Felix Hausdorff ( , ; November 8, 1868 – January 26, 1942) was a German
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, structure, space, models, and change.
History
On ...
who is considered to be one of the founders of modern
topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
and who contributed significantly to
set theory
Set theory is the branch of mathematical logic that studies 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 mathematics, is mostly conce ...
,
descriptive set theory,
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 mass and probability of events. These seemingly distinct concepts have many simil ...
, and
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 (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
.
Life became difficult for Hausdorff and his family after
Kristallnacht
() or the Night of Broken Glass, also called the November pogrom(s) (german: Novemberpogrome, ), was a pogrom against Jews carried out by the Nazi Party's (SA) paramilitary and (SS) paramilitary forces along with some participation from ...
in 1938. The next year he initiated efforts to emigrate to the United States, but was unable to make arrangements to receive a research fellowship. On 26 January 1942, Felix Hausdorff, along with his wife and his sister-in-law, died by suicide by taking an overdose of
veronal
Barbital (or barbitone), marketed under the brand names Veronal for the pure acid and Medinal for the sodium salt, was the first commercially available barbiturate. It was used as a sleeping aid (hypnotic) from 1903 until the mid-1950s. The chemic ...
, rather than comply with German orders to move to the Endenich camp, and there suffer the likely implications, about which he held no illusions.
Life
Childhood and youth
Hausdorff's father, the
Jewish
Jews ( he, יְהוּדִים, , ) or Jewish people are an ethnoreligious group and nation originating from the Israelites Israelite origins and kingdom: "The first act in the long drama of Jewish history is the age of the Israelites""The ...
merchant Louis Hausdorff (1843–1896), moved with his young family to
Leipzig
Leipzig ( , ; Upper Saxon: ) is the most populous city in the German state of Saxony. Leipzig's population of 605,407 inhabitants (1.1 million in the larger urban zone) as of 2021 places the city as Germany's eighth most populous, as wel ...
in the autumn of 1870, and over time worked at various companies, including a linen-and cotton goods factory. He was an educated man and had become a
Morenu
''Morenu'' ( he, מורנו, lit. "our teacher") is a customary religious title for a Jewish man with high religious education. This term has been used since the mid-14th Century and has a Talmudic origin. The title is generally considered a prer ...
at the age of 14. He wrote several treatises, including a long work on the
Aramaic
The Aramaic languages, short Aramaic ( syc, ܐܪܡܝܐ, Arāmāyā; oar, 𐤀𐤓𐤌𐤉𐤀; arc, 𐡀𐡓𐡌𐡉𐡀; tmr, אֲרָמִית), are a language family containing many varieties (languages and dialects) that originated in ...
translations of the Bible from the perspective of
Talmud
The Talmud (; he, , Talmūḏ) is the central text of Rabbinic Judaism and the primary source of Jewish religious law (''halakha'') and Jewish theology. Until the advent of modernity, in nearly all Jewish communities, the Talmud was the cente ...
ic law.
Hausdorff's mother, Hedwig (1848–1902), who is also referred to in various documents as Johanna, came from the Jewish Tietz family. From another branch of this family came
Hermann Tietz
Hermann Tietz (born 29 April 1837, in Birnbaum an der Warthe near Posen (today Międzychód, Poland), died on 3 May 1907 in Berlin) was a German-Jewish merchant, co-founder of the Tietz Department Store. He was buried in the Weißensee Cemet ...
, founder of the first department store, and later co-owner of the department store chain called "Hermann Tietz". During the period of Nazi dictatorship the name was "Aryanised" to
Hertie.
From 1878 to 1887 Felix Hausdorff attended the Nicolai School in Leipzig, a facility that had a reputation as a hotbed of humanistic education. He was an excellent student, class leader for many years and often recited self-written Latin or German poems at school celebrations.
In his later years of high school, choosing a main subject of study was not easy for Hausdorff. Magda Dierkesmann, who was often a guest in the home of Hausdorff in the years 1926–1932, reported in 1967 that:
He decided to study the natural sciences, and in his graduating class of 1887 he was the only one who achieved the highest possible grade.
Degree, doctorate and Habilitation
From 1887 to 1891 Hausdorff studied
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
and
astronom
An astronomer is a scientist in the field of astronomy who focuses their studies on a specific question or field outside the scope of Earth. They observe astronomical objects such as stars, planets, moons, comets and galaxies – in either obse ...
y, mainly in his native city of Leipzig, interrupted by one semester in
Freiburg
Freiburg im Breisgau (; abbreviated as Freiburg i. Br. or Freiburg i. B.; Low Alemannic: ''Friburg im Brisgau''), commonly referred to as Freiburg, is an independent city in Baden-Württemberg, Germany. With a population of about 230,000 (as o ...
(summer 1888) and
Berlin
Berlin ( , ) is the capital and largest city of Germany by both area and population. Its 3.7 million inhabitants make it the European Union's most populous city, according to population within city limits. One of Germany's sixteen constitue ...
(winter 1888/1889). Surviving testimony from other students depict him as an extremely versatile and interested young man, who, in addition to the mathematical and astronomical lectures, attended lectures in
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
,
chemistry
Chemistry is the science, scientific study of the properties and behavior of matter. It is a natural science that covers the Chemical element, elements that make up matter to the chemical compound, compounds made of atoms, molecules and ions ...
and
geography
Geography (from Greek: , ''geographia''. Combination of Greek words ‘Geo’ (The Earth) and ‘Graphien’ (to describe), literally "earth description") is a field of science devoted to the study of the lands, features, inhabitants, and ...
, and also lectures on
philosophy
Philosophy (from , ) is the systematized study of general and fundamental questions, such as those about existence, reason, knowledge, values, mind, and language. Such questions are often posed as problems to be studied or resolved. Some ...
and
history of philosophy
Philosophy (from , ) is the systematized study of general and fundamental questions, such as those about existence, reason, knowledge, values, mind, and language. Such questions are often posed as problems to be studied or resolved. Some ...
, as well as on issues of
language
Language is a structured system of communication. The structure of a language is its grammar and the free components are its vocabulary. Languages are the primary means by which humans communicate, and may be conveyed through a variety of met ...
,
literature
Literature is any collection of written work, but it is also used more narrowly for writings specifically considered to be an art form, especially prose fiction, drama, and poetry. In recent centuries, the definition has expanded to include ...
and
social sciences
Social science is one of the branches of science, devoted to the study of societies and the relationships among individuals within those societies. The term was formerly used to refer to the field of sociology, the original "science of soci ...
. In Leipzig he attended lectures on the
history of music
Although definitions of music vary wildly throughout the world, every known culture partakes in it, and it is thus considered a cultural universal. The origins of music remain highly contentious; commentators often relate it to the origin of ...
from musicologist
Oscar Paul
Oscar Paul (8 April 183618 April 1898) was a German musicologist and a music writer, critic, and teacher.
Biography
Oscar Paul was born in Freiwaldau in Silesia (now Gozdnica in the Województwo lubuskie of the Poland). He studied at Görlitz ...
. His early love of music lasted a lifetime; in Hausdorff's home he held impressive musical evenings with the landlord at the piano, according to witness statements made by various participants. Even as a student in Leipzig, he was an admirer and connoisseur of the music of
Richard Wagner
Wilhelm Richard Wagner ( ; ; 22 May 181313 February 1883) was a German composer, theatre director, polemicist, and conductor who is chiefly known for his operas (or, as some of his mature works were later known, "music dramas"). Unlike most op ...
.
In later semesters of his studies, Hausdorff was close to
Heinrich Bruns
Ernst Heinrich Bruns (4 September 1848 – 23 September 1919) was a German mathematician and astronomer, who also contributed to the development of the field of theoretical geodesy.
Early life
Heinrich Bruns was born on 4 September 1848 in Be ...
(1848–1919).
Bruns was professor of astronomy and director of the observatory at the University of Leipzig. Under his supervision, Hausdorff graduated in 1891 with a work on the theory of astronomical refraction of light in the atmosphere. Two publications on the same subject followed, and in 1895 his
Habilitation
Habilitation is the highest university degree, or the procedure by which it is achieved, in many European countries. The candidate fulfills a university's set criteria of excellence in research, teaching and further education, usually including a ...
also followed with a thesis on the absorbance of light in the atmosphere. These early astronomical works of Hausdorff, despite their excellent mathematical formulation, were ultimately of little importance to the scientific community.
For one, the underlying idea of Bruns was later shown to not be viable (there was a need for refraction observations near the astronomical horizon, and as Julius Bauschinger would show, this could not be obtained with the required accuracy). And further, the progress in the direct measurement of atmospheric data (from
weather balloon ascents) has since made the painstaking accuracy of this data from refraction observations unnecessary. In the time between defending his PhD and his Habilitation, Hausdorff completed his yearlong military requirement, and worked for two years as a
human computer
The term "computer", in use from the early 17th century (the first known written reference dates from 1613), meant "one who computes": a person performing mathematical calculations, before electronic computers became commercially available. Ala ...
at the
observatory
An observatory is a location used for observing terrestrial, marine, or celestial events. Astronomy, climatology/meteorology, geophysical, oceanography and volcanology are examples of disciplines for which observatories have been constructed. His ...
in Leipzig.
Lecturer in Leipzig
After his Habilitation, Hausdorff became a lecturer at the University of Leipzig where he began extensive teaching in a variety of mathematical areas. In addition to teaching and research in mathematics, he also pursued his literary and philosophical inclinations. A man of varied interests, he often associated with a number of famous writers, artists and publishers such as
Hermann Conradi Hermann or Herrmann may refer to:
* Hermann (name), list of people with this name
* Arminius, chieftain of the Germanic Cherusci tribe in the 1st century, known as Hermann in the German language
* Éditions Hermann, French publisher
* Hermann, Miss ...
,
Richard Dehmel
Richard Fedor Leopold Dehmel (18 November 1863 – 8 February 1920) was a German poet and writer.
Life
A forester's son, Richard Dehmel was born in Hermsdorf near Wendisch Buchholz (now a part of Münchehofe) in the Brandenburg Province, Ki ...
,
Otto Erich Hartleben
Otto Erich Hartleben (3 June 1864 – in Clausthal; 11 February 1905 in Salò) was a German poet and dramatist from Clausthal, known for his translation of ''Pierrot Lunaire''.
Childhood, Education and Marriage
Orphaned as a child, Hartlebe ...
,
Gustav Kirstein
Gustav Kirstein (born 24 February 1870 in Berlin; died 14 February 1934 in Leipzig) was a German publisher, writer, and art collector of Jewish descent.
Life
Kirstein was the son of a medical doctor. He first studied pharmacy, graduated, worked ...
,
Max Klinger,
Max Reger
Johann Baptist Joseph Maximilian Reger (19 March 187311 May 1916) was a German composer, pianist, organist, conductor, and academic teacher. He worked as a concert pianist, as a musical director at the Paulinerkirche, Leipzig, Leipzig University ...
and
Frank Wedekind
Benjamin Franklin Wedekind (July 24, 1864 – March 9, 1918) was a German playwright. His work, which often criticizes bourgeois attitudes (particularly towards sex), is considered to anticipate expressionism and was influential in the de ...
. The years of 1897 to 1904 mark the high point of his literary and philosophical creativity, during which time 18 of his 22 pseudonymous works were published, including a book of poetry, a play, an epistemological book and a volume of
aphorism
An aphorism (from Greek ἀφορισμός: ''aphorismos'', denoting 'delimitation', 'distinction', and 'definition') is a concise, terse, laconic, or memorable expression of a general truth or principle. Aphorisms are often handed down by tra ...
s.
In 1899 Hausdorff married Charlotte Goldschmidt, the daughter of Jewish doctor Siegismund Goldschmidt. Her stepmother was the famous suffragist and preschool teacher
Henriette Goldschmidt
Henriette Goldschmidt (1825–1920) was a History of the Jews in Germany, German Jewish Feminism in Germany, feminist, pedagogist and social worker. She was one of the founders of the German Association of Female Citizens, German Women's Associati ...
. Hausdorff's only child, his daughter Lenore (Nora), was born in 1900; she survived the era of National Socialism and enjoyed a long life, dying in Bonn in 1991.
First professorship
In December 1901 Hausdorff was appointed as adjunct associate professor at the University of Leipzig. An often-repeated
factoid
A factoid is either an invented or assumed statement presented as a fact, ''or'' a true but brief or trivial item of news or information.
The term was coined in 1973 by American writer Norman Mailer to mean a piece of information that becomes ac ...
, that Hausdorff got a call from
Göttingen
Göttingen (, , ; nds, Chöttingen) 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. At the end of 2019, t ...
and rejected it, cannot be verified and is most likely wrong. After considering Hausdorff's application to Leipzig, the Dean Kirchner felt compelled to make the following addition to the very positive vote from his colleagues, written by Heinrich Bruns:
This quote emphasizes the undisguised
anti-Semitism
Antisemitism (also spelled anti-semitism or anti-Semitism) is hostility to, prejudice towards, or discrimination against Jews. A person who holds such positions is called an antisemite. Antisemitism is considered to be a form of racism.
Antis ...
present, which especially took a sharp upturn throughout the German Reich after the
stock market crash of 1873. Leipzig was a focus of anti-Semitic sentiment, especially among the student body, which may well be the reason that Hausdorff did not feel at ease in Leipzig. Another contributing factor may also have been the stresses due to the hierarchical posturing of the Leipzig professors.
After his Habilitation, Hausdorff wrote other works on
optics
Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultraviole ...
, on
non-Euclidean geometry
In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geo ...
, and on
hypercomplex number
In mathematics, hypercomplex number is a traditional term for an element of a finite-dimensional unital algebra over the field of real numbers.
The study of hypercomplex numbers in the late 19th century forms the basis of modern group represent ...
systems, as well as two papers on
probability theory
Probability theory 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 expressing it through a set o ...
. However, his main area of work soon became set theory, especially the theory of
ordered set
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary r ...
s. Initially, it was only out of philosophical interest that Hausdorff began to study
Georg Cantor
Georg Ferdinand Ludwig Philipp Cantor ( , ; – January 6, 1918) was a German mathematician. He played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of ...
's work, beginning around 1897, but already in 1901 Hausdorff began lecturing on set theory. His was one of the first ever lectures on set theory; only
Ernst Zermelo
Ernst Friedrich Ferdinand Zermelo (, ; 27 July 187121 May 1953) was a German logician and mathematician, whose work has major implications for the foundations of mathematics. He is known for his role in developing Zermelo–Fraenkel axiomatic se ...
's lectures in Göttingen College during the winter of 1900/1901 were earlier. That same year, he published his first paper on order types in which he examined a generalization of
well-ordering
In mathematics, a well-order (or well-ordering or well-order relation) on a set ''S'' is a total order on ''S'' with the property that every non-empty subset of ''S'' has a least element in this ordering. The set ''S'' together with the well-o ...
s called
graded order types
Grade most commonly refers to:
* Grade (education), a measurement of a student's performance
* Grade, the number of the year a student has reached in a given educational stage
* Grade (slope), the steepness of a slope
Grade or grading may also r ...
, where a
linear order
In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X:
# a \leq a ( reflexive ...
is graded if no two of its segments share the same
order type
In mathematics, especially in set theory, two ordered sets and are said to have the same order type if they are order isomorphic, that is, if there exists a bijection (each element pairs with exactly one in the other set) f\colon X \to Y such ...
. He generalized the
Cantor–Bernstein theorem
In set theory and order theory, the Cantor–Bernstein theorem states that the cardinality of the second type class, the class of Countable set, countable order types, equals the cardinality of the continuum. It was used by Felix Hausdorff and nam ...
, which said the collection of countable order types has the
cardinality of the continuum
In set theory, the cardinality of the continuum is the cardinality or "size" of the set of real numbers \mathbb R, sometimes called the continuum. It is an infinite cardinal number and is denoted by \mathfrak c (lowercase fraktur "c") or , \mathb ...
and showed that the collection of all graded types of an
idempotent
Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of pl ...
cardinality has a cardinality of 2
.
For the summer semester of 1910 Hausdorff was appointed as professor to the
University of Bonn
The Rhenish Friedrich Wilhelm University of Bonn (german: Rheinische Friedrich-Wilhelms-Universität Bonn) is a public research university located in Bonn, North Rhine-Westphalia, Germany. It was founded in its present form as the ( en, Rhine U ...
. There he began a lecture series on set theory, which he substantially revised and expanded for the summer semester of 1912.
In the summer of 1912 he also began work on his magnum opus, the book ''Basics of set theory''. It was completed in
Greifswald
Greifswald (), officially the University and Hanseatic City of Greifswald (german: Universitäts- und Hansestadt Greifswald, Low German: ''Griepswoold'') is the fourth-largest city in the German state of Mecklenburg-Western Pomerania after Rostoc ...
, where Hausdorff had been appointed for the summer semester as full professor in 1913, and was released in April 1914.
The
University of Greifswald
The University of Greifswald (; german: Universität Greifswald), formerly also known as “Ernst-Moritz-Arndt University of Greifswald“, is a public research university located in Greifswald, Germany, in the state of Mecklenburg-Western Pom ...
was the smallest of the Prussian universities. The mathematical institute there was also small; during the summer of 1916 and the winter of 1916/17, Hausdorff was the only mathematician in Greifswald. This meant that he was almost fully occupied in teaching basic courses. It was thus a substantial improvement for his academic career when Hausdorff was appointed in 1921 to Bonn. There he was free to teach about wider ranges of topics, and often lectured on his latest research. He gave a particularly noteworthy lecture on probability theory (NL Hausdorff: Capsule 21: Fasz 64) in the summer semester of 1923, in which he grounded the theory of probability in measure-theoretic axiomatic theory, ten years before
A. N. Kolmogorov's "Basic concepts of probability theory" (reprinted in full in the collected works, Volume V). In Bonn, Hausdorff was friends and colleagues with
Eduard Study
Eduard Study ( ), more properly Christian Hugo Eduard Study (March 23, 1862 – January 6, 1930), was a German mathematician known for work on invariant theory of ternary forms (1889) and for the study of spherical trigonometry. He is also known f ...
, and later with
Otto Toeplitz
Otto Toeplitz (1 August 1881 – 15 February 1940) was a German mathematician working in functional analysis., reprinted in
Life and work
Toeplitz was born to a Jewish family of mathematicians. Both his father and grandfather were ''Gymnasiu ...
, who were both
outstanding mathematicians.
Under the Nazi dictatorship and suicide
After the takeover by the
National Socialist
Nazism ( ; german: Nazismus), the common name in English for National Socialism (german: Nationalsozialismus, ), is the far-right totalitarian political ideology and practices associated with Adolf Hitler and the Nazi Party (NSDAP) in Na ...
party,
anti-Semitism
Antisemitism (also spelled anti-semitism or anti-Semitism) is hostility to, prejudice towards, or discrimination against Jews. A person who holds such positions is called an antisemite. Antisemitism is considered to be a form of racism.
Antis ...
became state doctrine. Hausdorff was not initially concerned by the "
Law for the Restoration of the Professional Civil Service
The Law for the Restoration of the Professional Hitler Service (german: Gesetz zur Wiederherstellung des Berufsbeamtentums, shortened to ''Berufsbeamtengesetz''), also known as Civil Service Law, Civil Service Restoration Act, and Law to Re-es ...
", adopted in 1933, because he had been a German public servant since before 1914. However, he was not completely spared, as one of his lectures was interrupted by National Socialist student officials. In the winter semester of 1934/1935, there was a working session of the National Socialist German Student Union (NSDStB) at the University of Bonn, which chose "Race and Ethnicity" as their theme for the semester. Hausdorff cancelled his 1934/1935 winter semester Calculus III course on 20 November, and it is assumed that the choice of theme was related to the cancellation of Hausdorff's class, since in his long career as a university lecturer he had always taught his courses through to their end.
On March 31, 1935, after some back and forth, Hausdorff was finally given emeritus status. No words of thanks were given for his 40 years of successful work in the German higher education system.
His
academic legacy shows that Hausdorff was still working mathematically during these increasingly difficult times, and continued to follow current developments of interest.
He wrote, in addition to the expanded edition of his work on set theory, seven works on topology and descriptive set theory. These were published in Polish magazines: one in ''
Studia Mathematica'', the others in ''
Fundamenta Mathematicae
''Fundamenta Mathematicae'' is a peer-reviewed scientific journal of mathematics with a special focus on the foundations of mathematics, concentrating on set theory, mathematical logic, topology and its interactions with algebra, and dynamical syst ...
''.
He was supported at this time by
Erich Bessel-Hagen
Erich Bessel-Hagen (12 September 1898 in Charlottenburg – 29 March 1946 in Bonn) was a German mathematician and a historian of mathematics.
Erich Paul Werner Bessel-Hagen was born in 1898 in Charlottenburg, a suburb, later a district in Berlin. ...
, a loyal friend to the Hausdorff family who obtained books and magazines from the academic library, which Hausdorff was no longer allowed to enter.
A great deal is known about the humiliations to which Hausdorff and his family especially were exposed to after
Kristallnacht
() or the Night of Broken Glass, also called the November pogrom(s) (german: Novemberpogrome, ), was a pogrom against Jews carried out by the Nazi Party's (SA) paramilitary and (SS) paramilitary forces along with some participation from ...
in 1938. There are many sources, including the letters of Bessel-Hagen.
In 1939, Hausdorff asked the mathematician
Richard Courant
Richard Courant (January 8, 1888 – January 27, 1972) was a German American mathematician. He is best known by the general public for the book '' What is Mathematics?'', co-written with Herbert Robbins. His research focused on the areas of real ...
, in vain, for a research fellowship to be able to emigrate into the USA.
In mid-1941, the Bonn Jews began to be deported to the "Monastery for Eternal Adoration" in
Endenich
Endenich is a neighborhood in the western part of Bonn, Germany. Before 1904 it was an independent municipality. The village of Endenich was founded in the 8th century, and was first mentioned in 804 as ''Antiniche''. Today, about 12,000 people liv ...
,
Bonn
The federal city of Bonn ( lat, Bonna) is a city on the banks of the Rhine in the German state of North Rhine-Westphalia, with a population of over 300,000. About south-southeast of Cologne, Bonn is in the southernmost part of the Rhine-Ruhr r ...
, from which the nuns had been expelled. Transports to death camps in the east occurred later. After Hausdorff, his wife, and his wife's sister, Edith Pappenheim (who was living with them), were ordered in January 1942 to move to the Endenich camp, the three died by suicide on 26 January 1942 by taking an overdose of
veronal
Barbital (or barbitone), marketed under the brand names Veronal for the pure acid and Medinal for the sodium salt, was the first commercially available barbiturate. It was used as a sleeping aid (hypnotic) from 1903 until the mid-1950s. The chemic ...
. Their final resting place is located on the
Poppelsdorfer cemetery in Bonn. In the time between their placement in temporary camps and his suicide, he gave his handwritten ''
Nachlass
''Nachlass'' (, older spelling ''Nachlaß'') is a German word, used in academia to describe the collection of manuscripts, notes, correspondence, and so on left behind when a scholar dies. The word is a compound in German: ''nach'' means "after ...
'' to the Egyptologist and presbyter
Hans Bonnet, who saved as much of them as possible, even despite the destruction of his house by a bomb.
Some of his fellow Jews may have had illusions about the camp Endenich, but not Hausdorff. In the estate of Bessel-Hagen, E. Neuenschwander discovered the farewell letter that Hausdorff wrote to his lawyer Hans Wollstein, who was also Jewish. Here is the beginning and end of the letter:
After thanking friends and, in great composure, expressing his last wishes regarding his funeral and his will, Hausdorff writes:
Unfortunately, this desire was not fulfilled. Hausdorff's lawyer, Wollstein, was murdered in
Auschwitz
Auschwitz concentration camp ( (); also or ) was a complex of over 40 concentration and extermination camps operated by Nazi Germany in occupied Poland (in a portion annexed into Germany in 1939) during World War II and the Holocaust. It con ...
.
Hausdorff's library was sold by his son-in-law and sole heir, Arthur König. The portions of Hausdorff's ''
Nachlass
''Nachlass'' (, older spelling ''Nachlaß'') is a German word, used in academia to describe the collection of manuscripts, notes, correspondence, and so on left behind when a scholar dies. The word is a compound in German: ''nach'' means "after ...
'' which could be saved by Hans Bonnet are now in the University and State Library of Bonn. The ''Nachlass'' is catalogued.
Work and reception
Hausdorff as philosopher and writer (Paul Mongré)
Hausdorff's volume of aphorisms, published in 1897, was his first work published under the pseudonym Paul Mongré. It is entitled ''Sant' Ilario: Thoughts from the landscape of Zarathustra''. The subtitle plays first on the fact that Hausdorff had completed his book during a recovery stay on the Ligurian coast by Genoa and that in this same area, Friedrich Nietzsche wrote the first two parts of ''Thus Spoke Zarathustra''; he also alludes to his spiritual closeness to Nietzsche. In an article on Sant' Ilario in the weekly paper
Die Zukunft
''Die Zukunft'' ("''The Future''") has been the name of three newspapers.
''Die Zukunft'' was a German social-democratic weekly (1892–1923) founded and edited by Maximilian Harden. It published allegations of homosexuality of Philipp, Prince o ...
, Hausdorff acknowledged in
expressis verbis his debt to Nietzsche.
Hausdorff was not trying to copy or even exceed Nietzsche. "Of Nietzsche imitation no trace", says a contemporary review. He follows Nietzsche in an attempt to liberate individual thinking, to take the liberty of questioning outdated standards. Hausdorff maintained critical distance to the late works of Nietzsche. In his essay on the book
''The Will to Power'' compiled from notes left in the Nietzsche Archive he says:
His critical standard he took from Nietzsche himself,
In 1898—also under the pseudonym Paul Mongré—Hausdorff published an epistemological experiment titled ''Chaos in cosmic selection''. The critique of metaphysics put forward in this book had its starting point in Hausdorff's confrontation with Nietzsche's idea of eternal recurrence. Ultimately, it is about destroying ''any'' kind of metaphysics. Of the world itself, of the ''transcendent core of the world''—as Hausdorff puts it—we know nothing and we can know nothing. We must assume "the world itself" as undetermined and undeterminable, as mere chaos. The world of our experience, our cosmos, is the result of the selections that we have made and will always instinctively make according to our capacity for understanding. Seen from that chaos, all other frameworks, other cosmos, are conceivable. That is to say, from the world of our cosmos, one cannot draw any conclusions about the transcendent world.
In 1904, in the magazine The New Rundschau, Hausdorff's play appeared, the one-act play ''The doctor in his honor''. It is a crude satire on the duel and on the traditional concepts of honor and nobility of the Prussian officer corps, which in the developing bourgeois society were increasingly anachronistic. ''The doctor in his honor'' was Hausdorff's most popular literary work. In 1914–1918 there were numerous performances in more than thirty cities. Hausdorff later wrote an epilogue to the play, but it was not performed at that time. Only in 2006 did this epilogue have its premier at the annual meeting of the German Mathematical Society in Bonn.
Besides the works mentioned above, Hausdorff also wrote numerous essays that appeared in some of the leading literary magazines of the time. He also wrote a book of poems, ''Ecstasy'' (1900). Some of his poems were set to music by Austrian composer
Joseph Marx
Joseph Rupert Rudolf Marx (11 May 1882 – 3 September 1964) was an Austrian composer, teacher and critic.
Life and career
Marx was born in Graz and pursued studies in philosophy, art history, German studies, and music at Graz University, earnin ...
.
Theory of ordered sets
Hausdorff's entrance into a thorough study of ordered sets was prompted in part by Cantor's continuum problem: where should the
cardinal number
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. Th ...
be placed in the sequence
? In a letter to Hilbert on 29 September 1904, he speaks of this problem, "it has plagued me almost like
monomania
In 19th-century psychiatry, monomania (from Greek , one, and , meaning "madness" or "frenzy") was a form of partial insanity conceived as single psychological obsession in an otherwise sound mind.
Types
Monomania may refer to:
* De Clerambaul ...
". Hausdorff saw a new strategy to attack the problem in the set
. Cantor had suspected
, but had only been able to show that
. While
is the "number" of possible
well-ordering
In mathematics, a well-order (or well-ordering or well-order relation) on a set ''S'' is a total order on ''S'' with the property that every non-empty subset of ''S'' has a least element in this ordering. The set ''S'' together with the well-o ...
s of a
countable set
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; ...
,
had now emerged as the "number" of all possible orders of such an amount. It was natural, therefore, to study systems that are more specific than orders, but more general than well-orderings. Hausdorff did just that in his first volume of 1901, with the publication of theoretical studies of "graded sets". However, we know from the results of
Kurt Gödel and
Paul Cohen
Paul Joseph Cohen (April 2, 1934 – March 23, 2007) was an American mathematician. He is best known for his proofs that the continuum hypothesis and the axiom of choice are independent from Zermelo–Fraenkel set theory, for which he was award ...
that this strategy to solve the continuum problem is just as ineffectual as Cantor's strategy, which was aimed at generalizing the
Cantor–Bendixson principle from
closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a cl ...
s to general uncountable sets.
In 1904 Hausdorff published the recursion named after him, which states that for each non-limit ordinal
we have
This formula was, together with a later notion called cofinality introduced by Hausdorff, the basis for all further results for
Aleph exponentiation. Hausdorff's excellent knowledge of recurrence formulas of this kind also empowered him to uncover an error in
Julius König's lecture 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 Nevanlinna Prize (to be rename ...
in 1904 in
Heidelberg
Heidelberg (; Palatine German language, Palatine German: ''Heidlberg'') is a city in the States of Germany, German state of Baden-Württemberg, situated on the river Neckar in south-west Germany. As of the 2016 census, its population was 159,914 ...
. There König had argued that the continuum cannot be well-ordered, so its cardinality is not an Aleph at all, and thus caused a great stir. The fact that it was Hausdorff who clarified the mistake carries a special significance, since a false impression of the events in Heidelberg lasted for over 50 years.
In the years 1906–1909 Hausdorff did his groundbreaking and fundamental work on ordered sets. Of fundamental importance to the whole theory is the concept of
cofinality, which Hausdorff introduced. An ordinal is called regular if it is cofinal with any smaller ordinal; otherwise it is called singular. Hausdorff's question, whether there are regular numbers which index a limit ordinal, was the starting point for the theory of inaccessible cardinals. Hausdorff had already noticed that such numbers, if they exist, must be of "exorbitant size".
The following theorem due to Hausdorff is also of fundamental importance: for each unbounded and ordered dense set
there are two uniquely determined regular initial numbers
so that
is cofinal with
and coinitial with
(where * denotes the inverse order). This theorem provides, for example, a technique to characterize elements and gaps in ordered sets.
If
is a predetermined set of characters (element and gap characters), the question arises whether there are ordered sets whose character set is exactly
. One can easily find a necessary condition for
, but Hausdorff was also able to show that this condition is sufficient. For this one needs a rich reservoir of ordered sets, which Hausdorff was also able to create with his theory of general products and powers. In this reservoir can be found interesting structures like the Hausdorff
normal-types, in connection with which Hausdorff first formulated the
generalized continuum hypothesis
In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states that
or equivalently, that
In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to ...
. Hausdorff's
-sets formed the starting point for the study of the important model theory of
saturated structure
In mathematical logic, and particularly in its subfield model theory, a saturated model ''M'' is one that realizes as many complete types as may be "reasonably expected" given its size. For example, an ultrapower model of the hyperreals is \al ...
.
Hausdorff's general products and powers of cardinalities led him to study the concept of partially ordered set. The question of whether any ordered subset of a partially ordered set is contained in a maximal ordered subset was answered in the positive by Hausdorff using the well-ordering theorem. This is the
Hausdorff maximal principle
In mathematics, the Hausdorff maximal principle is an alternate and earlier formulation of Zorn's lemma proved by Felix Hausdorff in 1914 (Moore 1982:168). It states that in any partially ordered set, every totally ordered subset is contained ...
, which follows from either the well-ordering theorem or the axiom of choice, and as it turned out, is also equivalent to the axiom of choice.
Writing in 1908,
Arthur Moritz Schoenflies
Arthur Moritz Schoenflies (; 17 April 1853 – 27 May 1928), sometimes written as Schönflies, was a German mathematician, known for his contributions to the application of group theory to crystallography, and for work in topology.
Schoenflies ...
found in his report on set theory that the newer theory of ordered sets (i.e., that which occurred after Cantor's extensions) was almost exclusively due to Hausdorff.
The "Magnum Opus": "Principles of set theory"
According to previous notions, set theory included not only the general set theory and the theory of sets of points, but also dimension and measure theory. Hausdorff's textbook was the first to present all of set theory in this broad sense, systematically and with full proofs. Hausdorff was aware of how easily the human mind can err while also seeking for rigor and truth, so in the preface of his work he promises:
This book went far beyond its masterful portrayal of already-known concepts. It also contained a series of important original contributions by the author.
The first few chapters deal with the basic concepts of general set theory. In the beginning Hausdorff provides a detailed set algebra with some pioneering new concepts (differences chain, set rings and set fields,
- and
-systems). The introductory paragraphs on sets and their connections included, for example, the modern set-theoretic notion of functions. Chapters 3 to 5 discussed the classical theory of cardinal numbers, order types and ordinals, and in the sixth chapter "Relations between ordered and well-ordered sets" Hausdorff presents, among other things, the most important results of his own research on ordered sets.
In the chapters on "point sets"—the topological chapters—Hausdorff developed for the first time, based on the known neighborhood axioms, a systematic theory of topological spaces, where in addition he added the separation axiom later named after him. This theory emerges from a comprehensive synthesis of earlier approaches of other mathematicians and Hausdorff's own reflections on the problem of space. The concepts and theorems of classical point set theory
are—as far as possible—transferred to the general case, and thus become part of the newly created general or set-theoretic topology. But Hausdorff not only performed this "translation work", but he also developed basic construction methods of topology such as core formation (open core,
self-dense core) and shell formation (
closure), and he works through the fundamental importance of the concept of an open set (called "area" by him) and of the concept of compactness introduced by Fréchet. He also founded and developed the theory of the connected set, particularly through the introduction of the terms "component" and "quasi-component".
With the first Hausdorff countability axiom, and eventually the second, the considered spaces were gradually further specialized. A large class of spaces satisfying the countable first axiom are
metric space
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
s. They were introduced in 1906 by Fréchet under the name "classes (E)". The term "metric space" comes from Hausdorff. In ''Principles'', he developed the theory of metric spaces and systematically enriched it through a series of new concepts:
Hausdorff metric In mathematics, the Hausdorff distance, or Hausdorff metric, also called Pompeiu–Hausdorff distance, measures how far two subsets of a metric space are from each other. It turns the set of non-empty compact subsets of a metric space into a metri ...
,
complete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
,
total boundedness In topology and related branches of mathematics, total-boundedness is a generalization of compactness for circumstances in which a set is not necessarily closed. A totally bounded set can be covered by finitely many subsets of every fixed “size ...
,
-connectivity, reducible sets. Fréchet's work is not particularly famous; only through Hausdorff's ''Principles'' did metric spaces become common knowledge to mathematicians.
The chapter on illustrations and the final chapter of ''Principles'' on measure and integration theory are enriched by the generality of the material and the originality of presentation. Hausdorff's mention of the importance of measure theory for
probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...
had great historical effect, despite its laconic brevity. One finds in this chapter the first correct proof of the
strong law of large numbers
In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials shou ...
of
Émile Borel
Félix Édouard Justin Émile Borel (; 7 January 1871 – 3 February 1956) was a French mathematician and politician. As a mathematician, he was known for his founding work in the areas of measure theory and probability.
Biography
Borel was ...
. Finally, the appendix contains the single most spectacular result of the whole book, namely Hausdorff's theorem that one cannot define a volume for all bounded subsets of
for
. The proof is based on Hausdorff's
paradoxical ball decomposition, whose production requires the axiom of choice.
During the 20th century, it became the standard to build mathematical theories on axiomatic set theory. The creation of axiomatically-founded generalized theories, such as general topology, served among other things to single out the common structural core for various specific cases or regions and then set up an abstract theory, which contained all these parts as special cases. This brought a great success in the form of simplification and harmonization, and ultimately brought with itself an economy of thought. Hausdorff himself highlighted this aspect in the ''Principles''. In the topological chapter, the basic concepts are methodologically a pioneering effort, and they paved the way for the development of modern mathematics.
''Principles of set theory'' appeared in April 1914, on the eve of the First World War, which dramatically affected scientific life in Europe. Under these circumstances, the effects Hausdorff's book on mathematical thought would not be seen for five to six years after its appearance. After the war, a new generation of young researchers set forth to expand on the abundant suggestions that were included in this work. Undoubtedly, topology was the primary focus of attention. The journal ''
Fundamenta Mathematicae
''Fundamenta Mathematicae'' is a peer-reviewed scientific journal of mathematics with a special focus on the foundations of mathematics, concentrating on set theory, mathematical logic, topology and its interactions with algebra, and dynamical syst ...
'', founded in Poland in 1920, played a special role in the reception of Hausdorff's ideas. It was one of the first mathematical journals with special emphasis on set theory, topology, the theory of real functions, measure and integration theory, functional analysis, logic, and foundations of mathematics. Across this spectrum, a special focus was placed on topology. Hausdorff's ''Principles'' was cited in the very first volume of
Fundamenta Mathematicae
''Fundamenta Mathematicae'' is a peer-reviewed scientific journal of mathematics with a special focus on the foundations of mathematics, concentrating on set theory, mathematical logic, topology and its interactions with algebra, and dynamical syst ...
, and through citation counting its influence continued at a remarkable rate. Of the 558 works (Hausdorff's own three works not included), which appeared in the first twenty volumes of Fundamenta Mathematicae from 1920 to 1933, 88 of them cite ''Principles''. One must also take into account the fact that, as Hausdorff's ideas became increasingly commonplace, so too were they used in a number of works that did not cite them explicitly.
The Russian topological school, founded by
Paul Alexandroff and
Paul Urysohn, was based heavily on Hausdorff's ''Principles''. This is shown by the surviving correspondence in Hausdorff's
Nachlass
''Nachlass'' (, older spelling ''Nachlaß'') is a German word, used in academia to describe the collection of manuscripts, notes, correspondence, and so on left behind when a scholar dies. The word is a compound in German: ''nach'' means "after ...
with Urysohn, and especially Alexandroff and Urysohn's ''Mémoire sur les multiplicités Cantoriennes'', a work the size of a book, in which Urysohn developed dimension theory and ''Principles'' is cited no fewer than 60 times.
After the Second World War there was a strong demand for Hausdorff's book, and there were three reprints at Chelsea from 1949, 1965 and 1978.
Descriptive set theory, measure theory and analysis
In 1916, Alexandroff and Hausdorff independently solved the continuum problem for Borel sets: Every Borel set in a complete separable metric space is either countable or has the cardinality of the continuum. This result generalizes the
Cantor–Bendixson theorem In descriptive set theory, a subset of a Polish space has the perfect set property if it is either countable or has a nonempty perfect subset (Kechris 1995, p. 150). Note that having the perfect set property is not the same as being a p ...
that such a statement holds for the closed sets of
. For linear
sets
William Henry Young
William Henry Young FRS (London, 20 October 1863 – Lausanne, 7 July 1942) was an English mathematician. Young was educated at City of London School and Peterhouse, Cambridge. He worked on measure theory, Fourier series, differential ca ...
had proved the result in 1903, for
sets Hausdorff obtained a corresponding result in 1914 in ''Principles''. The theorem of Alexandroff and Hausdorff was a strong impetus for further development of descriptive set theory.
Among the publications of Hausdorff in his time at Greifswald the work ''Dimension and outer measure'' from 1919 is particularly outstanding. In this work, the concepts were introduced which are now known as
Hausdorff measure
In mathematics, Hausdorff measure is a generalization of the traditional notions of area and volume to non-integer dimensions, specifically fractals and their Hausdorff dimensions. It is a type of outer measure, named for Felix Hausdorff, that as ...
and the
Hausdorff dimension
In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was first introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of ...
. It has remained highly topical and in later years has been one of the most cited mathematical works from the decade of 1910 to 1920.
The concept of Hausdorff dimension is useful for the characterization and comparison of "highly rugged quantities". The concepts of ''Dimension and outer measure'' have experienced applications and further developments in many areas such as in the theory of dynamical systems, geometric measure theory, the theory of self-similar sets and fractals, the theory of stochastic processes, harmonic analysis, potential theory, and number theory.
Significant analytical work of Hausdorff occurred in his second time at Bonn. In ''Summation methods and moment sequences I'' in 1921, he developed a whole class of summation methods for divergent series, which today are called
Hausdorff method
Hausdorff may refer to:
* A Hausdorff space, when used as an adjective, as in "the real line is Hausdorff"
* Felix Hausdorff (1868–1942), the German mathematician after whom Hausdorff spaces are named
* Hausdorff dimension, a measure theoretic c ...
s. In
Hardy's classic ''Divergent Series'', an entire chapter is devoted to the Hausdorff method. The classical methods of
Hölder and
Cesàro proved to be special cases of the Hausdorff method. Every Hausdorff method is given by a moment sequence; in this context Hausdorff gave an elegant solution of the moment problem for a finite interval, bypassing the theory of continued fractions. In his paper ''Moment problems for a finite interval'' of 1923 he treated more special moment problems, such as those with certain restrictions for generating density
, for instance