Hahn was appointed to the teaching staff in Vienna as a privatdozent in 1905 after submitting his habilitation thesis. In session 1905-06 Hahn substituted for Otto Stolz at Innsbruck. He taught at Czernowitz in Austria-Hungary from 1909 to 1914 as an extraordinary professor; Czernowitz was renamed Cernauti when it became part of Romania after World War I, and then Chernovtsy after it became part of the USSR in 1940. Hahn served in the Austro-Hungarian army in World War I and was severely wounded. In 1916 he moved to Bonn where he was appointed as an extraordinary professor teaching there until 1920 having been appointed to a chair in 1917. He returned to a chair in Vienna in 1921. His three most famous students at Vienna were Karl Menger who was awarded his doctorate in 1924, Witold Hurewicz who was awarded his doctorate in 1926, and Kurt Gödel who was awarded his doctorate in 1929.
Menger writes about Hahn's contributions befoe World War I:-
Hahn's first results were contributions to the classical calculus of variations. He then turned to the study of real functions and set functions, especially integrals. He further published a fundamental paper on non-Archimedean systems, and early recognised the significance of Fréchet's abstract spaces. In a paper introducing local connectedness he characterised the sets which a point can traverse in a continuous motion; this is, the continuous images of a time interval or a segment (now often called Peano continua). The paper is a classic of the early set-theoretical geometry.As Menger explains, Hahn was a pioneer in set theory and functional analysis. However to many mathematicians he is best remembered for the Hahn-Banach theorem which we mention again below. He also made important contributions to the calculus of variations, mostly between 1903 and 1913, developing ideas of Weierstrass. He was interested in real analysis, writing on a variety of different topics in that area. He considered properties of the derivative, the representation of functions by definite integrals, semicontinuous functions, and separately continuous functions of several variables. In 1923 he introduced what today is known as the Hahn sequence space. He also wrote two books on real functions, Theorie der reellen Funktionen Ⓣ (1921), and Reelle Funktionen Ⓣ (1932).
Hahn wrote four papers on functional analysis. These include a report on integral equation he wrote in 1911, his modification of Hellinger's theory of invariants of quadratic forms, in which he dispensed with the use of the Hellinger integral, and his work on duality in Banach spaces, culminating with his proof of the Hahn-Banach theorem in 1927. He wrote papers on the theory of curves including one which gave a rigorous proof of the Jordan's theorem for simple closed polygons which he based on Veblen's geometrical axioms. Other papers in this area characterise topological spaces that are continuous images of a line segment and related to this topic is what is now known as the Hahn-Mazurkiewicz theorem. He also studied the theory of ordered abelian groups and ordered fields, initiating the theory in 1907 (see for example ).
Another area on which Hahn did research was measure theory. In this area he studied a construction of the Lebesgue integral as a limit of Riemann sums, an integral proposed by Borel around 1910, and worked on the theory of abstract measures, in particular product measures. Fourier analysis also interested Hahn, and he looked at singular integrals and orthogonal expansions investigating the validity of the Parseval relation in various circumstances. In some papers he looked at generalised harmonic analysis (independently of Norbert Wiener), and he also wrote a short note on Fejér summability.
Despite this wealth of deep papers on a wide range of mathematical topics, many people think of Hahn as a mathematical philosopher. During the 1920s Hahn, together with Philipp Frank, Otto Neurath, Moritz Schlick, founded the Vienna Circle of Logical Positivists, a discussion group of gifted scientists and philosophers who met regularly in Vienna. Menger writes on Hahn's role in the Vienna Circle in the article Hahn and the Vienna Circle.
Von Mises was also a member of the Circle:-
[Hahn] maintains that logic and mathematics are essentially tautological and say nothing about the external world. He admits, however, that the axiom of choice is not a tautology; whether we accept it or deny it, he says, depends on what we want the word "set" to mean. In his essay on the existence of infinity he says that there can be no absolute proof of consistency for an axiomatic system concerned with infinite sets.An example of Hahn's ideas on mathematical intuition are given in extracts we reproduce in the article Hahn's crisis in intuition.
However, Hahn's ideas on mathematical intuition are strongly criticised by Benoit Mandelbrot in an article he wrote in 1982. He writes that the editor of the English translation of Hahn's lectures (published in 1956):-
... states that [Hahn] discusses the cherished faculty of intuition, its role in mathematics, the nest of paradoxes it got us into, and how successful we have been in crawling out. My reaction is very different: Fractal geometry demonstrates that Hahn was dead wrong. Intuition is not invariable but can and must be trained to perform new tasks.Hahn received many honours for his achievements including the Lieban Prize in 1921. He was elected to the Austrian Academy of Sciences (Kaiserliche Akademie der Wissenschaften in Wien), and was made an honorary member of the Calcutta Mathematical Society.
Although Hahn died in 1934, he still managed to be an author of the book Set Functions which was published in 1948, fourteen years after his death. The book was co-authored by Arthur Rosenthal and continued to develop material which first appeared in Saks's Theory of the Integral (1937). Halmos writes:-
The book is divided into an introduction (containing an exposition of the relevant parts of set theory and point set topology) and five chapters, entitled (I) Additive and totally additive set functions, (II) Measure, (III) Measurable functions, (IV) Integration and (V) Differentiation. The authors' treatment is thorough, rigorous and exhaustive; it includes, with minor exceptions, the first four chapters of Saks's book and differs from those chapters mainly in the wealth of detail. The set functions studied are defined either in a perfectly abstract set, or else in metric spaces (and, in particular, Euclidean spaces); the intermediate cases of locally compact spaces, normal spaces and topological groups are not discussed.
Article by: J J O'Connor and E F Robertson
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