Hans Hahn (mathematician)
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Hans Hahn (; 27 September 1879 – 24 July 1934) was an
Austria Austria (, ; german: Österreich ), officially the Republic of Austria (german: Republik Österreich, links=no, ), is a landlocked A landlocked country is a country that does not have territory connected to an ocean or whose coastli ...

Austria
n
mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained ( ...

mathematician
and
philosopher A philosopher is someone who practices philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about reason, Metaphysics, existence, Epistemology, knowledge, Ethics, values, Philosophy of mind, ...

philosopher
who made contributions to
functional analysis 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis. Functional analysis is a branch of mathemat ...
,
topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structu ...

topology
,
set theory Set theory is the branch of that studies , 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 , is mostly concerned with those that are relevant to ...
, the
calculus of variations The calculus of variations is a field of mathematical analysis that uses variations, which are small changes in Function (mathematics), functions and functional (mathematics), functionals, to find maxima and minima of functionals: Map (mathematic ...
,
real analysis 200px, The first four partial sums of the Fourier series for a square wave. Fourier series are an important tool in real analysis.">square_wave.html" ;"title="Fourier series for a square wave">Fourier series for a square wave. Fourier series are a ...

real analysis
, and
order theory Order theory is a branch of mathematics which investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article intro ...
. In philosophy he was among the main
logical positivists Logical positivism, later called logical empiricism, and both of which together are also known as neopositivism, was a movement in Western philosophy Western philosophy refers to the philosophy, philosophical thought and work of the Western worl ...
of the
Vienna Circle The Vienna Circle (german: Wiener Kreis) of Logical Empiricism was a group of philosophers and scientists drawn from the Natural science, natural and Social Sciences, social sciences, logic and mathematics who met regularly from 1924 to 1936 at th ...
.


Biography

Born in
Vienna Vienna ( ; german: Wien ; bar, Wean, label= ) is the , largest city, and one of of . Vienna is Austria's , with about 2 million inhabitants (2.6 million within the , nearly one third of the country's population), and its , , and centre. It ...

Vienna
as the son of a higher government official of the k.k. Telegraphen-Korrespondenz-Bureau (since 1946 named "Austria Presse Agentur"), in 1898 Hahn became a student at the Universität Wien starting with a study of law. In 1899 he switched over to mathematics and spent some time at the universities of
Strasbourg Strasbourg (, , ; german: Straßburg ; gsw, label= , Strossburi , gsw, label= , Strossburig ) is the and largest city of the of and the of the . Located at the with Germany in the historic region of , it is the prefecture of the . I ...

Strasbourg
,
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Munich
and
Göttingen Göttingen (, , ; nds, Chöttingen) is a university city in Lower Saxony Lower Saxony (german: Niedersachsen ; nds, Neddersassen; stq, Läichsaksen) is a German state (''Land'') situated in Northern Germany, northwestern Germany. It is t ...
. In 1902 he took his Ph.D. in Vienna, on the subject "Zur Theorie der zweiten Variation einfacher Integrale". He was a student of
Gustav von Escherich
Gustav von Escherich
. He was appointed to the teaching staff (
Habilitation Habilitation is the procedure to achieve the highest university degree in many European countries in which the candidate fulfills certain criteria set by the university which require excellence in research, teaching, and further education. Its qua ...
) in Vienna in 1905. After 1905/1906 as a stand-in for Otto Stolz at
Innsbruck Innsbruck (; bar, Innschbruck, label=Bavarian language, Austro-Bavarian ) is the capital of Tyrol (state), Tyrol and the List of cities and towns in Austria, fifth-largest city in Austria. On the Inn (river), River Inn, at its junction with the W ...

Innsbruck
and some further years as a
Privatdozent ''Privatdozent'' (for men) or ''Privatdozentin'' (for women), abbreviated PD, P.D. or Priv.-Doz., is an academic title conferred at some European universities, especially in German-speaking countries, to someone who holds certain formal qualifica ...
in Vienna, he was nominated in 1909
Professor extraordinarius Academic ranks in Germany are the titles, relative importance and power of professors, researchers, and administrative personnel held in academia. Overview Appointment grades * (Pay grade: ''W3'' or ''W2'') * (''W3'') * (''W2'') * (''W2'', onl ...
in
Czernowitz Chernivtsi ( uk, Чернівці́} ; ro, Cernăuți; see also ) is a city in . It is situated on the upper course of the , and is the of (), which includes the Ukrainian part of . Chernivtsi is also the administrative center of and host ...

Czernowitz
, at that time a town within the empire of Austria. After joining the Austrian army in 1915, he was badly wounded in 1916 and became again
Professor extraordinarius Academic ranks in Germany are the titles, relative importance and power of professors, researchers, and administrative personnel held in academia. Overview Appointment grades * (Pay grade: ''W3'' or ''W2'') * (''W3'') * (''W2'') * (''W2'', onl ...
, now in
Bonn The Federal city The term federal city is a title for certain cities in Germany ) , image_map = , map_caption = , map_width = 250px , capital = Berlin , coordinates = , largest_city = capital , languages_type = Official langua ...

Bonn
. In 1917 he was nominated a regular
Professor Professor (commonly abbreviated as Prof.) is an academic An academy (Attic Greek: Ἀκαδήμεια; Koine Greek Ἀκαδημία) is an institution of secondary education, secondary or tertiary education, tertiary higher education, highe ...

Professor
there and in 1921 he returned to Vienna with this title, where he stayed until his rather early death in 1934 at the age of 54, following cancer surgery. He had married Eleonore ("Lilly") Minor in 1909 and they had a daughter, Nora (born 1910). He was also interested in philosophy, and was part of a discussion group concerning
Mach
Mach
's
positivism Positivism is a philosophical theory A philosophical theory or philosophical position''Dictionary of Theories'', Jennifer Bothamley is a view that attempts to explain or account for a particular problem in philosophy Philosophy (from ...
with
Otto Neurath Otto Karl Wilhelm Neurath (; 10 December 1882 – 22 December 1945) was an Austria Austria (, ; german: Österreich ), officially the Republic of Austria (german: Republik Österreich, links=no, ), is a landlocked Eastern Alps, East Alp ...

Otto Neurath
(who had married Hahn’s sister Olga Hahn-Neurath in 1912), and
Phillip FrankPhilipp Frank (March 20, 1884, Vienna, Austria-Hungary – July 21, 1966, Cambridge, Massachusetts, Cambridge, Massachusetts, United States) was a physicist, mathematician and also a philosopher during the first half of the 20th century. He was a log ...
prior to the First World War. In 1922, he helped arrange Moritz Schlick, Moritz Schlick's entry into the group, which led to the founding of the
Vienna Circle The Vienna Circle (german: Wiener Kreis) of Logical Empiricism was a group of philosophers and scientists drawn from the Natural science, natural and Social Sciences, social sciences, logic and mathematics who met regularly from 1924 to 1936 at th ...
, the group that was at the center of logical positivist thought in the 1920s. His most famous student was Kurt Gödel, whose Ph.D. thesis was completed in 1929. After Anschluss the fact that Hans Hahn had been of Jewish origin caused Gödel's difficulties with getting a position at The University of Vienna. Within the Vienna Circle, Hahn was also known (and controversial) for using his mathematical and philosophical work to study Extrasensory perception, psychic phenomena; according to Karl Menger he sometimes openly advocated further research into extrasensory perception while lecturing. Hahn's contributions to mathematics include the Hahn–Banach theorem and (independently of Stefan Banach, Banach and Hugo Steinhaus, Steinhaus) the uniform boundedness principle. Other theorems include: * the Hahn decomposition theorem; * the Hahn embedding theorem; * the Hahn–Kolmogorov theorem; * the Space-filling curve, Hahn–Mazurkiewicz theorem; * the Vitali–Hahn–Saks theorem. Hahn authored the book : according to Arthur Rosenthal, "''... (it) formed a great advance in the Theory of Real functions and had a great influence on the further development of this theory''". He was also a co-author of the book ''Set Functions'', published in 1948 by Arthur Rosenthal, fourteen years after his death in Vienna in 1934. In 1921 he received the Lieben Prize#Richard Lieben Prize, Richard Lieben Prize. In 1926 he was the president of the German Mathematical Society. In 1928 he was an Invited Speaker at the International Congress of Mathematicians in Bologna.


Publications

All his mathematical and philosophical works, except all books and all but one of his book reviews, are published in the three volumes , and of his "''Collected papers''". * (freely available at the Internet Archive). * * * *. *.


See also

*Domain (mathematical analysis) *Mathematical analysis


Notes


External links

* * {{DEFAULTSORT:Hahn, Hans 1879 births 1934 deaths Austrian Jews Scientists from Vienna 20th-century mathematicians Vienna Circle Topologists Set theorists Austrian agnostics Austrian mathematicians Jewish agnostics Jewish scientists Austro-Hungarian mathematicians Chernivtsi University faculty