Hans Hahn (; ; 27 September 1879 – 24 July 1934) was an Austrian mathematician and philosopher who made contributions to

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 (for example, Inner product space#Definition, inner product, Norm (mathematics ...

topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...

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 ...

, the

calculus of variations The calculus of variations (or variational calculus) 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 f ...

real analysis In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include co ...

, and

order theory Order theory is a branch of mathematics that 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 intr ...

. In philosophy he was among the main logical positivists of the Vienna Circle.

Biography

Born in

Vienna Vienna ( ; ; ) is the capital city, capital, List of largest cities in Austria, most populous city, and one of Federal states of Austria, nine federal states of Austria. It is Austria's primate city, with just over two million inhabitants. ...

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 ( , ; ; ) is the Prefectures in France, prefecture and largest city of the Grand Est Regions of France, region of Geography of France, eastern France, in the historic region of Alsace. It is the prefecture of the Bas-Rhin Departmen ...

Munich Munich is the capital and most populous city of Bavaria, Germany. As of 30 November 2024, its population was 1,604,384, making it the third-largest city in Germany after Berlin and Hamburg. Munich is the largest city in Germany that is no ...

and

Göttingen Göttingen (, ; ; ) 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. According to the 2022 German census, 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. He was appointed to the teaching staff ( Habilitation) in Vienna in 1905. After 1905/1906 as a stand-in for Otto Stolz at the

University of Innsbruck The University of Innsbruck (; ) is a public research university in Innsbruck, the capital of the Austrian federal state of Tyrol (state), Tyrol, founded on October 15, 1669. It is the largest education facility in the Austrian States of Austria, ...

and some further years as a Privatdozent in Vienna, he was nominated in 1909 Professor extraordinarius in 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, now in

Bonn Bonn () is a federal city in the German state of North Rhine-Westphalia, located on the banks of the Rhine. With a population exceeding 300,000, it lies about south-southeast of Cologne, in the southernmost part of the Rhine-Ruhr region. This ...

. In 1917 he was nominated a regular

Professor Professor (commonly abbreviated as Prof.) is an Academy, academic rank at university, universities and other tertiary education, post-secondary education and research institutions in most countries. Literally, ''professor'' derives from Latin ...

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 The Mach number (M or Ma), often only Mach, (; ) is a dimensionless quantity in fluid dynamics representing the ratio of flow velocity past a Boundary (thermodynamic), boundary to the local speed of sound. It is named after the Austrian physi ...

positivism Positivism is a philosophical school that holds that all genuine knowledge is either true by definition or positivemeaning '' a posteriori'' facts derived by reason and logic from sensory experience.John J. Macionis, Linda M. Gerber, ''Soci ...

with Otto Neurath (who had married Hahn’s sister Olga Hahn-Neurath in 1912), and Phillip Frank prior to 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 helped arrange Moritz Schlick's entry into the group, which led to the founding of the Vienna Circle, the group that was at the center of

logical positivist Logical positivism, also known as logical empiricism or neo-positivism, was a philosophical movement, in the empiricist tradition, that sought to formulate a scientific philosophy in which philosophical discourse would be, in the perception of ...

thought in the 1920s. His most famous student was

Kurt Gödel Kurt Friedrich Gödel ( ; ; April 28, 1906 – January 14, 1978) was a logician, mathematician, and philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel profoundly ...

, whose Ph.D. thesis was completed in 1929. After

Anschluss The (, or , ), also known as the (, ), was the annexation of the Federal State of Austria into Nazi Germany on 12 March 1938. The idea of an (a united Austria and Germany that would form a "German Question, Greater Germany") arose after t ...

the fact that Hans Hahn had been of partial

Jewish Jews (, , ), or the Jewish people, are an ethnoreligious group and nation, originating from the Israelites of History of ancient Israel and Judah, ancient Israel and Judah. They also traditionally adhere to Judaism. Jewish ethnicity, rel ...

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 psychic phenomena; according to Karl Menger he sometimes openly advocated further research into extrasensory perception while lecturing. Politically Hahn was a socialist and was chairman of the Association of Socialist University Teachers. He contributed to the Social Democrat magazine '' Der Kampf''. Hahn's contributions to mathematics include the

Hahn–Banach theorem In functional analysis, the Hahn–Banach theorem is a central result that allows the extension of bounded linear functionals defined on a vector subspace of some vector space to the whole space. The theorem also shows that there are sufficient ...

and (independently of Banach and Steinhaus) the uniform boundedness principle. Other theorems include: * the Hahn decomposition theorem; * the Hahn embedding theorem; * the Hahn–Kolmogorov theorem; * the 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 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 The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU). The Fields Medals, the IMU Abacus Medal (known before ...

in Bologna. Hahn died in 1934 following surgery for a newly diagnosed cancer. He was only 54 and his death came as a shock to friends and colleagues.

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 The Internet Archive is an American 501(c)(3) organization, non-profit organization founded in 1996 by Brewster Kahle that runs a digital library website, archive.org. It provides free access to collections of digitized media including web ...

). * * * *. *.

Notes

External links

* * {{DEFAULTSORT:Hahn, Hans 1879 births 1934 deaths Austrian Jews Scientists from Vienna 20th-century Austrian mathematicians Vienna Circle Topologists Set theorists Austrian agnostics Austrian socialists Jewish agnostics Jewish scientists Mathematicians from Austria-Hungary Academic staff of Chernivtsi University Functional analysts Presidents of the German Mathematical Society

Biography

Publications

See also

Notes

External links