
In game theory, a player's strategy in a game is a complete plan of action for whatever situation might arise; this fully determines the player's behaviour...
, anticipating 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,...
's more complete treatment of few years later. As a result he is regarded as one of the early founders of modern game theory
Game theory
Game theory is a mathematical method for analyzing calculated circumstances, such as in games, where a person’s success is based upon the choices of others...
. As a result of his work on infinite games Steinhaus, together with another of his students', Jan Mycielski
Jan Mycielski
Jan Mycielski is a Polish-American mathematician, a professor emeritus of mathematics at the University of Colorado at Boulder....
, proposed the Axiom of determinacy
Axiom of determinacy
The axiom of determinacy is a possible axiom for set theory introduced by Jan Mycielski and Hugo Steinhaus in 1962. It refers to certain two-person games of length ω with perfect information...
.
Steinhaus was also among the early contributors and founders of the probability theory, which at the time was still in its infancy and was not even considered to be an actual part of mathematics. He provided the first axiom
Axiom (disambiguation)
An axiom is a self-evident proposition in mathematics and epistemology.Axiom may also refer to:- Music :* Axiom , a 1970s Australian rock band featuring Brian Cadd and Glenn Shorrock...
atic measure-theoretic description of coin-tossing, which was to influence the full of axiomatization of probability by the Russian mathematician Andrey Kolmogorov
Andrey Kolmogorov
Andrey Nikolaevich Kolmogorov was a Soviet mathematician, preeminent in the 20th century, who advanced various scientific fields, among them probability theory, topology, intuitionistic logic, turbulence, classical mechanics and computational complexity.-Early life:Kolmogorov was born at Tambov...
a decade later. He was also the first to offer precise definitions of what it means for two events to be "independent", as well as for what it means for a random variable
Random variable
In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...
to be "uniformly distributed
Uniform distribution (continuous)
In probability theory and statistics, the continuous uniform distribution or rectangular distribution is a family of probability distributions such that for each member of the family, all intervals of the same length on the distribution's support are equally probable. The support is defined by...
".
While in hiding during World War II Steinhaus worked on the so-called "cake cutting problem"; how to divide a given resource in a manner which is "fair" according to precisely defined criteria, such as proportionality
Proportional (fair division)
Proportional division or simple fair division is the original and simplest problem in fair division. Fair division problems are also called cake-cutting problems. A proportional division of a cake between N people would ensure each of them got at least 1/N of the cake by their own valuation...
and envy-free
Envy-free
In mathematical sociology and especially game theory, envy-free is a property of certain fair division algorithms for a divisible heterogeneous good over which different players may have different preferences....
.
Steinhaus was also the first person to conjecture the Ham sandwich theorem
Ham sandwich theorem
In measure theory, a branch of mathematics, the ham sandwich theorem, also called the Stone–Tukey theorem after Arthur H. Stone and John Tukey, states that given measurable "objects" in -dimensional space, it is possible to divide all of them in half with a single -dimensional hyperplane...
, and one of the first to propose the method of k-means clustering
Legacy
Polish Mathematical Society
The Polish Mathematical Society began in Kraków, Poland in 1917. It was originally simply called the Mathematical Society. It was officially constituted on April 2, 1919.Hugo Steinhaus, Stefan Banach and Otto Nikodym were among the founders....
. He was a member of PAU (the Polish Academy of Learning
Polish Academy of Learning
The Polish Academy of Arts and Sciences or Polish Academy of Learning , headquartered in Kraków, is one of two institutions in contemporary Poland having the nature of an academy of sciences....
) and PAN (the Polish Academy of Sciences
Polish Academy of Sciences
The Polish Academy of Sciences, headquartered in Warsaw, is one of two Polish institutions having the nature of an academy of sciences.-History:...
), PTM (the Polish Mathematical Society
Polish Mathematical Society
The Polish Mathematical Society began in Kraków, Poland in 1917. It was originally simply called the Mathematical Society. It was officially constituted on April 2, 1919.Hugo Steinhaus, Stefan Banach and Otto Nikodym were among the founders....
), the Wrocławskie Towarzystwo Naukowe (Wrocław Scientific Society) as well as many international scientific societies and science academies.
Steinhaus also published one of the first articles in Fundamenta Mathematicae
Fundamenta Mathematicae
Fundamenta Mathematicae is a scientific journal of mathematics with a special focus on the foundations of mathematics. At present, it concentrates on papers devoted to set theory, mathematical logic, topology and its interactions with algebra, and dynamical systems...
, in 1921. He also co-founded Studia Mathematica
Studia Mathematica
Studia Mathematica is a Polish mathematics journal published since 1929. It was founded by Stefan Banach and Hugo Steinhaus. It accepts papers written in English, French, German, or Russian, primarily in functional analysis and related areas such as probability theory.- External links :* *...
along with Stefan Banach
Stefan Banach
Stefan Banach was a Polish mathematician who worked in interwar Poland and in Soviet Ukraine. He is generally considered to have been one of the 20th century's most important and influential mathematicians....
(1929), and Zastosowania matematyki (Applications of Mathematics, 1953), Colloquium Mathematicum, and Monografie Matematyczne (Mathematical Monographs).
He received honorary doctorate degrees from Warsaw University (1958), Wrocław Medical Academy (1961), Poznań University (1963) and Wrocław University (1965).
Steinhaus had full command of several foreign languages and was interestingly, known, for his aphorism
Aphorism
An aphorism is an original thought, spoken or written in a laconic and memorable form.The term was first used in the Aphorisms of Hippocrates...
s, to the point that a booklet of his most famous ones in Polish
Polish language
Polish is a language of the Lechitic subgroup of West Slavic languages, used throughout Poland and by Polish minorities in other countries...
, French
French language
French is a Romance language spoken as a first language in France, the Romandy region in Switzerland, Wallonia and Brussels in Belgium, Monaco, the regions of Quebec and Acadia in Canada, and by various communities elsewhere. Second-language speakers of French are distributed throughout many parts...
and Latin has been published posthumously.
In 2002, the Polish Academy of Sciences and Wrocław University sponsored "2002, The Year of Hugo Steinhaus", to celebrate his contributions to Polish and world science.
Notable mathematician Mark Kac
Mark Kac
Mark Kac was a Polish mathematician. His main interest was probability theory. His question, "Can one hear the shape of a drum?" set off research into spectral theory, with the idea of understanding the extent to which the spectrum allows one to read back the geometry. Kac completed his Ph.D...
, Steinhaus's student, wrote:
Chief works
- Czym jest, a czym nie jest matematyka (What Mathematics Is, and What It Is Not, 1923).
- Sur le principe de la condensation de la singularités (with Banach, 1927)
- Theorie der Orthogonalreihen (with Stefan KaczmarzStefan KaczmarzStefan Kaczmarz was a Polish mathematician. His Kaczmarz method provided the basis for many modern imaging technologies, including the CAT scan....
, 1935). - Kalejdoskop matematyczny (Mathematical Snapshots, 1939).
- Taksonomia wrocławska (A Wroclaw Taxonomy; with others, 1951).
- Sur la liaison et la division des points d'un ensemble fini (On uniting and separating the points of a finite set, with others, 1951).
- Sto zadań (One Hundred Problems In Elementary Mathematics, 1964).
- Orzeł czy reszka (Heads or Tails, 1961).
- Słownik racjonalny (A Rational Dictionary, 1980).
See also
- Freiling's axiom of symmetryFreiling's axiom of symmetryFreiling's axiom of symmetry is a set-theoretic axiom proposed by Chris Freiling. It is based on intuition of Stuart Davidsonbut the mathematics behind it goes back to Wacław Sierpiński....
- One-seventh area triangleOne-seventh area triangleIn plane geometry, a triangle ABC contains a triangle of one-seventh area of ABC formed as follows: the sides of this triangle lie on lines p, q, r whereA more general result is known as Routh's theorem.-References:...
- Steinhaus–Johnson–Trotter algorithm
- Steinhaus polygon notation
- Steinhaus theorem
Further reading
- Kazimierz KuratowskiKazimierz KuratowskiKazimierz Kuratowski was a Polish mathematician and logician. He was one of the leading representatives of the Warsaw School of Mathematics.-Biography and studies:...
, A Half Century of Polish Mathematics: Remembrances and Reflections, Oxford, Pergamon PressPergamon PressPergamon Press was an Oxford-based publishing house, founded by Paul Rosbaud and Robert Maxwell, which published scientific and medical books and journals. It is now an imprint of Elsevier....
, 1980, ISBN 978-0-08-023046-7, pp. 173–79 et passim. - Hugo Steinhaus, Mathematical Snapshots, second edition, Oxford, 1951, blurbBlurbA blurb is a short summary or some words of praise accompanying a creative work, usually used on books without giving away any details, that is usually referring to the words on the back of the book jacket but also commonly seen on DVD and video cases, web portals, and news websites.- History :The...
.