Years Ago

David E. Rowe, Editor

Felix Hausdorff: ‘‘We Wish for You Better Times’’ CHARLOTTE K. SIMMONS

Years Ago features essays by historians and mathematicians that take us back in time. Whether addressing special topics or general trends, individual mathematicians or ‘‘schools’’ (as in schools of fish), the idea is always the same: to shed new light on the mathematics of the past. Submissions are welcome.

â Submissions should be uploaded to http://tmin.edmgr.com or sent directly to David E. Rowe, [email protected]

‘‘Despite an extraordinarily productive career, perhaps unparalleled in intellectual and creative breadth, this multitalented figure only received scant attention from biographers or historians of mathematics.’’—David Rowe

H

ausdorff spaces, Hausdorff distance, the Hausdorff metric, the Hausdorff measure, Hausdorff dimension: Hausdorff is a familiar name to mathematicians. Still, the larger dimension of Hausdorff’s multifaceted intellectual life has long been overlooked or forgotten. That, however, began to change in the late 1990s following a special exhibition held in Bonn to commemorate the fiftieth anniversary of his death. It was during this time that a team of experts began to assemble his collected works and to document his remarkable life and career. This undertaking has involved the efforts of no fewer than sixteen mathematicians, four historians, two literary scholars, a philosopher, and one astronomer during the last 15 years [70, p. 36]. Although still underway, this Hausdorff editorial project has already accomplished one great task, namely to ensure that Felix Hausdorff’s intellectual achievements remain accessible to future generations. In the decades before the infamous Nazi ‘‘Law for the Restoration of the Professional Civil Service,’’ Jewish mathematicians had become an integral part of the Germanspeaking mathematical world and consequently mathematics was among the most heavily impacted of the sciences [5, Preface]. Some 144 German Jews who taught mathematics at institutions of higher education were forced out of their jobs after the Nazis seized power. In Michael Golomb’s words, ‘‘Most of them emigrated, but some of them lost their lives’’ [20]. Sadly, Felix Hausdorff (1868-1942) was among the latter. On the last night of his life, the evening before he and his family were to report to an internment camp, Hausdorff wrote to his attorney [16, p. 101]: When you receive these lines, we three will have solved the problem in another way—in the way which you have constantly tried to dissuade us from…. What has happened here against the Jews in the last months awakens well-founded fears that we will never again be allowed to experience conditions we can bear…. Forgive us for causing you trouble even after our deaths; I am sure you will do what you can do (and which perhaps is not very much). And forgive us our desertion! We wish you and all our friends that you may experience better times. Yours faithfully, Felix Hausdorff

 2014 Springer Science+Business Media New York DOI 10.1007/s00283-014-9474-0

The editorial board for the Hausdorff Edition initially consisted of Egbert Brieskorn (project director), Walter Purkert (project coordinator), Friedrich Hirzebruch, Reinhold Remmert, and Erhard Scholz. Working under their direction, an interdisciplinary team of scholars from Germany, Switzerland, Russia, the Czech Republic, and Austria has been assembling Felix Hausdorff—Gesammelte Werke [70, p. 36]. Although the volumes of Gesammelte Werke are in Hausdorff’s native German, each includes one or more commentaries in English. Jeremy Gray points out that these commentaries will be missed in a ‘‘casual perusal by monoglot English readers,’’ because they are not generally listed in the indices [p. 473]. Several other articles in English have appeared in the last decade in conjunction with this project (see, e.g., [55, 68, 69, and 71]). This article offers an overview of Hausdorff’s life and mathematical work with numerous references to the secondary literature available in English. It aims therefore to provide English readers with a glimpse of this unusually polyfaceted man, an astronomer, litterateur, philosopher, and ‘‘creative and productive mathematician of the first rank’’ whose last wish was that his friends would experience better times [64, p. xv].

(Photograph taken by Ludwig Hogrefe, Godesberg, on June 14, 1924; available at http://www.hcm.uni-bonn.de/abouthcm/mathematics-in-bonn/history-of-mathematics-in-bonn/ about-felix-hausdorff/.)

The Early Years ‘‘Fruitful is everything that occurs less than twice, every tree growing in its soil and reaching up to its sky, every smile that belongs to only one face, every thought that is only once right, every experience that breathes forth the heartstrengthening smell of the individual!’’—Paul Mongre´, in Sant’ Illario [58] Hausdorff was born in Breslau in 1868 to wealthy Jewish parents, Louis and Johanna Hausdorff. His father, a THE MATHEMATICAL INTELLIGENCER

businessman who managed various linen and cotton shops, provided his son with ‘‘a life shielded from financial worries’’ [16, p. 85]. Although his educational background was ‘‘in many ways typical for a child from a middle-class family with high aspirations,’’ Hausdorff was an outstanding student with broad interests and finished first in his Nicolai Gymnasium graduating class in 1887 [68, p. 38]. Although he was often selected to read his poetry at school convocations and dreamed of becoming a composer, with much urging from his father he listed on his graduation his intended field of study as the natural sciences. Nonetheless, Hausdorff never lost his passion for music, and throughout his life he invited friends to his home to entertain them at his piano. Hausdorff took a broad array of classes at the University of Leipzig, including philosophy, languages, literature, the history of socialism, the history of music, and the relationship between mental disorders and crime [68, p. 39]. Eventually he focused on applied mathematics and astronomy, and with the exception of one semester each at Berlin and Freiburg, he stayed at Leipzig through the completion of his doctorate and his Habilitation. Hausdorff wrote his dissertation under the direction of Heinrich Bruns, director of the Leipzig astronomical observatory, on the topic of refraction of light in the atmosphere. After completing his dissertation in 1891, he was employed in the Leipzig observatory as a ‘‘calculator’’ [62, p. 124]. After publishing two more manuscripts and a year of voluntary military service, Hausdorff completed his Habilitation at Leipzig in 1895 in mathematics and astronomy with a thesis on the absorption of light in the atmosphere [13, p. 1]. As a Privatdozent in Leipzig, Hausdorff taught a variety of classes on topics including analytic and projective geometry, analysis, mathematical statistics, probability theory, and actuarial science [62, p. 124]. Despite his heavy teaching load, he not only published several papers in astronomy, optics, and diverse mathematical topics between 1896 and 1901, but he also pursued his literary interests. Indeed, he published twenty-two works (including a book of poetry, a play, a book on epistemology, and a book of aphorisms) between 1897 and 1904 under the pseudonym Paul Mongre´ [68, p. 39]. Mongre´ was a ‘‘brilliant essayist’’ who was ‘‘capable of combining meticulous detail-work with verve and passion,’’ and many of his works ‘‘enjoyed high critical acclaim in their day’’ [16, p. 85, and 70, p. 36]. Hausdorff’s play, Der Arzt seiner Ehre, was performed nearly 300 times between 1904 and 1912 in more than 30 cities, including Berlin, Budapest, Frankfurt, Munich, Prague, and Zurich [68, p. 41]. He ‘‘moved in a milieu of Leipzig intellectuals and artists,’’ interacting with noteworthy poets, composers, and sculptors [71, p. 1]. In June 1912, Hausdorff attended a banquet in Berlin with ‘‘the cre`me de la cre`me of the Berlin theatrical scene,’’ including Max Reinhardt, Felix Holla¨nder, and Arthur Kah€ ane. Interestingly, in the Handbuch des judischen Wissens, a reference book on Jewish culture and history published in Berlin in 1936, Hausdorff was not among the 46 Jewish mathematicians listed; Paul Mongre´, however, did make the list of 100 Jewish philosophers and received a second mention as an author of philosophy, lyrics, and drama [17, p. 264]. Hausdorff married Charlotte Goldschmidt, the daughter of a Jewish physician, in 1899. The following year, their first

and only child, Lenore, was born. Little else is known of his personal life with certainty.

The Move to Bonn ‘‘It would be difficult to name a volume in any field of mathematics, even in the unclouded domain of number € in clearness and theory, that surpasses the Grundzuge precision…Its most striking feature is that it is the work of art of a master.’’—Henry Blumberg Hausdorff was appointed as an ‘‘unofficial associate professor’’ at Leipzig in December of 1901. The Dean reported a vote of 22 to 7 on his appointment and noted that those who voted against him ‘‘did so because he is of the Jewish faith’’ [68, p. 42]. When offered an official associate professor position at Bonn University in the summer of 1910, Hausdorff gladly made the move. In a letter to Friedrich Engel, a former mathematical colleague at Leipzig, he later revealed, ‘‘In Bonn one has the feeling, even as a junior faculty member, of being formally accepted, a sense I could never bring myself to feel in Leipzig’’ [68, p. 42]. Although quite content in Bonn, Hausdorff accepted the higher position of Ordinarius (full professor) at the small Prussian provincial University of Greifswald in 1913. Although he was only able to teach elementary courses at Greifswald and at times was the only mathematician there, Hausdorff stayed until he was offered the same title at Bonn in 1921. € der Mengenlehre (‘‘Basics of His masterpiece, Grundzuge Set Theory’’), was published in 1914. After returning to Bonn, Hausdorff was able to incorporate his research into his courses. Remarkably, his lecture notes from his entire teaching career spanning from 1895 to 1933 have survived: ‘‘All the notes are from Hausdorff’s own hand, and all lectures are fully elaborated, ready for printing’’ [69, p. 132]. Consequently, we know that Hausdorff gave a course entitled ‘‘Divergente Reihen’’ in 1925 and lectured on material related to his publications of 1921 [34] and 1923 [36]. These papers on moments introduced summability methods that are now referred to as ‘‘Hausdorff methods.’’ As noted by Purkert, had Hausdorff published these course notes, his would have been the first monograph in the field, because Hardy’s Divergent Series did not appear until 1948 [69, p. 133]. Similarly, Hausdorff’s course notes on probability theory from 1923 contain original ideas, and proofs and his treatment seems to an extent to foreshadow that of Kolmogorov’s Foundations of the Theory of Probability of 1933 [71, p. 2]. In particular, ‘‘Hausdorff came very early to the idea that the right approach to the fundamentals of probability theory was to use measure theory,’’ but ‘‘fails to make any decisive steps beyond this recognition’’ [21, p. 473, and 47, p. 742]. In his final year of teaching in 1933, Hausdorff taught a course on algebraic topology, a field that ‘‘was just taking shape’’ [69, p. 133, and 76, p. 782]. G€ unter Bergmann, a student of Hausdorff’s during the early 1930s, provides an interesting glimpse into his classroom. Apparently, Hausdorff never consulted his notes during a lecture, but rather would leave them in a closed folder on his desk throughout each hour and a half class period and then he would pick up the folder on his way out of the room [69, p. 133].

€ In a review of the Grundzuge published in 1921, American mathematician Henry Blumberg praised Hausdorff’s writing [6, pp. 116, 129]: No one thoroughly acquainted with its contents could fail to withhold admiration for the happy choice and arrangement of subject matter, the careful diction, the smooth, vigorous and concise literary style, and the adaptable notation; above all things, however, for the highly pleasing unifications and generalizations and the harmonious weaving of numerous original results into the texture of the whole…as a treatise it is of the first rank. Hausdorff’s literary flair shines through in his mathematical writings, where he writes ‘‘with lucidity of style, not without an occasional glimmer of humor’’ [6, p. 118]. As examples of Hausdorff’s ‘‘spirited and colorful language,’’ Blumberg cites his reference to the rational numbers being ‘‘distributed over the entire line as a dust of more than microscopic fineness,’’ as well as his explanation that a proper subset of a set can have the same cardinality as the set itself: ‘‘A segment and an arbitrarily small partial segment, a kilometer and a millimeter, the sun’s globe and a drop of water have in this sense the ‘same number of points.’’’ Moreover, although it is perhaps not surprising that Mongre´’s work is filled with allusions to literature, philosophy, painting, and drama, the reference to Voltaire € is, as noted by Purkert, rather in the preface to Grundzuge unexpected [69, p. 131]. € was delayed The mathematical impact of the Grundzuge until the end of World War I but the work attracted intense interest soon thereafter, particularly in Poland and the Soviet Union [71, p. 2]. The first twenty volumes of a new Polish journal focusing on the foundations of mathematics (including set theory and topology) appeared between 1920 and 1933; 88 of the first 558 articles appearing in Fundamenta Mathematicae referred to Hausdorff’s Grun€ [68, p. 45]. Hausdorff’s influence on the Russian dzuge topological school was equally strong, as Pavel Urysohn’s papers on dimension theory from 1925 and 1926 (‘‘Me´moire sur les multiplicite´s Cantoriennes’’) reference the € 60 times. Today, Hausdorff is credited as havGrundzuge ing created the new mathematical discipline of General € [19, p. 11]. Indeed, a Topology by means of the Grundzuge total of 61 mathematicians from 17 countries united to celebrate his memory in Berlin in 1992, 50 years after his death.

The Hitler Era and Hausdorff’s Death ‘‘He certainly is a mathematician of very great merit and still quite active.’’—Richard Courant Hausdorff was agnostic and his wife had converted to the Lutheran faith as an adult and had been baptized, but their lives were interrupted by the activities of the Nazis all the same [53, p. 134, and 71, p. 2]. Hausdorff had been a German civil servant before 1914, and unlike so many of his fellow German mathematicians, including Felix Bernstein, Richard Courant, and Emmy Noether, was therefore not dismissed from his university position under the 1933 Law for the Restoration of the Professional Civil Service [72,

 2014 Springer Science+Business Media New York

p. 125]. Nonetheless, his classes were disrupted in 1934 by Nazi Student Union protests, causing him to dismiss classes for the only time in his long career other than during the German Revolution of 1918 to 1919. Hausdorff was ultimately forcibly retired on March 31, 1935, as an emeritus professor in Bonn, with ‘‘not a word of thanks from the then responsible authorities’’ for his ‘‘forty years of successful labor in German higher education’’ [68, p. 47]. Apparently, the only colleague who visited Hausdorff after his retirement was Erich Bessel-Hagen, who brought books and journals from the mathematics library for Hausdorff to read [72, p. 171]. As a Jew, Hausdorff was no longer allowed to enter the library. Nevertheless, Hausdorff continued his research, publishing a new enlarged edition of the Mengenlehre from 1927 and seven papers on topology and descriptive set theory [68, p. 47]. Because he was not allowed to publish in Germany, his last publications were in Fundamenta Mathematicae [65, p. 6]. Hausdorff’s final publication was in 1938, but he continued to work on mathematics, and his last paper is dated just 10 days before his death [44, p. 402]. In a manuscript written three months before his death, Hausdorff gave a proof of an inequality in probability theory that had been stated without proof in a 1940 note by probabilist Kai Lai Chung; Chung remarked in 2003, ‘‘It saddened me that he must have done it ‘to make TIME pass faster’’’ [44, p. 823]. Bessel-Hagen reported in April 1941, ‘‘Things go tolerably well with the Hausdorffs, even if they can’t escape from the vexation and the agitation over continual new anti-Semitic chicanery’’ [72, p. 457]. By August of that same year, however, his concern for his friends had grown [16, p. 101]: In a horrible alarm night, the Hausdorff’s building was hit by an incendiary bomb. Fortunately, it fell on a…spot in the stairwell, and it was easy to put it out; but the fright remains. Frau H. often looks pitiably bad. The Hausdorffs had been forced to begin wearing the Yellow Star by October, and Bessel-Hagen recorded [16, p. 101]: The Hausdorffs have been subjected to some unpleasantness; in particular, when the Jews go into the street now, they have to wear identifying signs! Further, they have been forced to sell their stock and exchange it for Reich’s Treasury Notes, of course with exchange losses. And other things that I can’t describe. The Hausdorff home was invaded in the middle of the night during Reichskristallnacht in November 1938. Intruders yelled at a shocked Hausdorff, who had just celebrated his 70th birthday, ‘‘There he is, the Head Rabbi. Just watch out. We are going to send you to Madagascar, where you can teach mathematics to the apes’’ [65, p. 6]. According to Lenore Hausdorff Ko¨nig, her father was ‘‘psychologically finished’’ after that incident [16, p. 99]. Hausdorff contacted Richard Courant in February 1939 to see if there was any hope of his securing a research fellowship in the United States. Unfortunately, emergency organizations gave priority to younger scholars because ‘‘[m]athematical results of older scholars, produced at an early stage in their careers, were only of relative and limited value to host countries such as the United States’’ [74, p. 190]. In May of that same year, Hermann Weyl received a letter on Hausdorff’s behalf from George Polya´ [74, p. 96]: THE MATHEMATICAL INTELLIGENCER

A case which is very near to me is Hausdorff. He had written a few lines to Schwerdtfeger, then to me. From that anybody who knows him realizes that he is in a very bad situation….He is over 70—and he is one of the nicest and most pleasant human beings I know—his direct and indirect students (through his book) are everywhere densely distributed. Isn’t there a chance of doing anything for him? Despite the efforts of Weyl, who described Hausdorff as ‘‘a man with a universal intellectual outlook, and a person of great culture and charm,’’ and others such as John von Neumann, there was not [68, p. 48]. The Hausdorff family received an order in January 1942 to report by the end of the month to the internment camp in Endenich, a suburb of Bonn [72, p. 458]. Hausdorff and his wife and sister-in-law took an overdose of Veronal on January 26, and BesselHagen was forced to report their deaths: It is a horrible feeling to have to watch such dear people go under in the wild flood without being able to move a finger. One feels so horribly cowardly and is constantly ashamed. And yet I don’t know what I could have done. Had the Hausdorffs reported as ordered, it is likely that they would have eventually been sent to the concentration camp Theresienstadt. It is estimated that at least 121,500 of the more than 141,000 Jews that passed through Theresienstadt either died there or were deported to an extermination camp such as Auschwitz where they perished [16, p. 105]. Sadly, Hausdorff’s attorney, the recipient of his final note, was one of these. At the time of her parents’ death, Lenore was married and living in Jena. With the help of friends, she was able to obtain identification documents with a false name in 1945 and escape to the Harz Mountains where she was hidden until the end of the war. One of her hosts, a pastor’s wife, told her plainly that she was hiding her ‘‘out of Christian duty and not out of Christian love’’ [16, p. 100]. Lenore lived to the age of 91.

Hausdorff’s Contributions to Mathematics before 1912 ‘‘Set theory is the foundation of all of mathematics.’’— Felix Hausdorff Consistent with his research interests during the period when he was completing his dissertation and Habilitation, Hausdorff’s first four publications were in the areas of astronomy and optics [72, p. 456]. These early astronomical works have been described as excellent in their mathematical presentation, but ‘‘of no further consequence’’ [68, p. 39]. Hausdorff’s interest soon turned increasingly toward mathematics, with publications between 1897 and 1901 in a variety of areas: insurance mathematics (1897), non-Euclidean geometry (1899), hypercomplex number systems (1900), and probability theory (1901). Alter ego Paul Mongre´ published the philosophical and epistemological book Das Chaos in kosmischer Auslese [59] in 1898, with the hope of ‘‘stimulat[ing] anew the participation of mathematicians in the epistemological problem and, conversely, the interest of philosophers in the fundamental questions of mathematics’’ [59, p. 23].

Hausdorff’s interest had turned to set theory by 1901, the area that would occupy his attention for many years to come and in which he would become, along with Ernst Zermelo, ‘‘the most important set-theorist working between 1898 and 1925’’ [62, p. 121]. The mathematical discipline of set theory was created by Georg Cantor between 1870 and 1900. Cantor proved in his 1874 publication in Crelle’s Journal that infinite sets, in particular the set of natural numbers and the real numbers, may have different cardinalities. Between 1879 and 1884, Cantor published a sixpart manuscript [9] entitled ‘‘On Infinite Linear Point-Sets’’ in Mathematische Annalen that Zermelo has characterized as the ‘‘quintessence of Cantor’s life’s work’’ [66, p. 53]. In part five, which appeared in 1883, Cantor defined a wellordered set [10, p. 168]: By a well-ordered set we understand any well-defined set whose elements are related by a well-determined given succession according to which there is a first element in the set and for any element (if it is not the last one) there is a certain next following element. Furthermore, for any finite or infinite set of elements there is a certain element which is the next following one for all these elements (except for the case that such an element which is the next following one to these elements does not exist). Cantor was convinced that the Well-Ordering Theorem is true and referred to it in this same paper as a ‘‘remarkable law of thought’’ [10, p. 169]. He had at this point developed an infinite sequence of infinite cardinal numbers (the alephs) and all cardinals belonging to the aleph-sequence relied on its validity [62, p. 122]. Cantor published his last paper on set theory in 1897, the same year that his results on the relations of point-set theory to his theory of ordinal numbers were recognized in an address by Adolf Hurwitz at the First International Congress of Mathematicians in Zurich [64, p. ix]. Although set theory was still regarded primarily as a tool for analysis and was not yet seen as a mathematical discipline in its own right, it was at least beginning to be viewed as mathematically legitimate [62, p. 122]. Cantor’s Problem von der Ma¨chtigkeit des Continuums was the first problem of the twenty-three listed by David Hilbert during his famous address at the Second International Congress of Mathematicians in Paris in 1900. Hilbert called for a proof of Cantor’s (Weak) Continuum Hypothesis stated in [8] that every infinite subset of R is denumerable or has the power of R, as well as of the Well-Ordering Theorem [64, p. ix]. During the summer of 1901, Hausdorff taught a course on set theory at the University of Leipzig with an enrollment of three students. Although Cantor had lectured at Halle for more than 40 years, he had never taught a course devoted entirely to set theory [64, p. 7]. Thus, Hausdorff’s was only the second instance of such a course, the first having been taught by Zermelo at the University of Go¨ttingen the previous year. Hausdorff’s lecture notes show that he largely followed Cantor’s papers of 1895 and 1897 (as Zermelo had done) and that he had been in contact with Cantor at some point [62, p. 126]. In fact, having learned from Cantor that Cantor had proved the cardinal number of the set of the denumerable order-types is at

most the cardinal number of the real numbers, Hausdorff proved that these cardinal numbers are in fact equal in his lecture on June 27, 1901. This was his first discovery in set theory, and his first paper in the field [24] was published in December by the Leipzig Academy of Sciences. Although his notes reflect that he had independently discovered his proof 2 days before learning that Felix Bernstein’s dissertation contained the same result, Hausdorff did not mention this in his publication and instead referred to the result as the Cantor-Bernstein Theorem [68, p. 42]. Some speculate that the philosophical interests of Paul Mongre´ led Hausdorff to Cantor and to set theory. Whatever his reasons, Hausdorff was about to ‘‘explode upon the scene and usurp the position of the era’s number one Cantorian’’ [64, p. xv, and 68, p. 39]. In his 1901 paper, About a Certain Kind of Ordered Sets, Hausdorff generalized the Cantor-Bernstein Theorem. He noted in the introduction, ‘‘[I]t is really only the special realm of ordinal numbers about which we are somewhat well informed; extremely little is known about general types, the types of non-well-ordered sets’’; moreover, with ‘‘a more detailed knowledge and classification of type…possibly…the old question about the cardinality of the continuum can be brought closer to a solution’’ [24, p. 460]. Hausdorff’s second publication on set theory [25] was actually a Sprechsaal (research announcement) published in the Jahresbericht der Deutschen Mathematiker-Vereinigung. ‘‘The Concept of Power in Set Theory’’ was the only Sprechsaal to appear in 1904 and announced that there was an error in Bernstein’s thesis [25, p. 571]: ‘‘The formula obtained by Herr F. Bernstein…is, therefore, to be considered unproved for the time being. Its correctness seems rather problematical since, as Herr J. Ko¨nig has shown, the paradoxical result that the cardinality of the continuum is not an aleph and that there are cardinal numbers that are greater than every aleph would follow from it.’’ Julius Ko¨nig, who had a reputation for ‘‘acuity and complete reliability,’’ had presented a proof that the Continuum Hypothesis is false at the Third International Congress of Mathematicians in Heidelberg in 1904 [64, p. 28]. Hausdorff, Cantor, Hilbert, Kurt Hensel, and Arthur Schoenflies met in Switzerland afterward to discuss the ‘‘proof.’’ Hausdorff wrote to Hilbert in September 1904: ‘‘After the Continuum Problem had plagued me at Wengen almost like an obsession…I of course looked first at Bernstein’s dissertation. The error is exactly in the suspected place…’’ [64, p. 25]. Having found Bernstein’s error, Hausdorff proceeded to present a generalized recursive formula in his 1904 publication that helped form the basis for all future results in aleph exponentiation and now bears his name [62, p. 128, and 68, p. 43]. Investigations into Order Types, a series of five papers that appeared in 1906 [27] and 1907 [28] in a Leipzig publication (Abhandlungen der Ko¨niglich Sa¨chsischen Gesellschaft der Wissenschaften zu Leipzig), is regarded as Hausdorff’s most seminal work on ordered sets. Hausdorff states in the introduction to the first paper: ‘‘As far as I know, all the essential results of these works are new, since the hitherto existing investigations of others refer almost exclusively to subsets of the linear continuum or to well 2014 Springer Science+Business Media New York

ordered sets’’ [64, p. 45]. Whereas Cantor had worked on denumerable order-types (cardinality Q0), Hausdorff considers order-types of cardinality Q1, using his newly developed method of exponentiation of order-types to construct Q1 different order types, which he then classifies into fifty species [62, pp. 128–129, and 63, p. 505]. He introduced several terms into the theory of ordered sets (e.g., homogenous and continuous order types), coined the term ‘‘transfinite induction,’’ and defined the concept of cofinality of sets [64, p. 37]. Although Hausdorff’s next quest was to determine ‘‘which of our 50 species are really found among types of the second infinite cardinality,’’ he soon realized he would have to be content with constructing representatives of 32 of these: ‘‘A definitive answer to this question cannot be given at this time as long as the question of the cardinality of the continuum is not settled’’ [27, p. 156]. Plotkin contends that by applying set theory to solve a current problem in analysis, Hausdorff was following in the footsteps of the theory’s creator in the fifth and last paper in the series [p. 99]. Cantor’s initial results in set theory were motivated by his interest in the uniqueness of the representation of functions by Fourier series [66, pp. 49–52]. As for Hausdorff, his intent was to ‘‘salvage a failed speculation of P. Dubois-Reymond,’’ a German analyst who published the book Die allgemeine Functionentheorie in 1882 [28, p. 543]. Considering the set of all ‘‘monotonically increasing functions of a positive real variable for which limx?+? f(x) = +?,’’ Hausdorff notes that ‘‘for any two functions f(x) and g(x) there exists a relation’’ between them defined as follows [28, pp. 105–107]: f(x) is infinitarily equal to g(x) when lim{f ðx Þ : gðxÞ} = k, finite and = 0, f(x) is infinitarily less than g(x) when lim{f ðx Þ : gðxÞ} = 0, f(x) is infinitarily greater than g(x) when lim{f ðx Þ : gðxÞ} = +?, f(x) is infinitarily incomparable with g(x) when f ðx Þ : gðxÞ has no limit. He then explains that whereas ‘‘a set of pairwise comparable functions can be ordered rather like a set of ordinary numbers ‘according to size,’ and when one identifies infinitarily equal functions with a single element, it has a definite order type in the sense of G. Cantor,’’ the same is not true for infinitarily incomparable functions. Rather, ‘‘all attempts to produce a simple (linearly) ordered set of elements in which each infinity occupies its specific place had to fail: the infinitary pantachie in the sense of Du Bois-Reymond does not exist.’’ Hausdorff further remarks, however, that ‘‘[t]here is no reason to reject the entire theory [of Du Bois-Reymond] because of the possibility of incomparable functions as G. Cantor has done,’’ and proceeds to redefine the term ‘‘pantachie’’ [28, p. 110]: Thus, if we designate it our task to connect the infinitary rank ordering as a whole with Cantor’s theory of order types, then nothing remains but to investigate the sets of pairwise comparable functions that are as comprehensive as possible: as comprehensive as possible in the THE MATHEMATICAL INTELLIGENCER

sense that such a set should not be extendible by functions that are comparable to all the functions in the set. Retaining the term of Du Bois, but abandoning the unsuccessful concept, we are going to call such a class of functions—for which first of all, of course, an existence proof must be furnished—an infinitary pantachie (so not ‘‘the’’ pantachie, but instead ‘‘a’’ pantachie), noting immediately that two incomparable functions belong to different pantachies in any case. Noting that the definition can be strengthened, Hausdorff continues, A set of functions that are pairwise in the relations [described] is called an ordered domain. An ordered domain which is not contained in any more comprehensive ordered domain is called a pantachie. Thus, an ordered domain or a pantachie contains neither infinitarily equal nor infinitarily incomparable functions. Hausdorff’s ‘‘first question’’ regarding pantachie types is their cardinality and he proves that each pantachie of monotonic functions has the cardinality of the continuum. In doing so, he became the first person to compute the cardinality of the set of real monotonic functions [64, p. 110]. Hausdorff’s pursuits later in the paper to define and generalize new order-types that have the essential properties of pantachies ultimately led him to formulate his Generalized Continuum Hypothesis [62, p. 133]. Hausdorff made his ‘‘auspicious debut’’ in the Mathematische Annalen in 1908 with his publication of ‘‘The Fundamentals of a Theory of Ordered Sets’’ [29], a work described as a combination of ‘‘the tutorial aspects of a primer on the basics with the demands of an advanced research monograph’’ [64, p. 182]. In a letter to Hilbert in July 1907, Hausdorff said that Cantor had urged him to submit a short synopsis of ‘‘Investigations into Order Types’’ to the Mathematische Annalen so that his work might reach a wider audience: Thus I take the liberty of asking whether you are inclined in principle to receive an article for the Annals in the range of 2-3 sheets entitled, say, ‘‘Theory of Order Types.’’ I would only like to spare myself the trouble in case, perhaps the editors of the Annals should be disposed from the start to exclude the field of set theory, which is nowadays so often challenged. (And with such medieval weapons!)…In the hope that you, dear Geheimrat, still consider ‘‘Cantorism,’’ which Poincare´ declared dead, as somewhat alive, and that a work that adds something new to set theory with regard to contents is not denied your interest, I am faithfully yours… Plotkin clarifies that the ‘‘2–3 sheets’’ that Hausdorff offered were really Bogen sheets, large sheets of paper that fold to produce eight pages each. Cantor also wrote to Hilbert in August of 1907 [64, p. 193]: I am glad to hear that Herr Zermelo is successfully working on set theory. Give him my best. I also consider the work of Hausdorff in the theory of types to be useful, thorough, and promising. For this reason, I arranged to meet with him at the last meeting of the three universities, Leipzig, Jena, and Halle, in Ko¨sen on June 28, and I suggested an investigation of a question that he appears to have completed successfully.

Hausdorff states ‘‘Cantor’s Aleph Hypothesis’’ in this work, which was later proven equivalent to the Generalized Continuum Hypothesis [29, p. 494, and 63, p. 510]. He begins the paper as follows [p. 435]: In what follows, a sustained introduction that is systematic and as general as possible to the still practically unknown field of simply ordered sets, a field developed by Herr G. Cantor, is attempted for the first time. Up to now, only the well-ordered sets and sets of reals have actually experienced a detailed treatment. On the whole, even my own earlier studies, of which nothing is assumed here, pursue a special direction; their principal subject matter is certain types that are distinguished by especially regular structure (homogeneity) and that have sequences up to the second infinite cardinality, and as far as generalizations are strived for, they are restricted to the nearest levels, those that correspond to the alephs with finite index. In the present article, lest it swell into a book, these special types definitely had to fall back into the role of occasional illustrative examples and general methods had to occupy the foreground. By this time Hausdorff had achieved his objective of developing a genuine theory of order-types, along with the necessary tools for such a theory [62, pp. 128, 134]. He had ‘‘attained recognition as a leader in the second generation of Cantorians’’ and Arthur Schoenflies declared in ‘‘Die Entwickelung der Lehre von den Punktmannigfaltigkeiten (1908): ‘‘We also owe the other advances in ordered sets that we possess to Hausdorff’’ [64, p. 181]. Hausdorff’s last major work on ordered sets, ‘‘Graduation by Final Behavior’’ [30], was published in 1909 and is the first publication to contain an application of a ‘‘maximal principle’’ in algebra [64, p. 260]. In this paper, Hausdorff ‘‘take[s] the liberty to return to the pantachie problem,’’ providing this time a purely set-theoretic proof that pantachies exist using Zermelo’s Well-Ordering Theorem [30, p. 302]. He proves the existence of a ‘‘rational pantachie,’’ a pantachie closed under rational operations and therefore maximal (although he does not use the term). Although his argument was general enough to prove that an arbitrary partially ordered set has a maximal linearly ordered subset, he did not work in this generality until his publication of 1914 [64, p. 261]. We now know that Hausdorff’s Maximal Principle is equivalent to the Axiom of Choice (see [61, p. 168]). The interested reader may find [60] and the commentaries by Vladimir Kanovei in [51, pp. 367–405] enlightening. It is interesting to note that Hausdorff published another paper in 1906 [26] that has been classified as one of ‘‘his few papers in algebra’’ and contains his contributions to the Baker-Campbell-Hausdorff formula for the exponential map of a Lie algebra [21, p. 473]. Specifically, given elements x and y of a noncommutative real or complex algebra, Hausdorff recursively determined the series C(x,y) satisfying exey = eC(x,y). This formula expresses Lie group multiplication in the local coordinates of the corresponding Lie algebra. This equation was first considered by Campbell in 1897, and Baker gave a formula for computing C(x,y) in 1905, but Hausdorff’s was more explicit and his proof was simpler. Hausdorff was also the first to recognize the

importance of the Jacobi identity in determining C(x,y): ‘‘It is possible that the Jacobi identity is very near to the core of our problem’’ [13, pp. 425, 557]. Hausdorff was torn between Sophus Lie and Bruns when he selected his doctoral advisor and often lectured on Lie groups as a professor [18, p. 80]. Apparently, Bruns had a more amiable nature.

Hausdorff’s Grundzu¨ge der Mengenlehre € der Mengenlehre is without any doubt ‘‘The Grundzuge Hausdorff’s most influential work: it is his opus magnum. It was a milestone on the path from classical mathematics of the 18th and 19th centuries to the socalled modern mathematics of structures.’’—Walter Purkert Hausdorff’s attention next turned to the mathematical discipline he referred to as ‘‘Topology,’’ or ‘‘Analysis Situs,’’ noting that the latter term is originally credited to Leibniz but was reintroduced by Riemann [41, p. 257]. By the summer of 1912, Hausdorff had formulated his concept of topological space using neighborhood systems and developed an axiomatic foundation for topological spaces [16, p. 114]. He began work on a monograph that would ‘‘secure his international reputation’’ and be responsible for ‘‘introduc[ing] a generation of mathematicians to set theory in the broadest sense of the term’’ [21, p. 472, and 71, p. 2]. He did not consider order-types again until the 1930s (see [42] and [51]), when he investigated the connections between ordertypes and topological spaces [63, p. 510]. € ‘‘[d]edicated to Roughly the first half of the Grundzuge, the creator of set theory Herr Georg Cantor in grateful admiration,’’ contains Hausdorff’s view of Cantorian set theory that he had developed during the previous decade, including his most important results on ordered sets [64, p. xiii]. In the next three chapters, Hausdorff defines topological spaces and systematically develops their basic properties, laying the foundation for what is now called ‘‘point-set topology,’’ ‘‘set-theoretic topology,’’ or ‘‘general topology.’’ In the final chapter, Hausdorff discusses measure theory and integration (considered part of set theory at the time) and gives an axiomatic presentation of measure theory [68, p. 44, and 71, p. 2]. His text was the first to deal systematically with all aspects of set theory (general set theory, point sets, and measure theory), and he did so with ‘‘masterful exposition…characterized throughout by originality, naturalness, and beauty’’ [6, p. 122]. Hausdorff begins Chapter VII with an explanation of the value of working in generality [6, p. 123]: Now a theory of spatial point sets would naturally have, in virtue of the numerous accompanying properties, a very special character, and if we wished to confine ourselves from the outset to this single case, we should be obliged to develop one theory for linear point sets, another for planar point sets, still another for spherical point sets, etc. Experience has shown that we may avoid this pleonasm and set up a more general theory comprehending not only the cases just mentioned but also other sets (in particular, Riemann surfaces, spaces of a finite or an infinite number of dimensions, sets of  2014 Springer Science+Business Media New York

curves, and sets of functions). And indeed, this gain in generality is associated not with increased complication, but on the contrary, with a considerable simplification, in that we utilize—at least for the leading features—only few and simple assumptions (axioms). Finally, we secure ourselves in this logical-deductive way against the errors into which our so-called intuition may lead us; this alleged source of knowledge—the heuristical value of which, of course, no one will impugn—has, as it happens, shown itself so frequently insufficient and unreliable in the more subtle parts of the theory of aggregates, that only after careful examination may we have faith in its apparent testimony. He then defines a topological space to be a set E together with a set of subsets Ux of E (called neighborhoods of x where x is in E), such that the neighborhoods satisfy the following axioms [31, p. 213]: (A) Every point x in E belongs to at least one neighborhood Ux, and every neighborhood Ux contains x. (B) If Ux and Vx are both neighborhoods of x, then there is a neighborhood Wx contained in both. (C) If the point y is in Ux, then there is a neighborhood Uy that is contained in Ux. (D) For distinct points x and y, there are two neighborhoods Ux, Uy with no common points. Having defined an open set G as a set ‘‘in which every point x 2 G has a neighborhood Ux ( G (the null set included),’’ Hausdorff notes that the following ‘‘sum and intersection axioms’’ are valid for open sets [41, pp. 258–259]: (1) The space E and the null set 0 are open. (2) The intersection of two open sets is open. (3) The sum of any number of open sets is open. In modern topology textbooks, a topological space is typically defined as a collection of subsets of E, having properties (1) to (3), and elements of the topology are called open sets. A collection of subsets of E having properties (A) and (B) forms a basis for a topology on E in modern terminology (e.g., see [1, pp. 24, 29]). A topological space having property (D) is now called a Hausdorff space. In fact, Hausdorff did not include (D) in his definition of topological space in his 1935 text, but instead listed it as one of five separation axioms, and commented that at least one of the separation axioms should be required of a topological space lest it become ‘‘altogether pathological’’ [41, pp. 260–261]. The separation axioms and the countability axioms, both of which Hausdorff used to investigate specialized topological spaces such as metric spaces and Euclidean spaces, are among the ‘‘signal achievements’’ of the € identified in [45]. Others on the list are HausGrundzuge dorff’s concept of connectivity and the decomposition of a topological space into its components, his concept of a metric on the space of all bounded and closed subsets of a metric space (now known as the Hausdorff metric), and his concept of a complete and totally bounded metric space

THE MATHEMATICAL INTELLIGENCER

[69, p. 131]. Moreover, in his efforts to extend classical point-set-theoretic ideas and results from Rn to his newly defined topological spaces, Hausdorff introduced several new concepts (e.g., interior and closure operations and relative topologies), while further developing others (e.g., open sets and compactness) [6, p. 126, and 68, p. 44]. In his thesis of 1906, Maurice Fre´chet developed the notion of a limit based on an axiomization of convergent sequences and discussed a type of ‘‘L–space’’ on which a distance function could be defined [61, p. 235]. Hausdorff renamed these metric spaces (metrischer Raum) and gave the first comprehensive treatment of their theory, contributing several new ideas along the way (e.g., q-connectedness and reducible sets) [68, p. 44, and 71, p. 2]. According to Blumberg, ‘‘[i]n the carefully planned march from the abstract theory in the direction of greater specialization, Hausdorff gives repeated evidence of his mathematical-esthetic insight’’ [p. 123]. Whereas Fre´chet’s work had been largely ignored by the mathematical community, metric spaces were met with ‘‘uni€ [69, p. 130]. versal acceptance’’ after the Grundzuge In their 1935 book, Topologie, Paul Alexandroff and Heinz Hopf remark that topology includes the study of Borel sets, analytic sets (called A-sets at the time), and projective sets, which they refer to as ‘‘descriptive point set theory’’ [2, p. 19]. This was the first time the term was used in print, although they credit an article that appeared in the Journal de Math in 1905 by Henri Lebesgue as the origin of the field: Sur les fonctions repre´sentables analytiquement. Moreover, they regard what is now known as the Alexandroff-Hausdorff theorem to be the starting point for the further } development of this area of mathematics. In the Grundzuge, Hausdorff coined the term ‘‘Borel sets’’ which he defined as ‘‘those [sets] which are generated from open sets or closed sets by the formation of unions and intersections of sequences’’ [pp. 305, 466]. Two years later, he and Alexandroff published different proofs of a result that they had independently discovered, namely, that every Borel set in a complete separable metric space is either at most countable or has the cardinality of the continuum [55, p. 4–5]. The material related to descriptive set theory in the } Grundzuge is contained in Chapters VIII and IX, and includes a study of Borel sets in conjunction with metric spaces, an introduction to Hausdorff’s ‘‘reducible sets’’ (which can be represented as the sum of differences of descending normally ordered closed sets), and a formulation of the theory of Baire functions from a set-theoretic (rather than a function-theoretic) point of view [6, p. 127, and 55, p. 5]. Hausdorff continued to think about descrip} tive set theory beyond the Grundzuge, publishing seven papers in this area between 1916 and 1937. Fundamenta Mathematicae ranked among the leading mathematical journals from the first issue in 1920, and € clearly demonstrated that the Grundzuge had inspired numerous young researchers such as Alexandroff. The € was used as a standard reference by many Grundzuge authors publishing in the journal on various aspects of set theory, including several who wrote about descriptive set theory: Alexandroff, Stefan Banach, Kazimierz Kuratowski, Adolf Lindenbaum, Nikolai Lusin, Stefan Mazurkiewicz,

John von Neumann, Waclaw Sierpinski, and Alfred Tarski. Although attention soon shifted from Borel sets to analytic and projective sets after the publication of the AlexandroffHausdorff theorem, and Hausdorff’s visibility in the field declined through the years, Hausdorff is still listed among the founders of descriptive set theory [55, pp. 6–7]. € contains an introduction to Chapter X of the Grundzuge Lebesgue’s theory of measures and integration from Hausdorff’s point of view that has been characterized as impressive both for the generality of the approach and the originality of presentation [68, p. 44]. In particular, Hausdorff’s proof of Borel’s Strong Law of Numbers is the first that is ‘‘complete, correct, and explicit’’ [3, p. 186]. Hausdorff presents a reformulation of the Strong Law after first proposing a definition of probability couched in terms of measureable sets [31, pp. 416–417]: We remark that many theorems concerning the measure of point-sets appear more intuitively, if one expresses them in the language of probability. If two sets P and M are measureable, and M in particular is of positive measure, then one can define, by means of the quotient f(P)/f(M) if P ( M, or more generally by f(P\M)/f(M), the probability that a point of M belongs to P. Barone and Novikoff characterize Hausdorff’s comments on the connection between probability and measure theory as historically significant, and note that he also gives a ‘‘scrupulously clear’’ expression of independent events, countable additivity, conditional probability, and sample space [p. 177]. Moreover, unlike Borel’s ‘‘proof,’’ Hausdorff’s does not reference the Central Limit Theorem. € culminates with an appendix that contains The Grundzuge an example of a nonmeasurable set that Hausdorff referred to as ‘‘remarkable’’ and others have referenced as the most spectacular result in the book [68, p. 44]. Using the Axiom of Choice, Hausdorff constructs a ‘‘paradoxical’’ decomposition of the 2dimensional sphere as a disjoint union of sets, thereby anticipating the Banach-Tarski paradoxical decomposition of the three-sphere given in 1924. He settles a problem that originated in Lebesgue’s thesis of 1902 and is referred to today as the finitely additive measure problem in Rn, namely that a finitely additive measure invariant under congruences cannot be defined on all bounded subsets of Rn, for n C 3 [44, p.11, and 55, p. 5]. In Hausdorff’s words, ‘‘The proof rests on the remarkable fact that a half sphere and a third of a sphere may be congruent’’ [31, p. 469].

Hausdorff’s Contributions to Mathematics Beyond the Grundzu¨ge ‘‘Hausdorff took up questions in real analysis now informed by the new ‘‘basic features’’ of general set theory. His introduction of what are now called Hausdorff measure and Hausdorff dimension became of longlasting importance in the theory of dynamical systems, geometric measure theory and the study of ‘fractals,’ which aroused broad and even popular interest in the last third of the 20th century.’’—Erhard Scholz Duda [14] notes that although it was not Hausdorff’s main field of interest, he ‘‘embraced also the theory of

probability’’ and ‘‘one cannot help [getting] a feeling of immense richness and originality of his mind’’ from his works on the subject. His 1901 publication [23] is credited as having drawn attention to the importance of the concept of conditional probability and introducing the terminology ‘‘relative probability,’’ as well as the notation that was used by Andrey Kolmogorov and others until the middle of the twentieth century [68, p. 42, and 73, p. 7]. In an unpublished manuscript of 1915, Hausdorff defined a system of orthogonal functions that were independently discovered and published by Hans Rademacher in 1922; these are now known as Rademacher functions [47, p. 748]. Among the novelties found in Hausdorff’s (unpublished) lecture notes for his 1923 probability course is his treatment of the generalized Chebyshev-Markov limit theorem using Marcel Riesz’s approach to the moment problem [47, pp. 733–734]. Hausdorff, in fact, spent many years working on criteria for the solvability and determination of moment problems. Indeed, apart from his publications of 1921 [34] and 1923 [36], there are hundreds of pages in his posthumous papers dated between 1917 and 1924 devoted to this topic [68, p. 46]. Harald Crame´r notes in his autobiography that he ‘‘had the good luck to be allowed to see’’ Hausdorff’s lecture notes, and ‘‘these had a great influence on my subsequent work in the field’’ [12, p. 512]. Although it is not known what sparked his initial interest in probability theory, notes on a lecture course that Hausdorff took from his advisor on the subject in 1906 are preserved in his posthumous papers. Bruns, in fact, published a successful book on probability theory in 1906 [47, p. 735]. Epple speculates that it was his ‘‘early encounter with statistics and the mathematics of chance’’ that later led Hausdorff to develop his approach to probability [18, p. 81]. This early exposure came in the guise of a teaching assignment given to Hausdorff by the faculty at Leipzig while he was a Privatdozent. When Hausdorff was considered for a position at Go¨ttingen in astronomy in 1897, Bruns wrote to Felix Klein: ‘‘[I]n addition, we have given him the theoretical lecture courses on insurance mathematics. For this…task he will be able to rely on the specific dispositions of his race (not baptised).’’ Hausdorff was not offered the position. In that same year, Hausdorff published a paper [22] introducing the variance of an insurer’s losses as a measure of risk and calculated variance of loss for various types of life insurance. His results began to appear in the textbook literature immediately thereafter, and variance of loss is still a fundamental concept in the evaluation of insurance plans with fixed coverages and premiums today [68, p. 42]. Additionally, Hausdorff gave the first correct proof of Hattendorff’s Theorem, which has been described as one of the classical theorems of life insurance mathematics [56, p. 799]. This theorem states that the losses in successive years on a life insurance policy have mean zero and are uncorrelated. Following his seminal publication ‘‘Dimension and Outer Measure’’ of 1919 introducing the concept that is now called the Hausdorff dimension, Hausdorff wrote several hundred (unpublished) pages on topological dimension theory during the next two decades [44, p. 53]. Since Benoıˆt Mandelbrot’s discovery of fractals in the 1970s  2014 Springer Science+Business Media New York

and the appearance of his The Fractal Geometry of Nature (1982), the 1919 publication [32] has certainly been the most frequently cited of Hausdorff’s papers in the popular scientific literature [16, p. 114]. Its ‘‘enormous after-effects’’ in the research community are described by Chatterji in [44, p. 50], who concludes that ‘‘[i]t would be impossible, if not foolish, to try to give explicit individual references to all the papers which stemmed directly or indirectly’’ from this one. The Hausdorff measure and Hausdorff dimension have found applications in numerous areas of mathematics, including coding theory, dynamical systems, ergodic theory, geometric measure theory, harmonic analysis, number theory, potential theory, and stochastic processes [68, p. 46]. Whereas the Hausdorff dimension of an ndimensional vector space is n, its novelty and usefulness lies in the fact that it assumes noninteger values on sets that are irregular or jagged, such as the Cantor ternary set. In the words of Mandelbrot, ‘‘[c]louds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line’’ [57, p. 1]. Hausdorff’s interest in generalizing the geometric concepts of length, area, and volume was already evident in € [44, p. 50]. In a 1914 publication [11], the Grundzuge Constantin Carathe´odory constructed a linear measure that generalizes the concept of the length of curves in Rq, and briefly mentioned the possibility of using p-dimensional measures in Rq to generalize p-dimensional volumes of subsets of Rq (where 1 B p B q) [44, pp. 45–46]. Hausdorff begins his 1919 paper by praising Carathe´odory’s ‘‘exceptionally simple and general measure theory’’ before presenting what he describes as his ‘‘own small contribution’’ [15, p. 75]. For each subset A of Rq, he defines a unique extended real number L(A) as follows [44, pp. 44– 45]. Let X be any collection of bounded subsets of Rq with the usual q-dimensional Euclidean distance such that for every  [ 0, every subset of Rq can be covered by a countable number of sets U in X with diameter d ðU Þ\ . Also assume that each U in X has been assigned a nonnegative real number l(U). Then L(A)nP is defined to be l ð A;  Þ; where l ð A;  Þ ¼ inf lim o n  1 lðUn Þ j A  S !0 U ; U 2 X and d ð U Þ\ : n n n n1 Hausdorff first observes that the association A ? L(A) enjoys the following properties and therefore defines an outer measure (in the sense of Carathe´odory) on the family of subsets of Rq: (1) If B ( A then L(B) B L(A). P (2) If A = A1 [ A2 [ , then Lð AÞ  n  1 LðAn Þ. Although Hausdorff doesn’t explicitly do so, it is best to assume that ; 2 X and l ð;; Þ ¼ 0 for each  [44, p. 44]. As Hausdorff noted at the end of his paper, his definition can in fact be generalized to any complete separable metric space, and L is actually a metric outer measure [13, p. 233]. Taking X to be the collection of all bounded subsets of R and l(U) = d(U) in the above construction yields Carathe´odory’s linear measure, and Hausdorff remarks that the details of Carathe´odory’s proof apply in this more general context. He then proceeds to construct a family of regular THE MATHEMATICAL INTELLIGENCER

metric outer measures Lk that are generalizations of Carathe´odory’s p-dimensional measures, where k : [0, ?) ? [0, ?) is any continuous and strictly increasing function with k(0) = 0 and l(U) = k(d(U)) in the above construction. We now refer to Lk as the Hausdorff measure associated with the Hausdorff function k [44, pp. 45–46]. Given a Hausdorff function k and A ( Rq, Hausdorff defines the ‘‘fractional dimension’’ of A to be [k] if 0 \ Lk(A) \ ?. Note that the definition is given in terms of equivalence classes of Hausdorff functions, where [k] = [l] if there exists a, b, and  [ 0 such that 0 \ a B lðxÞ kðxÞ B b \ ? for some x 2 (0, ). Hausdorff wonders for which Hausdorff functions k there exists a subset of Rq of dimension [k], and constructs a bounded perfect nondense subset A of real numbers of dimension [k] for those k that are strictly concave with lim x!1 kðx Þ ¼ 1. Hausdorff’s construction of A is analogous to the classical construction of the Cantor ternary set in [0,1], but he partitions [0,1] into more than three parts initially and varies the length of the intervals removed at each stage of the process; the Cantor ternary set is obtained when k(x) = xp where p = ln(2)/ln(3). In fact, the concept of dimension referred to as Hausdorff dimension is actually slightly less refined than Hausdorff’s original definition in that the only Hausdorff functions used are k(x) = xp where p [ 0 [44, pp. 46–49]; it first appeared in a paper of Abram Besicovitch in 1929, whose contributions to dimension theory are best described by Mandelbrot: ‘‘For a long time, Besicovitch was the author or the co-author of nearly every paper on this subject. While Hausdorff is the father of non-standard dimension, Besicovitch made himself its mother’’ [57, p. 365]. Hausdorff published papers in 1921 [34], 1923 [36], and 1930 [38] containing ‘‘profound results of the theory of summability of divergent series’’ [13, p. 571]. The concepts of Hausdorff matrices, Hausdorff mean, Hausdorff method, and Hausdorff summation emanated from these works. Widder gives the following statement of the ‘‘F. Hausdorff moment problem’’ in his 1945 review of Shohat and Tamarkin’s 1943 monograph, The Theory of Moments: Given a sequence of real numbers l1, l2,…, find a non-decreasing R1 function w(t) such that ln ¼ 0 t n d w ðtÞ for all natural numbers n [77, p. 860]. Having developed his summation methods for divergent series in [34], Hausdorff showed in [36] that the moment problem has a solution if and only if {ln}nC0 is completely monotonic. Interestingly, this monograph was the inaugural volume for the American Mathematical Society Mathematical Surveys and Monographs series, and Widder predicts that it ‘‘may well establish the tone for the series’’, as it presents in a comprehensible manner ‘‘that branch of mathematics which has recently grown up about a problem posed and solved by T. J. Stieltjes.’’ Stieltjes originally posed the moment problem for the interval [0,?) rather than [0,1], asking whether the distribution of mass along [0,?) (i.e., w(t)) can be determined if the moments of all orders are known. His method of proof involved continued fractions, which Hausdorff circumvented by using his summability results. As Butzer aptly observes in regard to William Henry Young of the Hausdorff-Young inequality, whereas ‘‘some fifty [of Young’s papers] are devoted to Fourier analysis,’’ of

[Hausdorff’s] 42 publications only one is devoted to Fourier analysis’’ [p. 114]. Yet, this ‘‘one’’ publication [35] appeared in 1923 and ‘‘made a major contribution to the emergence of functional analysis in the 1920s,’’ for it was in this work that Hausdorff generalized the Riesz-Fischer theorem to Lp spaces [68, p. 46]. According to the Riesz-Fischer Theorem, if the squares of the coefficients of a trigonometric series 12 a0 þ P Pn¼1 ða2n cosð2nxÞ þ bn sinðnxÞÞ form a convergent series n=1 (an + bn), then the trigonometric series is the Fourier series of a function whose square is summable. Whereas Young tackled the case when ‘‘square summable’’ is replaced by ‘‘q + 1 summable’’ where q is an odd integer in a manuscript published in 1912–1913, Hausdorff settled the question in general and gave the theorem its present form [7, pp. 16–17]. The (generalized) Riesz-Fischer Theorem established the relationship between Lp function spaces and lq series of Fourier coefficients for 1p þ q1 ¼ 1, thereby ‘‘open[ing] the path for later developments in harmonic analysis on topological groups’’ [71, p. 2]. As for the exposition in Hausdorff’s paper, Chatterji admits in his commentary in [44, p. 185] that ‘‘there is hardly anything that can be further simplified or that needs any supplementary explanation.’’ Hausdorff received a visit in the summer of 1924 from Russian mathematicians Alexandroff and Urysohn, who € had been inspired by Hausdorff’s Grundzuge and his subsequent work on dimension theory. Their own contributions to the theory of topological spaces include Alexandroff’s formulation of the general axioms of a topological space, their results on compact and locally compact spaces, and their formulation of topological dimension, particularly as applied to countably compact spaces [4, p. 10]. The Go¨ttingen mathematicians were very impressed by the work of these young mathematicians and invited them for a visit, where they worked on their theory of dimension and algebraic topology with Emmy Noether. Hausdorff was fascinated by their results and the evenings for the trio were a mix of topological discussion and music [4, p. 14]. Interestingly, Alexandroff had temporarily abandoned mathematics after failing to solve the Continuum Hypothesis, and had spent time lecturing on foreign literature and directing a theatre company. Hausdorff expressed concern about the pair’s routine of swimming in dangerous spots such as the Rhine. In August of 1924, Urysohn tragically drowned in the Atlantic Ocean. During the visit, the three had discussed a problem posed by Fre´chet in 1910, namely whether there exists a separable universal space for the class of all separable metric spaces (i.e., a separable metric space containing isometric images of all separable metric spaces) [48, p. 766]. Both Hausdorff and Urysohn independently discovered such a space shortly thereafter. Upon Urysohn’s death, Hausdorff elected not to publish his own construction but encouraged Alexandroff to publish Urysohn’s instead [69, p. 134]. € der In 1927, the ‘‘second edition’’ of the Grundzuge Mengenlehre [37] was published under the title Mengenlehre. In actuality, this text was a new book, as Hausdorff explains in the introduction [41, p. 5]: In this case, there was an additional restriction that the new edition had to be substantially curtailed in length…this would have necessitated a revision down to

the smallest detail, and I found a complete rewriting of the book preferable. I thought it might be easiest to sacrifice…most of the theory of ordered sets, a subject that stands somewhat by itself, as well as the introduction to Lebesgue’s theory of integration, which does not lack for exposition elsewhere. What is more regretted is the abandonment, owing to the necessity of saving space, of the topological point of view in point-set theory, which seems to have attracted many people to the first edition of this book; in this new edition, I have restricted myself to the simpler theory of metric spaces and have given only a quick survey of topological spaces, which is a rather inadequate substitute. It seems clear from his posthumous papers that Hausdorff had originally planned to expand and revise the material on measure theory and integration in the Mengenlehre [45, p. 798]. Realizing that whereas there had been few books € was pubon Lebesgue integration when the Grundzuge lished, the subject now did ‘‘not lack for exposition,’’ he chose to focus instead on a new and emerging discipline. In Purkert’s assessment, the historical significance of the Mengenlehre is that it was the first monograph on descriptive set theory [69, p. 132]. As had been the case with the € Grundzuge, the Mengenlehre stimulated an abundance of research contributions to the area from the Polish mathematical community during the next decade [18, p. 91]. The second edition of Mengenlehre appeared in 1935, the year of Hausdorff’s forced retirement. He could only lament in the introduction that, although ‘‘an actual revision of this book’’ would have been desirable, ‘‘circumstances have prevented my doing this’’ [41, p. 6]. In the early 1930’s, Hausdorff ‘‘wound up his work on analysis,’’ publishing his last paper [40] on the subject in 1932 [13, p. 608]. Hausdorff’s manuscript appeared the same year as the first monograph of functional analysis, written by Stefan Banach, and has been characterized as the first brief review of the subject. Hausdorff included such topics as normed spaces, Banach spaces, the theorems of Banach-Steinhaus and Hahn-Banach, compact operators, weak convergence, and l p-spaces. In this same paper Hausdorff used the Well-Ordering Theorem to give the first proof of a crucial result in linear algebra: Every vector space has a basis [61, p. 228]. Moreover, Oswald Teichm€ uller built on Hausdorff’s result to prove that every Hilbert space has an orthonormal basis. Hausdorff was interested in extending maps, and both his first and last publications on metric spaces (apart from € the Grundzuge) were devoted to this topic [48, p. 565]. In 1919, he published two new proofs [33] of the Tietze extension theorem, which states that a continuous realvalued function from a closed subset A of a metric space X can be extended to a continuous function on X. Hausdorff had found the proof published by Carathe´odory in 1918 (but attributed to Harald Bohr) ‘‘a little artificial’’ [52, p. 162]. Hausdorff proved [39] in 1930 that a metric that is compatible with d on a closed subset A of a metric space (X,d) can be extended to X without altering the topology. Although simpler proofs of this theorem were found later, they rely on the Axiom of Choice, whereas Hausdorff’s proof does not [19, p. 293]. Hausdorff’s final publication [43]  2014 Springer Science+Business Media New York

in 1938 contains a proof of the following result: Given a continuous function f from a closed subset A of a metric space X onto a metric space N, there is a metric space Y containing N as a closed metric subspace and a continuous function F: X ? N extending f such that its restriction to X - A is a homeomorphism onto Y -N. Moreover, if f is a homeomorphism, then F is a homeomorphism [52, p. 164]. Much has been revealed through a careful examination of Hausdorff’s unpublished work. For example, Alexandroff discovered and published the Alexandroff long line, a space that is frequently used as a counterexample in topology, in 1924. In fact, Hausdorff had discovered it independently in 1915. Similarly, the (unpublished) separable universal metric space constructed by Hausdorff in 1924 was rediscovered independently in 2004 by A. M. Vershik, who demonstrated its significance to probabilistic metric spaces [69, p. 134]. Additionally, a manuscript written by Hausdorff in 1933 on homomorphisms of homological groups contains one of the first known uses of a commutative diagram. Although there is still much to learn about Hausdorff’s contributions to mathematics, it is hoped that what is presented here is enough to convince the reader that Czy_z’s characterization is fitting: ‘‘paradoxical, connected, and harmonic’’ [13, p. 659].

Hausdorff Remembered ‘‘There are many ways to forget someone. But how can we best keep our memory of [Hausdorff]?…One way is to maintain his scientific legacy. Another way is to seek access to a person who possessed great charm, numerous talents and a great love of life.’’—Eugen Eichhorn, International Conference in Memory of Felix Hausdorff The Hausdorff Center for Mathematics opened in Bonn in 2007, and more than sixty professors of mathematics are currently associated with it. Likewise, the Ernst Moritz Arndt Universita¨t Greifswald has honored Hausdorff’s memory with the Felix Hausdorff International Meeting Centre. Tourists may visit the Hausdorff home in Greifswald or their home in Bonn on Hausdorffstrasse, where they will find a memorial plaque on the street sign: Professor Doctor Felix Hausdorff, born 1868, Jewish Mathematician in Bonn, driven to suicide on the 26th of January, 1942, by the Nazi regime. A recent visitor to Hausdorff’s grave in Poppelsdorf Cemetery in Bonn reports that it appears that Hausdorff is still ‘‘largely forgotten,’’ for his grave ‘‘is covered with dust and green moss and small stones placed there by visitors who do remember’’ [54, p. 52]. Hausdorff’s home in Bonn is ‘‘alive with music once again,’’ though, having been purchased by fellow music enthusiasts. Upon Hausdorff’s death, his posthumous papers passed to a friend of the family who did his best to preserve them [68, p. 48]: [I]n December 1944 a bomb explosion destroyed my house and the manuscripts were mired in rubble from a collapsed wall. I dug them out without being able to pay attention to their order and certainly without saving them all. Then in January 1945 I had to leave Bonn… When I returned in the summer of 1946 almost all the THE MATHEMATICAL INTELLIGENCER

furniture had disappeared, but the papers of Hausdorff were essentially intact. They were worthless for treasure hunters. Nevertheless, they suffered losses and the remaining scattered pages were mixed together more than ever. Former student G€ unter Bergmann began ordering and cataloguing in 1964 the surviving 25,978 pages of this collection, consisting of letters, lecture notes, and manuscripts. At the request of Lenore Hausdorff Ko¨nig, the Hausdorff Nachlass was then donated to the library of the University of Bonn [69, pp. 127, 132]. Purkert has subsequently prepared a 553-page detailed inventory of the Nachlass [67]. Eight of the planned ten volumes of Gesammelte Werke have already appeared: Analysis, Algebra, and Number € der Mengenlehre (2002), PhiloTheory (2001), Grundzuge sophical Work (2004), Astronomy, Optics, and Probability Theory (2006), Descriptive Set Theory and Topology (2008), Literary Work (2010), Correspondence (2012), and General Set Theory (2013). The latter actually appears as Volume IA of the Gesammelte Werke [44–51] and contains an essay on Hausdorff as a teacher, as well as a list of the subject matter of the lecture notes appearing in the Nachlass evaluated by content and originality [69, p. 133]. Sadly, two original members of the editorial board for the Hausdorff Edition did not live to see the completion of the Gesammelte Werke. Hirzebruch died unexpectedly of a brain hemorrhage in 2012 and Brieskorn passed away in July 2013. The final volumes, Biography and Geometry, Space, and Time, are expected to appear by 2015. Meanwhile, Hausdorff is currently being honored in an exhibition that has been presented in several German cities, Israel, and the United States: ‘‘Transcending Tradition: Jewish Mathematicians in German-Speaking Academic Culture’’ and a companion publication [5]. Included in the exhibition are previously unpublished documents regarding Hausdorff’s forced retirement from Bonn and his passport application when he unsuccessfully attempted to emigrate; the passport is stamped with ‘‘J’’ for ‘‘Jew’’ and Hausdorff’s name is recorded as ‘‘Felix Israel Hausdorff’’ [75, p. 91]. In keeping with Eichhorn’s suggestion in 1996 that we should ‘‘seek access’’ to Hausdorff to best honor his memory, we hope we will soon know even more about the man who has been referred to as ‘‘one of the most remarkable individuals to appear in the first decades of the twentieth century’’ [68, p. 37]. University of Central Oklahoma Edmond, OK USA e-mail: [email protected]

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