Max Dehn, Kurt Gödel, and the Trans-Siberian Escape Route John W. Dawson Jr. This article contains the text of an invited address prepared for a special session on the exodus of math- ematicians from Nazi-occupied territories, held in Vienna in mid-September 2001 as part of a joint meet- ing of the Deutsche Mathematiker-Vereinigung and the Österreichische Mathematische Gesellschaft. Aware- ness of how difficult it was for those caught up in the rise of Nazism to escape from the terrors they experienced was reinforced by the terrorist attacks of September 11, which prevented the author’s at- tendance at the conference. He is grateful to Professor Karl Sigmund of the University of Vienna for having read the paper in his absence and to Professor Michael Drmota for granting permission for its reprinting here. It originally appeared in the April 2002 issue of the Internationale Mathematische Nachrichten. he careers of Max Dehn and Kurt Gödel Hilbert’s famous list [16]; both published important followed very different trajectories. Yet papers on decision problems; and both, by force Dehn and Gödel were linked by one his- of circumstance, emigrated to America via the torical circumstance: They were the only trans-Siberian railway. Tmathematicians of stature to flee the The disparity between the situations of Dehn and scourge of Nazism via the trans-Siberian railway. Gödel prior to their emigration exemplifies the di- The stories of their escapes and the contrasts in versity of backgrounds among the mathematicians their situations before and after their emigration who fled Hitler. The circumstances of their escapes exemplify both the perils and the limited range of highlight the dislocations, difficulties, and dan- opportunities that confronted intellectual refugees gers such emigrés faced. And the contrast in their of the Holocaust. subsequent careers in America is illustrative of In 1940 Max Dehn and Kurt Gödel each left Eu- the range of institutions in the United States that rope, never to return. Dehn was then a distin- provided havens for intellectual refugees. guished topologist nearing the end of his academic career, while Gödel was a young Privatdozent who Dehn’s European Career had only recently burst into prominence for his star- As yet there is no full-length biography of Max tling discoveries in mathematical logic. Dehn was Dehn, nor a collective edition of all of his pub- a Jew. Gödel was not. And their personalities were lished works. But several shorter articles provide starkly opposed: Whereas Dehn was an outgoing, details of his life and mathematical accomplish- generous man, esteemed by students and col- ments. For the present brief survey I have drawn leagues alike for his humanity, his breadth of in- primarily on [13], [15], and, especially, the chapter tellectual and cultural interests, and his love and on Hilbert’s third problem in [16]. knowledge of the outdoors, Gödel was a reclusive Dehn was born November 13, 1878, in Hamburg, hypochondriac who had few close friends, worked one of eight children of a physician, Maximilian in isolation, and suffered recurrent bouts of men- Moses Dehn. According to Max’s son Helmut, the tal illness. Nevertheless, in a few respects their ca- family were secularized Jews who “lived by princi- reers were similar: Both solved problems on ples that some…would call ‘good Christian’”, and who did not think of themselves as Jewish until the John W. Dawson Jr. is professor of mathematics at Penn- Nazis came to power [16, p. 118]. After graduating sylvania State University. His e-mail address is from the Gymnasium in Hamburg, Max went first [email protected]. to Freiburg and later to Göttingen, where he 1068 NOTICES OF THE AMS VOLUME 49, NUMBER 9 received his doctorate in 1900 under Hilbert’s su- For a time he received a pension and traveled to pervision. In his dissertation he established that the various European countries to lecture. He also con- Archimedean postulate is essential in order to tinued to publish, including an important paper that prove in neutral geometry that the sum of the an- appeared in 1938, in which he introduced the no- gles of a triangle does not exceed 180◦ (Legendre’s tion now referred to as “Dehn twists”. By 1936, theorem). however, he had prudently sent his children out of Later that same year, soon after Hilbert’s address reach of the Nazis, his son Helmut to the United on “Problems of Mathematics” at the International States and his daughters Maria and Eva to a board- Congress of Mathematicians in Paris (and before the ing school in Kent, England, where Dehn himself appearance of its printed version, in which the list taught from January to April of 1938. of problems was expanded from ten to twenty- Later that spring Dehn returned to Frankfurt — three), Dehn established a related result that solved a fateful act, as it turned out, for on November 11, the third of the published problems (one of those 1938 (the morning after Kristallnacht) he was ar- left unstated during the lecture [8]): By exhibiting rested by Nazi agents and taken to a local deten- two tetrahedra with the same base and height that tion center. Providentially, however, he was re- are neither equidecomposable into finite, congru- leased later that day, so many having been rounded ent parts nor equicomplementable by such parts up that there was no place to hold them all. to produce two polyhedra that are equidecom- Subject to imminent re-arrest and deportation, posable, he demonstrated that the Archimedean Dehn and his wife immediately fled to Bad Hom- postulate is also needed in order to prove that two burg, where they were given shelter by his friend tetrahedra of equal base and height have equal and colleague Willi Hartner; and there, in the com- volumes. pany of Hartner and Siegel, Dehn celebrated his six- For his solution of Hilbert’s third problem Dehn tieth birthday. Hartner recalled the occasion years was awarded his Habilitation at Münster, where he later in a newspaper tribute to Dehn [9]: “Unfor- served as a Privatdozent from 1901 until 1911. In gettable for those who saw him at the time was his 1907 he was coauthor with Poul Heegaard of the calmness, his philosophical composure. For the influential survey article “Analysis situs” in the conversations centered not on the events of the day, Enzyklopädie der mathematischen Wissenschaften. but on the relationship of mathematics to art, on In 1910 he introduced the so-called “Dehn dia- problems of archaeology, and finally on the con- grams” for groups and published a fundamental cept of humanity of Confucius.” paper on the topology of 3-dimensional space, Once the brutal initial phase of the pogrom in which included the result that has since come to Frankfurt ended, Dehn and his wife, with the as- be known as “Dehn’s lemma” (though with a proof sistance of Albert Magnus (son of Dehn’s student later seen to be faulty) and the technique now and colleague, Wilhelm Magnus), managed to escape called “Dehn surgery”. That paper also introduced by train through Frankfurt to Hamburg, where they the word and conjugacy (decision) problems for hid for a few weeks at the home of one of Dehn’s groups, which Dehn explored further in two sub- older sisters who had been left unmolested be- sequent papers, the second of which employed an cause of her age. From there, with further help algorithm now named after him. from Siegel and “a Danish colleague and former stu- From 1911–1913 Dehn was Extraordinarius at dent of Dehn’s” [15] — perhaps Jakob Nielsen — a Kiel, and from 1913–1921 Ordinarius at Breslau. On way was found for the Dehns to escape to Denmark August 23, 1912, he married Toni Landau, who and from there to Norway. In January 1939 they bore him three children during their years in Bres- reached Copenhagen, and not long afterward Dehn lau. In 1914 Dehn published a proof that a trefoil secured a temporary position at the Technische knot is not continuously deformable into its mir- Hochschule in Trondheim as a replacement for ror image—an important early result in knot the- Viggo Brun, who was then on leave. ory. Then, from 1915–1918, his work was inter- Until March 1, 1940, when the Nazis invaded Nor- rupted by army service. way, the Dehns were relatively safe. Financially, In 1921 Dehn succeeded Ludwig Bieberbach as however, their situation was precarious. Before Ordinarius at Frankfurt, and the following year he leaving Germany Dehn had been forced to sell his founded a seminar there on the history of mathe- library and much of his furniture at great loss. He matics, whose history and significance, as well as was, of course, paid by the Hochschule in Trond- Dehn’s leadership role in it, is poignantly recounted heim, and from the university in Frankfurt he some- in the memoir by Siegel cited above [13]. Dehn con- how managed to obtain an official leave of ab- tinued to direct the seminar until 1935, when, at sence, valid from April 1, 1939, until June 30, 1940, age 56, the Nazis forced him to retire (later than that enabled his pension payments to continue. most, due to his earlier war service). They were credited, however, to an account in Ham- After his removal from the university Dehn con- burg from which disbursements could only be tinued to live in Frankfurt for another three years. made to parties within the Reich, so that he was OCTOBER 2002 NOTICES OF THE AMS 1069 unable to pay storage charges on what little furni- Shortly after his return to Austria in the spring ture and other personal effects he had been able of 1934 Gödel suffered a serious bout of depres- to ship to London.